scheduling techniques to optimise rail operations · mahmoud masoud (b.sc., dip.or, msc) a thesis...

Scheduling Techniques to Optimise

Rail Operations

Mahmoud Masoud

(B.Sc., Dip.OR, MSc)

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

2012

Principal Supervisor: Professor Erhan Kozan

Associate Supervisor: Associate Professor Geoff Kent

Mathematical Sciences Discipline Faculty of Science and Technology

Queensland University of Technology

Australia

I

Statement of Original Authorship

The work contained in this dissertation has not been previously

submitted for degree at any tertiary educational institution. To the best

of my knowledge and belief, this dissertation contains no materials

previously published or written by another person except where due

reference is made.

Signed……………….

Date…………………

II

Acknowledgements

First and foremost, I sincerely thank Allah, my God, the Most Gracious, Most Merciful for enabling me to complete my Ph.D. successfully and for often putting so many good people in my way.

I would like to express my sincere gratitude to my principal supervisor Professor Erhan Kozan for the continuous support of my PhD study and

research, for his patience, motivation, enthusiasm and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my PhD study.

I would like also to express my most sincere gratitude to my associated supervisor A/ Professor Geoff Kent, for his comments and suggestions

which are very important contribution. Special thanks also go to all staff members of mathematical science

discipline, sugarcane research staff at Queensland University of Technology (QUT). I would like to acknowledge the funding support of the Sugar Research and Development Corporation (SRDC).

I’m indebted to my mother and father who gave my support and encouragement. I’m greatly indebted to my brothers, Gamal and Ayman,

who supported me during my study and their kids, Ahmed, Takwa, Sara, Esraa, Jana and Goudy. Also I would like to thank my uncle Mohamed Mahgoub for his support during my study.

Finally, my wife Amira and my daughter Retaj have been, always, my

pillar and my guiding light, and I thank them.

III

Glossary

MIP Mixed Integer Programming

CP Constraint Programming

OPL Optimisation Programming Language

SMS Single Machine Scheduling

PMS Parallel Machine Scheduling

FSS Flow Shop Scheduling

JSS Job Shop Scheduling

OSS Open Shop Scheduling

GSS Group Shop Scheduling

BPMJSS Blocking Parallel-Machine Job Shop Scheduling

ULBJSS Unlimited Buffer Job Shop Scheduling

LBJSS Limited Buffer Job Shop Scheduling

BJSS Blocking Job Scheduling Shop

CJSS Classical Job Shop Scheduling

LBGSS Limited Buffer Group Shop Scheduling

BGSS Blocking Group shop Scheduling

BFSS Blocking Flow Shop Scheduling

UkLBFSS Unlimited Buffer Flow Shop Scheduling

LBFSS Limited Buffer Flow Shop Scheduling

BFS Best First Search

SBS Slice Based Search

LDS Limited Discrepancy Search

DDS Depth-bound Discrepancy Search

IDFS Interleaved Depth First Search

SSS Standard Search Strategy

DSS Dichotomic Search Strategy

TSCE Terminal Segment Conflict Elimination

ISCE Intermediate Segment Conflicts Elimination

SBD Segment Blocking Determination

RCE Rail Conflict Elimination

CA Computing Acceleration

IV

Abstract

In Australia, railway systems play a vital role in transporting the sugarcane crop

from farms to mills. The sugarcane transport system is very complex and uses daily

schedules, consisting of a set of locomotives runs, to satisfy the requirements of the

mill and harvesters. The total cost of sugarcane transport operations is very high;

over 35% of the total cost of sugarcane production in Australia is incurred in cane

transport. Efficient schedules for sugarcane transport can reduce the cost and limit

the negative effects that this system can have on the raw sugar production

system. There are several benefits to formulating the train scheduling problem as a

blocking parallel-machine job shop scheduling (BPMJSS) problem, namely to

prevent two trains passing in one section at the same time; to keep the train activities

(operations) in sequence during each run (trip) by applying precedence constraints;

to pass the trains on one section in the correct order (priorities of passing trains) by

applying disjunctive constraints; and, to ease passing trains by solving rail conflicts

by applying blocking constraints and Parallel Machine Scheduling. Therefore, the

sugarcane rail operations are formulated as BPMJSS problem.

A mixed integer programming and constraint programming approaches are used to

describe the BPMJSS problem. The model is solved by the integration of constraint

programming, mixed integer programming and search techniques. The optimality

performance is tested by Optimization Programming Language (OPL) and CPLEX

software on small and large size instances based on specific criteria. A real life

problem is used to verify and validate the approach. Constructive heuristics and new

metaheuristics including simulated annealing and tabu search are proposed to solve

this complex and NP-hard scheduling problem and produce a more efficient

scheduling system. Innovative hybrid and hyper metaheuristic techniques are

developed and coded using C# language to improve the solutions quality and CPU

time. Hybrid techniques depend on integrating heuristic and metaheuristic techniques

consecutively, while hyper techniques are the complete integration between different

metaheuristic techniques, heuristic techniques, or both.

V

Publications Arising from this PhD Research

Masoud, M., Kozan, E., & Kent, G. (2011). A job-shop scheduling approach for

optimising sugarcane rail operations. Flexible Services and

Manufacturing Journal; 23(2):181-196.

Masoud, M., Kozan, E., & Kent, G. (2010a). Scheduling techniques to optimise

sugarcane rail Systems. ASOR Bulletin; 29:25-34.

Masoud M., Kozan E., & Kent, G. (2010b). A constraint programming approach to

optimise sugarcane rail operations, Proceedings of the 11th Asia Pacific

Industrial Engineering and Management Systems Conference 2010, 147:1-7,

Malaysia. (Outstanding Student Paper Award)

Masoud, M., Kozan, E., & Kent, G. (2010c). A comprehensive approach for

scheduling single track railways. The Annual Conference on Statistics,

Computer Sciences and Operations Research, Egypt, Cairo, 45,19-30.

Masoud, M., Kozan, E., & Kent, G. (under review) A new approach to automatically

producing schedules for cane railways. Australian Society of Sugar Cane

Technologies.

Masoud, M., Kozan, E., & Kent, G.. (under review) Hybrid/hyper metaheuristic

techniques for optimising sugarcane rail operations. Computer and

Operations research.

VI

Contents

Statement of Original Authorship………………………..………………...………….....……I

Acknowledgements…………………………..……………..…………………………...… II

Glossary………………………………………..………………….……………....……..….III

Abstract……………………………………………………...…………………..………..…IV

Publications Arising from this PhD Research…………………………………………..........V

Content………………………………………………………………...…………..…...........VI

List of Tables…………………………………….……………..………......….........XII

List of Figures……………………………………………...…………………….….…......XV

Chapter 1: Introduction and the Research Problems………………………..……….......1

1.1 Introduction........................................................................................................................................2

1.2 Research Problem .............................................................................................................................3

1.2.1 Background.......................................................................................................................3

1.2.2 Sugarcane Transportation Systems.....................................................................................6

1.2.2.1 The Sugarcane Rail Transport..............................................................................7

1.2.2.2 The Cane Transport and Harvesting Integration...................................................8

1.2.2.3 The Sugarcane Road Transport Systems............................................................11

1.2.3 Complexity of the Sugarcane Rail Transport Systems.....................................................13

1.2.3.1 Rail Complexity.............................................................................................13

1.2.3.2 Sugarcane Systems Complexity....................................................................16

1.2.4 Research Questions............................................................................................................21

1.3 Contribution and Significance of the Study.....................................................................................24

1.4 Outline of the Thesis........................................................................................................................26

Chapter 2: Scheduling Theory Review……………………………………………..….....28

2.1 Introduction .....................................................................................................................................29

2.2 Scheduling Classifications..............................................................................................................29

VII

2.2.1 Job Characteristic.............................................................................................................30

2.2.2 Machine Environment....................................................................................................30

2.2.2.1 Single Machine Scheduling (SMS)..........................................................32

2.2.2.2 Parallel Machine Scheduling (PMS)........................................................33

2.2.2.3 Flow Shop Scheduling (FSS)...................................................................34

2.2.2.4 Job Shop Scheduling (JSS)......................................................................34

2.2.2.5 Open Shop Scheduling (OSS).................................................................36

2.2.2.6 Group Shop Scheduling (GSS)...............................................................36

2.2.3 Optimality Criteria...........................................................................................................37

2.2.4 Concluding Remarks........................................................................................................38

2.3 Job Shop Scheduling (JSS) Solution Techniques............................................................................39

2.3.1 Disjunctive Graph Technique...........................................................................................41

2.3.1.1 Heuristic Techniques................................................................................46

2.3.1.1.1 The Shifting Bottleneck Algorithm................................................46

2.3.1.1.2 Dispatching Rules...........................................................................46

2.3.2 Mixed Integer Programming Approach for JSS Problem.................................................47

2.3.2.1 A hyper Branch and Bound Technique........................................................48

2.3.3 Constraint Programming Approach for the JSS Problem..................................................53

2.4 Scheduling Theory and Railway Systems in Literature..................................................................56

2.5 Conclusion.......................................................................................................................................60

Chapter 3: Modelling the Sugarcane Rail Transport Problem……………….…….....62

3.1 Introduction....................................................................................................................................64

3.2 Segment and Section Blocking Types............................................................................................65

3.2.1 Blocking Segment Types..................................................................................................65

3.2.1.1 Blocking Terminal Segments............................................................................66

3.2.1.2 Blocking Intermediate Segments......................................................................66

3.2.2 Blocking Section Types.....................................................................................................68

3.3 Description of the Blocking Segment Models of Sugarcane Rail System......................................69

3.3.1 Blocking Segment MIP Model of the Sugarcane Rail System..........................................76

VIII

3.3.2 Blocking Segment CP Model of the Sugarcane Rail System............................................83

3.4 Description of the Blocking Section Models of Sugarcane Rail System.......................................89

3.4.1 Blocking Section MIP Model of Sugarcane Rail System..................................................90

3.4.2 Blocking Section CP Model of Sugarcane Rail System....................................................93

3.5 Inclusion of the Delivery and Collection Time Constraints to the Model......................................95

3.5.1 Delivering Delay Constraints..............................................................................................99

3.5.2 Collecting Delay Constraints..............................................................................................101

3.6 Sugarcane Rail System as a Dynamic System...............................................................................102

3.7 Conclusion.....................................................................................................................................104

Chapter 4: Solution Approach…………………………………………………….........105

4.1 Introduction....................................................................................................................................107

4.2 Constraint Satisfaction (CS) Techniques.......................................................................................109

4.2.1 Constraint Propagation.....................................................................................................109

4.2.1.1 Node Consistency..............................................................................................111

4.2.1.2 Arc Consistencies..............................................................................................112

4.2.1.3 Bounds Consistency...........................................................................................115

4.2.1.4 Path Consistency................................................................................................115

4.2.2 Search Process..................................................................................................................116

4.2.2.1 Variable and Value Ordering Heuristics.............................................................117

4.2.2.2 Search Techniques..............................................................................................119

4.2.2.3 Global Constraint................................................................................................124

4.3 Proposed Algorithms......................................................................................................................125

4.3.1 Collecting and Delivering Conflict Elimination...............................................................126

4.3.1.1 Terminal Segment Conflict Elimination..............................................................127

4.3.1.2 Intermediate Segment Conflict Elimination........................................................130

4.3.2 Algorithms for Solving Train Conflicts.............................................................................133

4.3.3 Computing Acceleration (CA) Algorithms.........................................................................137

4.4 Conclusion......................................................................................................................................139

IX

Chapter 5: Computational Results of CP and MIP Models. ………….…...…....140

5.1 Introduction..............................................................................................................................143

5.2 A Case Study for Testing Blocking Segment CP and MIP Models.........................................143

5.2.1 Input Data...............................................................................................................144

5.2.2 Results of Makespan Minimisation Objective........................................................146

5.2.2.1 Constraint Programming Model Results..................................................146

5.2.2.1.1 Standard Constraint Programming Results...........................146

5.2.2.1.2 Results of Computing Acceleration (CA) Algorithms..........149

5.2.2.1.3 Train and Runs Scheduling for CP Model............................151

5.2.2.1.4 Solutions Analysis by Search Tree........................................152

5.2.2.2 MIP Model Results..................................................................................156

5.2.2.2.1 Result of Standard MIP Model................................................156

5.2.2.2.2 Results of Computing Acceleration (CA)...............................158

5.2.2.2.3 Train Scheduling Results.........................................................160

5.2.2.3 Comparisons of CP and MIP Models using Makespan Criterion............161

5.2.3 Results of Total Waiting Time Minimisation Objective ........................................163

5.2.3.1 Constraint Programming (CP) Model .......................................................163

5.2.3.1.1 Results of Standard CP.........................................................163

5.2.3.1.2 Results of Computing Acceleration (CA) Algorithms…….164

5.2.3.1.3 Train Scheduling Results......................................................165

5.2.3.2 Mixed Integer Programming (MIP) Model ................................................166

5.2.3.2.1 Result of Standard MIP Model.............................................166

5.2.3.2.2 Results of Computing Acceleration (CA) Algorithms…….167

5.2.3.2.3 Train Scheduling Results......................................................168

5.2.3.3 Comparisons of CP and MIP Models of Total Waiting Time Criterion….169

5.3 A large Scale Case Study for Testing Blocking Segment CP and MIP Models.......................171

5.3.1 Results of Makespan Minimisation Objective.........................................................175

X

5.3.2 Results of Total Waiting Time Minimisation Objective................................................177

5.4 Sensitivity Analysis of Blocking Segment MIP Sugarcane Rail Model.......................................178

5.4.1 Small Rail Systems......................................................................................................179

5.4.2 Large Rail Systems.......................................................................................................184

5.5 Blocking Section MIP and CP Results..........................................................................................186

5.5.1 The comparisons between the Blocking Segment and Sections Models......................187

5.6 The Results of Inclusion of the Delivery and Collection Time Constraints .................................189

5.7 Conclusion.....................................................................................................................................193

Chapter 6: Metaheuristic Techniques……………………………………………....….194

6.1 Introduction...................................................................................................................................197

6.2 Neighbourhood Structure..............................................................................................................198

6.2.1 Adjacent Pairwise Interchange (API) ..........................................................................200

6.2.2 Non-Adjacent Pairwise Interchange (NAPI)................................................................201

6.2.3 Extraction and Forward Shifted Reinsertion (EFSR)……………………...................201

6.2.4 Extraction and Backward Shifted Reinsertion (EBSR)................................................202

6.3 Neighbourhood Structure in the Railway Scheduling Problem ...................................................202

6.4 Simulated Annealing (SA) Technique ..........................................................................................207

6.4.1 Simulated Annealing Technique for Solving Job Shop Scheduling Problem…….....208

6.4.2 New Simulated Annealing Algorithms for Sugarcane Rail Cases.............................210

6.5 Tabu Search (TS) Technique.........................................................................................................213

6.5.1 Tabu Search Technique for Solving Job Shop Scheduling Problem............................213

6.5.2 A new Tabu Search Technique for Sugarcane Rail Cases............................................215

6.6 Metaheuristic Results of Sugarcane Rail Systems.........................................................................216

6.6.1 TS and SA Results by Changing Number of Trains........................................................219

6.7 Hybrid Metaheuristic Techniques for Sugarcane Rail Systems....................................................221

6.7.1 Hybrid SA/TS Technique.................................................................................................222

6.7.2 Hybrid TS/SA Technique .................................................................................................226

6.7.3 Hybrid Techniques Result for Sugarcane Rail Systems.....................................................229

6.7.4 Analysis of Hybrid Techniques..........................................................................................232

XI

6.8 Hyper Metaheuristic Techniques for Sugarcane Rail Transport Systems...............................234

6.8.1 Hyper SA/TS Technique..............................................................................................235

6.8.2 Hyper TS/SA Technique..............................................................................................238

6.8.3 Hyper Techniques Result for Sugarcane Rail Systems................................................241

6.8.4 Analysis of Hyper Techniques.....................................................................................244

6.9 Hybrid and Hyper Metaheuristic Techniques Test Cases........................................................246

6.10 Hyper and Hybrid Metaheuristic Technique (TS/SA) and MIP.............................................247

6.11 Study Analysis of Elements of Metaheuristic Techniques.....................................................249

6.12 Inclusion of Delivery and Collection Time Constraints for a Hyper TS/SA Results.............252

6.13 Conclusion..............................................................................................................................257

Chapter 7: Conclusions and Future Work……………………..………………..... 258

7.1 Introduction..............................................................................................................................259

7.2 Theoretical Contributions.........................................................................................................259

7.3 Practical Contributions.............................................................................................................262

7.4 Future Work..............................................................................................................................263

References ………………………………………………………………...………......265

Appendix A………………………………………………………………………….....276

Appendix B…………………………………………………………………………….316

Appendix C………………………………………………………………………….…320

XII

List of Tables

Table 1.1: Decision matrix of the rail network shown in Figure 1.5………………………………...20

Table 2.1: 3 jobs and 3 machines problem…………………………………………………….……...43

Table 2.2: Solving n/1/Cmax problem for the three machines M1, M2, and M3 at the final

level……...51

Table 5.1: The distances and the sectional running times of the sections…………………..........…144

Table 5.2: Siding capacity and siding allotments…………………………………………….......….145

Table 5.3: Train number, capacity and speed…………………………………………………..........145

Table 5.4: SSS and DSS results for standard CP model to optimise makespan…………......……...148

Table 5.5: SSS and DSS results for CA algorithms to optimise makespan………………......…….150

Table 5.6: Start and finish times of train run for CP model to optimise makespan…………......….151

Table 5.7: The standard CP model solutions before obtaining the optimal solution of

Makespan………………………………………………………………………………….……......153

Table 5.8: SSS and DSS results for standard MIP model to optimise makespan ………….....…..157

Table 5.9: SSS and DSS results for CA algorithms for MIP to optimise makespan ……….......…159

Table 5.10: Start and finish times of train runs for MIP to optimise makespan……………….......160

Table 5.11: Makespan results of standard MIP and CP and CA algorithms using SSS and

DSS………………………………………………………………………………………................162

Table 5.12: SSS and DSS results for Standard CP to optimise the total waiting time……….....….163

Table 5.13: Solution of CA algorithms for CP to optimise the total waiting time……………........164

Table 5.14: Start and finish times of train runs to optimise the total waiting time…………….......165

Table 5.15: SSS and DSS results for standard MIP model to optimise the total waiting time.........166

Table 5.16: SSS and DSS results of CA algorithms for MIP to optimise the total waiting

time………………………………......................………………………………………….....…....167

Table 5.17: Start and finish times of train runs of MIP to optimise the total waiting time….....…168

Table 5.18: Standard MIP and CP results using SSS and DSS to optimise the total waiting

time………………………………………………………………………………………....……...170

Table 5.19: MIP and CP results using SSS and DSS for CA algorithms………….…………....…170

Table 5.20: The distance between sidings in the extension case study……………..……….....….173

Table 5.21: Sidings capacity and allotments in the extension case study…………………....……174

XIII

Table 5.22: Train number, capacity and speed in the extension case study…………………....….175

Table 5.23: SSS results using makespan for the larger case study………………….............…..........175

Table 5.24: DSS results using makespan for the larger case study……………..……......….…...…..176

Table 5.25: Start and finish times of train runs using the makespan objective function….................176

Table 5.26: SSS results using total waiting time for the larger case study…………….…….........…177

Table 5.27: DSS results using total waiting time for the larger case study.………….……............…177

Table 5.28: Start and finish times of train runs using the total waiting time objective……................178

Table 5.29: Results of sensitivity analysis of total waiting time (a small rail)…………….......….....180

Table 5.30: Results of sensitivity study of makespan (a small rail)………………………........…….183

Table 5.31: Sensitivity analysis of changing some variables in the system using total waiting

time……………………………………………………………………………………...…........……185

Table 5.32: Sensitivity study for makespan criterion……………………………………........……...186

Table 5.33: Makespan and total waiting time results for the blocking section MIP model…...........187

Table 5.34: Makespan and total waiting time results for the blocking section CP model……..........187

Table 5.35: Train runs for seven harvester model……………………………………………............191

Table 6.1: Simulated annealing result for Example 2.1……………………………………...........…210

Table 6.2: TS technique result for example 2.1…………………………………………..........….....214

Table 6.3: TS and SA results for sugarcane rail transport system under section blocking

constraint……………………………………………………………………………………..............217

Table 6.4: Comparison of makespan of TS, SA and hybrid techniques…………………........…......230

Table 6.5: Metaheuristic and hyper techniques results of different cases………………….........…...242

Table 6.6: Metaheuristic, hybrid, hyper, and MIP results of different tested cases………….............248

Table 6.7: Train operations for delivering and collecting empty and full bins....................................254

Table A1: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level

1…………......277


2a...….......…..279


2b……............281

XIV


2c….........…...282


3a….….....….284


3bi……..........286


3bii….............287


3ci….........…..289


3cii……..........291


3ciii...............292


4ai….............294


4aii……........295

Table A13: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in

level4bi…..............297


4biia…..........298


4biib…..........300


4ci…….........301


4cii…......…..302


5aia...............304

XV


5aib…..........305


5aiia….........306


5aiib..….......307


5bia…...........308


5bib...............310


5biia........…..311


5cii…............312


6aii…............313


6bia…...........314

Table B1: simulated annealing result for Example 2.1…………………………......………..............319

Table C1: Tabu search technique result for example 2.1…………………………………….............329

List of Figures

XVI

Figure 1.1: Value chain linkages in the sugar industry (Milford, 2002)………………………........…..4

Figure 1.2: The length of train less than the length of section………………………………........…..14

Figure 1.3: The length of train greater than the length of section………………………………..........15

Figure 1.4: The segment length is greater than the train length………….………………………........15

Figure 1.5: Delivering and collecting point at the middle and the end of the branch (sugarcane)....... 16

Figure 1.6: Delivering and collecting point at the end of the branch (Coal mining)…………….........17

Figure 1.7: Research plan…………………………………………………………...…………........…27

Figure 2.1: Scheduling problem types…………………………………………………………........…31

Figure 2.2: single machine scheduling problem; n jobs processed on one machine………….….........32

Figure 2.3: Parallel machine scheduling problem; n jobs processed on m parallel machines…….......33

Figure 2.4: Flow shop problem; n jobs processed on m machines……………………………….........34

Figure 2.5: Job shop problem; n jobs processed on m machines………………………………...........35

Figure 2.6: Open shop problem; n jobs processed on m machines……….……………………...........36

Figure 2.7: Group shop scheduling problem; n jobs include different groups on m machines…..........37

Figure 2.8 Job shop solution techniques………………………………………………………........…40

Figure 2.9: Conjunctive arcs for 5 jobs and 5 machines….…………………………….………..........42

Figure 2.10: Disjunctives arcs for 5 jobs and 5 machines…………….……………………........…….42

Figure 2.11: The disjunctive graph for 3 jobs and 3 machines……………….………………….........43

Figure 2.12: The first feasible solution for the 3 jobs and 3 machines example…………….….…......44

Figure 2.13: The Gantt chart for the first feasible solution…………………………………............…45

Figure 2.14: The second feasible solution for 3 jobs and 3 machines example………………........….45

Figure 2.15: The Gantt chart for the second feasible solution…………………………………...........46 Figure 2.16: Disjunctive graph for the final solution…………………………………………….........51

Figure 2.17: Optimal solutions of MR1R, MR2R, and MR3R at the final level for n/1/CRmax

Rproblem……............51

Figure 2.18: The complete branching procedure of the optimal solution………………………..........52

Figure 3.1: MIP and CP approaches……………………………...………………………………........64

Figure 3.2: Two trains travelling in the same direction. One requires the blocking terminal

segment………………………………………………………………………............…...66

XVII

Figure 3.3: Two trains travelling in the same direction. One requires the blocking intermediate

segment……………………………………………………........………………………...67

Figure 3.4: Two trains travelling in different directions. One requires the blocking intermediate

segment………………………………………...……………….........…………………...68

Figure 3.5: Train requires the blocking section s……………………….……........…………...……...69

Figure 3.6: A simple cane rail network with three sidings for delivering and collecting….........…….70

Figure 3.7: A single cane rail network after applying model…………………………………........….72

Figure 3.8: Rail branch includes passing loop…………..………………………………...…........…..73

Figure 3.9: Rail siding includes passing point……………………………………………...........……74

Figure 3.10: MIP and CP models structure………………………………………………........………75

Figure 3.11: A single cane rail network with three sidings can allow passing trains………......…......90

Figure 3.12: Numerical case…………………………………………………………........……….….98

Figure 3.13: Visit times of siding A for one day………………………………………........…..……..98

Figure 3.14: Sugarcane rail system as a dynamic system………………………………........………103

Figure 4.1: The order of nodes that will be expanded by DFS strategy………………….........……..120

Figure 4.2: Limited discrepancy search…………………….....………………………….........…….122

Figure 4.3: Siding at the beginning of terminal segment…………………………………....….........127

Figure 4.4: Siding at the end of terminal segment…………………………………………........…...128

Figure 4.5: Siding at the middle side of terminal segment…………………………………..............128

Figure 4.6: Terminal segment conflict elimination algorithm……………………………….............129

Figure 4.7: Siding at the beginning of intermediate segment………………………………..............130

Figure 4.8: Siding at the end of intermediate segment………………………………...……..............131

Figure 4.9: Siding at the middle side of intermediate segment……………………………..…..........131

Figure 4.10: A rail conflict case…………………………………………………………............…..135

Figure 4.11: A section conflict case using a delayed train technique……….....................................136

Figure 4.12: A section conflict case using a slow train technique…………………….....….............137

Figure 5.1: A realistic test case study (small sector of rail network of Kalamia mill)……….............143

Figure 5.2: CPU time of SSS and DSS for the standard and CA of the CP using Makespan..............151

Figure 5.3:Scheduling 5 trains on 11 segments using CP with the makespan of 2664……................152

Figure 5.4: Solutions analysis by search tree………………………….…………………...…...........155

XVIII

Figure 5.5: CPU time of standard and CA algorithms for MIP to optimise Makespan.......................160

Figure 5.6: Scheduling 5 trains on 11 segments using MIP with the makespan of 2664…….............161

Figure 5.7: CPU time of standard and CA algorithms for CP to optimise the total waiting time........165

Figure 5.8: CP for scheduling 5 trains on 11 segments with the total waiting time of 1984…….......166

Figure 5.9: CPU time of SSS and DSS for the standard and the CA algorithms of MIP to optimise the

total waiting time……………………………………………………………………………..............168

Figure 5.10: MIP for scheduling 5 trains on 11 segments with the Total waiting time of 1984..........169

Figure 5.11: Larger case study: bigger part of rail network of Kalamia mill……........………...........172

Figure 5.12: SSS and DSS results of different cases using a small rail to optimise TWT……...........181

Figure 5.13: SSS and DSS results of different problems using a small rail to optimise makespan....184

Figure 5.14: Blocking segment and section results using makespan and total waiting time…...........188

Figure 5.15: Harvester usage for seven harvester model……………………………………........….189

Figure 5.16: Mill yard stock chart for seven the harvester model……………………..….........…….190

Figure 5.17: Train utilisation for the seven harvester model…………………………………...........190

Figure 6.1: API technique for 10 jobs………………………………………………………….........201

Figure 6.2: NAPI technique for 10 jobs………………………………………………………..........201

Figure 6.3: EFSR technique for 10 jobs………………………………………………………..........202

Figure 6.4: EBSR technique for 10 jobs…………………………......................................................202

Figure 6.5: A single cane rail network with 22 sections…………………………….………….........203

Figure 6.6: Transition matrix for a single railway includes 22 sections………………………..........204

Figure 6.7: Case of three trains require the same section……………………………….....................207

Figure 6.8: Disjunctive graph of final solution for the 3 jobs and 3 machine example………...........209

Figure 6.9: Disjunctive graph of final solution for the 3 jobs and 3 machines example…...…...........214

Figure 6.10: Metaheuristic techniques using makespan for different cases …………...……….........218

Figure 6.11: CPU time of TS and SA for different cases………….…………………….…......…….219

Figure 6.12a: Makespan of metaheuristics on 10 sections…………………......…………............….220

Figure 6.12b: Makespan of metaheuristics on 15 sections…………...........................….......……….220

Figure 6.12c: Makespan of metaheuristics on 20 sections…………..............................……........….221

Figure 6.13: Hybrid SA/TS technique…………..............................………………...…............…….223

XIX

Figure 6.14: The detailed hybrid SA/TS technique………………..........................………….......….225

Figure 6.15: Hybrid TS/SA technique…………..............................…………………...……….........226

Figure 6.16: The detailed hybrid TS /SA technique…………..............................…..………….........228

Figure 6.17: Comparison of TS, SA and hybrid techniques using makespan…………......................231

Figure 6.18: CPU time of TS, SA and hybrid techniques for different cases…………......................232

Figure 6.19a: Makespan of hybrid techniques on 15 sections…………………………....…….........233

Figure 6.19b: Makespan of hybrid techniques on 20 sections……………………...……............….233

Figure 6.19c: Makespan of hybrid techniques on 25 sections……………………………….............233

Figure 6.19d: Makespan of hybrid techniques on 30 sections……………………………….............233

Figure 6.20: Hyper SA/TS technique………………………………………………..………........….235

Figure 6.21: The detailed SA/TS hyper Technique…………………….……………………….........237

Figure 6.22: Hyper TS/SA technique…………………………………………………..……….........238

Figure 6.23: The detailed hyper TS/SA Technique………………….………………………….........240

Table 6.24: Metaheuristic and hyper techniques results of different cases………………........….….243

Figure 6.25: Comparison of CPU time of TS and SA and hyper techniques………........…………...244

Figure 6.26a: Makespan of hyper techniques on 15 sections………………………………........…..245

Figure 6.26b: Makespan hyper techniques on 20 sections………………….……………........……..245

Figure 6.26c: Makespan of hyper techniques on 25 sections……………………….………........…..245

Figure 6.26d: Makespan of hyper techniques on 30 sections………………………….………..........245

Figure 6.27: Hybrid and hyper techniques results using makespan for some tested cases……..........246

Figure 6.28: CPU time of some tested cases using hybrid and hyper techniques………........………247

Figure 6.29: Effect of α value on the hyper SA/TS solution……………………………....................249

Figure 6.30: Effect of α value on the hyper SA/TS solution…………………………………............249

Figure 6.31: Effect of number of iterations on hyper TS/SA solution………………………….........250

Figure 6.32: Effect of number of iterations on CPU time of hyper TS/SA solution…………............250

Figure 6.33: The average of makespan of hyper TS/SA case study 30/12 with different runs........251

Figure 6.34: A sample train schedule of seven trains, 45 sections and 15 harvesters.........................253

Figure A1: Conjunctive graph for the 3/3/G/Cmax problem………………………….………….........276

Figure A2: Conjunctive graph for the 3/3/G/Cmax problem……………………….………….............277

XX

Figure A3: Optimal solutions of M1, M2, and M3 in level 1for n/1/Cmax

problem……………........…277

Figure A4: The branching procedure at level 1……………………………………...........………….278

Figure A5: Conjunctive graph for the 3/3/G/Cmax problem…………………………........…………..279

Figure A6: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level

2a.........……..279

Figure A7: The branching procedure at level 2a ……………………………………….....................280

Figure A8: Disjunctive graph for the level 2b ……………………..………………….......................280


2b……...........281

Figure A10: The branching procedure at level 2b…………………………………….…...................281

Figure A11: Disjunctive graph for the level 2c…………………………………….….......................282


2c….............282

Figure A13: The branching procedure at level 2c…………………………………............................283

Figure A14: The complete branching procedure at level 2………………………………….........….283

Figure A15: Disjunctive graph for the level 3a…………………………………………....................284

Figure A16: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 3a

…............284

Figure A17: The branching procedure at level 3a…………………………………….………...........285

Figure A18: Disjunctive graph for the level 3bi…………………………………………...................285

Figure A19: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 3bi

…...........286

Figure A20: The branching procedure at level 3bi…………………………………………........…...286

Figure A21: Disjunctive graph for the level 3bii………………………...…………...…............…...287


3bii.….........287

Figure A23: The branching procedure at level 3bii…………………………………………........….288

Figure A24: Disjunctive graph for the level 3ci………………………………………………...........288

XXI


3ci…........…289

Figure A26: The branching procedure at level 3ci………………………………………...................290

Figure A27: Disjunctive graph for the level 3cii……………………………...…………...................290


3cii…...........291

Figure A29: The branching procedure at level 3cii……………………………………............……..291

Figure A30: Disjunctive graph for the level 3ciii……………………………………….....................292


3ciii…..........292

Figure A32: The complete branching procedure at level4……………………………………...........293

Figure A33: Disjunctive graph for the level 4ai…………………………………............…………...293

Figure A34: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 4ai

…...........294

Figure A35: The complete branching procedure at level4ai…………………………........................294

Figure A36: Disjunctive graph for the level 4aii………………………………………......................295


4aii…...........295

Figure A38: The complete branching procedure at level4aii……………………………...................296

Figure A39: Disjunctive graph for the level 4bi………………………………………………...........296


4bi…............297

Figure A41: The complete branching procedure at level 4bi………………………….......................297

Figure A42: Disjunctive graph for the level 4biia………………………………….……...................298


4biia.............298

Figure A44: The complete branching procedure at level 4biia…………………………....................299

Figure A45: Disjunctive graph for the level 4biib……………………………………...........….…...299


4biib…........300

XXII

Figure A47: Disjunctive graph for the level 4ci…………………………………………...................300


4ci…............301

Figure A49: Disjunctive graph for the level 4cii………………………………………...……...........301

Figure A50: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 4cii

…..........302

Figure A51: The complete branching procedure at level 4cii…………………………………..........303

Figure A52: Disjunctive graph for the level 5aia…………………………………….........................303


5aia….........304

Figure A54: Disjunctive graph for the level 5aib…………………………………..……...................304


5aib…..........305

Figure A56: Disjunctive graph for the level 5aiia………………………………………...................306


5aiia….........306

Figure A58: Disjunctive graph for the level 5aiib………………………………………....................307


5aiib….........307

Figure A60: Disjunctive graph for the level 5bia…………………………………………........…….308


5bia…..........308

Figure A62: Disjunctive graph for the level 5bib………………………………………....…........….309


5bib…..........310

Figure A64: Disjunctive graph for the level 5biia………………………………..…........….…….....310


5biia….........311

Figure A66. Disjunctive graph for the level 5cii……………………………………...………...........311

XXIII


5cii…...........312

Figure A68. Disjunctive graph for the level 6aii……………..…………………………....................312


6aii…...........313

Figure A70: Disjunctive graph for the level 6bia…………………………………………........…….313


6bia.............314

Figure A72: The complete branching procedure at level 6bia……………………...…………...…...315

Figure B1: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example…...........316

Figure B2. Disjunctive graph of an initial solution for the 3 jobs and 3 machines example…...........317

Figure B3: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example…...........317

Figure B4: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example...............318

Figure B5: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example…......….319

Figure C1: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example..….........320

Figure C2: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example…...........320








Figure C10: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example….........327

Figure C11: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example….........328

1

Chapter 1

Introduction and the Research Problems

Chapter Outline

1.1 Introduction...............................................................................................................2

1.2 Research Problem .....................................................................................................3

1.2.1 Background.................................................................................................3

1.2.2 Sugarcane Transportation Systems...................................................................6

1.2.2.1 The Sugarcane Rail Transport...............................................................7

1.2.2.2 The Cane Transport and Harvesting Integration...................................8

1.2.2.3 The Sugarcane Road Transport Systems............................................11

1.2.3 Complexity of the Sugarcane Rail Transport Systems....................................13

1.2.3.1 Rail Complexity...........................................................................13

1.2.3.2 Sugarcane Systems Complexity......................................................16

1.2.4 Research Questions...........................................................................................21

1.3 Contribution and Significance of the Study......................................................................24

1.4 Outline of the Thesis.........................................................................................................26

2

1.1 Introduction Railway systems in Australia play a vital role in transporting the sugarcane crop

between farms and mills. About 90% of the total Australian crop is transported in

this way. The sugarcane transport system is very complex and uses a daily schedule

of train runs to meet the needs of both the harvesters and the mill. An efficient

sugarcane transport schedule can reduce costs and minimise the negative effects on

the overall sugar production system.

The sugarcane rail transport system has a significant effect on the performance of the

sugar production process and its overall costs. Potential negative effects of a poor

rail transport system include: stopping the supply of cane to mills, causing

interruptions to the raw sugar production process; delaying the arrival of sugarcane

to the mill; allowing cane to deteriorate and lose sugar quality; delaying the arrival of

empty cane bins to harvesters causing the harvesters to waste time and money

waiting for empty bins; increasing cane production costs through inefficiencies in the

sugarcane transport system itself, and in the harvesting and raw sugar production

processes.

Many researchers have developed optimization models to improve the efficiency of

the railway and sugarcane transport system. However, extensive work is still

required to develop new techniques of scheduling and achieve optimal solutions for

the sugarcane rail transport problem.

This research aims to produce effective techniques for producing efficient schedules

for the sugar rail transport system by developing new scheduling models based on

Mixed Integer Programming (MIP) and Constraint Programming (CP). This

research treats the train scheduling problem as a blocking parallel-machine job shop

scheduling (BPMJSS) problem and develops a novel job shop scheduling (JSS)

approach to solve it. This approach integrates mixed integer programming and

constraint programming to produce efficient schedules to solve the sugarcane rail

transport scheduling problem in a reasonable time. The new models include the

requirements of all major elements of the cane transport system to reduce the overall

cost and optimise the performance of the system. New metaheuristics including

simulated annealing and tabu search are proposed to solve this complex and NP-hard

3

scheduling problem. Hybrid and hyper techniques are developed using the

integration of the heuristic and metaheuristic techniques to improve the solution

quality and Computer Processing Unit (CPU) time. Optimisation Programming

Language (OPL), CPLEX software and C# are used to develop the code of the

proposed models and the solution techniques.

This chapter introduces the background to this research, the main research problem

and significance of the study. The main research question has been articulated

through investigative questions in Section 1.2 while the significance of the study is

highlighted in Section 1.3. A brief outline of the complete document is provided in

section 1.4.

1.2 Research Problem 1.2.1 Background

The value chain of the Australian sugar industry is illustrated in Figure 1.1 and

consists of seven main stages: sugarcane growing; sugarcane harvesting; sugarcane

transport; raw sugar production (at the sugar mill); raw sugar storage; shipping and

refining.

The sugarcane transport system in Australia is an important element in the

production of raw sugar. Most of the sugarcane crop is transported from farms to

mills by rail. A cane railway network is mainly single track which includes many

branches, lines and sidings. The cane railway system performs two main tasks:

delivering empty bins from the mill to the harvesters at sidings; and collecting the

full bins of cane from harvesters and transporting them to the mill. From the

perspective of the transport system, the mill serves the function of converting full

bins into empty bins while the harvesters convert empty bins to full bins.

4

Material flow

Information and planning

Figure 1.1: Value chain linkages in the sugar industry (Milford, 2002)

Sidings, where the empty and full bins are delivered and collected, have a finite

capacity that cannot be exceeded. However, each siding has a daily allotment of bins

(the number, determined by the mill, of bins to be filled by the harvesters) that often

exceeds the capacity of the siding, hence requiring multiple deliveries and

collections. Each train can haul a limited number of empty and full bins depending

on its capacity. Safety conditions require that empty and full bins are not mixed on

the train. As a result, all empty deliveries must take place before full collections.

This transport scenario significantly impacts on the overall cost of the sugarcane

production system and represents over 35% of the total cost of sugar production in

Australia.

5

Sugarcane rail transport systems are very complicated systems and these systems

face many challenges which directly and indirectly affect the overall cost or the

performance of sugarcane system. The challenges include:

Limited harvesting hours. In Australia, most harvesting operations occur

during daylight hours while the transport system and mills operate

continuously. As a result, the cane bins are used for temporary storage of the

harvested crop at sidings and at the mill. Long storage times reduce

sugarcane quality and crop profitability.

Minimising the number of locomotive runs to reduce operating costs.

Reducing the time between harvesting and crushing (cane age) to keep sugarcane quality high.

Scheduling passing of trains on the single rail track to improve performance

of the railway system.

Maintaining a non-interrupted supply of full bins to the mill and empty bins to the harvesters.

Many changing circumstances. A generic model needs to be designed to deal

with unexpected events such as ever-changing numbers of harvesters, bins

and trains.

An Automatic Cane Railway Scheduling System (ACRSS) was first developed at

James Cook University in the 1970s to produce daily schedules (Pinkney and Everitt,

1997). This software solved the railway scheduling problem by dividing the problem

into two parts: a routing problem that produced train runs, and a sequencing problem

to determine the train run times to satisfy harvester and mill requirements. Further

refinement of the solution was then undertaken by adding train runs to produce a

daily schedule to satisfy the objective function. This function considered issues such

as the number of cane bins, cane age, a non-interrupted supply of full bins to the

mill, and the capacity of the sidings.

ACRSS has since been upgraded to be more efficient and easy to use than it was

initially, but it still has many limitations. Firstly, this scheduler depends on iterative

techniques which produce feasible, but not optimal, solutions. ACRSS does not

include all sugarcane transport system constraints such as train passing constraints,

6

having two harvesters at one siding, and different speed and load limits on different

sections of track. The required manual modifications to the produced schedule take a

long time to undertake and cannot be used to prepare operating schedules on a daily

basis.

1.2.2 Sugarcane Transportation Systems The sugarcane transport system is a complex system that includes a large number of

variables and elements. These elements work together to achieve the main system

objectives which aim to satisfy both mill and harvester requirements and improve the

efficiency of the system in terms of low overall costs. These costs include delay,

congestion, operating and maintenance costs.

The main focus for recent studies into how to reduce the overall cost of the

sugarcane transport systems can be divided into: the sugarcane rail transport

systems; The Cane Transport and Harvesting Integration, in particular transportation

and harvesting; and road transport systems.

Integration of the transport and harvesting elements in the value chain of the

Australian sugar industry can increase the performance efficiency of the transport

system. For example, optimising the delivery and collection times throughout the

rail system requires information about harvester start times, harvesting rate, the

number of harvesters in the systems, and their locations. Harvester breakdown can

interrupt the transport system.

Cane is transported from the farms to the sidings using trucks. It is useful therefore

to study the earlier work on cane road transport systems to develop new models

which optimise the delivery and collection times between harvester locations and

sidings. This optimisation can further optimise the rail transport system between

sidings and the mill. Developing cane road system models and integrating these with

the rail models and road models, is beyond the domain of this current study. These

issues can be undertaken as future research.

7

1.2.2.1 The Sugarcane Rail Transport

The Australian sugarcane rail transport system is a unique system. Rail is uncommon

outside Australia for transporting sugarcane from farms to mill. Therefore, there are

very few studies that relate directly to sugarcane rail systems.

Everitt and Pinkney (1999) described an integrated set of tools to manage the

performance of a cane transport scheduling system. This work was designed to

achieve integration between the schedule simulation program, namely the animated

cane transport scheduling system (ACTSS), the schedule generating program,

namely the automatic cane railway scheduling system (ACRSS), and Traffic Officer

Tools (ToTools), (McWhinney & Penridge, 1991). ACRSS software used the data

of railway networks and harvesting patterns to produce efficient schedules for

reducing the number of locomotive movements. ACTSS software gave flexibility to

modify and develop the schedules. ToTools allowed arranging and managing all

daily operations in the system such as the distribution of harvesters to help the traffic

officer. Since all these software programs used the same data and similar methods of

processing, the integration process between them saved a lot of time in modelling

and improved the performance of the system. In addition, using the global

positioning system (GPS) in transport systems gave more data such as trip times,

speeds of trains, and shunt times at sidings. This data helped to better define system

parameters so that ACRSS and ACTSS could produce more effective schedules.

Martin et al. (2001) proposed using Constraints Logic Programming (CLP) to solve

cane railway scheduling problems. CLP is an artificial technique using the ECLiPSe

language to solve large-scale problems. ECLiPSe is a Prolog language with extra

features to suit for the cane railway system. This technique produced daily schedules

to minimize the number of trains and their runs, satisfy the constraints of the system

such as siding capacity and the characteristics of bins on a train.

Higgins and Davies (2005) introduced a simulation model for the capacity of sugar

cane transport systems in Australia. This capacity was determined by estimating

some variables such as the number of trains used in the system and their movements,

number of bins, and the time spent waiting for the empty bins at farms. The

transportation modelling divides into two main types, the first of which is the static

8

model. In static models all variables are assessed and assumed beforehand. This

applies to variables such as the amount of time spent between cutting and crushing

cane, the demand for bins and trains, and the time spent waiting for empty bins at the

farms. This model has been previously used in research to develop a transport

system. The second type of transportation modelling is the dynamic model. Dynamic

models produce many schedules which are suitable for handling unexpected events

in the transport system. These models integrate with the fundamental basics of

stochastic simulation which help these models to be more flexible and easy to use.

Furthermore, these models help in finding the link between all parameters in the

system of sugar cane transport such as the average time spent between cutting and

crushing of cane, train movements per day, and time spent waiting for bins at siding.

1.2.2.2 The Cane Transport and Harvesting Integration The integration between sugarcane value chain elements, particularly transport and

harvesting elements, was the main focus for the most recent studies on how to reduce

the overall cost of the system. Many techniques and methods were used to achieve

this purpose.

Higgins et al. (2007) concentrated on the sugar value chain elements, especially in

the harvesting and transport sectors. The integration between the value chain

elements improves the performance of the sugar production system and reduces the

overall cost. This research focused on logistical and non-logistical integration

between the value chain elements. Mathematical techniques were used to improve

the integration between value chain elements. This study included many countries

such as Brazil, South Africa, Thailand and Australia as case studies and exposed

many techniques which were used for each sector.

Grimley and Horton (1997) proposed a mathematical model to reduce the total cost

of the harvest and transport systems using optimization techniques. They analysed

the whole system to understand all components such as transportation networks,

costs for all parts of the system, and the combination of groups of harvesters. In the

next step, they defined the interactions and relations between some sectors such as

transport, harvesting, crushing, inventory, and their effects on the total cost. Then

they used mathematical programming and operations research techniques to obtain

9

the best solutions based on the analysis of collected data. They developed two

models: first, the rotation model to determine the harvesters’ schedules, transport

capacity, and mill’s needs to keep continuous supply; and second, the daily model to

process the rotation model output to produce optimal schedules of trains and bin

shifts. The two models were formulated by mixed integer programs and used AMPL

modelling language and CPLEX solvers. The model of Grimley and Horton did not

prove practical for use by sugar mills and is not in use.

Higgins et al. (2004a) developed a modelling framework to improve the sectors of

transport and harvesting efficiency via two case studies – one of Plane Creek and the

other of Mourilyan in North Queensland. The development of a modelling

framework depended on the integration between activities of the harvesting and

transport sectors in the sugarcane industry. The study found that the location of

farms affects the number of train movements and cane bin demand which has a huge

effect on the overall transport cost and on other components in the system.

Higgins (2004) concentrated on the reduction of costs of harvesting sugar cane and

transport by building a model for optimising siding use by harvesting groups. This

model was able to achieve best utilization of rail capacity, reducing the movement of

harvesters between the sidings, and achieving satisfaction and fairness for growers.

He used some advanced methods which were very useful in producing efficient

siding schedules and at the same time kept the mills operating without interruption.

These methods depended on metaheuristic techniques such as tabu search. Tabu

search is more efficient than other local search heuristics because of its flexibility in

treating any expected or unexpected changes of data or of constraints to the system.

Higgins et al. (2004b) designed models for three regions in Australia to produce

optimal sugarcane harvest schedules. These models depended on coordination

between growers, harvesters, and mills. They developed software that was used to

help the implementation process. These models achieved an increase in profits for all

three regions.

Higgins and Laredo (2006) developed a modelling framework to integrate the

transport and harvesting sectors to reduce the total cost of production. The two

sectors have many elements which can integrate inside the framework such as siding

and harvester rosters, and transport capacity planning. They used P-Median and

10

spatial clustering formulations to build these models. Neighbourhood search

techniques were used to solve their model. This work was applied to a sugar area

located in the north-east of Australia and achieved good results in reducing the

overall cost of the system. In addition, Higgins and Laredo could reduce the number

of sidings and harvesters by increasing the period of harvesting to 24 hours per day.

Iannoni et al. (2006) applied discrete simulation techniques to manage and analyse

the performance of the harvesting and transportation sectors in Brazil. Colin (2009)

used mathematical programming (quadratic programming) to develop a new model

for plantation planning and scheduling. The main aim of this model was to increase

the efficiency and competitiveness of Brazilian agro industries (integrating the

agricultural and industrial operations) such as sugarcane, orange and wood and

reduce the deviation from the constructed plan.

Salassi and Barker (2008) developed a framework to reduce harvest costs by

integrating the transport and harvesting sectors in Louisiana. They proposed a

mathematical model to minimise the total waiting time at the mill. This model

considered transport and harvesting variables and the delivery times of the crop.

Barker (2007) designed a linear programming model to obtain optimal harvest

schedules for groups of farms by combining the harvest units. Minimising the

waiting times at the mill can reduce the overall costs of the production system. The

coordination of harvest schedules through groups of farms has improved the

transport efficiency.

Kaewtrakulpong (2008) used multi objective optimisation techniques to improve the

efficiency of harvesting and transport systems. He used an integer programming

formulation to define the truck allocations and simulated the harvesting and transport

system. In this research he clarified the relation between mechanical harvesting and

the transportation processes in Thailand and optimised the harvesting groups’

allocations to reduce the total cost of the production. The old system in Thailand was

based on delivery directly from farms to mills.

Many papers have been presented to increase the efficiency and profitability of the

cane supply chain. These studies concentrated on the integration of farmers and

millers (Le Gal et al., 2004). Lejars et al. (2008) proposed a decision support

11

approach to improve the management of sugarcane supply through the mill area.

This approach depended on the MAGI simulation tool, modelling tool for sugarcane

supply from field to sugar mill, ( Le Gal et al., 2003). In this paper, they suggested

some ways to increase efficiency such as rearrangement of mill areas or changing

cane delivery allocation rules to increase the total sugar production and the total

revenue. This approach has been implemented for two mills in Reunion and one mill

in South Africa. Integration of supply chain elements such as the mill crushing rate,

harvest mechanism, and transport capacities can improve sugarcane supply

management, increase the mill sugar production and achieve high quality sugarcane

in terms of the recoverable value of sugar as a percentage of the sugarcane mass (Le

Gal et al., 2009; Le Gal et al., 2008).

Grunow et al. (2007) designed optimisation models to keep a constant supply of

sugarcane while minimising the associated costs. They divided the sugarcane supply

problem into three levels: plantation planning, harvest scheduling, and harvesters

groups and machines. They found the optimal decision for each level by using

Mixed Integer Linear Programming (MILP) to decrease the costs.

1.2.2.3 The Sugarcane Road Transport Systems

The road transport systems have very different constraints to rail transport

systems. Developing models for such cane road systems could be useful for future

project to optimise the delivery and collection time between harvester and siding.

Models have been developed in earlier work by a variety of researchers. The main

llimitation of these models is there is no integration between these models and the

rail transport models to optimise the sugacane transport systems.

A simulation model was developed by Hahn and Riberio (1999) with a heuristic

technique to improve the performance of the sugarcane road transport system. They

used a software package, System Simulator for the Transport of Sugarcane (SSTS) to

solve this model including a minimal allocation heuristic technique. Díaz and Pérez

(2000) arranged and optimized sugar cane harvest and road transport processes using

a simulation optimization approach. This technique was based on the combination

of simulation modelling, response surfaces and optimization techniques to determine

the best solution. In this paper the response surface method was used to find average

http://www.sciencedirect.com/science/article/pii/S0168169907001822#bbib3

12

trip times for sugar cane transport. They used the output of a simulation model as an

initial input for optimization methods to reach the near optimal solution.

Lopez et al. (2006) designed a mathematical model to reduce the daily cost of sugar

cane road transport from farms to the mill in Cuba. The model estimated the

capacity of the transport system which helped the supply of full bins to mills to be

continuous without interruptions. The mathematical formulation of this model

depended on a mixed integer linear programming method. The advantage of this

model was how it dealt with changes which happen from one day to the next. For

instance, the number of working hours and means of cutting and transportation may

all change so the model may need reformulation daily. Also, the number of farms,

harvesting machines and transportation means may change, so the decision variables

will change. Therefore, they used a tailor-made software package to solve these

problems with a high number of variables, thus making the daily update easier and

saving a lot of time. This paper also examined two different methods for the

transportation of cane: direct transportation between farms and mills, and

transportation of cane from farms to intermediate stores at stations to be cleaned and

subsequently carried to mills. The choice between these methods depended on the

location of farms and mills.

Arjona et al. (2001) developed a simulation model of sugarcane harvesting and road

transportation systems including processes from cutting the crop at farms to its

movment to mills. This model aimed to reduce the number of harvesters, and all

transport means without increasing the end time of all stages of the system. They

considered all activities of the system independent and used SIMACT (Simulation of

Activities) software to design this model. SIMACT software was written in the C++

language. The results of this paper showed that this model made the system more

profitable and more efficient.

Higgins (2006) built a new model of a sugar cane road transport system. This model

was based on a mixed integer programming method. He solved this model using two

metaheuristic techniques: tabu search (TS) and variable neighbourhood search

(VNS). He produced transport schedules to reduce waiting times in the transport

system and at the mill. In this research, he proved that this model was more efficient

13

than manual methods used by a mill traffic officer when he applied this model in the

Maryborough region in Australia.

Chetthamrongchai et al. (2001) designed a new road transport scheduling system for

sugarcane in Thailand. Loading stations were constructed between farms and mills to

reduce the cost of cane transport especially for the small sugarcane growers. These

areas are located close to the growers in order to ease the supply of cane to the mills.

1.2.3 Complexity of the Sugarcane Rail Transport Systems The current sugarcane rail transport system has two issues which make it complex-

the first issue is rail complexity and the second issue is the complexity of the

sugarcane system itself. Rail complexity refers to the large number of conflict points

that exist throughout the rail network and impede the safe passing of trains. The

complexity of the sugarcane system itself derives from the fact that there are

numerous points situated throughout the rail network which serve dual roles as

sidings for delivering and collection and passing loops. This complexity causes

inefficiencies in system operations. Sections 1.2.3.1 and 1.2.3.2 examine theses

complexity issues in detail.

1.2.3.1 Rail Complexity Single track railway systems use blocking constraints to achieve safe operations.

These blocking constraints work by preventing more than one train from occupying a

track section, a length of track between two key points in the track network as

explained in Chapter 4, at the same time and help in resolving conflicts throughout

the railway network. These constraints work with many types of railways such as

mining or freight railways. However, the blocking constraints are not always

effective in a cane railway network. Some sections are too short and the distance

between two sidings can be less than the length of the train. As a result, in this case,

blocking a section is not sufficient to satisfy safety requirements and this may cause

accidents. Additionally, cane railway networks include many segments (branches or

lines). Some of these branches do not have passing loops. As a result, complete

branches are sometimes used as substitute passing loops. A segment blocking

approach is considered in the proposed models in Chapter 4 (Modelling the

14

Sugarcane Rail Transport Systems Problem) to deal with this type of railway. All

sections in a segment are blocked at the same time preventing other trains from

simultaneously using them at the same time.

The different situations throughout the rail network of the sugarcane system are

shown in Figures 1.2, 1.3 and 1.4. Two sections are included in Figure 1.2, where the

length of the train is less than the length of each section. The two sections work as

sidings for delivery and collection. In that case, the blocking section constraints can

be applied correctly when the length of train is less than the length of the section.

Section length Train head Train tail Section occupation time by train length

Section1

The length

of train k

Section 2

Time

Figure 1.2: The length of train less than the length of section

Figure 1.3 shows that the train length is greater than each section length. In that case,

the blocking section constraints are not sufficient enough to meet the safety

requirements, where the train length is greater than each section length. As a result,

the segment blocking constraints are required to solve any conflicts and satisfy the

safety requirements.

15

Section length Train head Train tail Section occupation time by train length

Section 1 Section 2 The length of

train k

Time

Figure 1.3: The length of train greater than the length of section

Figure 1.4 shows that section 1, section 2 and section 3 are included in segment1,

and the length of the segment1 is greater than the length of the train. As a result, the

blocking segment constraints can be applied successfully to solve the conflict in

Figure 1.3. Segment length Train head Train tail Segment occupation time by train length

Section 1

Segment 1

The length of

Section 2 train k

Section 3

Time

Figure 1.4: The segment length is greater than the train length

Sections 1, 2 and 3 in Figure 1.4 cannot be joined as one long section if sections 1

and 3 contain two sidings where bins need to be delivered or collected. The two

sections must be considered as two separated sections included within one segment

16

and applying blocking segment constraints. This issue is discussed in more detail in

the blocking segment models in Chapter 4.

1.2.3.2 Sugarcane Systems Complexity The biggest difference between a sugarcane rail system and other rail systems such

as coal mining is in delivering and collecting operations. Sugarcane systems have

many sidings in the middle of rail network branches which work as delivering and

collecting points (Figure 1.5). These sidings are blocked during the delivering and

collection operations. The blocked sidings in the middle of the rail network increase

the number of conflict points and reduce passing opportunities.

Figure 1.5: Delivering and collecting point at the middle and the end of the branch (sugarcane)

The coal rail network on the other hand generally has sidings for delivering and

collecting at the end of the network only (Figure 1.6) and the train travels to the end

of the line and returns. While these sidings are also blocked during the delivering

and collecting operations, the blocked sidings do not increase the conflict points

because the blocked siding at the end does not work as a passing section for other

trains.

17

Figure 1.6: Delivering and collecting point at the end of the branch (coal mining)

Having sidings at the end of rail network branches reduces the number of options for

the train after delivering and collecting to one. This option is for the train to return

because the branch has ended and has no further sections. As a result, the number of

the model constraints can be reduced as well. On the other hand, having sidings in

the middle of rail network branches provides options for the train.

These options are that the train delivers:

all empties to the siding and then has to return.

all empties to the siding but the train has to collect full bins from some

sidings on the way back.

some empties at the siding and delivers the remaining empty bins to other

sidings further along the branch.

Other options include the train collecting:

full bins and its capacity is reached.

some full bins from one siding and another siding at the same branch recalls

it to collect full bins without its delivering any empty bins.

18

Additional issues such as arrival time, departure time, and waiting time to deliver

and collect at each siding make the decision making even more complex. The

combination of all these options, at all sidings, can produce a final set of operations

for each train during its run. The combination of all sets of operations for all runs for

all trains can then produce an optimal schedule for the entire system.

Increasing the number of sidings through each branch increases the number of

decision points (delivering and collecting points) which in turn increases the number

of conflict points throughout the rail network making the system even more

complicated. A good decision at each siding can optimise the efficiency of the entire

sugarcane system.

An optimal decision for one run can be produced by evaluating (D1×D2×…DS)

decisions, where Ds is the average number of decisions at each section s and S is the

total number of rail sections, including the siding points. This optimal decision for

one run covering 13 sections and one mill as illustrated in Figure 1.5 can be

produced by evaluating all combinations of the number of decisions, and the number

of sections, including the mill. An objective function can be used to evaluate these

decisions. The total combinations of the decisions in the outbound and inbound

directions which are explained in Table 1.1 and 14 sections (13 sections+ 1 mill) are

calculated as follows:

(D1×D2×…D14)= (10×8×11×8×12×8×7×12×7×8×8×12×8×8)

=1.95×1013 decisions / run.

The total number of the decisions and their evaluated combinations which produce

an optimal decision for the system, including the number of trains and runs is

calculated from:

The total combination of the decisions= ((D1×D2×…DS) ) ×K×R, (1.1)

Where K is the total number of trains and R is the total number of runs for each train.

19

Equation (1.1) indicates that increasing the number of sections or sidings in the

system can increase the number of decisions sharply. Additionally, the numbers of

trains and runs are significant factors for increasing the complexity of the sugarcane

rail system. For instance, for a system including 2 trains and 5 runs for each train for

the network shown in Figure 1.5, the total decisions are:

1.95×1013 × 2×5 =1.95×1014 decisions.

Table 1.1 describes the decision matrix of the sugarcane system based on the

decisions of the train at each section in Figure 1.5. The two directions, outbound and

inbound, for each train are shown in Table 1.1, where the delivering and collecting

operations are in the outbound and inbound directions respectively, so as to reduce

the operating time of the trains. The decision of whether or not to visit other sidings

is taken at each siding but not in a single section. The decision to begin a new run is

made at the mill only.

This analysis of the complexity of the sugarcane rail transport system indicates that

the optimisation problem is strongly NP-hard and needs computational complexity to

produce the optimal solution of an efficient schedule.

20

Table 1.1: Decision matrix of the rail network shown in Figure 1.5

D: there is a decision

n/a: not applicable

Dec

ision

Ty

pe

Options

Outbound direction

Inbound direction

M s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 M s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13

Del

iver

ing

and

colle

ctin

g ac

tiviti

es

No activities D D D D D D D D D n/a D D D n/a D D D D D D n/a D n/a n/a D D D n/a

Only collecting D n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a D n/a D n/a n/a D n/a D n/a D n/a D

Only delivering n/a n/a D n/a D n/a n/a D n/a D n/a D n/a D D n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

Delivering & collecting n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

Vis

it tim

e Arrival time n/a D D D D D D D D D D D D D D D D D D D D D D D D D D D

Departure time D D D D D D D D D D D D D D n/a D D D D D D D D D D D D D

Waiting time D D D D D D D D D n/a D D D n/a D D D D D D D D D D D D D D

Run

de

cisi

on

Visit other sidings D n/a D n/a D n/a n/a D n/a n/a n/a D n/a n/a n/a n/a n/a n/a D n/a n/a D n/a D n/a D n/a D

New run n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a D n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a

21

1.2.4 Research Questions

The complexity of the sugarcane rail transport system means there are many research

questions to be investigated in this thesis.

Question 1 Is a mathematical formulation of the sugarcane transport system necessary? The sugarcane transport system is a complex system that involves many variables

and elements. Mathematical modelling provides many benefits in terms of accuracy

of results, prediction of urgent and future problems and subsequent solutions which

are faster than some other techniques.

The following questions are therefore investigated:

Why do we need a quantitative model?

How can the sugarcane transport problem be formulated as an application of

operations research?

How can the best solution be obtained for the sugarcane rail transport problem?

Question 2

What are the capabilities and accuracy of Constraint Programming (CP) and Mixed

Integer Programming (MIP) techniques to solve sugarcane transport system

problems under job shop scheduling fundamentals?

This study investigates the ability of CP and MIP techniques by designing new

models which have capabilities to produce efficient schedules, and so improve the

performance of the sugarcane transport system and reduce overall costs.

As such, the following questions are investigated:

What is the quality of the solution?

What is the CPU time of the solution?

22

Question 3

What is the impact of integrating constraint programming search techniques with

mixed integer programming and constraint programming models on the quality of

the solution of the sugarcane transport system problem?

This research investigates the impact of integrating CP search techniques such as

Best First Search Technique (BFS), Depth-First Search Strategy (DFS), Slice Based

Search (SBS), Limited Discrepancy Search (LDS), Depth-bound Discrepancy Search

(DDS), Interleaved Depth First Search (IDFS), Standard Search Strategy (SSS) and

Dichotomic Search Strategy (DSS) with mixed integer programming and constraint

programming models on the quality of problem solutions. The sugarcane system

includes an extensive number of variables. Therefore, constructive techniques are

required to deal with all variables and produce good solutions in a reasonable time.

The following questions are investigated:

What is the quality of the solution?

What is the CPU time of the solution?

What are the effects on the overall cost of the sugarcane rail transport

system when the new model is applied?

Are the necessary safety conditions achieved during transporting of bins

between farms and mill, without increasing the transport cost?

Question 4 How can the new models achieve flexibility and deal with new or urgent situations in

the sugarcane transport system?

The sugarcane transport system is a dynamic system because new and sometimes

urgent situations arise daily such as changes in the number of bins, trains and labour.

For this reason, flexible models are required which can deal with these changes and

achieve the objectives. For example, if the number of the trains is reduced, the need

to minimise the total completion time has a high priority, while if the number of

trains is large, the priority of the objective function will be to minimise the total

waiting time. These models can obtain solutions for cases where the objective

function changes between criteria such as minimising makespan, total waiting time,

23

and total completion time of the train run per day. Therefore the following are

investigated:

What are the effects of the changing objective function on selecting the

solution technique?

What are the impacts of changing of the objective function on the optimality

of solution or CPU time?

What is the impact of changing the objective function on the overall costs of

the system?

What are the impacts of changing the number of trains and harvesters on the

optimality of solution or CPU time?

Question 5 What is the impact of hybrid and hyper metaheuristic techniques on the solution for

the sugarcane transport system problem, in particular for large scale problems?

Exact solution techniques cannot solve large-scale problems in a reasonable time.

Therefore, metaheuristic techniques such as simulated annealing (SA) and tabu

search (TS) are used to reduce the CPU time of large-scale problems. Hybrid and

hyper metaheuristic techniques are proposed in this research to improve the solutions

of SA and TS.

Therefore the following will be investigated:

What is the impact of the SA technique on the quality of the solution?

What is the impact of the SA technique on the CPU time of the solution?

What is the impact of the TS technique on the quality of the solution?

What is the impact of the TS technique on the CPU time of the solution?

What are the impacts of hybrid metaheuristic techniques such as hybrid

SA/TS and hybrid TS/SA hybrid techniques on the quality and CPU time of

the solution?

What are the impacts of hyper metaheuristic techniques such as hyper SA/TS

and hyper TS/SA on the quality and CPU time of the solution?

24

1.3 Contribution and Significance of the Study As already identified, the sugarcane transport system has a significant effect on the

sugar production process and its overall cost. In Australia, the total cost of

sugarcane transport operations is high, so it is important for this study to develop

efficient schedules for the sugarcane transport system to optimise its performance.

The application of optimal schedules to the sugarcane rail transport system will:

Ensure a continuous supply of full bins to the mill, and empty bins to the

harvesters.

Remove train scheduling conflicts.

Minimise the time between sugarcane harvests at farms and the crushing

operation at the mill (the cane age) by optimising delivery and collection

times.

Minimise the waiting time of trains at sidings so that trip time is

decreased between farms and mills.

Maximise the capacity of the railway system.

Minimise the number of locomotive shifts and so reduce the number of

crews.

New mathematical models are developed using the Mixed Integer Programming

(MIP) and Constraint Programming (CP) approaches. MIP and CP approaches

under Blocking Parallel Job Shop Scheduling (BPJSS) fundamentals will be used to

solve the sugarcane transport system problem. Other applications of these models

include systems such as the coal mining rail transport system. This study focuses on

many types of integration of MIP and CP search techniques. Heuristic and

metaheuristic techniques are developed to solve large scale problems. The

integration types and solution techniques followed in this project are outlined below:

i. Mixed Integer Programming MIP: The MIP formulation as a model and

CP search techniques are integrated to combine the power of both to

improve the solution and the solutions’ CPU time, especially in large-

scale problems. MIP models are developed using two different blocking

constraints, segment and section blocking, and the parallel job shop

scheduling approach. OPL solver will be used to obtain a solution for

25

MIP to minimise makespan. The CPLEX MIP Solver will be used to minimise the

total waiting time as an objective function. This Solver includes the CPLEX MIP

algorithm and a branch and bound algorithm which uses cooperation between a

simplex algorithm and constraint-based domain reduction.

ii. Constraint programming CP: The CP formulation as a model and CP search

techniques as solution techniques are used. CP models are developed under

segment and section blocking constraints. OPL Solver and Scheduler Solver will

be used to obtain solutions.

iii. Standard Search Strategy (SSS) and Dichotomic Search Strategy (DSS) are

integrated with all CP search techniques such as BFS, DFS, SBS, LDS, DDS, and

IDFS with the MIP and CP models to reduce the CPU time.

iv. Algorithms are developed to solve the rail network issues such as algorithms for

collecting and delivering conflict elimination, algorithms for solving train

conflict, and algorithms for computing acceleration.

v. Metaheuristic techniques, SA and TS, are adapted to solve large-scale problems.

The C# language is used to develop the metaheuristic code.

vi. New neighbourhood techniques are developed based on the section and train

neighbourhoods

vii. Hybrid and hyper metaheuristic techniques, hybrid SA/TS, hybrid TS/SA, hyper

SA/TS and hyper TS/SA, are developed to improve the quality and the CPU time

of the metaheuristic solutions. Two hyper techniques were developed using C#

language.

The new models have the flexibility of changing the objective function based on the

priority of the objectives in the sugarcane transport system. For example the

objective function in this research can minimise makespan, total waiting time or total

completion time based on the needs of the system.

26

1.4 Outline of the Thesis

The main research plan of this thesis is shown in Figure 1.7 and is explained in

details in seven chapters. Chapter 1 describes the research problem. Chapter 2

presents the overview of the main scheduling approaches, in particular job shop

scheduling, and different solution techniques, such as disjunctive graph and hybrid

branch and bound technique, to obtain optimal solutions. Chapter 3 describes

Constraint Programming (CP) and Mixed Integer Programming (MIP) models for

the sugarcane rail transport system, including train scheduling and sugarcane system

constraints. The extensions of the mathematical models are presented in Chapter 3 to

optimise the real visit times at the sidings for real sugarcane rail transport systems,

avoiding any interruption in the supply of bins to harvesters. Chapter 4 shows the

different solution techniques and how they can be applied to solve the sugarcane rail

transport system under blocking constraints. Many new algorithms and heuristics are

developed to solve the proposed mathematical models. Chapter 5 presents the

computational results of the CP and MIP models for sugarcane rail transport systems

under blocking constraints. Optimization Programming Language (OPL) and

CPLEX are used to obtain results from the CP and MIP models. This chapter

develops a timetable for the sugarcane rail transport system. Chapter 6 reports the

metaheuristic techniques of simulated annealing (SA) and tabu search (TS) that are

used to solve the sugarcane rail transport scheduling problem. The hybrid and hyper

metaheuristic techniques developed to improve the quality and CPU time of the

solutions are also presented here. Visual Studio C # language was used for coding

the search problem and obtaining the results. Finally, conclusions and

recommendations for future work are presented in Chapter 7.

This chapter has briefly outlined the background to the research problem and has

demonstrated the need for new models that optimise the performance of the

sugarcane system which in turn reduces the total cost of the transport process

between farms and mills. The set of investigative questions along with the main

research problem have been identified. The significance of the research has also been

highlighted.

27

Figure 1.7: Research plan

Sugarcane Rail Transport System

Scheduling Approaches

Job Shop Scheduling Approach

Constraints Programming CP

Metaheuristic Techniques

Hybrid Techniques

Applicable for small size problems

Mixed Integer Programming MIP

Disjunctive Graph

Applicable for small size problems

Simulated Annealing

(SA)

Tabu Search (TS)

Final Outputs and Results

Justification of Final Model for Real Life Problems

Improve the solutions

Verify and validate

Applicable for large size problems

Improve the solutions

Branch and Bound

Hybrid TS/SA

Hybrid SA/TS

Hyper TS/SA

Hyper SA/TS

Blocking Segment Model

Blocking Section Model

Blocking Segment Model

Blocking Section Model

CPLEX Software

OPL Software

C# Language

Search Techniques

MIP Models CP Models

Dichotomy Search Strategy

Best First Search

Depth First Search

Slice Based Search

Depth Bound Discrepancy Search

Interleaved Depth First Search

Standard Search Strategy

Hybrid and Hyper Techniques

Published paper Published paper Published paper

Published paper

Paper under review

Paper under review

28

Chapter 2

Scheduling Theory Review

Chapter Outline

2.1 Introduction .....................................................................................................................29

2.2 Scheduling Classifications................................................................................................29

2.2.1 Job Characteristic..............................................................................................30

2.2.2 Machine Environment.......................................................................................30

2.2.2.1 Single Machine Scheduling (SMS)..............................................32

2.2.2.2 Parallel Machine Scheduling (PMS)............................................33

2.2.2.3 Flow Shop Scheduling (FSS).......................................................34

2.2.2.4 Job Shop Scheduling (JSS)...........................................................34

2.2.2.5 Open Shop Scheduling (OSS)......................................................36

2.2.2. 6 Group Shop Scheduling (GSS)....................................................36

2.2.3 Optimality Criteria............................................................................................37

2.2.4 Concluding Remarks..........................................................................................38

2.3 Job Shop Scheduling (JSS) Solution Techniques.............................................................39

2.3.1 Disjunctive Graph Technique...........................................................................41

2.3.1.1 Heuristic Techniques......................................................................46

2.3.1.1.1 The Shifting Bottleneck Algorithm.................................46

2.3.1.1.2 Dispatching Rules............................................................46

2.3.2 Mixed Integer Programming Approach for JSS Problem..................................47

2.3.2.1 A hyper Branch and Bound Technique...........................................48

2.3.3 Constraint Programming Approach for the JSS Problem...................................53

2.4 Scheduling Theory and Railway Systems in Literature...................................................56

2.5 Conclusion........................................................................................................................60

29

2.1 Introduction

In this research, scheduling theory, job shop scheduling in particular, is adapted to

scheduling the sugarcane rail system. Many problems in real life are classified as

scheduling problems; where scheduling is defined as the allocation of resources over

time to perform a collection of tasks (Baker, 1974). Each resource (machine) has its

own features and configurations while each task (job) can be described by its

processing time, resource requirement, start and finish time. Resources can be

equated to machines in a job shop scenario and tasks can be equated to jobs.

Scheduling makes decisions by optimising one or more objectives. These decisions

are called solutions. The easiest approach to obtain the best solution is to evaluate all

solutions and then select the best. The main problem with this approach is that the

number of tested solutions grows exponentially with the size of the problem (in a

combinatorial optimisation problem). As a result, the computation time of this

approach increases as the size of the problem grows, making it harder to solve these

problems in reasonable time, even in small cases. Many heuristic and metaheuristic

techniques have been developed to solve large-scale problems. The main core of

these techniques is to reduce the solution space to reach good solutions in a short

time. Many scheduling problems are hard to solve and are called NP-hard

(nondeterministic polynomial time), where no algorithm can solve these problems in

polynomial time (the solution can be provided in predetermined and finite number of

steps) (Lawler et al., 1993).

.

2.2 Scheduling Classifications

In the scheduling environment, there are many forms and classifications which

include three main elements: job characteristics; machine environment and

optimality criteria (Pinedo, 2008).

30

2.2.1 Job Characteristic The first element in the scheduling problem is job characteristic where for each job j

the following features can be obtained;

Processing time (gij): the required time for processing job j on machine i.

Ready time (rj): the arrival time of job j in the system for processing.

Due date (di): the time by which job j should be completed.

Weight (wi): the weight (wi) is the factor for determining the priority of execution

for each job i relative to the other jobs in the system

Additionally, its setup time does or does not allow any pre-emption. Pre-emption

(the processing of any job on a machine that can be interrupted at any time and that

allows for the processing of another job on the same machine) is allowed or not,

depending on arrival time, and departure time of other jobs. These job features can

affect the definitions of the scheduling problem types and the complexity of each

type. For example, jobs which depend on processing time only are easier to deal

with than those which depend on processing time and due date. That is, increasing

the number of features of any job can raise the complexity of the scheduling

problem. Furthermore, increasing the number of jobs inside the scheduling problem

can increase the complexity of the problem. For example, dealing with problems that

include a limited number of jobs is easier than dealing with problems that include an

unlimited number of jobs.

2.2.2 Machine Environment

The machine environment includes the machines in the system and the features of

these machines. Figure 2.1 illustrates the three main machine types.

31

Figure 2.1: Scheduling problem types

Open Shop Scheduling

OSS

Flow Shop Scheduling FSS

Single Machine

Scheduling SMS

Group Shop Scheduling

GSS

Limited Buffer

Flow Shop Scheduling

LBFSS

Unlimited Buffer

Flow Shop Scheduling ULBFSS

Multi Machine

Scheduling MMS

Job Shop Scheduling

JSS

Blocking Job

Scheduling Shop

BJSS

Classical Job Shop

Scheduling CJSS

Blocking Group shop Scheduling

BGSS

Limited Buffer Job

Shop Scheduling

LBJSS

Blocking Flow Shop Scheduling

BFSS

Scheduling Approaches

Limited Buffer Group

Shop Scheduling

LBGSS

Unlimited Buffer Job

Shop Scheduling ULBJSS

Parallel Machine

Scheduling PMS

32

• Single Machine Scheduling (SMS): includes one machine or single

production unit.

• Parallel Machine Scheduling (PMS): A parallel machine system includes

identical machines that are working in parallel. Each job can be processed by

any one of the free machines.

• Multi Machine Scheduling (MMS): includes many scheduling forms such as

flow shop scheduling, job shop scheduling (JSS), open shop scheduling

(OSS) and group shop scheduling (GSS).

Each type is shown as follows:

2.2.2.1 Single Machine Scheduling (SMS)

Single machine scheduling has n jobs; j=1... n to be processed on a single machine

m1 with the same route as shown in Figure 2.2.

Figure 2.2: single machine scheduling problem; n jobs processed on one machine

The general formulation for the single machine scheduling problem for minimising

the total weighted completion time is:

Min 1

n

j jj

w C=∑

Where, CRj R and wRjR are completion time and weight respectively for the j P

th Pjob.

j1

j2

j3

• •

jn

m1

Ready time

33

2.2.2.2 Parallel Machine Scheduling (PMS)

Parallel machine scheduling has n jobs; j=1... n to be processed on m parallel

machines with the processing time gj for each job j as shown in Figure 2.3.

Figure 2.3: Parallel machine scheduling problem; n jobs processed on m parallel machines

The makespan (Cmax ), the completion time of the last operation of last job in the

system, as an objective function of the parallel machine scheduling problem

(P││Cmax) , is formulated as:

1

1,.., .n

maxj

y g C i mi j j=

≤ ∀ =∑

Where,

1 0

if job j is processed on machine iyi j otherwise

=

j1

j2

j3

• •

jn

m1

m2

m3

m4

mm

•

•

Ready time

34

2.2.2.3 Flow Shop Scheduling (FSS)

All jobs in flow shop scheduling have to be processed on all machines in the same

sequence. n jobs are processed on m machines in a fixed order and as a result all jobs

have the same route as shown in Figure 2.4.

Figure 2.4: Flow shop problem; n jobs processed on m machines

There are two main types of flow shop scheduling problems that depend on the

buffer capacity between successive machines. The two types are: flow shop with

unlimited buffer capacity and flow shop with limited buffer capacity. A flow shop

with unlimited buffer capacity (Fm|| Cmax) is suitable for many practical applications

especially when the products are small. The flow shop is called permutation if the

order of jobs on the machines does not change.

Flow shop with limited buffer capacity (Fm| block | Cmax) means storage space

between two successive machines is limited. A flow shop with a limited buffer

capacity means that machine (m) might be blocked when it finishes job (j) because

(j) cannot find empty space while waiting for processing on the next machine. This

means (j) must remain with (m). This can prevent other jobs waiting in the queue for

(m) from being processed on (m).The flow shop scheduling problem is a special case

of job shop scheduling problems.

2.2.2.4 Job Shop Scheduling (JSS)

The job shop scheduling problem is NP hard and one of the combinatorial problems

in the scheduling field (Lenstra et al., 1979). The JSS problem assigns machines to

operations over time, where processing times (duration of tasks) are independent of

m1

j1

j2

j3

jn

• •

Ready time

m2

m3

m4

mm

• •

35

the schedule which optimises the given objective. In job shop scheduling problems,

each job has a predefined route which may be different from other job routes as

shown in Figure 2.5.

The classical JSS problem can be described as follows for a set of machines M where

M= {m1, m2, m3, m4…mm} and a set of jobs J, {j1, j2, j3…jn}. Each job ji∈ J has a

set of k operations where oi={o1i, o2

i, o3i…ok

i} and is processed in the same order, in

non-interrupted time (non-preemptive) on a unique machine. Each machine can

process, at most, one operation at a time. All operations follow the precedence

constraints, where the preceding operation has to finish before starting the

succeeding operation. Many objectives can be achieved by solving the job shop

scheduling problem, such as minimising total tardiness, mean tardiness, total flow

time and mean flow time, but minimising the completion time of the last operation,

or the makespan, is the most popular. These objectives are described in Section

2. 2.3 in detail.

Figure 2.5: Job shop problem; n jobs processed on m machines

The two main JSS problem types are blocking job shop problems and classical job

shop problems where the blocking constraints are not applied. Each machine is

blocked during the period of processing any operation, until this operation leaves

that machine. Limited buffers and unlimited buffers have to be considered while

applying the blocking constraints.

m1

m2

m3

m4

mm

j1: o1, o2, o3, o4... ok

m1

m2

m4

m3

mm

m2

m3

m4

mm

m2

m1

m3

m4

mm

j2: o1, o2, o3, o4... ok

j3: o1, o2, o3, o4... ok

• •

• •

• •

• •

m1

Ready time

• •

• •

• •

• •

• •

• •

jn: o1, o2, o3, o4... ok

36

2.2.2.5 Open Shop Scheduling (OSS)

The open shop scheduling (OSS) problems are similar to those in job shop

scheduling, but there are no precedence constraints between the operations of the

same job (no order between operations). This means no fixed order of the job

operations as shown in Figure 2.6. Gonzales and Sahni (1976) have proved NP-

hardness for the OSS problem.

Figure 2.6: Open shop problem; n jobs processed on m machines

2.2.2. 6 Group Shop Scheduling (GSS)

In group shop scheduling (GSS) problems, a set of operations that is processed on a

set of machines for each job is divided into a set of groups G1, G2…Gk, where k is

the number of groups. There are no restrictions on operations in the same group,

while the relations between operations of the different groups follow the priorities

and the constraints between the groups (see Figure 2.7). If the number of groups

equals one, then the GSS problem becomes an open shop scheduling problem, while

if each group includes one operation, the problem becomes a job shop scheduling

problem. As both JSS and OSS are special cases of GSS, GSS is an NP hard

problem. Blocking constraints can be applied with limited buffer capacity in group

shop problems.

m1

m2

m3

m4

mm

j1: o1, o3, o4, o2... ok

m1

m2

m4

m3

mm

m2

m3

m4

mm

m2

m1

m3

m4

mm

j2: o3, o2, o4, o1... ok

j3: o1, o4, o2, o3... ok

• •

• •

• •

• •

m1

Ready time

• •

• •

• •

• •

• •

• •

jn: o3, o2, o1, o4... ok

37

Figure 2.7: Group shop scheduling problem; n jobs include different groups on m machines

2. 2.3 Optimality Criteria

In scheduling problems, there are many different optimality criteria (objective

functions) that change from one application to another. These criteria depend on the

job features. Criteria are divided into:

Criteria that depend on the processing time

Completion time (Cj): The finish time of processing job j in the system.

Flow time (Fj): The time taken to complete job j; Fj=Cj-rj .

Maximum completion time (makespan); Cmax=max Cj for all j.

Mean flow time ( F ) = 1

1 n

jj

Fn =∑

Maximum flow time (Fmax) = maxj {Fj}.

Criteria that depend on the processing time and on the due date

Lateness (Lj): the time by which the completion time exceeds the due date of job

j; Lj=Cj-dj

Tardiness (Tj): Tj=max{Cj-dj,0} where tardiness is never negative.

m1

m2

m3

m4

mm

j1: o1, o3, o4, o2... ok

m1

m2

m4

m3

mm

m2

m3

m4

mm

m2

m1

m3

m4

mm

ko ...1o ,4o, 2o ,3o: 2j

ko ...3o ,2o, 4o ,1o: 3j

• •

• •

• •

• •

m1

Ready time

• •

• •

• •

• •

• •

• •

ko ...4o ,1o, 2o ,3o: nj

G1 G2 Gk

38

Maximum lateness (Lmax): Lmax=max{Lj} for all j

Total weighted completion time = 1

n

jw Cj j

=∑

Total weighted tardiness = 1

n

jw Tj j

=∑

Number of tardy jobs (NT): 1

( ),n

T jj

N Tα=

=∑

where

0, 0( )

1, 0j

jj

if TT

if Tα

== >

2.2.4: Concluding Remarks

Scheduling theory can be applied to solve many problems in real applications by

identifying the three elements of job characteristics, machine environment and

optimality criteria. In this case it is applied to a single track railway scheduling

problem by identifying the job as a train run and the machine as a single section

track. The job characteristic in train scheduling will be applied where each train has

a ready time before starting the trip; and a sectional running time (processing time)

and due time for delivering and collecting the empty and full bins to and from the

siding. These characteristics will be prioritized (weighted) to determine the

importance of train trips for the sidings.

In a machine environment, the techniques to solve a single track railway scheduling

problem should have four main abilities: prevent two trains passing in one section at

a time; keep the train activities (operations) in sequence during each run (trip); pass

the trains on one section in correct order (priorities of passing trains); and ease

passing trains by solving rail conflicts.

The SMS depends on using one machine and multi jobs. As the research problem

involves multi machines (section), the SMS cannot be used individually to solve the

research problem and will be discarded. The nature of the FSS is different from the

research problem because in FSS the machines are sequenced and used in one

direction, while the machines (sections) in a single track railway scheduling problem

will be used in two directions (outbound and inbound). The OSS technique is

unsuitable for the research problem where there are no precedence constraints

39

between the operations of the same job (no order between operations). This means

the train activities (operations) will not be in a sequence during each trip (run). The

same problem arises in GSS where there are no restrictions on operations in the same

group.

A blocking parallel-machine job shop scheduling (BPMJSS) will be used to

formulate the research problem where the blocking constraints, MPS, and JSS are

integrated to solve the single track railway scheduling problem. The JSS prevents

two trains passing in one section at a time and the section availability for a passing

train can be taken into account by applying the blocking constraints. The JSS

includes the precedence and disjunctive constraints which keeps the train activities

(operations) in sequence during each run (trip) and allows for more than one train

pass on one section, in the correct order. PMS helps solve the rail conflicts by easing

passing trains.

Makespan and the total waiting time are used in this research as optimality criteria to

reduce the total cost of the rail transport system. The two criteria are suitable and

satisfy the aims of this research.

2.3 Job Shop Scheduling (JSS) Solution Techniques

.

The JSS problem is computationally difficult and many techniques have been

developed to solve it. Researchers have used many different approaches to

formulate the JSS problem such as integer programming (IP), mixed integer

programming (MIP), disjunctive graphs, dynamic programming (DP) and constraint

programming (CP) formulations. Many solution techniques have also been proposed

over the last few decades to solve the JSS problem. These techniques are divided

into exact solution techniques (branch and bound, dynamic programming) and

approximate solution techniques (heuristic, metaheuristic, and search techniques).

(Brucker et al., 1994; Artigues et al.,2009)

The JSS problem is formulated in this research in three main formulations; mixed

integer programming, constraint programming and disjunctive graph techniques.

These techniques have a high ability to generate accurate solutions in reasonable

time using approximate solution techniques and exact solution technique (Puget &

40

Lustig, 2001; Foccaci et al., 2002). Additionally, these techniques are suitable for

the research problem. Figure 2.8 shows the job shop solution techniques including

the different mathematical formulations.

Figure 2.8 Job shop solution techniques

Job Shop Solution approach

Heuristic Techniques

Branch and Bound

The Shifting Bottleneck

Dispatching Rules

Meta Heuristic Techniques

Tabu Search

Simulated Annealing

Search Techniques

Best First Search

Depth First Search

Slice Based Search

Limited Discrepancy Search

Depth Bound Discrepancy Search

Interleaved Depth First Search

Standard Search Strategy

Dichotomy Search Strategy

First Input First Output Technique

Longest Processing Time

Critical Ratio

Earliest Due Date

Shortest Processing Time

Minimum Slack Rule

Disjunctive Graph

Mixed integer Programming

Constraint Programming

Dynamic Programming

Mathematical formulations

41

2.3.1 Disjunctive Graph Technique

This section defines the disjunctive graph model G= (O, A, E) where O is a set of

nodes, A is a set of directed arcs (conjunctions) and E is a set of undirected arcs

(disjunctions). The classical JSS problem can be formulated by the disjunctive graph

model to minimise the makespan as an objective function. In the disjunctive graph

model, the makespan value is determined by the critical path.

A critical path in a job shop schedule consists of a set of operations in which the first

operation starts at time 0 and the last operation finishes at the makespan. The

completion time of an operation on the critical path is equal to the starting time of

the next operation on the path.

To find the solution, the sequence of operations on the critical path is explored to

identify alternative solutions that have minimal impact on the makespan. A

disjunctive graph shows all feasible solutions and the critical path in JSS problems

and can be used to minimise the makespan. The approach can be illustrated as

follows:

The set of all job operations is defined by the set O of nodes. There are two dummy

nodes 0 and F where 0 is the start node and F is the end node. The maximum number

of operations is O = J×M, where J is the total number of jobs and M is the total

number of machines. The set of precedence constraints between consecutive

operations of the same job is represented by the set A of conjunction arcs (where the

order of operations must be preserved) shown as solid lines in Figure 2.9. In a

feasible solution (a schedule), a conjunction arc from operation k to operation j (k →

j) with processing time gk must satisfy the precedence constraint tk + gk ≤ tj where tk

is the start time of operation k, and tj is the start time of operation j.

42

Figure 2.9: Conjunctive arcs for 5 jobs and 5 machines

The sequence of job operations executed on the same machine is represented by the

set E of disjunctions where E=1

Mi

Ei= and iE is the subset of disjunctive pair-arcs

(where the order of operations can be swapped) shown as dotted lines in Figure 2.10

corresponding to machine i (Adams et al., 1988). Disjunctive arc k―j which has the

processing times gRkR and gRjR has two constraints and one of them needs to be satisfied;

tRkR + gRkR ≤ tRjR or tRjR + gRjR ≤ tRk Rwhere k and j are operations for two different jobs.

Figure 2.10: Disjunctives arcs for 5 jobs and 5 machines

0

1 2 3 4 5

6

17

7

18

8

20

F

M1

9

13

10 11 12

14 15 16

M2 M3 M4 M5

M1 M2 M4

M1 M3 M4 M5

M4 M1 M3 M2

M5 M1

19

M3 M4

0

1 2 3 4 5

6

17

7

18

8

20

F

M1

9

13

10 11 12

14 15 16

M2 M3 M4 M5

M1 M2 M4

M1 M3 M4 M5

M4 M1 M3 M2

M5 M1

19 M3

M4

J1

J2

J3

J4

J5

J1

J2

J3

J4

J5

43

Cyclic selection, which does not include any conjunctive arcs, is used for sequencing

machines. A cyclic selection Bi is a feasible solution for machine i. A complete

selection of operations set B includes the union of selections Bi; i∈M. The makespan

is the length of the longest path between the two operations k and j. The job shop

scheduling problem solution is then a complete selection B⊂E in a disjunctive graph

that minimises the length of the longest path (critical path). Example 2.1 explains the

disjunctive graph model by using the 3/3/G/Cmax (the system includes 3 jobs are

processed on 3 machines without interrupted time to minimize the makespan) as a

JSS problem.

Example 2.1

The machine sequence and the processing time for each job are

shown in Table 2.1 below.

Table 2.1: 3 jobs and 3 machines problem

Processing time Machine Operation Job 4 3 6 1 2 3 1 2 3 JR1 4 6 2 1 3 2 4 5 6 JR2 3 6 4 2 1 3 7 8 9 JR3

Figure 2.11 shows the disjunctive arcs set E;

E=ER1R ER2R ER3R;

ER1R={(1,4), (4,1),(1,8),(8,1),(4,8),(8,4)} ER2R={(2,6), (6,2),(2,7),(7,2),(6,7),(7,6)}

ER3R={(3,5), (5,3),(3,9),(9,3),(5,9),(9,5)}

Figure 2.11: The disjunctive graph for 3 jobs and 3 machines

0

1 2 3

7 8

10

J1

M1

4 5 6

M2 M3

M1 M3 M2

M2 M1 9 M3

J2

J3

4

3

6

6

4

3

2

6

4

44

This example uses two feasible solutions to explain the

disjunctive graph. A critical path at any graph Ds will be a

feasible solution, where the critical path is the length of the

longest path from the start of operation 0 to the finish of

operation 10 (0 and 10 are dummy operations). This length is

the makespan Cmax. All operations at the critical path are

critical operations as well. Figure 2.12 shows the first feasible

solution for Example 2.1, where the complete selection S is:

S=S1 S2 S3;

SR1R= {(4, 1), (1, 8)}

SR2R= {(7, 2), (2, 6)}

SR3R= {(5, 9), (9, 3)}

Figure 2.12: The first feasible solution for the 3 jobs and 3 machines example

Makespan=24

Figure 2.13 shows that the critical path is

{0→4→1→8→9→3→10} with length (makespan CRmaxR) equals

24.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3 M1

4

4

3

6

2

4

6

3

6

45

Figure 2.13: The Gantt chart for the first feasible solution

Figure 2.14 shows the second feasible solution of Example

2.1, where the complete selection S is:

S=S1 S2 S3;

SR1R= {(4, 8), (1, 8)}

SR2R= {(7, 2), (2, 6)}

SR3R= {(5, 9), (9, 3)}

Figure 2.14: The second feasible solution for 3 jobs and 3 machines example

Makespan=23

In Figure 2.15, The critical path is {0→4→8→1→2→3→10} with length (makespan CRmaxR ) 23.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3 M1

4

3

6

4

6

3

6

2

4

46

Figure 2.15: The Gantt chart for the second feasible solution

Disjunctive graphs in example 2.1 show that any operation i has two types of

predecessors and successors. Job predecessor and successor are defined by JP[i] and

JS[i] and their machine predecessor and successor are defined by MP[i] and MS[j].

Job predecessor and successor are arcs of the conjunctive graph Ds while machine

predecessor and successor are disjunctive arcs. The final solution of the JSS problem

depends on the two types of predecessors and successors for each operation.

2.3.1.1 Heuristic Techniques

2.3.1.1.1 The Shifting Bottleneck Algorithm

Adams et al (1988) proposed the shifting bottleneck algorithm as an effective

technique for solving the job shop scheduling problem. This algorithm depends on

selecting the machine that significantly reduces the makespan value. This machine is

called a bottleneck machine and its algorithm produces the sequence of operations in

the set of machines to determine the bottleneck machine every time.

.

2.3.1.1.2 Dispatching Rules

Many dispatching rules have been proposed to solve the job shop scheduling

problem. These techniques are faster than many other heuristic techniques and

sometimes are very effective in solving job shop scheduling problems. In this section

some of the priority rules are summarized.

47

The first rule is the shortest processing time “SPT” rule. The main core of this

technique is the ascending order of the processing time, pi, of the job i where:

p1 < p2 < p3 <…. pn and n is the total number of jobs. This technique performs well

with minimising tardy jobs (Chang et al., 1996 ; Rajendran & Holthaus, 1999).

The second rule is the weighted shortest processing time, WSPT. In this technique,

each job i has a weight wi where; p1/w1 < p2/w2 < p3/w3 <…. Pn/wn.

The next rule is called the critical ratio CR. This technique includes the ascending

order of the ratio; CR= (di –system time)/ri, where ri is the remaining processing

time. This technique is more efficient for many scheduling problems (Moser &

Engell, 1992; Pierreval & Mebarki, 1997; Rose, 2002; Abu-Suleiman et al., 2005).

Another rule is the longest processing time LPT which depends on the descending

order of the processing time of all jobs. Additionally there is the first input first

output rule (FIFO) and the minimum slack rule (MS).

The last rule is the earliest due date (EDD) where EDD depends on the processing

time and due date di of job operations, where d1 < d2 < d3 <…. dn. This rule orders

the due date of operations of each job in ascending order. This rule performs well for

minimizing the tardy rate (Chang et al., 1996) and the mean tardiness (Jeong & Kim,

1998).

2.3.2 Mixed Integer Programming Approach for JSS Problem

Manne (1960) proposed a mixed integer programming formulation to solve the job

shop scheduling problem. Formulating the job shop scheduling problem requires two

main constraint types to be included in the main model: precedence constraints

(Equation 2 below) and disjunctive constraints (Equations 3 and 4 below).

Precedence constraints ensure that operation o of job j∈J on machine i ∈M must

finish before operation o+1 of the same job starts on the same machine. Disjunctive

constraints ensure jobs j and k are processed on machine i in the correct order. Either

job j on machine i precedes job k on the same machine where job k cannot use this

machine before job j leaves it, or job k on machine i precedes job j on the same

machine where job j cannot use this machine before job k leaves it. The main model

for the job shop scheduling problem under a makespan Cmax optimisation criterion is:

48

Model Notations

tijo: start time of operation o of job j on machine i.

gijo : processing time of operation o of job j on machine i.

yijk =1: if job j precedes job k on machine i.

yijk =0 : otherwise

Z: a big number.

Cmax: makespan.

Model formulation

Minimise CRmaxR (2.1)

Subject to

.( 1)t g t o job j on machine iijo ijo ij o+ ≤ ∀ ∈+ (2.2)

( ) 1 . .t t g Z y o j and o kijo iko iko ijk ′≥ + + − ∀ ∈ ∈′ ′ (2.3)

( ) - . .t t g Z y o j and o kiko ijo ijo ijk ′≥ + ∀ ∈ ∈′ (2.4)

.maxC t g o jijo ijo≥ + ∀ ∈

(2.5)

0tijo ≥ and 0.gijo ≥

(2.6)

2.3.2.1 A hyper Branch and Bound Technique

While exact techniques give an optimal solution for some problems, they are very

time consuming even in small cases. In this section, a hyper technique which

depends on the integration of a branch and bound method and the n/1/Lmax

algorithm is described to solve a job shop scheduling problem (Pinedo, 2008).

Dynamic programming to JSS is not within the scope of this study because DP is

difficult to apply for large scale problems.

49

To obtain an optimal solution for the job shop scheduling problem, branch and

bound techniques can be used. A branching method is used to generate all active

schedules and select the schedule with the minimum makespan (or the optimal

schedule by another criterion). A feasible schedule is called active if no operations

can be completed earlier and there are no delaying operations. This method is based

on complete enumeration, and as a result is very time consuming, especially with

large-scale problems. The branching method is described as following.

Notations α: the set of all operations for which all predecessors have already been scheduled.

sij : earliest possible starting time of operation(i,j) є α, where,

i is number of machine and j is number of job.

pij: processing time of job j on machine i.

α΄: a subset of set α.

Step 1: (initial conditions)

α: first operation of each job.

sij=0 for all (i,j) є α.

Step2: (machine selection)

Calculate t(α) for the current partial schedule (schedule during the

new branching process).

t(α)=min{ sij+pij} ; (i,j) є α.

i*: the machine on which the minimum is achieved.

Step 3: (branching)

α΄: includes all operations (i*,j) on machine i* where si*j< t(α).

For (i*,j) є α΄ then

Extend partial schedule by scheduling the next operation

on machine i*.

For each such selection then

Delete (i*,j) from α.

Add job successor of (i*,j) to α. Return to step 2.

End

50

The branching method explains the execution of the branching scheme. From Step 3,

the current partial schedule at any node is used to produce new branches from this

node. The partial schedule can be determined by the disjunctive arcs produced

during the branching process. The number of branches is determined by the number

of operations in α΄. A branch of any node depends on the choice of an operation

(i*,j) Єα΄ as the next on machine i*. Nodes at the very bottom of the tree can

produce all the active schedules. A lower bound of makespan can be identified by

the length of the critical path in the disjunctive graph. A better lower bound is

calculated after producing all disjunctive arcs. The n/1/Lmax problem (n jobs, 1

machine and maximum lateness objective) is solved to avoid processing more than

one operation on the same machine at the same time, where not all disjunctive arcs

are yet produced, and to obtain a better lower bound. In this problem, the maximum

lateness, Lmax, is calculated to process n jobs on one machine. The next algorithm is

used to solve the n/1/Lmax problem which is classified as strongly NP-hard.

The n/1/Lmax algorithm is integrated with the branch and bound method to solve a job

shop scheduling problem.

Step1: calculate the lower bound (LB) from the longest path from the source node

to the sink node.

Step 2: calculate the earliest starting time, sij for all operations (i,j) n machine i by

calculating the longest path from the source to operation(i,j) in the disjunctive

graph.

Step 3: calculate the longest path from (i,j) to sink in the graph, Φij, and then

calculate the due date dij=LB- Φij+pij

Step4: solve the single machine problem under the conditions of release dates, with

no pre-emption to minimise the lateness Li.

Step 5: repeat this algorithm for all machines to calculate L1, L2,...Lm. A better lower

bound can be obtained by using LB*= LB +max Li, where i=1 to m.

The largest value LB* is a lower bound at any node of the graph. The

makespan can be obtained by the minimum lower bound.

51

Solving a job shop problem using a hyper branch and bound technique as an exact

technique is still very time consuming especially for large-scale problems. Example

2.1 is a 3/3/G/Cmax problem that is strongly NP –hard.

By applying the hyper branch and bound technique to Example 2.1, the final solution

for this example is obtained at level 6. Figure 2.16 shows the complete disjunctive

graph for the final solution. All steps taken in Example 2.1 are shown completely in

Appendix A. The complete solution shows that these techniques are time consuming

even in small cases.

Figure 2.16: Disjunctive graph for the final solution

The lower bound LB=L{0,((1,4),(1,1),(2,2),(3,3),(3,9),10}=21

Table 2.2 and Figure 2.17 illustrate the better lower bound.

Table 2.2: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rat the final level

MR1 MR2 MR3 JR1 JR2 JR3 JR1 JR2 JR3 JR1 JR2 JR3

SR11R=4 SR12R=0 SR13R=8 SR21R=8 SR22R=11 SR23R=0 SR31R=11 SR32R=4 SR33R=17

ΦR11R=17 ΦR12R=21 ΦR13R=1

0

ΦR21R=1

3

ΦR22R=2 ΦR23R=1

6

ΦR31R=1

0

ΦR32R=1

6

ΦR33R=4

dR11R=8 dR12R=4 dR13R=14 dR21R=11 dR22R=21 dR23R=8 dR31R=17 dR32R=11 dR33R=21

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

52

Optimal solution for M1 Optimal solution for M2 Optimal solution for M3

J2 J1 J3 J3 J1 J2 J2 J1 J3

4 8 14 time 3 8 11 13 time 4 10 11 17 21 time

L1=0 L2=0 L3=0

Figure 2.17: Optimal solutions of M1, M2, and M3 at the final level for n/1/Cmax problem

The new lower bound LB*=21+max {0, 0, 0} =21.

The branching stops at node (3, 3). Figure 2.18 presents the complete search tree

where the lower bounds are represented in the boxes at each level.

Level 0

Level 1

2a 2b 2c Level 2 3a 3bi 3bii 3ci 3cii 3ciii

Level 3 4ai 4aii 4bi 4biia 4biib 4ci 4cii

Level 4

Level 5 5aia 5aib 5ai 5aiib 5bia 5bib 5biia 5cii

Level 6

The optimal solution Figure 2.18: The complete branching procedure of the optimal solution

Nodes at the very bottom of the tree correspond to all active schedules, where there

is complete selection. The solution provided in Figure 2.18 indicates that the optimal

solution is in the lower level and makespan=21.

As seen from Example 2.1, the exact techniques are time consuming. There are many

steps in obtaining the final solution. Therefore there is an urgent need to develop

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

23

2, 6

26 29 24

3, 3 3, 5

24

3, 3 3, 9 2, 6

24

2, 2 2, 6

29

23 24 24 28 21 24 23 30

3, 3 3, 3 21 21

53

techniques which provide good solutions in a reasonable time. Heuristic techniques

such as the shifting bottleneck algorithm and dispatching rules obtain quick solutions

for NP hard problems and these are described in section 2.3.5. The metaheuristic

techniques such as simulated annealing and tabu search that deal with NP hard

problems and obtain good solutions in a reasonable time are examined in Chapter 6.

2.3.3 Constraint Programming Approach for the JSS Problem

The field of constraint programming (CP) is relatively new with the basic concepts

developed in the field of artificial intelligence in the 1970s and then further advanced

in computer science in the mid 1980s. CP was first discussed internationally in 1995

and since then has become an important technique that complements traditional

mathematical programming technologies to optimise many business activities (Sadeh

& Fox, 1996; Sadeh et al., 1995; El Sakkout & Wallace, 2000).

The formulation of any problem using CP can be defined by three elements: (S,

D, C), where S is a set of variables, {s1,s2,s3,...,sn }; D is a set of variable domains,

where D(si ) = Di is the domain of variable si; and C is a set of constraints which

gives the values that variables can take, where ci∈C is a constraint of some

variables and can be written as:

ci (si, sj, sk,... ) ⊆ Di × Dj × Dk ×…

Formulating the job shop scheduling problem in constraint programming is relatively

simple. Each operation can be expressed as a variable where the start time of each

operation is defined by the determined domain. With these variables, the precedence

and disjunctive constraints can be defined easily (Equations 2 and 3 below

respectively). Makespan (Cmax) is presented to restrict the completion times of all

operations. A schedule with minimal makespan will be obtained. A formulation of

the JSS problem using the CP tool of Optimisation Programming Language (OPL) is

as follows:

54

Minimise Cmax (2.7)

Subject to

task [i, j, o].start + Duration [i, j, o] preceded task [i, j, o+1].start ∀o∈j (2.8)

{task [i, j, o].start ≤ task [i, k, o'].start + Duration [i, k, o'] } ∨

{task [i,k, o'].start ≤

task [i, j, o].start + Duration [i, j, o]} ∀o∈j and o'∈j (2.9)

Cmax ≥

task [i, j, o].start + Duration [i, j, o]

∀o∈ j (2.10)

task [i, j, o].start≥0 and Duration [i, j, o]≥0 (2.11)

Duration [i, j, o] represents the processing time of operation o of job j on machine i

and task [i, j, o].start means the start time of operation o of job j on machine i. The

JSS problem solution depends on solving a sequence of constraint satisfaction

problems related to different upper bounds for the makespan. The constraint

programming technique assumes there is an upper bound Y “makespan” and finds the

start time (t) for each operation o where t is in the initial domain [0, Y-d (o)]; d (o) is

the processing time of operation o; and the precedence and resource constraints are

satisfied. Nuijten and Le Pape (1998) used the ILOG Scheduler to obtain a solution

by looking for the earliest start time, the latest start time, the earliest completion time

and the latest completion time inside the domain to minimise the makespan. The

main aim of using CP in the scheduling field is to reduce computation times and

reduce the variable domains using the constraints during the search procedure. This

process is called constraint propagation. Variable and value ordering heuristics and

backtracking techniques have been applied to capture the JSS problem search space

and reduce the search process time (Sadeh & Fox, 1996; Sadeh et al., 1995).

Cheng and Smith (1997) applied a constraint programming approach for the JSS

problem, especially Precedence Constraint Posting (PCP) and multi PCP using the

benchmark. An edge-guessing algorithm was used to achieve an optimal solution for

the operation sequences for small-scale JSS problems and a near-optimal solution for

large-scale problems (Belhadji & Isli, 1998). This algorithm integrated constraint

55

propagation with a problem decomposition approach to solve the JSS problems

easily in a short time (Dorndorf et al., 2002).

The efficiency of Constraint Logic Programming (CLP) as an approach for solving

JSS problems individually or integrated with other techniques is described in much

research. Trentesaux et al. (2001) presented two approaches: CLP and Distributed

Problem Solving (DPS) which are inherited from artificial intelligence (AI) and

distributed artificial intelligence (DAI) respectively to solve the JSS problem. The

results for JSS problems have been obtained using Constraint Handling in Prolog

(CHIP) for a CLP and Distributed Production Scheduling System (DPSS) for a DPS.

The optimality for DPS cannot be proved for all cases, while in CLP it can be proved

for small-scale problems and some large-scale problems. The integration of CLP and

DPSS improved the quality of the JSS problem solution.

The neural network was constructed using constraint programming to improve its

performance while looking for the solution to the JSS problem. An adaptive neural

network reduced the computational time, especially for large-scale JSS problems,

and improved the quality of the solutions to be near-optimal solutions (Yang, 2006;

Yang et al., 2010).

Constraint Integer Programming (CIP) has been proposed as a new approach to solve

combinatorial problems, in particular the JSS problem (Achterberg, 2007a, 2007b;

Achterberg et al., 2008). Most solvers for CP, MIP and Satisfiability (SAT)

problems (mathematical logic problems) work by dividing the main problem into

smaller sub problems, and then obtaining all the potential solutions. However, these

solvers differ in the method of processing or solving the sub problems. The CIP

approach integrates the three techniques CP, MIP and SAT. The CIP approach

depends on a branch and bound technique as a minimum approach that is used in the

three areas, MIP, CP, and SAT. Additionally, algorithms such as Linear

Programming LP relaxations and cutting plane separators work with branch and

bound techniques in MIP. Constraint specific domain propagation algorithms are

used in CP, SAT and conflict analysis as in modern SAT solvers.

New models were proposed to solve JSS problems (Hentenryck & Michel, 2005;

Watson & Beck, 2008; Barba et al., 2009) that depended on the integration of

56

constraint satisfaction techniques and local search techniques to improve the quality

of the solution in large-scale problems. This research modelled the JSS problem as a

constraint satisfaction problem which was solved using local search techniques and

designing new neighbourhood structures. Dovier et al. (2009) designed empirical

comparisons between CLP and Answer Set Programming (ASP) using combinatorial

problems to select the most efficient approach for each problem. The solution

approaches classification by its efficiency in solving the different problems. It is

useful for future research or for applications related to CLP and ASP.

CLP Techniques are used for solving JSS problems in three ways. The first way uses

pure CLP as an individual approach and provides good results in reasonable time for

small cases. However, results for large-scale problems are not sufficiently stable to

obtain the optimal solution. The second way makes comparisons between the CLP

and other approaches to identify the advantages of CLP in solving many different

cases in job shop scheduling problems. It also allows the researcher to know which

approach is going to obtain the most efficient solution. The last way CLP techniques

are used for solving JSS problems is by integrating CLP with other techniques such

as Integer Linear Programming ILP, MIP, and local search techniques to solve large-

scale problems for optimal solutions in reasonable time.

2.4 Scheduling Theory and Railway Systems in Literature

Railway systems are very complicated particularly if they are single track systems.

Many studies and researchers have analysed and solved major problems to improve

the efficiency of such systems. Most studies have concentrated on railway operation

development or optimisation of train schedules, and improving railway capacity.

These studies have analysed the single track rail system as an NP-hard problem, so

that finding a solution demands special techniques. These techniques and

mathematical models were developed using scheduling theory to optimise system

performance and solve NP-hard problems in the transport sector such as reducing

delays and solving conflicts.

Mixed integer programming and other mathematical programming models were

proposed to optimise the train scheduling system. Yang et al. (2010) developed a

57

mixed integer programming model for a single track rail to solve the conflicts on

tracks. A Branch and bound technique was used to solve the model. Higgins and

Kozan (1997) introduced a mathematical model to study the impact of timetable

changes and railroad infrastructure changes so as to optimise train schedules on a

single track rail. Their approach was to change the number of trains and the number

of sidings, where the solution for 31 trains and 14 sidings was obtained in a

reasonable time. Higgins et al. (1996a, 1996b) designed a mathematical model to

optimise the number and location of sidings on a single track rail to reduce the

delays caused by congestion. They used a decomposition approach where the model

was divided into two sub models, one to optimise track segment lengths and arrival

and departure times, and the other to determine the optimal train schedule. This

approach was useful in that it reduced the calculation time especially for large-scale

problems. Jeong and Kim (2011) developed a model for sequencing the delivery

operations in two main train movement directions (inbound and outbound) to obtain

optimal solutions for transfer by the rail crane. Probability models were developed to

determine the knock-on delays of trains caused by rail conflicts, (Marinova &

Viegasb, 2001; Yuan and Hansen, 2007; Murali et al., 2009). Meng and Zhou (2011)

suggested a new mathematical model and stochastic programming approach to

optimise single track train movement to reduce the total train delay time.

Branch and bound techniques were proposed with a local search technique to

improve the solutions of the rail system optimization problem. Corman et al. (2010)

developed some algorithms to improve a real-time traffic management system by

integrating rescheduling methods and local rerouting strategies in a tabu search

technique. The modified branch and bound method obtained a solution with a short

computation time. Neighbourhood structures for train rerouting are used in this

research and the Dutch railway network was used as a practical case. A discrete

event optimisation model and a branch and bound method were developed to solve a

single track train scheduling problem as a blocking job shop problem (Ariano et al.,

2007; Zhou & Zhong, 2007). Solving conflicts through a single track rail was the

main aim of these studies. A branch and bound method reduced the total time of

train trips and improved the accuracy of the solution to be near-optimal, as well as to

decrease computation times. The optimisation techniques increase the efficiency of

solutions but require long computation times for large-scale cases.

58

Constraint programming techniques were used to develop new models for the train

scheduling problem. Rodriguez (2007) developed a new model for the routing and

scheduling of trains by using constraint programming and based this model on a

simulator to calculate train run time. The main aim of Rodriguez’s model was to

solve conflicts and reduce delays at junctions by integrating a decision support

system and constraint programming. The model improved train system performance

for some real cases in a reasonable time. Abril et al. (2008) formulated the railway-

scheduling problem as constraint satisfaction problems which were defined as NP-

complete and solved by division into sub-problems. Search techniques used in this

research were the depth-first search technique and a tree partition method. All

techniques depended on graph partitioning where the domain of the main problem

was divided into sub domains where each domain has the same number of nodes.

Sato et al. (2007) suggested a shunting scheduling method in a railway depot which

was very suitable for many dynamic systems. The flexible job shop scheduling

problem formulation and constraint programming techniques as solution techniques

were used to achieve good results in a reasonable time. Salido and Barber (2009)

developed new heuristic techniques to solve periodic train scheduling problems some

of which were based on mathematical programming methods such as linear

programming and local search, while other techniques were based on railway

topological characteristics and constraint programming techniques. Spanish railways

were used as a case study in this project which aimed to maximise the capacity of the

rail line. CPLEX software was used and obtained results which were close to optimal

solutions.

Disjunctive graph models were proposed and metaheuristic techniques used to obtain

near optimal solutions. Burdett and Kozan (2008) proposed a novel hybrid job shop

approach to solve a train scheduling problem and designed a disjunctive graph model

of train operations and used metaheuristic techniques such as simulated annealing.

This approach achieved good results for many objective criteria and improved train

movements. Burdett and Kozan (2010) developed a new constructive technique, the

job shop approach to solve train- scheduling problems. This solution depended on a

disjunctive graph model and constructive solution methods and achieved better

results than metaheuristic techniques for different criteria such as minimizing a

http://portal.acm.org/author_page.cfm?id=81324487572&coll=GUIDE&dl=GUIDE&trk=0&CFID=75043835&CFTOKEN=83272179

http://www.hindawi.com/81037853.html


59

makespan objective. Mascis et al. (2002) studied the job shop scheduling problem

with blocking, where the operations of each job must be processed without any

interruption. They used a generalization of the disjunctive graph to formulate the

problem. Higgins and Kozan (1997) proposed a better mathematical model to

optimise a train schedule on a single track rail to solve train conflicts by introducing

metaheuristic techniques such as genetic algorithms, tabu search and hybrid

algorithms. CPU time was calculated to compare these techniques for different

problems. Liu and Kozan (2009, 2011) represented the train-scheduling problem as a

blocking parallel machine job shop scheduling problem. Here, they solved the

blocking problem for single and multiple railway tracks and extended the classical

disjunctive techniques and proposed a new heuristic method, named the feasibility

satisfaction procedure. The new technique was applicable to many complex real

situations in train crossing problems.

Heuristic techniques were used to increase the efficiency of railway performance.

Kozan and Burdett (2005) developed a new methodology to measure the capacity of

the railway system and increase the efficiency of railway performance. They

assessed factors or variables which have a significant effect on railway capability

such as length and weight of trains, stopping rules for trains, and the distance

between two stations. Also proposed was a new definition and estimation of train

travelling times between two points which had a significant impact on railway

capacity. Burdett and Kozan (2006) extended their previous railway capacity

modelling to estimate the capacity for different railway systems. Firstly, they

developed techniques for railway lines to improve the level of performance.

Secondly, they developed more complex techniques to solve railway network

problems. The numerical results for the case study showed the positive impact of

these techniques on the performance of railway systems.

Mixed integer programming, disjunctive graphs and constraint programming

approaches have been used to develop new models for railway scheduling problems

particularly with respect to single track rails. Many solution techniques such as

branch and bound, heuristic and metaheuristic techniques were used to solve the

developed models.

60

2.5 Conclusion

This research uses job shop scheduling techniques to solve the sugarcane rail

transport problem. There are several benefits in treating the train scheduling problem

as a blocking parallel-machine job shop scheduling (BPMJSS) technique, namely;

BPMJSS technique prevents two trains operating on one section of track at the same

time, because it considers that only one train (job in a conventional job shop) can

operate on a track section (machine in a conventional job shop) at the same time. The

availability of each section is taken into account where some sections or segments

are unavailable for use by some trains during a specific time; these sections are

called blocking sections. Secondly, the precedence constraints, where the preceding

operation has to finish before starting the next operation, will be applied to the train

scheduling to keep the train activities (operations) in sequence during each run (trip).

Disjunctive constraints in JSS ensure two jobs are processed on one machine, and in

the correct order. These constraints can be applied to train scheduling where more

than one train passes on one section in the correct order. This serves to reduce the

waiting time and solve rail conflicts. PMS is integrated with JSS to ease passing

trains and help solve rail conflicts as well. The JSS technique looks more promising

for finding better solutions, in reasonable computational times, than alternative

methods do.

Many solution techniques have been used to solve the job shop scheduling problem.

JSS was formulated using mixed integer programming, disjunctive graph and

constraint programming. The hyper branch and bound technique is an exact solution

technique which obtains the optimal solution for the job shop scheduling problem.

This technique is time consuming and requires a large number of iterations to

achieve optimality. Other techniques such as the shifting bottleneck algorithm or

dispatching rules are quick to obtain solutions, but the accuracy of these techniques

is not high. Metaheuristic techniques will be described in Chapter 6.

The integration of the railway operations and the sugarcane system will be the main

point of this current research. Designing new models which include all sugarcane

61

system constraints (train and siding capacity, delivery and collection constraints) and

the railway operation will have significant benefits in reducing the overall cost and

achieving safe conditions during the passing of trains.

Integrating the railway operations and the sugarcane system involves two main

sections, namely; rail operation constraints related to passing trains, and capacity

constraints. A blocking parallel-machine job shop scheduling approach is used for

the sugarcane rail problem since the fundamentals of the job shop technique are

suitable for train scheduling problems. The blocking sections and segments will be

used. Mixed integer programming and constraint programming are also used to

formulate and solve this large scale transport problem. This section provides an

insight into the proposed methodology, which will be described in detail in the

following chapter.

62

Chapter 3

Modelling the Sugarcane Rail Transport Problem

Chapter Outline

.

3.1 Introduction....................................................... ...............................................................64

3.2 Segment and Section Blocking Types..............................................................................65

3.2.1 Blocking Segment Types....................................................................................65

3.2.1.1 Blocking Terminal Segments..............................................................66

3.2.1.2 Blocking Intermediate Segments........................................................66

3.2.2 Blocking Section Types.......................................................................................68

3.3 Description of the Blocking Segment Models of Sugarcane Rail System........................69

3.3.1 Blocking Segment MIP Model of the Sugarcane Rail System............................76

3.3.2 Blocking Segment CP Model of the Sugarcane Rail System..............................83

3.4 Description of the Blocking Section Models of Sugarcane Rail System..........................89

3.4.1 Blocking Section MIP Model of Sugarcane Rail System....................................90

3.4.2 Blocking Section CP Model of Sugarcane Rail System......................................93

3.5 Inclusion of the Delivery and Collection Time Constraints to the Model........................95

3.5.1 Delivering Delay Constraints................................................................................99

3.5.2 Collecting Delay Constraints...............................................................................101

3.6 Sugarcane Rail System as a Dynamic System................................................................102

3.7 Conclusion......................................................................................................................104

63

Publications Arising from Chapter 3


optimising sugarcane rail operations. Flexible Services and Manufacturing

Journal; 23(2):181-196.



64

3.1 Introduction

Job Shop Scheduling (JSS) techniques were used to develop new models for the

sugarcane rail transport system using two main approaches: Constraint Programming

(CP) and Mixed Integer Programming (MIP). MIP focuses on the objective function

to improve its value using linear relaxation techniques, which eliminate suboptimal

solutions. On the other hand, the CP approach focuses on satisfying the constraints

using consistency algorithms or filtering algorithms, as explained in Chapter 4, to

remove infeasible candidate solutions as shown in Figure 3.1. Filtering algorithms

are useful to discover infeasible solutions for solving JSS problems, while some

other approaches are time consuming because they keep running without discovering

the infeasible solutions. Some problems are best handled by MIP, others by CP,

while the integration of the MIP and CP solution techniques can solve some harder

problems (Hentenryck, 2002).

Focuses Using To

On Prune

Focuses Using To

On Eliminate

Figure 3.1: MIP and CP approaches

The train scheduling problem is formulated in this research as a blocking parallel-

machine job shop scheduling problem (BPMJSS). In this research, the rail track

sections are defined as machines and the train runs as jobs. Each job (train run)

contains a number of operations determined by the start time and processing time at

each machine (a single section of track). Some machines (sections) may be single

units (a single track) or have several alternative units (parallel track sections,

including sidings or passing loops). As discussed in Section 2.2.4, there are several

benefits in treating the train scheduling problem as a blocking parallel-machine job

shop scheduling (BPMJSS) problem, namely to prevent two trains passing in one

section at the same time; to keep the train activities (operations) in sequence during

each run (trip) by applying the precedence constraints; to pass the trains on one

Suboptimal Solution MIP

CP

Objective Function

Linear Relaxation

Constraints

Filtering Algorithms

Infeasible Candidate Solutions

65

section in the correct order (priorities of passing trains) by applying disjunctive

constraints; and, to ease passing trains by solving rail conflicts by applying blocking

constraints and PMS (Parallel Machine Scheduling).

MIP and CP models are developed using blocking segment and blocking section

constraints. Blocking segment models in Section 3.3 are applicable for sugarcane rail

systems, while blocking section models in Section 3.4 can be applied to other

applications such as coal mining rail systems (Masoud et al., 2011). The blocking

section models can be applicable to sugarcane rail systems after adding assumptions

and modifications to the rail network infrastructure of the sugarcane system, and as

explained in Section 3.4.

In this research, Blocking segment constraints are proposed to develop the sugarcane

rail transport models because many branches in sugarcane rail networks do not

contain passing loops and because some section lengths are shorter than the train

length. In spite of these models including some assumptions to improve the

performance of the blocking segment model reducing the total waiting time value of

each train run, this value is still high.

New models using blocking section constraints are presented in Section 3.4 to reduce

the waiting time value and other objectives. The blocking section models can be

applicable to sugarcane rail systems after adding assumptions and modifications to

the rail network infrastructure of the sugarcane system, and as explained in Section

3.4. The blocking section models in can be applied to other applications as well such

as coal mining rail systems (Masoud et al., 2011).

3.2 Segment and Section Blocking Types

3.2.1 Blocking Segment Types

A segment blocking approach is used to develop the sugarcane railway system. All

sections in a segment are blocked at the same time and prevent other trains from

simultaneously using them. Two main segment types are included in the rail

network: a terminal segment (end of the network with no segments following it) and

66

an intermediate segment (has another segment following it). Blocking segment types

will be clarified using different cases.

3.2.1.1 Blocking Terminal Segments

All operations are executed continuously in terminal segments without any

interruption. Trains that require the terminal segment take the outbound direction.

Developing an algorithm to solve the blocking for that segment is therefore quite

straightforward. Figure 3.4 shows blocking terminal segment B being used by train

k1 with k2 waiting at segment A to use segment B, as shown in Figure 3.2. Terminal

segment C can be used by k1 or k2 to solve any conflicts for the two train.

Figure 3.2: Two trains travelling in the same direction. One requires the blocking terminal segment

3.2.1.2 Blocking Intermediate Segments

The situation is more complicated at intermediate segments of the track. At any

intermediate segment, operations in the outbound and inbound directions cannot be

implemented continuously if the train used the intermediate segment to go further in

the outbound direction to another segment. Therefore, the train would complete the

operations of the inbound direction on the intermediate segment after visiting this

other segment. As a result, dealing with the intermediate segment as one segment

and blocking this segment would cause a significant loss in the use of this segment

and reduce the efficiency of the rail network because any intermediate segment

Blocking terminal segment B

Terminal segment C

k1

k2

Outbound direction

Inbound direction

Intermediate segment A

67

might be blocked for a long time. For that reason, any intermediate segment has

been expressed mathematically as two segments for the two different directions:

outbound and inbound, as detailed in Section 3.3. There are two scenarios for

blocking intermediate segments:

Scenario one

Two trains k1 and k2 have the same direction, inbound, so in this case one segment

(inbound) will be used by the two trains. That is, the blocking constraints are

applied to one segment and will be quite straightforward. In Figure 3.3, train k2 uses

segment A and train k1 requires the same segment in the same direction. As a result

the blocking constraint can be applied easily for segment A and train k1 has to wait

for segment A to be unblocked before continuing into segment A.

Blocking segment

Figure 3.3: Two trains travelling in the same direction. One requires the blocking intermediate segment

Scenario two

The two trains k1 and k2 have different directions, inbound and outbound

respectively. Segment A is occupied by train k2 and is therefore a blocked segment,

while at the same time train k1 requires segment A, as shown in Figure 3.4.

Terminal segment B

Terminal segment C

k1

k2

Outbound direction

Inbound direction


68

Blocking segment

Figure 3.4: Two trains travelling in different directions. One requires the blocking intermediate segment

All blocking segment types are modelled mathematically and explained in detail in

Section 3.3. New algorithms for solving train conflicts are proposed in Section 4.3.2.

3.2.2 Blocking Section Types

In the blocking section approach, two trains (k1 and k2) can use one segment at a

time but no more than one train can occupy one section at a time. Blocking section

constraints can be applied correctly when the length of the train is less than the

length of the section, as explained in Section 1.2.3.1.

Figure 3.5 shows that the two trains k1 and k2 have two different directions, inbound

and outbound respectively. Section s is occupied by train k1 and is therefore a

blocked segment, but at the same time, train k2 requires section s.

One operation is executed on section s at a time and the index for this operation is

the same in the two travelling directions, inbound and outbound, as explained in the

blocking section models in Section 3.4.

Because segments can consist of a number of sections, applying a blocking section

constraint is easier than applying a blocking segment constraint that entails the

blocking of all sections in that segment.

Terminal segment B

Terminal segment C

k1

k2

Outbound direction

Inbound direction


69

Section s

Figure 3.5: Train requires the blocking section s

3.3 Description of the Blocking Segment Models of Sugarcane Rail System

Two main models are developed in this research for the rail transport systems using

the constraint programming technique (CP) and the mixed integer programming

technique (MIP). These models tackle and solve many scheduling train problems

throughout different types of rail networks. Each model consists of two main parts:

the objective functions and the constraints. The models’ objective functions deal

with minimising the makespan objective and the total waiting time separately. The

models’ constraints include rail operation constraints and sugarcane system

constraints. Rail operation constraints are related to trains passing on the rail network

and include precedence, order of rail segments, train runs, passing priority, and

blocking constraints. The sugarcane system constraints include train capacity, siding

capacity, mill capacity, empty and full bin requirements, harvester rates, and

harvesting times.

The model constructs train runs that consist of a series of operations. These

operations involve activities such as traversing a track section and delivering and

collecting bins. The train runs are modelled as whole trips, generally involving

leaving the mill, delivering empty bins to sidings during the outbound part of the

run, collecting full bins from sidings during the inbound part of the run and returning

to the mill. As such, two operations generally will be conducted on each track

section, each run. One of them will be implemented in the outbound direction and

the other operation will be executed in the inbound direction.

Terminal segment B

Terminal segment C

k1 k2

Outbound direction

Blocking section s


Inbound direction

70

The main elements of a railway network are shown in Figure 3.6. The sample rail

network in Figure 3.6 has seven sections, where sections S2, S5 and S7 contain

sidings (for storing empty and full bins for the harvesters). Three segments are

included in the figure. Two segments are defined as terminal while one segment is

defined as intermediate. In the terminal segments, the train can travel outbound and

then immediately travel inbound without any interruption. For intermediate

segments, however, the train may go to visit other segments like terminal segment 2

or terminal segment 3 after traversing intermediate segment 1 in the outbound

direction, and come back in the inbound direction to visit the intermediate segment

again. Operations can be conducted continuously in the two directions on terminal

segments while there can be an interruption between the operations of the outbound

direction and the inbound direction on intermediate segments.

Figure 3.6: A simple cane rail network with three sidings for delivering and collecting

If blocking constraints were simply applied to the intermediate segment as they can

be to a terminal segment, the segment would be blocked during the time period of

implementing the outbound and inbound operations on it. The blocked time period

includes the interruption period between the operations of the outbound and the

inbound directions when the train is in either segment 2 or segment 3. As a result,

the utilisation efficiency of this intermediate segment will be reduced because it will

Mill

S1

Terminal segment 2

Intermediate segment 1

Terminal segment 3

S2 S3 S4 S5

S6

S7

Outbound direction

Inbound direction

k1 k2

Junction

Siding

71

be blocked for a longer time. Blocking a terminal segment does not affect utilisation

efficiency since the operations of these segments are conducted continuously.

Example 3.1 examines these issues.

Example 3.1

Assume that the train k1 run includes the path, Mill-S1-S2-S3-S6-S7-

S7-S6-S3-S2-S1-Mill, where outbound operations O1-O2-O3 are

implemented on the intermediate segment. Outbound operations 4-5

and inbound operations O6-O7 are implemented on terminal segment

3 continuously. Inbound operations O8-O9-O10 are implemented on

the intermediate segment. Blocking the intermediate segment during

use of k1, to satisfy the safety conditions, means this segment is

blocked during the time period of implementing the operations O1-

O2-O3-O8-O9-O10. This period includes the time of operations O4-

O5-O6-O7 since the operations of each segment should be

implemented and conducted continuously. As a result, if the train k2

requires the intermediate segment, the waiting time of the train k2 to

use the intermediate segment equals the time period of implementing

the operations O1-O2-O3-O4-O5-O6-O7-O8-O9-O10 by train k1

which affect the utilisation efficiency of that segment.

The proposed model increases the utilisation efficiency of segments by assuming

that intermediate segments have been given separate segment numbers for the

outbound and inbound directions (segments 1 and 4 in Figure 3.7). Segment 1

includes the outbound operations O1-O2-O3 and segment 4 includes the inbound

operations O8-O9-O10.

72

Figure 3.7: A single cane rail network after applying model

In the outbound and inbound directions of train k1, segment1 and segment 4 are

blocked respectively. As a result, the waiting time of train k2 is reduced to equal the

time period of implementing the outbound operations O1-O2-O3 on segment 1 by

train k1. If k1 is in the inbound direction, the waiting time of k2 to catch segment 4 is

equal to the time period of the inbound operations O8-O9-O10 of train k1. Blocking

segment 1 does not automatically mean the intermediate segment is blocked

completely, since segment 4 can be used by another train k2 while train k1 is using

segment 1. This conflict is addressed by the proposed algorithms of the solution

approach in Chapter 4 by distinguishing between the segment types to prevent using

one physical segment by more than one train at the same time.

The efficiency of the blocking segment constraints can be increased further in the

proposed models by considering the passing loops in each rail branch. Any blocking

for any segment which includes a passing point can increase the waiting time of the

system and then increase the operating cost and decrease the efficiency of rail

section utilisation. Passing points can be passing loops (no activities; no delivering

and no collecting) or a big siding (delivering or collecting) with passing loop and can

allow for more than one train to pass at the same time.

Mill

S1

Terminal segment 2

Inbound segment 4

Terminal segment 3

S2 S3 S4 S5

S6

S7

Outbound direction

Inbound direction

k1 k2

Junction

Siding

Blocked segment

Outbound segment 1

73

Case a: Passing loop

Each branch which includes a passing loop can be divided into two segments to

increase the utilisation of this branch and reduce the waiting time of any train at this

branch. As a result, the blocking segment is applied to each part that does not include

a passing loop. For example in Figure 3.8, branch B1 includes a passing loop, so two

segments (A1 and A2) are included in it where each segment includes some sections.

The passing loop has two parallel tracks (C1 and C2) without storage, where C1 and

C2 have one section for each. Branch B2 has no passing loop so the whole branch is

considered as one segment.

Figure 3.8: Rail branch includes passing loop

Passing loop (no storage)

Rail segment A1 Rail segment A2

Rail section

Rail branch B1

Rail branch B2

Rail segment A3

Junction

Siding

C1

C2

Parallel track segments

74

Case b: Siding as a passing loop

Siding types in sugarcane rail system are divided into big sidings that can allow for

more than one train to pass and small sidings which do not allow for more than one

train to pass at the same time. Small sidings can have a double track section, where

one of them is used for cane storage and another for a passing train (only one train at

a time). The big sidings can have more than two tracks where one is used for cane

storage and other tracks for more than one train passing at a time (parallel machines).

Each track in this siding can be considered as one segment that includes one section

as shown in Figure 3.9.

Figure 3.9: Rail siding includes passing point

Figure 3.9 shows a big siding (storage with passing loop) that has a triple section

where each section can be a segment and allow for passing trains. Each segment

includes only one section where Segments 3, 4 and 5 include sections S3, S4, S5

respectively. Blocking segments in this case includes blocking sections.

Mill

S1 S2 S4 S9 S8

Single section

Siding with passing loop (triple section)

Siding

S5

S6

S7

Segment 1 Segment 2 Segment 5

Segment 3

Segment 4

S3

Siding

75

Concluding Remarks

Sections 3.3.1 and 3.3.2 show the proposed mixed integer programming and

constraint programming models for the sugarcane rail transport system using

segment constraints. Figure 3.10 shows the main structure of the MIP and CP

models, where the objective functions and the rail operation and sugarcane system

constraints are used to solve the sugarcane rail problem.

Figure 3.10: MIP and CP models structure

MIP and CP models

Minimising total waiting time

Minimising makespan

Precedence

Sugarcane system constraints

Runs order

Rail operation constraints

Objective functions

Constraints

Segments order

Passing priority

Blocking

Train

Siding capacity

Bin allotment

Ready time

76

3.3.1 Blocking Segment MIP Model of the Sugarcane Rail System

In this section mixed integer programming is used as a mathematical programming

technique to obtain the solution to the sugarcane rail transport problem. Two

objective functions are used separately in this model: minimising makespan and the

total waiting time. The sugarcane system is a dynamic system so many changes can

occur daily such as the number of the trains in the system or the number of working

sidings. The mixed integer programming model is designed to be more flexible so as

to change the objective function easily to solve any new situation in the system.

Notation

Maximum number of trains. K

Index of trains; k=1,2,…K., k'=1,2,…K. ,k k′

Maximum number of segments. E

Index of the segments; e=1,…E. e

Index of sections; s=1,2,…S.

Maximum number of sections.

s

S

Index of operations of each train.

1, 2, . .o O= …… , 1, 2, . .o O′ = ……

Maximum number of operations of each train.

,o o′

O

Index of runs of each train; r=1, 2…R., r'=1,2,…R.

Maximum number of train runs. ,r r′

R

Start time of train k in run r on section s on segment e.

k osr et

Ready time for all trains where all trains are at mill and ready to move. kη

Processing time of operation o of train k on section s on segment e.

koseg

A big positive number. V

Number of full bins collected from siding s by train k during

operation o and run r on segment e. k osr e

B

Number of empty bins delivered for siding s by train k during

operation o and run r on segment e. k osr e

α

Total allotment of siding s per day.

se

A

77

Capacity of train k of empty bins. kp

Capacity of train k of full bins.

kf

Siding capacity.

seC

Makespan.

maxC

1, if train assigned to section on segment during run . =

0, otherwise.k s e r

k sr e

X

1, and are processed on section on segment = during run and respectively, and train precedes train .

0, otherwise .

k k s er r k k′

′ ′

k k sr r e

Z ′ ′

1, train uses section on segment before segment during run .=

0,otherwise.k s e e r′

k s sr e e

β ′ ′

1, if run is assigned for train before run . =

0, otherwise. r k r′

krr

µ′

1, if run is assigned for train . =

0, otherwise. r k

kr

λ

1, if the operation of train requires section on segment during run 0, otherwise.

o k s e=

k osr e

q

1, if train requires section on segment , but operation of train = scheduled at the same section on the same segment during run .

0, otherwise.

k s e ok r

′

k k osr e

b ′

Definition of a track section and segment The single track railway includes a track section and segment where; Track section: A length of track between two key points in the track network.

- End of one siding or passing loop to the start of the next.

- The start of a siding or passing loop to the end of that siding or passing loop,

including the section of mainline parallel to it.

- From the start or end of a siding or passing loop to a junction.

Track segment: a length of each branch or line in the rail network.

78

The Objective Functions

Two criteria are used to optimise the new model: minimising the makespan (Cmax)

and minimising the total waiting time for the blocking segment (TWTBSG).

To minimise the makespan (Cmax ), the completion time of the last operation, is used

and the objective function is given by Equation (3.1):

1max ( ) max

S

k k Os kOs k Os kR E E R E RsMinC Min t g q λ

=

= +

∑ (3.1)

Equation 3.1 ensures the finish time of each operation for all trains during all runs

has to be less than the makespan.

To minimise the total waiting time for the blocking sections group (TWTBSG) for

all train runs in the system, Equation (3.2) is used.

Minimise: TWTBSG = IWT + OSWT + ISWT + RWT (3.2)

The components of this objective function are explained in detail below:

The initial waiting time (IWT) considers the waiting time before starting the first run

of each train, Equation (3.3).

IWT= ( ) 11 1 11min .

K

k s k kk

t sη λ=

− ∀∑ (3.3)

The segment waiting time is the waiting time after finishing operations on one

segment and starting operations on a new one by each train. This part includes the

waiting time in both outbound and inbound directions (OSWT and ISWT,

respectively) (Equation (3.4) and Equation (3.5)).

OSWT= ( ){ }1 11 1 1 1 1 1

K R E E S S

k s k k s k Os k k Os kOs k s sr e r r e r e r r e e r e ek r e e s sq t q t gλ λ β′ ′ ′′ ′ ′′ ′= = = = = =

− +∑∑∑∑∑∑

(3.4)

ISWT = { }( )1 11 1 1 1 1 1

( 1K R E E S S

k s k k s k Os k k Os kOs k s sr e r r e r e r r e e r e ek r e e s sq t q t gλ λ β′ ′ ′ ′′ ′ ′ ′′ ′= = = = = =

− + −∑∑∑∑∑∑

(3.5)

79

The run waiting time (RWT) is the waiting time between finishing one run and

starting a new one as shown in Equation (3.6).

RWT= ( ){ }(1) 1( 1) ( 1) ( 1) 11 1

E S

k s k k s k Os k k Os kOsr e r r r E r r E Ee sq t q t gλ λ

+ + += =− +∑∑ (3.6)

Constraints

Rail operation constraints Ready time

Equation (3.7) ensures that the start time of the first operation of the first run of train

k on the first segment and the first run has to be greater than or equal to the ready

time.

( )11 1 1min 0 , .k s k kt k sη λ− ≥ ∀ (3.7)

Precedence

Equation (3.8) ensures operation o+1 of train k cannot be processed before finishing

operation o for train k on section s on segment e.

( ) ( 1) ( 1) , , , , .k os kos k os k o s k o sr e e r e r e r et g q t q k r o s e+ ++ ≤ ∀ (3.8)

Where,

If

/ 2 o O≤ for each terminal segment, then the train is moving in the outbound

direction.

If / 2 o O> for each terminal segment, then the train is moving in the inbound

direction.

In the case of an intermediate segment, the direction of the train is determined by

whether the train is occupying an outbound or inbound segment.

Segments order

Equation (3.9) ensures segment e is processed before segment e'.

80

( ) ( )-1

, , , , , , ; .

k Os kOs k Os k o s k o s k s sr e e r e r e r e r e et g q t q V

k r o e e s s e e

β′ ′ ′ ′ ′′ ′ ′+ ≤ +

′ ′ ′ ′∀ ≠ (3.9)

Equation (3.10) ensures segment e' is processed before segment e.

( ) ( )-

, , , , , , ; .

k Os kOs k Os k os k os k s sr e e r e r e r e r e et g q t q V

k r o e e s s e e

β′ ′ ′ ′′ ′ ′ ′+ ≤

′ ′ ′∀ ≠ (3.10)

Equation (3.11) states the logic relation between decision variables.

1k o s k o s k s s k s sr e e r e e r e e r e e

q q β β′ ′ ′ ′′ ′ ′ ′+ − ≤ + (3.11)

Runs order

Equation (3.12) ensures that run r is assigned before run r+1for train k.

( ) 1 1 1

1; , , , , .

k Os kOs k Os k k os k os kr E E r E r r e r e rt g q t q

r R o e e s s

λ λ′ ′′ ′+ + ++ ≤

′ ′∀ ∈ −

(3.12)

Passing priority

Equations (3.13) and (3.14) ensure trains k and k' are processed on section s of

segment e in the correct order. Either train k precedes train k' where train k' cannot

use this segment before train k leaves it, or train k' precedes train k where train k

cannot use this segment before train k' leaves it.

( )11 1

1

, , , , , , , .

O S

k o s k s kos k k sr e r e e r r eo st t g V Z

k k K k k o O s S e E r r R

′ ′ ′′ ′= =

≥ + + −

′ ′ ′ ′∀ ∈ ≠ ∈ ∈ ∈ ∈

∑∑

(3.13)

81

( )11 1

, , , , , , , .

O S

k os k s k o s k k sr e r e e r r eo st t g V Z

k k K k k o O s S e E r r R

′ ′ ′ ′′ ′′= =≥ + −

′ ′ ′∀ ∈ ≠ ∈ ∈ ∈ ∈

∑∑

(3.14)

Equation (3.15) ensures each segment cannot process more than one train at the same

time.

1 11 1

E S

k s k s k s k k s k k sr e r e r e r r e r r ee sX and X X Z Z′ ′ ′′ ′ ′

= == + − ≤ +∑∑ (3.15)

Blocking

Equation (3.16) defines the blocking constraint at each segment, where, if the

segment is used by train k, all other trains have to wait until that segment is free

before they can use it.

1 1 1 1 1

, , , , .

K E S E S

k os k os kr e r e rk e s e st b q q tk o s k k s k o s kr e r e r e r

k r r o o

λ λ′= = = = =

≥′ ′ ′ ′ ′ ′′ ′ ′

′ ′∀

∑∑∑ ∑∑

(3.16)

Equation (3.17) ensures non-negativity.

0 , 0 , , . k os kosr e e

t g k s e≥ ≥ ∀ (3.17)

Train

Equations (3.18) and (3.19) are train capacity constraints. In equation (3.18), the

number of empty bins delivered for all sidings during run r has to be less than or

equal to the capacity of train k.

1 1 1 , .

E O S

k os k os kr e r e re o sq p k rkα λ

= = =≤ ∀∑∑∑ (3.18)

82

In equation (3.19), the number of full bins collected from all sidings during run r has

to be less than or equal to the capacity of train k.

1 1 1 , .

E O S

k os k os k kr e r e re o sq B f k rλ

= = =≤ ∀∑∑∑ (3.19)

Equations (3.20) and (3.21) ensure empty bin delivery is in the outbound direction

where o and o' are two operations at each siding in the two directions and o' precedes

o. Similarly, equations (3.23) and (3.24) ensure the collection of full bins is in the

inbound direction but o precedes o'.

If o' ≤ o then

, , , , .k o s kr ep k r o s eα ′ ′≤ ∀ (3.20)

0 , , , , .k o sr eB k r o s e′ ′= ∀ (3.21)

Else

, , , , .k o s kr eB f k r o s e′ ′≤ ∀ (3.22)

0k o sr eα ′ = (3.23)

End if

Siding capacity

Equation (3.24) ensures that the total number of empty and full bins at each siding is

less than the siding capacity during each operation o, where each operation is

addressed only for a delivery or collection but not both.

( ) Ck os k os k os k sr e r e r e r eB qα λ+ ≤ (3.24)

Bin Allotment

Equations (3.25) and (3.26) show the relation between the total bin allotments for

each siding and the number of empty and full bins delivered to and collected from

83

each siding. The total number of empty bins delivered to siding s has to be equal to

the allotment of the harvester operating at this siding.

1 1 1 1 .

K R E O

k os k os k sr e r e r ek r e oq A sα λ

= = = == ∀∑∑∑∑ (3.25)

Equation (3.26) ensures that the total number of full bins collected from siding s has

to be equal to the allotment of the harvester operating at this siding.

1 1 1 1 .

K R E O

k r e oq B A sk os k os k sr e r e r e

λ= = = =

= ∀∑∑∑∑ (3.26)

3.3.2 Blocking Segment CP Model of the Sugarcane Rail System

In this section, a CP approach is used to formulate the problem. CP has the ability to

model many different types of combinatorial optimisation problems. CP is better

than mathematical programming techniques individually which have restrictive

modelling powers. CP can produce more than one solution for many scheduling

problems. CP techniques have the ability to solve all feasibility problems and can

deal with conflicting objectives.

CP problem modelling depends on the CP software package used. There are many

different modelling languages that can be used in this field. ILOG’s OPL modelling

language has been used here to formulate the sugarcane rail transport problem

because OPL has a suitable set of constructs and statements for scheduling problems

to develop an effective CP model. OPL Unary resources can process only one

operation at a time and the segments and sections are modelled as Unary resources.

Each train run is a unary resource because a train can only conduct one run at a time.

Each operation is associated with a start time and duration. Additionally,

ActivityHasSelectResource is used in OPL to decide which unary resource will be

selected for processing operation o of train k and which run ε will be selected by

train k to start first. Also, each operation requires a unary resource to be processed

on it. This section is a summary of the sugarcane transport system model using CP in

the OPL language environment.

84

Additional Notation for CP model

w Index of segments; w = 1,2…E.

k oswB

ε Number of full bins collected from siding s by train k during operation

o and run ε on segment w

k oswεα

Number of empty bins delivered for siding s by train k during operation

o and run ε on segment w

swA

Total allotment of siding s per day

swC

Capacity of siding s on segment w

k oswtε Start time of train k in run ε on section s on segment w

koswg

Processing time of operation o of train k on section s on segment w

The Objective Functions

To minimise the makespan (Cmax ), the completion time of the last operation, is used

and the objective function is given by Equation (3.27):

Minimise Cmax = ( ) ( )( )max k Os kOsR E EMin k in trains t g+ (3.27)

To minimise the total waiting time for all train runs in the system, Equation (3.28) is

used:

Minimise: TWTBSG = IWT + OSWT + ISWT + RWT (3.28)

The components of this objective function are as follows:

The initial waiting time (IWT) is shown in Equation (3.29).

IWT= ( )1 11

K

k s kk

tε

η=

−∑ (3.29)

The waiting time in outbound and inbound directions, OSWT and ISWT is shown in

Equations (3.30) and (3.31) respectively.

OSWT= ( )111 1 1 1 1

K R E E S

k k Os k Osw w wk w w st t gε ε εε

′′= = = = =− +∑∑∑∑∑ (3.30)

85

ISWT = ( )111 1 1 1 1

K R E E S

k k Os kOsw w wk w w st t gε εε

′ ′′= = = = =− +∑∑∑∑∑ (3.31)

The run waiting time (RWT) is shown in Equation (3.32).

RWT=

( )1( 1) 11 1

E S

k s k Os kOsE Ew st t g

ε ε+= =− +∑∑

(3.32)

The Model Constraints

Ready time

Equation (3.33) ensures that the start time of first operation of train k on the first

segment and the first run has to be greater than or equal to the ready time.

( )11 1min 0 , .k s kt k sη− ≥ ∀ (3.33)

Precedence

Equation (3.34) ensures that operation o+1 of train k cannot be processed before

finishing operation o for train k on section s of segment w.

( 1)( ) , , , , .k os kos k o sw w wt g precedes t k o s wε ε

ε++ ∀ (3.34)

Segments Order

Equation (3.35) ensures segment w is processed before segment w' or, segment w'

before segment w.

, , ; , , , .

k Os kOs k os k Os kOs k osw w w w w wt g t t g t

k w w w w o s

ε ε ε ε

ε

′ ′ ′′ ′ ′+ ≤ ∨ + ≤

′ ′∀ ≠ (3.35)

Runs order Equation (3.36) ensures that train k selected run ε (unary resource) to be processed.

ActivityHasSelectResource (k, Trip, ε) kε⇔ , .k K Rε∀ ∈ ∈ (3.36)

86

Runs Order

Equation (3.37) ensures that run ε is assigned to train k before run ε+1.

1

, , , , , 1.

k Os kOs k osE E wt g t

k K s s S o O w E R

ε ε

ε

′+

+ ≤

′∀ ∈ ∈ ∈ ∈ ∈ − (3.37)

Passing Priority

Equation (3.38) ensures that trains k and k' are processed on section s of segment w

in the correct order. Either train k precedes train k' where train k' cannot use this

segment before train k leaves it, or train k' precedes train k where train k cannot use

this segment before train k' leaves it.

1 11 1 1 1

, , , , , , , , .

O S O S

k o s k s kos k os k s k o sw w w w w wo s o st t g t t g

k k K k k s S w E R o o O

ε ε ε ε

ε ε

′ ′ ′ ′ ′′ ′ ′= = = =≥ + ∨ ≥ +

′ ′ ′ ′∀ ∈ ≠ ∈ ∈ ∈ ∈

∑∑ ∑∑

(3.38)

Blocking

Segments are modelled as Unary resources, where each segment w will be a unary

resource in OPL language. OPL Unary resources can process only one train at a

time.

Train

Equations (3.39) and (3.40) are train capacity constraints. In equation (3.39), the

number of empty bins delivered for all sidings during run ε has to be less than or

equal to the capacity of train k.

1 1 1. ,

E O S

k os kww o sp k

εα ε

= = =≤ ∀∑∑∑ (3.39)

Equation (3.40) ensures that the number of full bins collected from all sidings during

run ε has to be less than or equal to the capacity of train k.

87

1 1 1. ,

E O S

k os kww o sB f k

εε

= = =≤ ∀∑∑∑ (3.40)

Equations (3.41) and (3.42) ensure empty bin delivery is in the outbound direction

where o and o' are two operations at each siding in the two directions and o' precedes

o. Similarly, equations (3.43) and (3.44) ensure the collection of full bins is in the

inbound direction but o precedes o'.

If o' ≤ o then

, , , , .k o s kwp k o w s

εα ε′ ′≤ ∀ (3.41)

0 , , , , .k o sw

B k o s wε

ε′ ′= ∀ (3.42)

Else

, , , , .k o s kwB f k o s w

εε′ ′≤ ∀ (3.43)

0k o swεα ′ = (3.44)

End if

Bins Allotment

Equations (3.45) and (3.46) show the relation between the total bin allotments for

each siding and the number of empty and full bins delivered to and collected from

each siding.

The total number of empty bins delivered to siding s has to be equal to the allotment

of the harvester operating at this siding.

1 1 1 1.

K R E O

k os sw wk w oA s

εεα

= = = == ∀∑∑∑∑ (3.45)

The total number of full bins collected from siding s has to be equal to the allotment

of the harvester operating at this siding.

1 1 1 1.

K R E O

k os sw wk w oB A s

εε= = = == ∀∑∑∑∑ (3.46)

88

Siding Capacity

Equation (3.47) ensures that the total number of empty and full bins at each siding is

less than the siding capacity during each operation o.

, , , , .k os k os sw w wB C k o s w

ε εα ε+ ≤ ∀ (3.47)

Concluding Remarks

CP model uses fewer constraints than the MIP model because CP can combine

several constraints to work as one constraint using the statement OR “∨ ” such as in

passing priorities constraints and runs order constraints. CP uses variable subscripts

instead of binary decision variables. For this reason, there are no binary decision

variables in CP, while the MIP model includes many of them such as k osr eq . Two

variable subscripts are used: sw and kε, where sw represents selecting section s on the

selected segment w to train k while kε represents the selected run ε of train k. For that

reason the notations of segment and run are changed in the CP model.

When implementing the CP code using OPL software, some commands can be used

as well by the following form under constraint programming fundamentals:

koswg variable will be defined as Duration [k, r, e, o, s]

Activity task [k, ε, w, o, s] is defined on Duration [k, r, e, o, s]

The variable k osr wt can be formulated by constraint programming as

task [k, ε, w, o ,s].start

Usually in OPL code we use precedes in CP formulation instead of ≤

(less than sign).

89

3.4 Description of the Blocking Section Models of Sugarcane Rail System

Blocking section models are proposed to improve the efficiency of the performance

of the sugarcane rail system by reducing the total waiting time and other objectives

such as makespan. The main issue is the blocking section is not applicable for the

current real sugarcane rail system, so some assumptions are proposed to apply the

blocking section models to this system. This section investigates whether the

blocking section model can reduce the makespan and total waiting time for the

sugarcane rail system (Masoud et al., 2010a). Here proposals are put forward to

develop the blocking sections for sugarcane rail models. These proposals include

physical changes and modelling approaches to reduce delays as follows:

i. Physically, More passing loops may be constructed in the track segments of

the sugarcane rail network or some sidings are working as passing loops.

Passing loops can ease the passage of passing trains and reduce the number

of conflict points throughout the rail network.

ii. In the developed models, the consecutive short sections may be combined to

form one section. This can help ensure that the length of the train is less than

the length of the section and so section blocking constraints can be applied.

iii. In the developed models, the sidings located in consecutive short sections

may be combined with their nearest siding to work as one siding. The

capacity of this siding would equal the combined capacities of the original

sidings and the total allotments of these sidings.

MIP and CP approaches have been integrated with blocking parallel job shop

scheduling (BPJSS) techniques to produce MIP and CP models for the sugarcane rail

system. The segment parameter has been removed from the MIP and CP models. In

these models, parallel track sections are called units and the parallel track sections

are defined as the same section. Some units are defined as sidings or passing loops.

Figure 3.11 shows a portion of a rail network where siding and siding with passing

loop (multi track section units) are included. Single sections in the rail network are

90

used only for one-train-at-a-time passing and they do not include any sidings for

delivery and collection.

Figure 3.11: A single cane rail network with three sidings can allow passing trains

3.4.1 Blocking Section MIP Model of Sugarcane Rail System

The blocking segment MIP model in section 3.3.1 has been modified and updated to

develop the blocking section MIP model. Five equations have been updated and

some notations have been removed as follows:

1- Precedence constraint (Equation 3.8) was updated.

2- Segments order constraints (Equations 3.9, 3.10, and 3.11) were removed.

3- Passing priority constraints (Equations 3.13, 3.14, and 3.15) were updated.

4- Blocking constraints (Equation 3.16) were modified.

5- The siding capacity and bin allotment constraints (Equations 3.18 - 3.26) were

modified with section unit notation.

Additional Notation

U Maximum number of units of sections s.

u Index of unit uth of section s; u=1,2…U.

1 : if section is processed by train before section

0 : k ssr

s k s

otherwiseϕ ′

′=

Mill

s1

s22

s6

s21

s23

s3 s51 s4

s71

s72

s73

Single section

Siding Junction

Multi section units

s24

91

Objective function

Two criteria have been developed for the blocking section MIP model; minimising

the makespan (Cmax) and minimising the total waiting time.

To minimise the makespan (Cmax ), Equation (3.48) is developed as follows:

( ) 1 1

, 1,.., .maxS U

k k os kos k os kR u u R u Rs uo k KMinC Min max t g q λ

= =∀ =

= +

∑∑ (3.48)

To minimise the total waiting time for the blocking section (TWTBS) for all train

runs in the system, Equation (3.49) is used.

TWTBS= IWT + OSWT + ISWT + RWT (3.49)

The initial waiting time (IWT) is the waiting time before starting the first run of each train as shown in Equation (3.50).

IWT= ( )1 11 1 11 ,

K

k s k k s ku ukt q s uη λ

=− ∀∑

(3.50)

The section waiting (SWT) time is the waiting time on each section where the next

section is blocking. This part includes the waiting time in outbound and inbound

directions (OSWT and ISWT, Equation (3.51) and (3.52) respectively).

OSWT= ( ){ }( 1) ( 1)1 1 1 1 1

K R S S U

k o s k k o s k os k k os kos k ssr u r r u r u r r u u rk r s s uq t q t gλ λ ϕ′ ′ ′+ +

′= = = = =− +∑∑∑∑∑

(3.51)

ISWT= ( ){ }( )( 1) ( 1) ( 1)1 1 1 1 1

1K R S S U

k os k k os k o s k k o s k o s k ssr u r r u r u r r u u rk r s s uq t q t gλ λ ϕ′ ′ ′ ′+ + +

′= = = = =− + −∑∑∑∑∑

(3.52)



RWT= ( )( ){ }(1) 1( 1) ( 1) ( 1)1 1max

S U

k s k k s k os k k os kosr u r r u r u r r u us uq t q t gλ λ

+ + += =− +∑∑

(3.53)

92

Precedence Equation (3.54) ensures operation o+1 of locomotive k cannot be processed before

finishing operation o for locomotive k for outbound locomotives.

( ) ( )( 1) ( 1) 1

., ., 1..., 1., , .

k os kos k os k k o s k ssk o sr u u r u r r u rr u krt g q q t V

k K r R o O s S u U

λ ϕλ ′+ ++ ≤ − −

∀ ∈ ∈ = − ∈ ∈

(3.54)

Equation (3.55) ensures operation o of locomotive k cannot be processed before

finishing operation (o+1) in locomotive k for inbound locomotives.

( )( )( 1) ( 1) 1

., ., 1..., 1., , .

k o s k o s k os k os kos k ssr u r u r u r u u rq t q t g V

k K r R o O s S u U

ϕ ′+ + ≤ − + −

∀ ∈ ∈ = − ∈ ∈

(3.55)

Passing priority

In equation (3.56), trains k and k' respectively are processed on the unit uth of section

s and locomotive k is processed after locomotive k'.

( )'1

, , ; ; ; , ; , .

k os k o s k o s k ks k ksr u r u u ut t g Z V Z

k k K k k s S u U r r R o o O

′ ′ ′ ′ ′′≥ + − −

′ ′ ′ ′∀ ∈ ≠ ∈ ∈ ∈ ∈ (3.56)

In equation (3.57), trains k and k' respectively are processed on the unit uth of section

s and locomotive k' after locomotive k.

( )1

, , , , , , o, .

k o s k os kos k ks k ksr u r u u u ut t g Z VZ

k k K k k s S u U r r R o O

′ ′ ′ ′′≥ + − −

′ ′ ′ ′∀ ∈ ≠ ∈ ∈ ∈ ∈ (3.57)

Blocking

Equation (3.58) is a blocking condition. Operation o of locomotive k requires the

unit uth of section s but the operation o' of locomotive k' is scheduled on the same

unit of the same section.

93

1 1 1 1 1

, , , , .

K S U S U

k o s k k s k o s k k os k os kr u r u r u r r u r u rk s u s ut b q q t

k r r o o

λ λ′ ′ ′ ′ ′ ′′ ′ ′′= = = = =≥

′ ′∀

∑∑∑ ∑∑ (3.58)

3.4.2 Blocking Section CP Model of Sugarcane Rail System

The units of each section are used as unary resources in the blocking section CP

model, meaning that no more than one train can use each unit of each section at a

time. The blocking segment CP model is modified and updated as follows:

In the blocking CP model, unit u on section s is a Unary resource in OPL language. ActivityHasSelectResource is used instead of binary variables.

As for the blocking section MIP model, the blocking section CP model requires

changes to the objective function, and precedence, passing priority and blocking

constraints from the blocking segment CP model.

The makespan and total waiting time are minimised as objective functions in the Blocking Section CP Model (Equations 3.59 and 3.60):

( ) 1 1

, 1,.., maxS U

k k os kosR u us uo k KMin C Min max t g

= =∀ =

= +

∑∑

(3.59)

TWTBS= IWT + OSWT + ISWT + RWT (3.60)

The initial waiting time (IWT) is the waiting time before starting the first run of each train as shown in Equation (3.61).

IWT= ( )111 ,

Ktk s kuk

s uη=

− ∀∑ (3.61)

The section waiting (SWT) time is the waiting time on each section where the next

section is blocked. This part includes the waiting time in outbound and inbound

directions (OSWT and ISWT, Equation (3.62) and (3.63) respectively).

OSWT= ( )( 1)1 1 1 1 1

K R S S U

k os kosk o s u uk s s ut t g

u εεε′+

′= = = = =

− +

∑∑∑∑∑

(3.62)

94

ISWT= ( ){ }( 1) ( 1)1 1 1 1 1

K R S S U

k os k o s k o su u uk s s ut t gε εε

′ ′+ +′= = = = =

− +∑∑∑∑∑

(3.63)



RWT= ( )( ){ }1( 1)1 1max

S U

k s k os kosu u us ut t g

ε ε+= =− +∑∑ (3.64)

Precedence

Equation (3.65) ensures operation o+1 of locomotive k cannot be processed before

finishing operation o for locomotive k for outbound locomotives or operation o of

locomotive k cannot be processed before finishing operation o+1 in locomotive k for

inbound locomotives.

( ) ( )( 1) ( 1)

., R., 1..., 1., , .

k os kos k o s k o s k os kosu u u u u ut g t t t g

k K o O s S u U

ε ε ε ε

ε

+ ++ ≤ ∨ ≤ +

∀ ∈ ∈ = − ∈ ∈

(3.65)

Passing priority In equation (3.66), trains k and k' respectively are processed on unit u of section s

and locomotive k is processed after locomotive k' or locomotive k' after locomotive

k.

( ) ( )

, , ; ; ; , ; , .

k os k o s k o s k o s k os kosu u u u ut t g t t g

k k K k k s S u U R o o O

ε ε ε ε

ε ε

′ ′ ′ ′ ′ ′′ ′≥ + ∨ ≥ +

′ ′ ′ ′∀ ∈ ≠ ∈ ∈ ∈ ∈ (3.66)

Blocking

Sections and units are modelled as Unary resources. OPL Unary resources can

process only one operation at a time. Each unit u of section s will be a unary resource

in OPL language.

95

3.5 Inclusion of the Delivery and Collection Time Constraints to the Model

The models developed in sections 3.3 and 3.4 examine many issues of train

scheduling with blocking constraints; single track railway conflicts; delivering and

collecting conflicts in the outbound and inbound directions; and train and siding

capacities during the delivering and collecting of bins. Results from these models are

described in Chapter 5. Constraints associated with delivery and collection times for

satisfying the empty and full bin requirements for each siding are not included in

these models.

The delay problems in satisfying the empty and full bin requirements for each siding

in the sugarcane rail system have been solved with additional constraints to optimise

the delivery and collection time. The time table needs to satisfy the harvesters’

requirements for a continuous supply of empty bins to harvest continuously and the

mill requirements for a continuous supply of full bins to crush continuously. In

addition, the number of bins in the system has to be minimised by spreading out the

empty bin deliveries to match the mill’s production of empty bins resulting from

emptying the full bins during crushing operations. Optimising the collection time for

each siding can reduce the time between harvesting and crushing (cane age) to keep

sugarcane quality high.

A specific scenario, developed to illustrate the main aims of a sugarcane rail

transport system is explained as follows:

This scenario assumes that the delivered empty bins during the last visit

to a siding during a day are provided for use the next day when the

harvester commences. The number of bins to deliver is determined by

considering the total allotment for each siding and the siding capacity.

The visits to each siding for one day are organised in the next steps.

96

Begin

Step1: First visit

Aim: The main aim of the first visit to a siding for the day is to deliver

empty bins which are to be used until the train’s next visit.

Additionally, during this first visit, the empty bins delivered the

previous day that have been filled are collected.

Time: The time of the first visit is optimised by considering the number

of empty bins delivered to the siding in the last visit of the previous

day, the harvester start time and the harvesting rate at the siding as

follows:

(first visit time – harvester start time)*harvester rate ≤ the number of

empty bins delivered in the last visit of the previous day.

Delivered empties: The number of delivered empties in the first visit

should be sufficient to satisfy the harvester’s needs without interruption

until the second visit to the siding. The maximum number of delivered

empty bins is limited by the siding capacity.

Step 2: Second and subsequent visits up to the last visit

Aim: the main aim of the second and subsequent visits before the last

visit is to deliver the empty bins to the siding that will be used until the

next visit. Additionally, the now-full bins at the siding will be collected.

Time: the time of the second and subsequent visits before the last visit is

optimised by considering the number of empty bins delivered to the

siding in the previous visit, the time of the previous visit and the

harvester rate at the siding as follows:

(second visit time – first visit time)*harvester rate ≤ the number of

empty bins delivered in the first visit of that day.

97

Delivered empties: the number of delivered empties in the second and

subsequent visits before the last visit should be sufficient to satisfy the

harvester’s needs until the next visit to the siding.

Repeat Step 2 until the siding allotment of empty bins has been

delivered

Step 3: Last visit

Aim: the main aim of the last visit is to deliver the empty bins to the

siding to be used the next day until the first visit and to collect the last

of the full bins (the siding’s allotment of full bins is completed) filled

today.

Time: the time of the last visit is after the completion of the harvest for

the day and is optimised by considering when the full bins are required

by the mill.

Delivered empties: the delivered empties in the last visit should be

sufficient to make the harvester work without interruption the next day

from harvester start time until the train’s first visit to the siding.

General Constraints

The siding capacity and the anticipated number of full bins at the siding

are taken into account during delivery of empty bins at the siding. The

allotment of the delivered empty bins equals the allotment of the

collected full bins each day.

End

A numerical example for the visit times and operations at siding A is shown in

Figure 3.12.

98

Harvester start=4 am

Harvester rate =10 bins/hour

Allotment 174 bins

Siding capacity 140 bins

Siding A

Figure 3.12: Numerical case

Figure 3.13 shows the results of the numerical case in Figure 3.12 using the specific

scenario to optimise the siding visit time. Given the siding capacity of 140 bins, no

more than 70 bins can be delivered at a time so that there is room for the next

delivery. The 34 empty bins delivered overnight will be consumed at 7:24 and so

that is the time chosen for the first visit. The 70 empty bins delivered in the first

visit will be consumed at 14:24 and so that the time chosen for the second visit. The

70 empty bins delivered in the second visit will be consumed at 21:24 and so that is

the time chosen for the last visit. The last visit time can be later than 21:24 since

delivery of empty bins is not time critical.

0 24 hours

harvester start first visit second visit last visit

Time 4 am 7:24 14:24 21:24(or later)

Delivered empties 70 70 34

Collected Fulls 34 70 70

Figure 3.13: Visit times of siding A for one day

Optimising the delivery and collection time for the sugarcane rail transport system

requires extra equations. All steps to optimise the siding visit time in the sugarcane

transport system were modelled using MIP.

Visits

99

Additional Notations for the Delivery and Collecting Time Constraints

rateseH Harvester rate at siding s in segment e.

startseH

Harvester start time at siding s in segment e.

1,There is a delivery at siding in segment during run by train .0,Otherwise.

s e r kNk osr e

=

New constraints were developed to update the MIP model to optimise the visit time

table to the sidings as follows:

3.5.1 Delivering Delay Constraints

- No Delays with the First Visit

To remove any Delivering Delay at each siding throughout the rail network, the

model has to distinguish between the following scenarios:

Sections are used for travelling through, and not delivering and not

collecting.

Some sections contain a siding, but there is no harvester at this siding. This

siding is only for train travelling

Sections are used for train travelling and including a siding and have

harvesters but maybe one of the trains uses it just for travelling so as to reach

another siding, so no delivering or collecting.


harvesters but the train wants collecting only and no delivering.


harvesters and the train needs to make deliveries.

Equations (3.67) and (3.68) ensure there are no harvesting delays at the time of the

first visit. Equation (3.67) is important for achieving Equation (3.68), where

Equation (3.67) ensures the section working as a siding and there is a delivery to the

siding (last scenario) . This will execute Equation (3.68) and without Equation

(3.67), Equation (3.68) cannot be applied correctly.

100

M *(1 ) , , , .k os k osR e R eN k K o O s S e Eα ≤ − ∀ ∈ ∈ ∈ ∈ (3.67)

( )1

60*60

, , , .

/ * *k os k os start rate k osR e e s s R ee et H H M N

k K o O s S e E

α

∀ ∈ ∈ ∈ ∈

≥ − −

(3.68)

Equation (3.68) ensures the number of delivered empties in the last run for siding s

in segment e is greater than or equal to the siding’s requirements until the time of the

first visit. The last run is used for the next day to ensure there are sufficient empty

bins for the period between harvester start time and the first visit the next day.

The time of the first visit, the harvester start time, and the harvesting rate are

considered for each siding s. Equation (3.68) will be ignored if there is no delivery at

the siding.

- No delaying Between Next Visits

Equations (3.69) and (3.70) ensure there are no harvesting delays, where the time of

the second visit for each siding depends on the number of empty bins delivered in

the first visit, and so on. Equations (3.69) and (3.70) ensure the delivered empty bins

in any run will be sufficient until the next visit. These equations work from the

second visit to the last visit for each siding.

*(1 ) , , , , .k os k osr e r eM N k K r R o O s S e Eα ≤ − ∀ ∈ ∈ ∈ ∈ ∈ (3.69)

( )( 1)

(( ) / 60*60 )* *

, 1, , , .

k os k os k os rate k osr e r e r e s r eet t H M N

k K r R o O s S e E

α+

≥ − −

∀ ∈ ∈ − ∈ ∈ ∈ (3.70)

The time between the visits are used by the harvester to fill the empty bins at the

siding, so the delivered number at any visit has to be sufficient for the harvester until

the next visit. Equation (3.69) operates as in Equation (3.67) to check if there is a

delivery to be made at the siding.

101

3.5.2 Collecting Delay Constraints

These constraints ensure that no bins are collected before they are filled.

The number of full bins collected depends on the harvesting rate at each siding and

the visit times. Equations (3.71) and (3.72) ensure that the number of full bins

collected is no more than the number of full bins produced at each siding.

( ) 1 1

( ) / 60*60 )*

, , , .

k os k os start ratee e s se eB t H H

k K o O s S e E

≤ −

∀ ∈ ∈ ∈ ∈ (3.71)

( )(( ) / 60*60 )*( 1)

, 1, , , .

rateseB t t Hk os k os k osr e r e r e

k K r R o O s S e E

≤ −+

∀ ∈ ∈ − ∈ ∈ ∈ (3.72)

The challenge, however, is to minimise the size of the bin fleet. It is easy to

simply add more empty and full bins at the mill at the start of the simulation to

make sure that the mill never runs out of empty bins to supply the harvesters or

full bins to crush. The challenge is to delay deliveries as long as possible to

reduce empty bin demand at the mill. To meet this challenge, the time of the

last visit to a siding after the completion of the harvest for the day (when time

constraints on deliveries are looser) is optimised by considering when the full

bins are required by the mill.

102

3.6 Sugarcane Rail System as a Dynamic System

The sugarcane transport system is a dynamic system because new and sometimes

urgent situations arise daily such as changes in the number of trains, sections and

harvesters. The sugarcane rail system is modelled as a blocking parallel-machine job

shop scheduling (BPMJSS) problem. BPMJSS can be applied for dynamic systems

considering some dynamic variables: section unexpected delay, harvester unexpected

Delay and adding new jobs in the system including the train unexpected delay. The

main reasons for unexpected delays are maintenance (stop working temporary) or

breakdown (stop working for a long time) process.

Two options are available to resolve the unexpected delays; repair or remove the

used element (harvester, loco or section) in the system. Three parameters have to be

constructed to identify the sections or harvesters unexpected delay; Index of

harvester or section delay; Delay time point (delay start time); Delay period (the time

between the start and finish delay time)

Adding new jobs in the system means adding new trains to the rail system where two

parameters are identified; the index and arrival time of each train. The rail system

operations will be rescheduled by updating three main lists; harvesters, sections and

trains as shown in Figure 3.14. Solution techniques of the sugarcane rail transport

system are very quick to reschedule the rail operations system in reasonable time.

103

Figure 3.14: Sugarcane rail system as a dynamic system

104

3.7 Conclusion

MIP and CP models have been developed to solve sugarcane rail transport system

problems by optimising the operational time throughout the rail system. The core of

the constraint programming approach is to satisfy all problem constraints, while MIP

focuses on improving the objective function. Both of these approaches are useful and

so the CP search techniques are integrated with the MIP model in the solution

approach in Chapter 4 to solve the sugarcane rail transport system problem. The CP

approach has the additional advantage that the number of constraints is less than the

number of MIP model constraints, since CP combines many constraints into one

constraint. Therefore, the memory capacity during implementing the solution code

can be saved in the CP model particularly in small-size problems. Many solution

techniques are described in Chapter 4 to solve the MIP and CP models in reasonable

time.

The models have been constructed in two parts. The first part, containing the rail

operation constraints and the basic bin delivery and collection constraints, is

extensively used in this thesis. The second part extends the models to optimise the

visit times at each siding. Optimising visit times removes any interruption to

delivering and collecting the empty and full bins to and from each siding.

105

Chapter 4

Solution Approach

Chapter Outline

4.1 Introduction.....................................................................................................................107

4.2 Constraint Satisfaction (CS) Techniques........................................................................109

4.2.1 Constraint Propagation.......................................................................................109

4.2.1.1 Node Consistency...............................................................................111

4.2.1.2 Arc Consistencies................................................................................112

4.2.1.3 Bounds Consistency............................................................................115

4.2.1.4 Path Consistency.................................................................................115

4.2.2 Search Process...................................................................................................116

4.2.2.1 Variable and Value Ordering Heuristics..............................................117

4.2.2.2 Search Techniques...............................................................................119

4.2.2.3 Global Constraint.................................................................................124

4.3 Proposed Algorithms......................................................................................................125

4.3.1 Collecting and Delivering Conflict Elimination................................................126

4.3.1.1 Terminal Segment Conflict Elimination...............................................127

4.3.1.2 Intermediate Segment Conflict Elimination.........................................130

4.3.2 Algorithms for Solving Train Conflicts..............................................................133

4.3.3 Computing Acceleration (CA) Algorithms..........................................................137

4.4 Conclusion......................................................................................................................139

106





Malaysia. (Outstanding Student Paper Award)

107

4.1 Introduction

In this research, mixed integer programming (MIP) and constraint programming

(CP) formulations of the sugarcane rail transport systems are solved using the

integration of job shop scheduling (JSS) techniques and constraints satisfaction

techniques. Constraints satisfaction has been used previously as a solution technique

for constraint programming problems. It has been applied in this research with mixed

integer programming, in particular for search techniques. Linear relaxation

techniques in Optimization Programming Language OPL and CPLEX software are

integrated with constraint satisfaction techniques during problem solving for the CP

and MIP models respectively. The linear relaxation technique focuses on the

objective function and improves its value by removing suboptimal solutions, while

constraint stratification techniques such as constraint propagation and search

techniques focus on the problem constraints and remove the infeasible values from

the constraints domain.

MIP uses an LP (Linear Programming) solver for the continuous relaxation of the

problem to obtain infeasibility or a lower bound for the objective function. CP on

the other hand uses constraints propagation to obtain infeasibility or a lower bound,

in the constraint variables. The search algorithms of the two techniques are only

slightly different. The algorithms in CP are similar to the Branch and Bound

algorithms which are used in MIP, (Haralick & Elliott, 1980; Bockmayr & Ksper,

1998). A search tree is used in most of the search algorithms, where each node

represents a decision variable and each branch represents a value assigned to the

variable. The integration of the MIP formulation and constraint satisfaction search

techniques of CP have been used in this research.

Integrating CP and MIP or Integer Linear Programming (ILP) approaches has

received attention from many researchers since the late 1990s to combine the

advantages of both and avoid their individual weaknesses (Puget & Lustig, 2001;

Foccaci et al., 2002; Mouret et al., 2009). Identifying their commonalities and

differences ensures integrity between these techniques. The CP and ILP techniques

were compared through three criteria: modelling ability, node processing and search

algorithm. The two approaches have the common concepts of fundamental

108

modelling such as decision variables, constraints on variables, and the search

algorithm.

Integration of integer programming and CP has been used to solve combinatorial

problems, especially scheduling problems, as opposed to using either of the two

approaches individually, (Milano, 2004). Artigues et al. (2009) designed a new

mathematical model to optimise employee timetabling by integrating ILP and CP.

The CP approach combined the LP relaxation into global constraint and reduced

cost-based filtering techniques. Optimization Programming Language (OPL) has

integrated an integer programming approach and CP to solve scheduling problems.

Hentenryck (2002, 1999) designed CP and integer programming models and showed

how OPL can integrate the two approaches to solve combinatorial optimization

problems easily and in reasonable time.

A hybrid method for solving planning and scheduling problems (Hooker, 2005; Beck

& Refalo, 2003) integrates Mixed Integer Linear Programming (MILP) and CP

through logic-based Bender decomposition to take advantage of both techniques to

minimise cost and/or makespan objectives. MILP solved the assignment using

CPLEX software and CP solved scheduling problems using ILOG Scheduler

software by obtaining an optimal solution for large-scale problems where Bender

decomposition algorithms gave a feasible solution. El Khayat et al. (2006) designed

a mathematical model and CP model for solving scheduling problems, especially job

shop scheduling problems, where machines and vehicles were both considered as

constraining resources. Mathematical models are time consuming for solving some

problems, especially some large-scale problems, while CP can provide solutions in

reasonable time. Integrating the two approaches produced optimal schedules for

many test case problems using ILOG OPL Studio.

In chapter 3 of this thesis, four models of sugarcane rail transport systems were

formulated as blocking parallel-machine job shop scheduling (BPMJSS) problems

using two approaches. MIP and CP. Each model included two main parts: an

objective function and constraints, where all problem constraints have to be satisfied

while improving the objective function. In this chapter, many solution techniques

are proposed including constraints satisfaction techniques which are integrated with

developed algorithms to solve many issues in the sugarcane rail transport system

109

models such as train conflicts and the delivery and collection of empty and full bins

(Masoud et al., 2010b).

Metaheuristics with hybrid (hybrid SA/TS and hybrid TS/SA) and hyper (hyper

SA/TS and hyper TS/SA) techniques are presented in Chapter 6 to solve large-scale

problems in reasonable time.

4.2 Constraint Satisfaction (CS) Techniques

Constraint satisfaction (CS) that deals with problems defined within the finite set of

possible values of each variable is the main technology used for solving

mathematical formulation problems. This ‘finite set’ is often called the domain of the

variables. Most constraints of industrial applications use a finite domain. CS depends

on two main techniques: constraint propagation and search process, where both are

integrated to solve the mathematical formulation problems.

The following sections show some of the aspects of CS including the techniques and

methods used to solve optimisation problems, especially sugarcane rail transport

system problems.

4.2.1 Constraint Propagation

Constraint propagation eliminates values from a variable’s domain that do not satisfy

the constraints. Propagation is often referred to as consistency maintenance and

includes node consistency (NC), arc consistency (AC) and path consistency. Many

techniques have been developed to achieve AC and NC (Tsang, 1993). The nature of

the consistency techniques is deterministic while the search process is non-

deterministic. The following Example 4.1 illustrates the main concept of constraint

propagation and how it works with the problem domain.

Example 4.1

Assume x, y, and z are three variables with integer values in the

interval [1, 6], with constraint y < z. The decision variable y can take a

value in [1, 6], so y has to be at least 1. While z > y, z=1 cannot satisfy

the constraint. Therefore 1 is removed from the domain of z; z∈[1,

6]. By the same method, the domain of y will be [1, 5], and there is no

110

change in the domain of x, because no constraint includes x at this

point. A new constraint can be added to the problem like, x = y×z.

The minimal possible value for y is 1, and for z is 2, so x has to be at

least 2. As a result, the domain of x will be reduced to [2, 6].

Moreover the maximum value of x is 5 and the minimum value of z is

2, while y which is equal to x/z, must be, at most, 3.

In Example 4.1, Constraint propagation reduced the domain of the variables to be

[ ]2,6x∈ , [ ]1,3y∈ and z∈[2, 6]. As a result, the new domains of x, y, and z have

become consistent with all problem constraints.

Consistency techniques were introduced by Montanari (1974) and Waltz (1975) in

the artificial intelligence area to improve the efficiency of picture recognition

programs. Picture recognition includes labelling all lines in a picture in a consistent

manner. A significant number of possible combinations can be reduced to consistent

combinations only. Consistency techniques effectively remove many inconsistent

assignments at an early stage and are applicable for complex problems. Example 4.2

describes how these techniques work and their application.

Example 4.2

Let x<y be a constraint of the variable x with the domain Dx=[4,8] and

y with the domain Dy=[3,6]. For some values of Dx, there are

inconsistent values in Dy which do not satisfy the constraint x<y and

vice versa. As a result, some values will be removed from the

respective domains without loss of any solution. So, the domains will

be Dx= [4, 5] and Dy= [5, 6].

In Example 4.2, the reduction of the domains of x and y does not remove all

inconsistent pairs necessarily, where x=5, and y=5 is still in the domains of x

and y. In spite of that, for each value of domain x, one can get a consistent

value for domain y and vice versa.

Many consistency techniques were proposed for constraint graphs in binary CS

problems to prune the search space. A consistency-enforcing algorithm makes any

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partial solution of a small sub network extensible to some surrounding network.

Thus, the potential inconsistency is detected as soon as possible. In the following

techniques, a binary CS problem is represented as a constraint graph. Each node is

labelled by the variable and the edge between two nodes corresponds to the binary

constraint binding the variables that label the nodes connected by the edge. A unary

constraint can be represented by the cycle edge and it can terminate at the same

node.

4.2.1.1 Node Consistency

Node consistency (NC) is the simplest consistency technique. The node representing

the variable y is node consistent if and only if for every value z in the current y

domain, each unary constraint on y is satisfied. A CS problem is node consistent if

and only if all variables are node consistent, i.e., all variables’ value in its domain

satisfy the constraints of these variables. The next paragraph illustrates how this

technique works.

If the domain Dy of the variable y includes a value “b” that does not satisfy the

unary constraint on y, then the instantiation of y to “b” will always result in

immediate failure. So, the node inconsistency can be removed by eliminating those

values from domain Dy of variable y which can not satisfy the unary constraint on y.

The main steps of NC are shown as follows:

Node Consistency Algorithm For each y in nodes(G) Do

For each z in the domain Dy of y Do

If any unary constraint on y is inconsistent with z Then

Delete z from Dy

End If

End For

End For

End NC

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4.2.1.2 Arc Consistencies

This strategy considers a constraint as a link between variables including the

bounded domains of these variables. To illustrate this method, consider the domains

of the variables x, y are Dx=Dy = {2, 3, 4, 5}, and with the simple constraint, x-1=y,

on variables x and y. So the constraint Cxy will be represented as the set {(3, 2), (4,

3), (5, 4)}, which is all the combinations of x and y that satisfy this constraint. The

constraints can be classified depending on the number of variables, where unary

constraints affect one variable. For example, y is an odd integer variable, so the node

consistency will remove all even numbers from y’s domain, in the other words, ∀ a

∈y, cy ≡odd is satisfied. Another type of constraint is a binary constraint, which

affects two variables. Generally, n-ary constraints have an effect on n variables. A

graph can be used to represent binary CS problems. To do that, nodes represent the

variables and the edge is the constraint between two variables if there is a constraint

between them.

Definition: The directed arc between x and y, arc(x, y), is called an arc constraint if

∀ u ∈Dx ∃ v∈Dy such that u and v satisfy the constraint Cxy. A CS problem is arc

consistent if and only if every arc(x, y), in its constraint graph is arc consistent.

From the constraint x-1=y, the value x=2 cannot satisfy the constraint, because there

is no value in Dy such that 2-1=y. As a result, any value u ∈ Dx that cannot satisfy

the constraint Cxy, can be removed from Dx. By the same previous idea of making

arc(x, y) consistent, any inconsistent values can be removed from y’s domain by

acting arc(x, y) consistent.

In the literature, many algorithms have been proposed to solve a binary CS problem

using arc consistent principles. The complexity for general constraints is O (d2c),

where d is the maximum domain size and c is the number of binary constraints in the

problem. It is clear that an arc, arc(x, y), can be made consistent by simply deleting

those values from the domain of x, Dx, for which there does not exist a

corresponding value in the domain of Dy such that the binary constraint between x

and y is satisfied (note that deleting of such values does not eliminate any solution of

the original CS problem). The algorithm of arc consistency includes the next main

steps as follows:

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Arc Consistency Algorithm For each x in Dx Do

If there is no such y in Dy such that (x, y) is consistent, Then

Delete x from Dx

End If

End For

End

Executing the previous technique for each arc just once is insufficient to make every

arc of the constraint graph consistent. Implementing an arc consistency algorithm

once reduces the domain of some variables xi, then each previously revised arc(xj, xi)

has to be revised again, since some of the values of the domain of xj may not be

consistent with any remaining values of the revised domain of xi.

In addition to the arc consistency algorithm, there are many algorithms demonstrated

as Arc Consistency-n (AC) named AC-1 to AC-7. These algorithms were developed

for binary constraints. Mohr and Henderson (1986) developed AC-4, the important

method of selecting values for a variable domain to support a value through another

variable’s domain. For example with the binary constraint x-1=y, between variable x

and y, (presented before in this section), 3 in Dy is supporting value 4 in Dx, 4 in Dy

is supporting value 5 in Dx, and so on. But there is no value supporting the value 2 in

Dx and for that reason, value 2 can be removed from Dx. The same procedure can be

applied to the directed arc(y, x).

For any value through a variable’s domain, AC-4 compiles and preserves a list of

supporting values from the domain of the other variables. If there are no supporting

values or all supporting values are removed from the domains of their variables, then

the supported value should also be deleted from the domain of its variable. The AC-4

is executed in the worst case time complexity O (d2c), where d is the maximum

domain size and the c is the number of binary constraints (Mohr & Henderson, 1986;

Deville & Hentenryck, 1991). Hentenryck et al. (1992) proposed the AC-5

algorithm which is executed in the worst case time complexity O (d2c). However for

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functional, anti-functional and monotonic classes of constraints, it provides a

specialized arc consistency algorithm of time complexity O (dc). Moreover, this

algorithm provides the user with an interface for applying consistency checks for

many types of constraints. This interface helps identify the difference between the

current variable domain and the previous domain of this variable, after the previous

activation of the constraint (Puget 1995). The AC-5 method was implemented by

ILOG Solver.

Bessiere et al. (1993) proposed AC-6 to improve the AC-4 algorithm. AC-6 is

executed in the worst case time complexity O (d2c); however, its space complexity is

decreased to O (dc). This algorithm keeps just one supporting value at a time, for

each value to be supported for each constraint. Another value can replace the

original, if this supporting value is removed. If no other supporting value remains,

the supported value can be removed. Bessiere et al. (1994) developed the AC-7

algorithm to improve the supporting value idea of AC-4 and AC-6. The proposed

AC-7 algorithm works in two directions and uses the constraint meta knowledge to

reduce the arc consistency computation. For instance, a value u ∈Dx which is

supported by a value v ∈ Dy also supports the value v.

Arc- consistency algorithms are efficient for general binary constraints. However, for

general n-ary constraints, these algorithms are time consuming. Mohr and Henderson

(1986) introduced the General Arc Consistency (GAC-4) algorithm which depends

on supporting values as in AC-4. The worst case time complexity is O (dnc), where n

is the maximum number of ary constraints. Puget (1998) and Regin (1994) proposed

specialised and faster algorithms for particular types of non-binary constraints, in

particular the all-different constraint. These algorithms consider the semantic of the

constraint so more pruning can be implemented when compared with a binary

constraint. Many consistency techniques have been applied to general CP problems.

However these techniques are very time consuming when applied to problems

including n-ary constraints, even if a simple consistency technique such as arc-

consistency is used (Lhomme,1993). Bound propagation is an alternative technique

to the n-ary constraints and is used for solving numeric CS problems. This bound

propagation is explained in the next section.

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4.2.1.3 Bounds Consistency

The bounds consistency method was proposed because arc-consistency is very time

consuming if used as a complete method. Bounds consistency propagates only the

bounds of the variable’s domain through other primitive constraints. If any of the

bounds has no support value through the other variable domains, it must be removed.

As a result, a constraint programming problem becomes bound consistent if each

constraint ci∈C is bound consistent as described in Example 4.3.

Example 4.3

Assume the constraint C includes some variables s1, s2, s3,...,sn.

Hence, C(s1,s2,s3,...,sn) is bound consistent if and only if for each

variable si; for all ui ∈ [min( Di ), max( Di )], and for all j ≠ i, there

is uj ∈ [min(min( Di ), max( Di )], and where c(u1, u2, u3,…un ) is

satisfied.

Generally, a constraint satisfaction problem is bounds consistent if each constraint is

bounds consistent. The bounds consistency method is implemented in time O (dc),

and is useful for a binary constraint.

4.2.1.4 Path Consistency

Arc and bounds consistency work on each constraint in turn. The path consistency

method tries to involve more than one constraint in turn as described in Example 4.4.

Example 4.4

A path (x, y, z), where x, y, z are variables, and there are binary

constraints cxz, czy, and cxy between (x, z) and (z, y), and (x, y)

respectively. The path (x, y, z) is a path constraint, if and only if,

xu D∈ , yv D∈ satisfy the constraint cxy, ∃ a value w ∈ Dz such that

(u, w)∈ cxz and (w, v)∈czy. Constraints are represented as a set of

values, some of which are discarded from domain z if values do not

satisfy constraint cxy.

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This type of consistency method is very time-consuming and achieved the worst case

time of O (d3c3).

4.2.2 Search Process Example 4.5 indicates there are many values still possible for each decision variable

and as a result, constraint propagation alone does not necessarily solve the problem

completely without other techniques. A search tree provides a solution with each

node representing a decision variable x, y, or z, and each branch expressing a

possible value for that variable.

Example 4.5

If the search starts with variable x, many branches can be created

from this node, each one taking one of the values in the domain x [2,

6]. Assume that the first branch assigns 2 to x. At this point,

constraints are automatically used to further reduce the variable

domains, if needed. Since the domain x has changed, the constraint

x=yz will be used and [ ]1,3y∈ and z∈[2, 6]. Propagation of this

constraint will reduce the domain y to 1 and the domain z to 2. Now,

all decision variables have been assigned a value, and the problem is

solved, so the search stops.

In some cases, a chosen node is inconsistent, which means one or more decision

variables have an empty domain. In this case no solution is possible and another

choice must be made. This means returning to the last choice made and removing all

consequences of the choice and then choosing another node. This process involves

backtracking through the tree to follow another branch. This process continues until

there is a solution or the entire tree is exhausted.

Search process can be adapted to solve optimisation problems. The initial value of

the objective function is stored as the initial solution. Each time a solution is found,

the value of its objective function is stored, and a new constraint is added to the

problem. The new value of the objective function must be strictly better than the one

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just found. This process involves looking only for solutions that improve the best

solution already found. The order of choosing nodes or variables is considered in

advance. This order between variables is important to improve search performance.

For example, the order may be defined according to which variable or node has the

least number of possible values in its domain. Section 4.2.2.1 describes the order of

the variables and value techniques used in the search process.

4.2.2.1 Variable and Value Ordering Heuristics The order of the variables and the values assigned to the variables in backtracking

have important roles in improving the efficiency of a search algorithm.

Variables Ordering

Variables chosen for instantiation have a significant impact on the number of

backtracks, where the number of backtracks increases the efficiency of constraint

satisfaction. Variables ordering can be static or dynamic. Static ordering means the

order of the variables is determined before the search begins, and is not changed

thereafter. Dynamic ordering means selecting the next variable (during the search)

based on the current state of the search. Dynamic ordering is not feasible for all

search algorithms, because it needs extra information on the variable domains. For

example, there is no extra information available during the search for a simple

backtrack algorithm (BT) to make a different ordering choice from the initial

ordering. However, in the forward checking (FC) algorithm, the current state

includes the domains of the variables as they have been pruned by the current set of

instantiations, so choosing the next variable is possible with this information.

Many heuristics have been developed for dynamic variable ordering. Haralick and

Elliot (1980) developed the Fail-First (FF) principle for dynamic variable ordering

which consists of selecting the next variable to minimise the expected length of each

branch and so minimise the search cost. Smith and Grant (1998) stated that it is not

necessary to minimise the length of branches as this process minimises the size of

the search tree. As a result, the FF principle is inefficient to use and is unsound.

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A common heuristic, often described as an implementation of the FF principle, is

choosing the next variable which has the fewest remaining alternatives for

instantiation. In other words, choosing the next variable equates to choosing that

which has the smallest current domain. The smallest domain heuristic works well for

many problems. Another heuristic used when all variables have the same number of

values, is choosing the variable that participates in most constraints. One heuristic

for the static ordering of variables is suitable for simple backtracking. This heuristic

depends on choosing the next variable that has the largest number of constraints with

the past variables.

Value Ordering

Once a variable is selected for instantiation, many values become available. The

order of these values has to be considered because it significantly affects the time to

find the first solution. However, if all solutions are required or there are no solutions,

value ordering is indifferent. The branches starting from each node of the search tree

will be rearranged by value ordering. As a result, the branch which leads to a

solution is searched earlier than other branches. That is if the CS problem has a

solution, and a correct value is assigned for each variable, then the solution can be

found without any backtracking. Therefore, in order to select a variable to

instantiate, choosing the first value of that variable is necessary. When the aim is to

find a solution, a value is chosen to succeed with no conflict. This Succeed – First

strategy (SF) can be applied to value ordering.

Another heuristic is to prefer those values that maximise the number of options

available. The AC-4 algorithm works well with this heuristic as it counts the

supporting values. It is possible to estimate the “promise” of each value by knowing

the domain sizes of the future variables after choosing the value. That is, the AC-4

algorithm can predict the upper bound on the number of possible solutions resulting

from the assignment of values to the variables. The value that has the highest number

of solutions has to be selected. Additionally, AC-4 can calculate the percentage of

values in the domains of the future variables that are no longer usable. The best

solution has the lowest cost. Another heuristic is one that depends on selecting the

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value that leads to CS problem solution. This strategy requires estimating the

difficulty of solving a CS problem.

4.2.2.2 Search Techniques

Many strategies are followed to find the solution through the search trees. Most are

prime methods that use a parallel branch and bound technique to solve combinatorial

optimisation problems. These methods obtain the optimal solution for many NP-hard

problems. However, some techniques used individually are time consuming and need

huge memory capacities to solve large-scale problems. Different search techniques

and strategies are applied to solve the two sugarcane rail transport system

formulation models, MIP and CP. The different search techniques such as Best First

Search Technique (BFS), Depth-First Search Strategy (DFS), Slice Based Search

(SBS), Limited Discrepancy Search (LDS), Depth-bound Discrepancy Search

(DDS), and Interleaved Depth First Search (IDFS) are integrated with the two main

search strategies, Standard Search Strategy (SSS) and Dichotomic Search Strategy

(DSS), to improve the accuracy and CPU time for the techniques’ solutions.

BFS- Best First Search Technique

Best First Search is a well-known technique for optimization problems and consists

of choosing the node with the best value (the relaxation) of an expression that is,

typically, the objective function. That is, the Best First Search maintains all the

nodes still needing to be explored and chosen. The next node to expand is that which

has the best lower bound. The selected node is then expanded into another set of

nodes that are inserted in the set of unexplored nodes in replacement of the selected

node. For the Best First Search to be effective, it is important to have a precise lower

bound, where the distance between the lower bound of a node and the best solution

obtained from that node, should be as small as possible.

The initial upper-bound on the makespan, in the job shop scheduling problem, is the

sum of the duration of all activities. If a solution is found with a lower makespan, the

new bound is added at the leaf node (terminated node). The search then continues as

if the solution state was actually a failure. Sierra and Varela (2008) used Best First

Search to reduce the search domain in solving the job shop scheduling problem with

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total flow time. The evaluation function, in these techniques namely f(s), is used to

determine the makespan of any generated solution path that extended the current

path of a node s.

The best first search technique starts with the root node which is in open

nodes(nodes are generated but have not been expanded yet), and then chooses the

node which has the best makespan value (minimum) in the open list(list includes

generated nodes but have not been expanded yet) for expansion. This node will be

removed from the open list and their children will be generated and the parents put in

the closed list. A new child will then be chosen for expansion. The closed and open

lists are checked and updated to add the best node, and remove the worst, depending

on the makespan value.

DFS-Depth-First Search Strategy (DFS)

The strategy followed for the depth-first search is, as its name implies, to search

“deeper” in the graph whenever possible. Depth-first search (DFS) starts at the root

node and proceeds by descending to its first descendant, and continues this process

to reach a leaf. The process requires backtracking to the parent of the leaf and

descending to its next descendant. This process continues until the root node is

revisited after all nodes or its descendants have been visited (Tsin, 2002).

Figure 4.1 shows that the order of visited nodes starts from the root A and visits in

order B, C, and then leaf D. Then DFS comes back to the parents C and B to visit E

and F consecutively. It then comes back to the root A to visit G and H. From node H,

it moves to visit I and J, and returns to node I to visit K and then node H to visit L.

A

H G B

C F L I

K

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Figure 4.1: The order of nodes that will be expanded by DFS strategy

The time and space analysis of DFS differs according to its application area. In

theoretical computer science, DFS is typically used to traverse an entire graph, and

takes time O (|V| + |E|), linear in the size of the graph. DFS may experience non-

termination when the length of a path in the search tree is infinite. Therefore, the

search is only performed to a limited depth, and due to limited memory availability,

one typically does not use data structures that keep track of the set of all previously

visited vertices. In this case, the time is still linear in the number of expanded

vertices and edges (Her and Ramakrishna 2007).

DFS lends itself efficiently to heuristic methods of choosing a likely-looking branch.

When an appropriate depth limit is not known, a priori, an iterative deepening depth-

first search applies DFS repeatedly with a sequence of increasing limits. In the

artificial intelligence mode of analysis, with a branching factor greater than one,

iterative deepening increases the running time by only a constant factor over the case

in which the correct depth limit is known. This increase is due to the geometric

growth of the number of nodes per level.

SBS-Slice Based Search

The SBS technique is effective for job shop problems and depends on the existence

of a good heuristic to obtain a solution. This method supposes that initially, there is a

small number of allowed discrepancies (the choices during the search process which

do not follow the heuristic) at the beginning of the search process which increases

during the exploration of the search tree until a solution is found or the whole search

tree has been explored.

LDS-Limited Discrepancy Search

The LDS is one of the complete search techniques that can provide an optimal

solution for many problems. This technique depends on building the search tree

with good heuristics (Harvey & Ginsberg, 1995). This technique supposes that the

J E D

http://en.wikipedia.org/wiki/Heuristics

http://en.wikipedia.org/wiki/Iterative_deepening_depth-first_search

http://en.wikipedia.org/wiki/Iterative_deepening_depth-first_search

http://en.wikipedia.org/wiki/Branching_factor

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first leaf visited is likely to be a solution. If not, then it is likely that the number of

mistakes along the path from the root to this leaf is small. The next leaf nodes that

have paths from the root and differ only in one choice from the initial path, will then

be visited. This process continues by visiting the leaves that have a higher

discrepancy than other leaves from the initial path as described in Example 4.6.

Example 4.6

Suppose that v0 is a node with ordered descendants v1, v2, v3 …vk. The

discrepancy of vj is the discrepancy of v0+j-1 for j=1, 2…k and the

discrepancy of the root node is 0. The search process in LDS technique

starts at the root node where the level of discrepancy is d=0 and proceeds to

its first descendant 1v whose discrepancy is not higher than d. This process

continues until a leaf is reached. Backtracking to the parent of the leaf is

required and the descent to its next existing descendant, which has a

discrepancy that is not higher than k is made. This process continues back to

the root node when all descendants not higher than k have been visited. Set

d=d+1 and repeat this process until back at the root node and all descendants

have been visited.

Figure 4.2 shows limited discrepancy search and the order of the visited nodes

(Hoeve, 2005). The black node of the search tree indicates the root; visited

nodes are red nodes; and red arcs indicate the parts of the tree that are newly

traversed.

a. discrepancy 0 b. discrepancy 1

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c. discrepancy 2 d. discrepancy 3

Figure 4.2: Limited discrepancy search This strategy identifies that for a discrepancy d>0, LDS visits leaves with a

discrepancy less than d. Leaves therefore are visited many times. Hence, if the

discrepancy of a node v is d, and the length of a path from v to a leaf is l, then the

descendants with a discrepancy between d-l and d will be considered. This revisiting

can be avoided by keeping track of the remaining depth to be searched. This

technique is called improved limited discrepancy search (ILDS). It is noted that DFS

and ILDS are complete search techniques that are not redundant where these

techniques have to visit all paths from the root to a leaf exactly once. LDS technique

is out of the domain of this research because it is time consuming.

DDS-Depth-bound Discrepancy Search The DDS technique supposes that the heuristic used to build the search tree is more

likely to make a mistake near the root node (Walsh, 1997; Beck &Perron, 2000).

DDS aims to remove the domain values due to propagation along a path from the

root to a leaf meaning discrepancies will be allowed only above a certain depth-

bound. At iteration i, the depth-bound is i-1, where under this depth, it is only

allowed to descendants which do not cause the discrepancy increase.

IDFS-Interleaved Depth First Search

Interleaved depth first search strategy works through DFS where IDFS prevents DFS

falling into mistakes. DFS sometimes visits bad nodes and cannot reach other new

nodes searching sequentially from left to right. IDFS on the other hand searches in

parallel many subtrees at a tree level that is called active. It searches in each subtree

to reach each leaf. If this leaf is the goal, the search process terminates. Otherwise,

the state of the current subtree is recorded and added to a list of active subtrees,

which will be taken later at the suspended point.

IDFS selects another subtree and repeats the previous process. Parallel DFS on

active subtrees is simulated by IDFS using a single processor. IDFS avoids

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completely falling into mistakes by distributing the search among the subtrees,

which are expected to include some good subtrees. For optimisation problems,

particularly minimisation, constraint satisfaction problems handle the associated

objective function using the two strategies of Standard Strategy (SSS) and

Dichotomic Search Strategy (DSS).

SSS-Standard Search Strategy

Standard Strategy (SSS) is uncomplicated and solves the optimisation problem as a

constraint satisfaction problem. The main steps of this strategy are firstly to ignore

the objective function and look for a feasible solution. The objective function is

evaluated at the first feasible solution point, and an upper bound is obtained.

The second step is to add a constraint to the original constraints and so obtain the

next feasible solution which is better or at least the same value (if multiple solutions

are wanted) as the objective value of the current solution.

DSS-Dichotomic Search Strategy

The DSS strategy is useful if the lower bound (W) of the problem is known in

advance. The main steps are firstly to evaluate the objective function value V at a

feasible solution: that is, evaluate the mid value Z given by (W + V)/2. The set of

constraints of the original problem is then added to a constraint f(x1,x2…..,xn) < Z,

where f(x1,x2…..,xn) is the objective function, and the CP problem is solved by

neglecting this function. In the next step, a feasible solution should be better than the

current solution, hence, the value of V is updated, and the strategy is continued with

a new value of the mid value Z. An unfeasible solution requires the updating of the

lower bound W and hence the mid value Z, so that the search strategy is executed as

above.

4.2.2.3 Global Constraint

The ability to express and reason with sophisticated constraints is a very important

feature of constraint programming. Example 4.7 shows the logic behind the global

constraint.

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Example 4.7

Z1∈[1, 3] (1), Z2∈[1,3] (2), Z3∈[1,3]

(3)

Z4∈[1, 3] (4) , and Zi ≠Zj (5)

This problem has no solution, since there are three values for four

variables. However, domains are not reduced by constraint

propagation on each constraint. In fact, a given constraint Zi ≠Zj, for

each value of Zi (say 1), and a value always exists in the domain of

the other variable Zj (say 3) that satisfies the constraint.

The conflict occurs when each elementary constraint Zi ≠Zj is treated separately

illustrating that there are just three values assigned to four variables. This global

analysis of the problem presents the new aspect called global constraint where the

reduction techniques are used to n-ary constraints. The set of inequality constraints

in the Example 5.7 can be changed by a global constraint in the form alldiff (Z1, Z2,

Z3, Z4). Many techniques are proposed to deal with the global constraint such as a

graph of theoretical approaches that calculates the arc consistency domain reductions

for the alldiff constraint in O (n25). A bound consistency algorithm was proposed by

Puget (1998) that runs in O (n * log (n)) and is very important in scheduling

problems, especially in terms of job shop or flow shop problems.

Many complementary algorithms in Section 4.3 are developed to integrate with

solution techniques in Section 4.2 to solve many issues during implementing the two

Models CP and MIP of sugarcane rail transport systems. Some of these algorithms

are developed using MIP formulation but can be applied easily to the CP.

4.3 Proposed Algorithms

The proposed algorithms are developed to solve many issues in the sugarcane rail

transport system models. Collecting and Delivering Conflict Elimination algorithms

are developed in this research to optimise the collecting and delivering operations.

This type of proposed algorithms includes Terminal Segment Conflict Elimination

(TSCE) and Intermediate Segment Conflict Elimination (ISCE) algorithms. The

second type of proposed algorithms is for solving railway conflicts and includes

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Segment Blocking Determination SBD and Rail Conflict Elimination RCE

algorithms. The last type consists of algorithm Computing Acceleration (CA)

Algorithms for Reducing Computing Time.

In spite of the proposed algorithms are developed to solve segment blocking models

of sugarcane rail system, all these algorithms can be adapted easily to the blocking

section models.

4.3.1 Collecting and Delivering Conflict Elimination

The main purpose of the sugarcane rail transport system is to satisfy the sidings’ and

mill’s requirements regarding empty and full bins respectively. So, optimising the

delivering and collecting operations is important for achieving this purpose at lowest

cost. Some assumptions are made when considering the sugarcane rail models and

the proposed algorithms.

The sugarcane rail transport system models assume that there are two operations, o

and o´, to be implemented on each section during train run r in the outbound and the

inbound directions respectively. The train outbound operations are assigned for

delivering empty bins to the sidings without any collecting, while the train inbound

operations are assigned for collecting without any delivering. No delivering or

collecting occurs if the section is a passing section and does not work as a siding

This manner of operating can reduce the total train operating cost and keep the

quality of the cane high by reducing the cane age (the time between the cane

harvesting at sidings and the crop crushing at the mill). Additionally, for safety

reasons, trains are, not allowed to have a mix of empty and full bins in tow.

Optimising collection and delivery of bins depends on the location of each siding in

the railway network, where some sidings are located in terminal segments and others

in intermediate segments. Distinguishing between the outbound direction and

inbound direction on each siding can help in assigning the delivery of empty bins to

the outbound operation on each siding and the collection of full bins to the inbound

operation at the same siding. So, some proposed algorithms can establish relations

between the outbound and inbound operations on each siding to enforce each train to

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either deliver or collect (or to do nothing in the case of passing sections). In terminal

segments, the outbound operation o´ at segment e is less than the inbound operation

o at segment e (o´ < o), while, in the intermediate segment, the scenario is different

because the outbound operation o´ and the inbound operation o will be in different

segments. As a result, we cannot ensure the operation o´ will be less than the

operation o or obtain direct relations between them because this depends on the

location of the siding in the segment. The terminal segment conflict and

intermediate segment conflict are solved using two proposed algorithms through

studying all conflict cases of delivering and collecting operations.

4.3.1.1 Terminal Segment Conflict Elimination

The main advantage of the terminal segment is all train operations can be executed

continuously without any interruption. This means, depending on the model

assumptions, there is one segment to implement the outbound and inbound

operations. The different possible locations of the siding in a terminal segment are

studied using the next three cases.

Case 1: The Siding at the Beginning of Terminal Segment

Figure 4.3 clarifies that segment A is terminal and includes 3 sections s1, s2, s3 and 6

operations will be executed continuously on it. Section s1 works as a siding andIS

located at the beginning of the segment A. The outbound operation is o´ = 1 and the

inbound operation is o= 6, and 1< 6. As a result, the location of the siding at the

beginning of the segment does not affect the inequality o´ < o, where the outbound

operation will be less than the inbound operation.

Segment A

Inbound direction

S1

S3

S2

128

Figure 4.3: Siding at the beginning of terminal segment

Case 2: The Siding at the End of Terminal Segment Figure 4.4 shows that the section s3 works as a siding and is located at the end of the

terminal segment A. The outbound operation is o´ = 3 and the inbound operation is

o= 4, and 3< 4. As a result, the inequality o´ < o is still satisfied.

Segment A

s1 s2 s3 Segment A Outbound direction

Figure 4.4: Siding at the end of terminal segment

Case 3: The Siding at the Middle of Terminal Segment Section s2 works as a siding and is located in the middle of the terminal segment A

(as shown in Figure 4.5). The outbound operation is o´ = 2 and the inbound

operation is o= 5, and 2< 5. As a result, the inequality o´ < o is still satisfied.

Segment A

Outbound direction

Inbound direction

Segment A

S2 S3 S1

Inbound direction

129

Figure 4.5: Siding at the middle side of terminal segment

Cases 1, 2 and 3 prove that changing the siding location at the terminal segment does

not affect the relation between the outbound and inbound operations and the

inequality o´ < o is satisfied. Terminal Segment Conflict Elimination (TSCE)

algorithm is proposed to solve the delivering and collecting conflicts at sidings at

terminal segments as illustrated in Figure 4.6.

Terminal Segment Conflict Elimination (TSCE) Algorithm Select train k; k є K

Select segment e ЄE //step 2

Select section s; s Є S

If s is a siding then

o´ and o are two operations will be

executed at siding s; o´ <> o

If o´ < o then // outbound direction

k o s kr e epα ′ ≤

0k o sr e e

B ′ =

k k k o sr r r e eα α α ′= −

go to step 2

Else // inbound direction

k o s kr e eB f′ ≤

0k o sr e e

α ′ =

k k k o sr r r e eB B B ′= −

go to step 2

Else

End if

Else

Segment A

Outbound direction

130

0k o sr e e

α ′ =

0k o sr e e

B ′ =

go to step 2

End if;

End.

4.3.1.2 Intermediate Segment Conflict Elimination

Optimising the delivering and collecting operations on each siding located at an

intermediate segment requires establishing a relationship between the two operations

o and o´. This relation depends on two scenarios. The first one is if the train visits

the intermediate segment and decides to return to the mill after delivering or

collecting, the intermediate segment becomes like a terminal segment and the TSCE

algorithm can be used to solve any conflict. The second scenario is if the train visits

the intermediate segment and then visits other segments, the intermediate segments

require new algorithm to solve any delivering and collecting conflicts. To develop a

solution for intermediate segment conflicts, the different siding locations in the

intermediate segment are studied using the next three cases. The next three cases

show that the inequality is not satisfied in all intermediate segment cases.

Case 1: The Siding at the Beginning of an Intermediate Segment

Figure 4.7 shows that the intermediate segment includes two segments A and B as

outbound direction and inbound directions respectively. Section s1 works as a siding

at the beginning of segment A and at the end of segment B. The outbound operation

o´ at segment A will be o´=1, while the inbound operation o at segment B will be

o=3. As a result o´ < o is satisfied.

Segment B

Inbound direction

S1 S3

S2

Figure 4.6: Terminal segment conflict elimination algorithm

131

Figure 4.7: Siding at the beginning of intermediate segment

Case 2: The Siding at the End of an Intermediate Segment Figure 4.8 shows that the siding is located at the end of segment A and the beginning

of segment B. The outbound operation o´ at segment A is o´=3, while the inbound

operation o at segment B is o=1. As a result, o´ > o and the TSCE algorithm can

not be applied for this case. So, another algorithm is required to solve any problem

related to this type of intermediate segment.

Segment B

s1 s2 s3 Segment A Outbound direction

Figure 4.8: Siding at the end of intermediate segment

Case 3: The Siding at the Middle of an Intermediate Segment

In this case, siding s2 works where o´ = o; o´ =2 and o =2. As a result we cannot

distinguish easily which operation is for the delivering and which one is for colleting

(as shown in Figure 4.9).

Segment A

Outbound direction

Inbound direction

Segment B

132

Figure 4.9: Siding at the middle side of intermediate segment

From the previous discussion, there is an urgent need for a new algorithm to solve all

possible cases for the intermediate segment and the new algorithm can integrate with

the TSCE algorithm for solving all delivering and collecting cases.

Intermediate Segment Conflicts Elimination (ISCE) Algorithm

From all the previous three cases relating to the intermediate segment, the inbound

segment always has to be less than the outbound segment, where segment A is

visited before the inbound segment B. So, the intermediate-segment-conflicts-

elimination algorithm distinguishes between the outbound operations and inbound

operations using the index of the segment.

The ISCE Algorithm supposes that segment e´ and e are two segments in outbound

and inbound directions respectively which represent an intermediate segment, where

e´ ≤ e. This algorithm assigns the operation of segment e´ for delivering and the

operation e for collecting. The ISCE Algorithm is described below:

Select train k; k є K //step 1

Select section s; s Є S //step 2

If s is a siding and located in two different segments

e´ and e “ from the assumption of the model” then

If e´ ≤ e then // outbound direction then

k o s kr e epα

′ ′≤

0k o sr e eB

′=

k k k o sr r r e eα α α

′ ′= −

go to step 2

Segment A

Outbound direction

S2 S3 S1

133

Else // inbound direction

k o s kr e eB f

′ ′≤

0k o sr e eα

′ ′=

k k k o sr r r e eB B B

′ ′= −

go to step 2

Else

End if

Else

0k o sr e eα

′ ′=

0k o sr e eB

′ ′=

go to step 2

End if;

End.

4.3.2 Algorithms for Solving Train Conflicts

As mentioned previously, the rail network includes two main types of segments,

terminal segments and intermediate segments. For solving any conflicts through the

rail networks, blocking segment cases are taken into consideration using two

algorithms. The two developed algorithms are integrated to solve all types of

segment conflicts through the rail network by applying blocking constraints and

some heuristics such as Shortest Processing Time (SPT) to improve the algorithm

solutions. The first algorithm, the Segment Blocking Determination (SBD)

algorithm, is to recognize the segment that requires blocking and then the second

algorithm, the Rail Conflicts Elimination (RCE) algorithm, applies blocking

constraints and eases the trains passing. Blocking segment types are described using

some cases and scenarios in Chapter 3.

134

The Two algorithms are integrated to solve all the types of segment conflicts and to

apply blocking constraints correctly. The SBD algorithm selects the segment that

needs to be blocked and the RCE algorithm applies the blocking constraints and

solves any segment conflicts.

Segment Blocking Determination (SBD) Algorithm

This algorithm depends on distinguishing the segment types and identifying which

segment requires the blocking constraints. Each intermediate segment includes two

different segments mathematically so as to distinguish between the intermediate

segments and other segments in the rail network. Segment Blocking Determination

SBD algorithm will be used. SBD uses the index of sections of segment to

distinguish the different segments. If any section is included in any two different

segments, it means this segment is intermediate and so blocking constraints can be

applied to both intermediate outbound and intermediate inbound segments.

Select trains k1and k2 є K.

Select e´ and e are two segments, where k1 uses e´ and k2 requires e.

Select section s; s Є S;

If e´ <> e then

If s Є e´ ” s is the last section in the e´.\” then

If s Є e ” s is the first section at segment e” then

e´ and e are one intermediate segment; e´ and e are the same physically.

Apply Rail conflict elimination (RCE) algorithm Else

e´ and e are different physically

Do not apply Rail conflict elimination (RCE) algorithm

Endif

Else

Endif

Else

There is a one segment as a terminal,

Apply the blocking constraints and Rail conflict elimination (RCE) algorithm

135

Endif;

SBD algorithm is integrated with the Rail Conflict Elimination (RCE) algorithm to

solve any railway conflicts through the rail network.

Rail Conflict Elimination (RCE)Algorithm

The RCE algorithm is developed for solving a track segment conflict throughout the

rail network. Each segment has many sections but no passing loops inside the

segment. As a result, the RCE Algorithm works on solving any conflicts in that

segment. A heuristic technique, the Shortest Processing Time (SPT), is used inside

this algorithm to reduce the waiting time of trains before starting that segment.

Oliveira (2001) used the SPT technique to resolve a section conflict. In Figure 4.10,

k and k' Є{1,...K} are two trains where k is travelling in the outbound direction and k'

is travelling in the inbound direction. e Є{1,...E} is a segment which includes the

conflict point. {s1, s2, s3, s4, s5} are some sections included in segment e. in Figure

4.10, start time of train k' is delayed to solve the conflict on segment e.

Figure 4.10: A rail conflict case

Let o and o' be two operations of trains k and k' on section s on segment e,

Let koseg and k o se

g ′ ′ be the processing time, and k osr et

and k o sr e

t ′ ′′be the start time of

o and o' consecutively.

136

Begin

If there is a conflict (e, d), where d Є [0...Total system time] then

Select the e which the (e, d) is the earliest to occur.

Calculate 1 1

O S

kos k ose r eo sg q

= =∑∑

and

1 1

O S

k o s k o se r eo sg q′ ′ ′ ′′

= =∑∑ the total processing time for k and

k' consecutively on segment e.

Apply the shortest processing time technique to all trains on each conflicted segment

If 1 1

1O S

kos k os k o s k o se r e r e r eo st q g q t qk s k osr e r e

′ ′ ′ ′′ ′= =

+ − <

∑∑

11 1

O S

k s k o s k o s k o s k os k osr e r e e r e r e r eo st q g q t q′ ′ ′ ′ ′ ′ ′′ ′ ′′= =

+ −

∑∑ then

o of train k will be selected to be scheduled first.

Else

o' of train k′ is selected first.

End if.

If the selected operation cannot in the correct order to execute then

Backtrack choosing another operation.

End if

End if

End.

The RCE can be adapted to solve a section conflict in the blocking section models as

shown in Figure 4.11. Train k' is delayed to solve the conflict on s1.

137

Figure 4.11: A section conflict case using a delayed train technique

The conflict point can be solved using another technique in Figure 4.12, where the

speed of train k is reduced to be slower.

Figure 4.12: A section conflict case using a slow train technique

4.3.3 Computing Acceleration (CA) Algorithms

138

Two algorithms, segment elimination and section elimination algorithms are

discussed in this section. The main objective of the first algorithm is to reduce the

total number of segments of the rail network. The second algorithm reduces the total

number of sections of rail network. These algorithms work together to simplify the

problem and reduce the computing time.

Segment Elimination Algorithm In this algorithm, the segments not in use during the day’s operations are removed

from the complete list segments. Lk is the visited segments list. L΄k is the non-

visited segments list.

Select train k; k Є K //step1

Lk=Ø

L'k=E // the total number of segments at beginning for all trains

Select segment e ЄE // step 2 If 1k o sr e e

q = then

Lk=Lk+1 //increase the number of visited segments

L'k= L'k - 1 //decreasing the number of non visited segments

go to step 2

Else go to step 2

End if Set k

newE = E - L'k go to step 1

End.

139

Section Elimination Algorithm The Section Elimination Algorithm reduces the number of segments in the model.

Each eliminated segment includes some sections which are eliminated at the same

time but there are still some sections remaining that are not used. This algorithm is

used to eliminate the unused sections in the used segments to further simplify the

model. The reduced number of segments from the Segment Elimination Algorithm is

integrated with Section Elimination Algorithm. By changing the domains of

segments and sections in the model, the total time of the calculations is substantially

reduced. Vk is the list of sections visited by train k. V'k is the unused sections list.

Select train k; k Є K //step1

Vk=Ø

V΄k= knewS // the total number of sections of new list of segments

which we had before from segment elimination algorithm

Select sections se ЄS on e segment where e Є knewE (segment elimination algorithm)

//step2

If k o sr e eq =1 then

Vk=Vk + 1 //increase the number of visited sections

V΄k= V΄k - 1 //decreasing the number of non visited sections

go to step 2

Else

go to step 2

End if

Vk = knew kS V΄ −

go to step 1

End.

4.4 Conclusion

140

This chapter describes many different techniques to solve the sugarcane rail transport

system problem. Search techniques used to solve the MIP and CP models in Chapter

4, were the integration of MIP model and constraint programming search techniques

that can produce good solutions in reasonable time. The proposed algorithms in this

chapter are used to solve all types of conflicts throughout the rail network and reduce

the CPU time of CP and MIP models code. The heuristic and metaheuristic

techniques applied to sugarcane rail transport systems are described in Chapter 6.

These solve large-scale problems in reasonable time.

140

Chapter 5

Computational Results of CP and MIP Models

Chapter Outline

5.1 Introduction.................................................................................................................... 143

5.2 A Case Study for Testing Blocking Segment CP and MIP Models...............................143

5.2.1 Input Data........................................................................................................144

5.2.2 Results of Makespan Minimisation Objective................................................146

5.2.2.1 Constraint Programming Model Results......................................... 146

5.2.2.1.1 Standard Constraint Programming Results.....................146

5.2.2.1.2 Results of Computing Acceleration (CA) Algorithms. ...149

5.2.2.1.3 Train and Runs Scheduling for CP Model.......................151

5.2.2.1.4 Solutions Analysis by Search Tree..................................152

5.2.2.2 MIP Model Results...........................................................................156

5.2.2.2.1 Result of Standard MIP Model......................................156

5.2.2.2.2 Results of Computing Acceleration (CA)......................158

5.2.2.2.3 Train Scheduling Results...............................................160

5.2.2.3 Comparisons of CP and MIP Models using Makespan Criterion...161

5.2.3 Results of Total Waiting Time Minimisation Objective ................................163

5.2.3.1 Constraint Programming (CP) Model .............................................163

5.2.3.1.1 Results of Standard CP...................................................163

5.2.3.1.2 Results of Computing Acceleration (CA) Algorithms...164

5.2.3.1.3 Train Scheduling Results................................................165

5.2.3.2 Mixed Integer Programming (MIP) Model .....................................166

5.2.3.2.1 Result of Standard MIP Model.......................................166

5.2.3.2.2 Results of Computing Acceleration (CA) Algorithms...167

141

5.2.3.2.3 Train Scheduling Results................................................168

5.2.3.3 Comparisons of CP and MIP Models of Total Waiting Time

Criterion.............................................................................................169

5.3 A large Scale Case Study for Testing Blocking Segment

CP and MIP Models…………………………………………………….…….……….171

5.3.1 Results of Makespan Minimisation Objective...............................................175

5.3.2 Results of Total Waiting Time Minimisation Objective..................................177

5.4 Sensitivity Analysis of Blocking Segment MIP Sugarcane Rail Model.........................178

5.4.1 Small Rail Systems.........................................................................................179

5.4.2 Large Rail Systems.........................................................................................184

5.5 Blocking Section MIP and CP Results...........................................................................186

5.5.1 The comparisons between the Blocking Segment and Sections Models........187

5.6 The Results of Inclusion of the Delivery and Collection Time Constraints ………......189

5.7 Conclusion......................................................................................................................193

142


Masoud, M., Kozan, E., & Kent, G. (2010c). A comprehensive approach for



Masoud, M., Kozan, E., & Kent, G. (under review). A new approach to

automatically producing schedules for cane railways. Australian Society of

Sugar Cane Technologies.

143

5.1 Introduction CP and MIP models of blocking segment and section constraints are solved using

OPL and CPLEX software respectively where the OPL solver using constraint

programming search techniques and the ILOG scheduler are integrated in the OPL

software. SSS and DSS were integrated with the different search techniques as well

to obtain solutions for the two models. The proposed algorithms in Chapter 4 are

included and used to solve the models. The results were obtained before and after

applying Computing Acceleration (CA) Algorithms to simplify the model Small and

large sizes of Kalamia Mill are used as case study for investigating the different

solution techniques. Makespan and the total waiting time are used as objective

functions. The results of Inclusion of the delivery and collection time constraints will

be presented in this chapter.

5.2 A Case Study for Testing Blocking Segment CP and MIP Models

A case study is examined to demonstrate and validate the CP and MIP models in this

research. A sector of the transport system at Kalamia Mill, near Ayr south of

Townsville, was used to obtain the optimal completion time of all runs of trains per

day. Figure 5.1 shows the distances between sidings, the siding capacities and the

siding allotments.

Figure 5.1: A realistic test case study (small sector of rail network of Kalamia Mill)

144

The case study involves 26 sections, 15 sidings, 11 segments and 5 trains. More

details are given in Tables 5.1, 5.2 and 5.3 for this case. All algorithms are included

and used to solve the models.

5.2.1 Input Data The three types of data used to identify the parameters of the sugarcane rail transport

system are rail data, harvest data and trains data.

Rail Data

Table 5.1 provides information about the distances between sidings and sectional

running time in seconds, which is used as input data. In this table each section

between two sidings, or two passing loops, or siding and conjunction point, or two

conjunction points takes a number as index number. Table 5.1: The distances and the sectional running times of the sections

Section Number

From To Distance (km)

Sectional running time (s)

1 Mill JN - 35 0 0 2 JN - 35 JN - 40 .48 58 3 JN - 40 HEA .32 39 4 JN - 40 JN - 39 1.59 192 5 JN - 39 JN - 41 4.22 508 6 JN - 41 Gainsford 2 .27 33 7 JN - 41 Gainsford 4 2.03 245 8 JN - 39 Kal Plains .98 118 9 Kal Plains JN 38 1.82 219 10 JN 38 Brandon1 .19 23 11 JN 38 Brandon3 .94 113 12 Brandon3 Brandon4 1.6 193 13 JN - 35 Chiverston 2 1.68 202 14 Chiverston 2 Chiv terminus 2.4 289 15 Mill Town PTS J2 .71 86 16 Town PTSJ2 JN37 .66 80 17 JN37 Lillesmers .36 43 18 JN37 Town 3 1.53 184 19 Town 3 Town Terminus 3.13 377 20 Town PTS J2 Mainline 1A 1.11 134 21 Mainline 1 a Mainline 1 .44 53 22 Mainline 1 Mainline 3 2.19 264 23 Mainline 3 JN33 1.07 129 24 JN33 Mainline 4A .32 39 25 Mainline 4A Mainline 4B .01 1 26 JN33 Central PTS J8 .13 16

145

Harvest Data

The harvest data such as siding capacity and siding allotments is shown in Table 5.2.

Each bin has a capacity of 6 tonnes. Most sidings have a capacity less than their

allotments. This means that these sidings need more than one run per day by one

train or more than one train.

Table 5.2: Siding capacity and siding allotments

Train Data

The number of trains, train name, maximum capacity of trains and speed are

illustrated in Table 5.3. The train capacity includes the maximum number of empty

bins and maximum number of full bins delivered by the train.

Table 5.3: Train number, capacity and speed

* 1 bin= 6 tonnes **8 km/h: kilometre per hour

Siding section number

Siding Capacity (bins)

Allotment (tones)

Allotment (bins)

3 Hea 120 - 6 Gainsford 2 140 1044 174 7 Gainsford 4 160 - - 8 Kal Plains 162 - -

10 Brandon 1 136 - - 11 Brandon 3 252 - - 12 Brandon 4 110 - - 13 Chiverston 2 128 - - 14 Chiv Terminus 114 676 114 17 Lillesmers 132 - - 18 Town 3 110 - - 19 Town Terminus 120 - - 20 Mainline 1 A 130 540 90 21 Main_Line 1 62 - - 22 Main_Line 3 148 780 130 24 Mainline 4A 100 - -

Total 3040 506

Train index

Train Max empty(bins) Max fulls (bins) Average speed(km/h)

1 Barratta 120 90 30 2 Rita_Island 120 124 30 3 Jarvisield 120 124 30 4 Norham 120 110 30 5 Kilrie 120 124 30

146

5.2.2 Results of Makespan Minimisation Objective

Minimising the makespan in the sugarcane rail transport system reduces the total

operating cost and helps implement all train runs in a specific time. The makespan

minimisation objective is achieved using the constraint programming and mixed

integer programming models. The makespan results of the two models are compared

in Section 5.1.2.3.

5.2.2.1 Constraint Programming Model Results

The main objective of the constraint programming model is to minimise the

makespan. The two cases of the constraint programming model are the standard

constraint programming and the integration of constraint programming model and

Computing Acceleration Algorithms.

5.2.2.1.1 Standard Constraint Programming Results

Standard Constraint Programming means using the original constraint programming

without using the computing acceleration algorithms. Table 5.4 shows results of SSS

for the CP model before applying the algorithms of Computing Acceleration (CA).

The number of variables and the number of constraints are reduced because the CP

formulation combines some constraints in one constraint as seen in the CP

formulation model. Table 5.4 shows the standard CP model results using the

integration of SSS strategy and other search techniques before applying algorithms.

Makespan and CPU time are calculated in second(s). The number of variables and

the constraints in the tested case are not small in number, so obtaining the solutions

is not easy. The choice point column shows the number of explored points that can

be branched to search the solutions, while the failure points column shows the

number of points that stop and cannot be extended in the search tree. The Solver

memory in kilo byte (kb) is presented to show the PC memory which the solution

techniques code needs in order to solve the problem. Table 5.4 illustrates that SSS

works well with all search techniques except BFS, in CP formulation. The DFS

technique gives results in a shorter CPU time than the other search techniques.

147

Table 5.4 shows the results of the integration of DSS and other search techniques for

the CP model before applying algorithms. All techniques using the DSS strategy

have given more reasonable results in a shorter time than the SSS strategy. The

number of Choice points in the DSS strategy is reduced by 9.8% of SSS, while the

number of failure points is reduced sharply by 99.9% of the Choice points in the

SSS. DFS achieved a good CPU time with the two strategies while the Best first

search gave a shorter time with DSS but did not give any result with the SSS. As a

result, the DFS technique is more suitable for solving the CP formulation model than

other techniques.

148

Table 5.4: SSS and DSS results for standard CP model to optimise makespan

Search technique

Variables number

Constraints

number

SSS DSS

Choice points

Failure points

Makespan (s)

CPU Time

(s)

Solver memory

(kb)

Choice points

Failure points

Makespan (s)

CPU Time

(s)

Solver memory

(kb) DFS

30253

380150

14021 14018 2664 100.68 136098 12657 10 2664 17.81 135716 SBS 14021 14018 2664 101.34 136508 12657 10 2664 18.06 136122 DDS 14021 14018 2664 102.61 136508 12657 10 2664 19.31 136122 BFS n/a n/a n/a n/a n/a 12681 12 2664 11.41 136098 IDFS 13657 13655 2664 106.64 136613 12657 10 2664 18.16 136122

149

SSS requires a longer time with the different search techniques than DSS with these

search techniques. Additionally, the BFS search technique does not work with the

SSS strategy while all search techniques work well with DSS. DFS provides better

results in a reasonable time with two strategies than other techniques can.

5.2.2.1.2 Results of Computing Acceleration (CA) Algorithms

Computing Acceleration (CA) algorithms are applied to the CP model to improve

the CPU time of the solutions. For that reason, the two algorithms reduced the

number of variables of the tested case by 92.5% before applying the algorithms. The

reduction of the variables caused a reducing of the constraints by 96.1%. As a result,

the number of choice points is reduced by more than 91%. The solver memory is

reduced by around 96% as well after applying the algorithms. Table 5.5 illustrates

the results of the integration of SSS and all search techniques for solving the CP

model after applying CA algorithms. All search techniques give an optimal solution

in good CPU time, except the BFS technique that does not work with SSS.

The DSS results with the search techniques are shown in Table 5.5. The DSS

strategy can be applied to all search techniques and obtains the same solution in

approximately the same CPU time. Table 5.5 results show the number of failure

points is reduced by DSS around 99% of the SSS failure points, while the number of

choice points is reduced generally by 8%.

The two strategies have the same solution and are closed to each other in CPU time.

The performance of some techniques is promising, while others may either require

extensive time or be unsuitable for this model with SSS, such as the BFS technique.

Figure 5.2 shows the comparisons of SSS and DSS before and after applying the CA

algorithms. These algorithms have a significant effect in reducing the CPU time.

Additionally, the DSS strategy works better than the SSS strategy in most cases and

uses the different search techniques.

150

Table 5.5: SSS and DSS results for CA algorithms to optimise makespan

Search technique

Variables

Constraints

SSS DSS

Choice points

Failure points

Makespan (s)

CPU time (s)

Solver memory

(kb)

Choice points

Failure points

Makespan (s)

CPU time (s)

Solver memory

(kb) DFS

2253

14706

2229 2226 2664 0.30 4433 2035 10 2664 0.13 4401 SBS 2229 2226 2664 0.28 4466 2035 10 2664 0.14 4433 DDS 2229 2226 2664 0.30 4466 2035 10 2664 0.13 4433 BFS n/a n/a n/a n/a n/a 1021 12 2664 0.11 4433 IDFS 2135 2133 2664 0.53 4502 2035 10 2664 0.14 4433

151

Figure 5.2: CPU time of SSS and DSS for the standard and CA of the CP using makespan

5.2.2.1.3 Train and Runs Scheduling for CP Model

Table 5.6 shows the exact start and finish times of train runs and assigns the runs to

trains. The trains 1, 3, and 4 start at time 0 and they are scheduled first, second and

third respectively, but all have the same priority to start. Train 5 starts fourth, and

train 2 starts fifth.

Table 5.6: Start and finish times of train run for CP model to optimise makespan

Train index

Run index

Siding number

Visit time (s)

Empties Fulls Run time (s)

Start time (s)

Finish time (s)

1 1 6 791 120 120 2414 0 1582 2 5 6 1873 54 54 1582 250 2664 3 2 14 491 112 112 982 0 982 4 3 20 220 90 90 1890 0 1976

22 537 10 20 5 4 22 1439 120 110 1074 86 2240

Total 5 - - 506 506 7942 0 2664

All trains have to deliver the full allotment of bins to each siding and collect the

same number of full bins from each siding. Table 5.6 also shows the actual visit

times for all trains at each siding, the number of sidings to be visited, the number of

empty and full bins to be delivered to or collected from each siding and train run

times. Train run time is defined as the time a train spends between starting the run at

the mill and returning to the mill after visiting the sidings.

0

2

4

6

8

10

12

DFS SBS DDS BFS IDFS

CPU

in se

cond

s

Search techniques

SSS of standard CP /10

DSS of standard CP/10

SSS of the CA algorithms

DSS of the CA algorithms

n/a n/a

152

Figure 5.3 shows the waiting time of trains while implementing their runs. A

distinction has been made between intermediate segments and terminal segments

(defined graphically in Figure 5.1). Intermediate segments include some operations

in both the inbound and outbound direction but the inbound direction operations may

be implemented after visiting other segments. As a result, two segment numbers

have been assigned to these segments. There can be no operations in other segments

between the outbound and inbound operations on terminal segments and so only one

segment number is assigned to these segments.

Figure 5.3: Scheduling 5 trains on 11 segments using CP with the makespan of 2664

The visit time in Table 5.6 is not optimised as real visit time because many variables

have not yet been considered in the CP model such as harvester start and harvester

rate. For that reason, Section 5.5 will introduce the optimising delivery and

collection operations in the sugarcane rail transport system including these variables.

5.2.2.1.4 Solutions Analysis by Search Tree

After applying the CP search techniques for the MIP and CP models, this section

explains how OPL uses the search tree techniques to obtain case study test solutions

and how the search tree is built by branching techniques. The case study was tested

using the CP model and four solutions were obtained before the optimal solution

using DFS with SSS. Table 5.7 shows all solutions before applying CA algorithms

and the spent CPU time to obtain each solution. The first solution was obtained in

9.48 seconds, while the optimal solution was obtained after 100 seconds.

153

Table 5.7: The standard CP model solutions before obtaining the optimal solution of makespan

Solutions variables constraints choice points

failure points

CPU time (s)

Makespan (s)

Solver memory

(kb) First sol. 30253

380150

4304 3731 9.48

3356

135.531

Second

sol. 30253 380150 8334 7260 11.02 3106 135.575

Third sol.

30253 380150 11388 11359 98.7 2914 135.985

Optimal sol.

30253 380150 14021 14018 100.68 2664 136.098

The search tree uses coloured nodes to express the node types inside OPL software.

For example, the red nodes are the failures, the solutions are green, the blue are the

explored choice points, white are the nodes created internally by ILOG Solver and

still unexplored, and the black nodes are pruned points that appear only while using

the LDS method. Figure 5.4 shows the stages of obtaining all solutions using the

search tree and SSS strategy. Each stage includes four sectors of the search tree of

each solution.

Stage 1: Discovering the fist solution: The makespan of the first solution (3356) is

found within (9.48) seconds where choice points are (4304) and failure points are

(3731).

a. Started discovering the nodes of tree b. More discovered nodes to reach first solution

c. Sector of search tree of first solution d. First solution is discovered

154

Stage 2: Discovering the second solution: The makespan of the second solution

(3106) is obtained by discovering around (8334) nodes and the number of failure

nodes is (7260) in the time (11.02).

e. Started discovering the second solution f. More discovered nodes to reach second solution

g. Sector of search tree of second solution h. Second solution is discovered

Stage 3: Discovering the third solution: The value of the third solution is decreased to be (2194) in (98.7) seconds after (11388) nodes are checked and (11359) were failure points.

i. Started discovering the third solution j. More discovered nodes to reach third solution

k. Sector of search tree of third solution l. Third solution is discovered

155

Final stage: The fourth solution: The fourth solution is optimal where makespan is

(2664) in CPU time (100.68). Seconds after (14021) discovered nodes and (14018)

were failure points.

m. Started discovering the fourth solution n. More discovered nodes to reach fourth solution

o. Sector of search tree of fourth solution p. Fourth solution is discovered

Figure 5.4: Solutions analysis by search tree

Conclusion on CP

The sugarcane railway operations are very complex and have a large number of

variables. The proposed scheduling model is very complicated but needs to be solved

in a reasonable time because of the dynamic nature of the system. The CP model

works well to solve the sugarcane rail transport problem using the integration of the

search techniques and the two search strategies. The BFS technique does not give a

solution with a standard search strategy. The integration of The DSS with the

different search techniques requires a shorter CPU time than the SSS with these

search techniques. The DSS is better to solve CP model than SSS.

156

5.2.2.2 MIP Model Results

5.2.2.2.1 Result of Standard MIP Model

The case study using the standard MIP model contains 26 sections, 11 segments, a

maximum of 10 operations for each segment, and 15 sidings. The total number of

trains is five and each train had one run to execute the tasks of delivering and

collecting the bins. The number of variables, before applying the Computing

Acceleration algorithms, is 32231 and the total number of constraints is 412407.

Table 5.8 shows the SSS strategy results with different search techniques to solve the

standard MIP model. The DFS technique works well with the SSS strategy and can

provide a solution in a reasonable time. The IDFS on the other hand is time

consuming and takes up to 9.5 hours (34422.09 seconds) to find a solution. SBS and

DDS techniques do not work with the SSS strategy and require a larger memory

capacity than DFS and BFS.

The DSS strategy is integrated with other search techniques in Table 5.8. All search

techniques work well with DSS and reach the optimal solution in a reasonable time.

The number of choice points and failure points is reduced, which makes the CPU

time shorter than SSS. Many nodes are ignored in the search tree, which means that

these nodes cannot provide better solutions than the current solutions.

The DFS results are more stable than other search techniques with SSS or DSS

strategies, where DFS can solve the proposed model and provide solutions in a

reasonable time. IDFS is time-consuming using SSS, even though it provides a

solution in a reasonable time using DSS.

157

Table 5.8: SSS and DSS results for standard MIP model to optimise makespan

Sear

ch

tech

niqu

e

Var

iabl

es

Con

stra

ints

SSS DSS

Choice points

Failure points

Makespan (s)

CPU time (s)

Solver memory

(kb)

Choice points

Failure points

Makespan (s)

CPU time (s)

Solver memory

(kb) DFS

32231

412407

59033 59030 2664 43.86 147734 30645 14 2664 26.09 147162

SBS n/a n/a n/a n/a n/a 30645 14 2664 26.58 147653 DDS n/a n/a n/a n/a n/a 30645 14 2664 26.86 147653 BFS 19429 31718 2664 116.81 151240 19423 14 2664 68.45 151212 IDFS 1033363 1055888 2664 34422.09 369480 30645 14 2664 25.67 147653

158

5.2.2.2.2 Results of Computing Acceleration (CA)

After applying Computing Acceleration (CA) algorithms (Segment and Section

Elimination Algorithms) to the model, the total number of sections is reduced from

26 to 11, the total number of segments from 11 to 3, the total number of operations

from 10 to 6 and the total number of sidings from 15 to 6. The number of trains and

runs has not changed. After applying Computing Acceleration (CA) Algorithms, the

total number of variables reduced from 32231 to 2611, where the reduction ratio is

around 91.8%. The total number of constraints reduced from 412407 to 15239,

where the reduction ratio is around 96%.

Table 5.9 shows the integration results of the search techniques and SSS strategy

after applying Computing Acceleration (CA). SSS works well in the small cases

(after applying CA algorithms) where it provides a faster solution with DFS and BFS

than before the algorithms were applied. The solution time using IDFS reduced

sharply from 9.5 hours, in the standard model, to be 0.53 computing acceleration

algorithms. The SBS and DDS techniques still do not work with the SSS strategy.

The Solver memory capacity reduced furtherafter applying CA algorithms than in

the standard model.

All search techniques work well with DSS to solve the proposed model as shown in

Table 5.9. CA algorithms with DSS reduce the number of choice and failure points

more than SSS. The solutions using DSS require the Solver memory capacity to be

less than SSS. CPU time of DSS and SSS with all search techniques is less than one

second. Applying BFS using SSS or DSS minimises the number of choice and

failure points more than by applying other search techniques. As a result, BFS CPU

time is shorter than other search techniques with the two strategies.

159

Table 5.9: SSS and DSS results for CA algorithms for MIP to optimise makespan

Sear

ch

tech

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e

Var

iabl

es

Con

stra

ints

SSS DSS

Choice points

Failure points

Makespan (s)

CPU time (s)

Solver memory

(kb)

Choice points

Failure points

Makespan (s)

CPU time (s)

Solver memory

(kb) DFS

2611

15239

4020 4018 2664 0.16 5057 2950 15 2664 0.19 4988

SBS n/a n/a n/a n/a n/a 2950 15 2664 0.17 5037 DDS n/a n/a n/a n/a n/a 2950 15 2664 0.19 5037 BFS 1494 1542 2664 0.11 5057 1488 14 2664 0.14 5049 IDFS 4614 4668 2664 0.52 5291 2950 15 2664 0.17 5037

160

Figure 5.5 shows the CPU time of the search techniques before and after applying

CA algorithms using SSS and DSS strategies. These graphs clearly identify that the

CPU time is reduced sharply after applying the algorithms and that SBS and DDS

are not applicable for the SSS strategies. IDFS results of standard MIP require

excessive time. The BFS technique requires the longest CPU time before the

algorithms are applied using DSS strategy.

Figure 5.5: CPU time of standard and CA algorithms for MIP to optimise makespan 5.2.2.2.3 Train Scheduling Results

Table 5.10 shows the start and finish times of train runs and the trains to which they

were assigned. Table 5.10 shows train 2 and 3 start time at zero. Trains 1 and 5 are

scheduled to be the third and the fourth and to start at time 250. Train 4 is the last to

start and commences its run at time 336. Figure 5.1 highlights three different rail

routes from the mill which means up to three trains can start simultaneously.

Table 5.10: Start and finish times of train runs for MIP to optimise makespan

Train index

Run index

Siding number

Visit time

Empty bins

Full bins Run time

Run start

time (s)

Run finish

time (s) 1 3 6 1873 54 54 2414 250 2664 2 1 6 791 120 120 1582 0 1582 3 2 14 491 112 112 982 0 982 4 5 22 1783 100 96 2250 336 2586 5 4 20 470 90 90 2250 250 2500

22 787 30 34 Total 5 - - 506 506 9478 2664

0 1 2 3 4 5 6 7 8 9

10 11 12


CPU

in se

cond

s

Search techniques

SSS of standard MIP /10

DSS of standard MIP /10

SSS of CA algorithms

DSS of CA algorithms

n/a n/a n/a n/a

3442.2

161

All trains have to deliver the full allotment of empty bins to each siding and collect

the same number of full bins from each siding. Table 5.10 shows the sidings visited

during each run, the visit time, the number of empty bins to be delivered and the

number of full bins to be collected from each siding as well as train run times. Train

run time is defined as the time that a train spends between starting the run at the mill

and returning to the mill after visiting the sidings. Figure 5.6 shows the detail of the

train run schedule as shown in Table 5.10.

Figure 5.6: Scheduling 5 trains on 11 segments using MIP with the makespan of 2664

5.2.2.3 Comparisons of CP and MIP Models using Makespan Criterion

Makespan results illustrate that DSS with search techniques is faster than standard

search techniques for both MIP and CP formulation models. Additionally, the

application of these algorithms helped to obtain faster solutions and also retained the

model solution as optimal. This section compares the MIP and CP models with the

SSS and DSS search techniques before and after applying CA algorithms. Table

5.11 illustrates the results of the SSS and DSS for the standard and the CA

algorithms for MIP and CP models.

DSS provides better solutions with all search techniques for CP standard model in

reasonable time than the standard MIP. SSS requires a longer time for the standard

CP model than the standard MIP model. The CA algorithms for the CP model

require a shorter time with all search techniques integrating with DSS than the MIP

model requires. CA algorithms for MIP models using SSS require a shorter time than

the CP model; however, MIP does not work with two search techniques, SBS and

DDS.

162

Table 5.11: Makespan results of standard MIP and CP and CA algorithms using SSS and DSS

Search technique

Standard MIP and CP CA algorithms for MIP and CP MIP CP MIP CP

SSS DSS SSS DSS SSS DSS SSS DSS

Mak

espa

n (s

)

CPU

Tim

e (s

)

Mak

espa

n (s

)

CPU

Tim

e (s

)

Mak

espa

n (s

)

CPU

Tim

e (s

)

Mak

espa

n (s

)

CPU

Ti

me(

s)

Mak

espa

n (s

)

CPU

Tim

e (s

)

Mak

espa

n (s

)

CPU

Tim

e (s

)

Mak

espa

n

(s)

CPU

Tim

e (s

)

Mak

espa

n (s

)

CPU

Tim

e (s

)

DFS 2664 43.86 2664 26.09 2664 100.68 2664 17.81 2664 0.16 2664 0.19 2664 0.30 2664 0.13 SBS n/a n/a 2664 26.58 2664 101.34 2664 18.06 n/a n/a 2664 0.17 2664 0.28 2664 0.14 DDS n/a n/a 2664 26.86 2664 102.61 2664 19.31 n/a n/a 2664 0.19 2664 0.30 2664 0.13 BFS 2664 116.81 2664 68.45 n/a n/a 2664 11.41 2664 0.11 2664 0.14 n/a n/a 2664 0.11 IDFS 2664 34422.09 2664 25.67 2664 106.64 2664 18.16 2664 0.52 2664 0.17 2664 0.53 2664 0.14

163

5.2.3 Results of Total Waiting Time Minimisation Objective

The two mathematical models (CP and MIP) are solved using the total waiting time

as an objective function. The solution techniques include the branch and bound

algorithm, simplex algorithm and constraint-based domain reduction. These

techniques are integrated to obtain good solutions using OPL and CPLEX software.

Additionally, the integration of mixed integer programming and constraint

programming search techniques can occur in OPL using Minimize with linear

relaxation. The case study involves 26 sections, 15 sidings, 11 segments and 5 trains.

5.2.3.1 Constraint Programming CP Model

5.2.3.1.1 Results of Standard CP

The standard CP model under the total waiting time criterion includes 30254

variables and 381539 constraints. A comparison of the integration of the different

search techniques and the two search strategies, SSS and DSS, in solving the

proposed model before applying CA algorithms, is examined in this section. Table

5.12 shows that SSS and DSS can give the same value of the total waiting time with

all search techniques except for the BFS technique, which it is not applicable to both

strategies and needs too much time to solve a small case.

Table 5.12: SSS and DSS results for Standard CP to optimise the total waiting time

Sear

ch

tech

niqu

e

SSS DSS

Cho

ice

poin

ts

Failu

re

poin

ts

Wai

ting

time

(s)

CPU

Tim

e (s

)

Mem

ory

solv

er(k

b)

Cho

ice

poin

ts

Failu

re

poin

ts

Wai

ting

time

(s)

CPU

Tim

e (s

)

Mem

ory

solv

er(k

b)

DFS 12657 0 1984 707 979848 12657 0 1984 699 979848 SBS 12657 0 1984 717 980266 12657 0 1984 700 980179 DDS 12657 0 1984 698 980274 12657 0 1984 699 980270 BFS n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a IDFS 12657 0 1984 715 980242 12657 0 1984 699 980258

Generally, SSS with search techniques spend longer time than DSS to produce the

solutions except DDS which requires longer time with DSS than SSS.

164


After applying Computing Acceleration (CA) algorithms to the CP model, it was

found that the algorithms reduced the total number of sections from 26 to 11, the

total number of segments from 11 to 3, the total number of operations from 10 to 6

and the total number of sidings from 15 to 6. The number of trains and runs was not

changed. After applying CA algorithms, the total number of variables reduced from

30253 to 2254 and the total number of constraints reduced from 380150 to 14823.

Table 5.13 shows a comparison of results for all search techniques and the two

search strategies of CA algorithms. This table shows that SSS and DSS work well in

the small cases with all search techniques except BFS, which is not applicable and

requires too much time. After applying CA algorithms, SSS with the search

techniques requires shorter CPU time than DSS, however the CPU time of both

techniques are close to each other. SSS and DSS are able to obtain a good solution in

reasonable time of CA algorithms.

Table 5.13: Solution of CA algorithms for CP to optimise the total waiting time

Sear

ch

tech

niqu

e

SSS DSS

Cho

ice

poin

ts

Failu

re

poin

ts

Tota

l w

aitin

g tim

e (s

)

CPU

Tim

e (s

)

Mem

ory

solv

er(k

b)

Cho

ice

poin

ts

Failu

re

poin

ts

Tota

l w

aitin

g tim

e (s

)

CPU

Tim

e (s

)

Mem

ory

solv

er(k

b)

DFS 997 0 1984 1.17 15032 997 0 1984 1.20 15028 SBS 997 0 1984 1.19 15127 997 0 1984 1.20 15060 DDS 997 0 1984 1.22 15131 997 0 1984 1.19 15060 BFS n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a IDFS 997 0 1984 1.17 15064 997 0 1984 1.19 15064

Comparisons of the standard model and CA algorithms using the two strategies to

optimise the total waiting time are shown in Figure 5.7. The CPU time and memory

solver capacity is reduced sharply after applying the algorithms for the two

strategies. The search technique BFS is not applicable to both strategies. Choice

points of the algorithms are fewer than in the standard model, however, the failure

points are zero which affected the search time positively.

165

Figure 5.7: CPU time of standard and CA algorithms for CP to optimise the total waiting time

5.2.3.1.3 Train Scheduling Results

Table 5.20 shows the start and finish times of train runs and the trains to which they

were assigned. Table 5.14 shows that trains 2, 3 and 5 start at time zero. A train 4 is

scheduled to be the fourth and starts at time 86. Train 1 is the last to start and

commences its run at time 250. Figure 5.8 highlights that because there are three

different rail routes from the mill, the three trains start at the same time, zero.

Table 5.14: Start and finish times of train runs to optimise the total waiting time

Train index

Run index

Siding number

Visit time

Empty bins

Full bins Total run time

Run start

time (s)

Run finish

time (s) 1 3 6 1840 120 90 2414 250 2664 2 1 6 758 54 84 1582 0 1582 3 2 14 202 112 112 982 0 982 4 5 20 988 90 - 1890 86 1976

22 1175 10 110 5 4 20 854 - 90 1074 0 1074

22 273 120 20 Total 5 - - 506 506 9478 2664

* s: second ** kb: kilo byte All trains have to deliver the full allotment of bins to each siding and collect the

same number of full bins from each siding. Table 5.14 shows the sidings visited

during each run, the visit time, the number of empty bins to be delivered and the

number of full bins to be collected from each siding, as well as train run times.

Figure 5.8 shows the detail of the train run schedule shown in Table 5.14.

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

6 6.5

7 7.5


CPU

in se

cond

s

Search technique

SSS of standard CP/100

DSS of standard CP/100



n/a n/a n/a n/a

166

Figure 5.8: CP for scheduling 5 trains on 11 segments with the total waiting time of 1984

5.2.3.2 Mixed Integer Programming (MIP) Model

5.2.3.2.1 Result of Standard MIP Model

The standard MIP model using the total waiting time criterion includes 32231

variables and 412407 constraints. Table 5.15 shows a comparison of results for all

search techniques and search strategies to solve the proposed model of Standard

MIP. Table 5.15 shows that SSS and DSS can provide the same value of the total

waiting time with all search techniques except SBS and DDS search techniques,

where both require too much time to solve a small case using the two strategies SSS

and DSS. The CPU time results show that SSS requires a longer time than DSS with

DFS and IDFS while SSS with BFS requires a shorter time than DSS with BFS. The

BFS technique works well with both search strategies and provides a better solution

in a reasonable time than other search techniques. The IDFS requires more than one

hour to obtain a solution. The used solver memory capacity to obtain the solutions is

shown in Table 5.15.

Table 5.15: SSS and DSS results for standard MIP model to optimise the total waiting time

Sear

ch

tech

niqu

e

SSS DSS

Cho

ice

poin

ts

Failu

re

poin

ts

Wai

ting

time

(s)

CPU

Tim

e (s

)

Solv

er

mem

ory

(kb)

Cho

ice

poin

ts

Failu

re

poin

ts

Wai

ting

time

(s)

CPU

Tim

e (s

)

Solv

er

mem

ory

(kb)

DFS 36017 33010 1984 1928 211414 36017 33010 1984 1919 211426 SBS n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a DDS n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a BFS 3028 3060 1984 1740 211237 3028 3060 1984 1744 211245 IDFS 12789 13273 1984 4502 211526 12789 13273 1984 4482 211534

167


Computing Acceleration (CA) algorithms are applied to the MIP model to reduce the

total number of sections from 26 to 11, the total number of segments from 11 to 3,

the total number of operations from 10 to 6, and the total number of sidings from 15

to 6. Therefore, the total number of variables is reduced from 32231 to 2611 and the

total number of constraints reduced from 412407 to 15239. The number of trains and

runs is not changed. The used solver memory capacity to obtain the solutions is

reduced from 690468 kb to 8070 kb bytes.

Table 5.16 compares the results for all search techniques and the two search

strategies of CA algorithms. SSS and DSS work well with all search techniques in

the small cases, where both provide the same solution in nearly the same time.

Table 5.16: SSS and DSS results of CA algorithms for MIP to optimise the total waiting time

Sear

ch

tech

niqu

e

SSS DSS

Cho

ice

poin

ts

Failu

re

poin

ts

Wai

ting

time

(s)

CPU

Tim

e (s

)

Solv

er

mem

ory

(kb)

Cho

ice

poin

ts

Failu

re

poin

ts

Wai

ting

time

(s)

CPU

Tim

e (s

)

Solv

er

mem

ory

(kb)

DFS 2093 2090 1984 2.72 8026 2093 2090 1984 2.70 8030 SBS 4200 4204 1984 3.45 8066 4200 4204 1984 3.45 8066 DDS 2176 2198 1984 3.66 8597 2176 2198 1984 3.63 8525 BFS 528 536 1984 2.44 8002 528 536 1984 2.42 8018 IDFS 579 616 1984 3.77 8018 579 616 1984 3.77 8022

The effect of applying the CA algorithms on CPU for the two strategies is shown in

Figure 5.9 where the CPU time is reduced sharply. The search techniques SBS and

DDS are not applicable for the SSS strategies while these techniques work well with

the DSS strategy.

168

Figure 5.9: CPU time of SSS and DSS for the standard and the CA algorithms of MIP to optimise the total waiting time

5.2.3.2.3 Train Scheduling Results

This section shows the trains scheduling and the visit times of all sidings working on

that day. The trains scheduling using the total waiting time objective is the same in

the MIP and CP models. Therefore, MIP model results are introduced only in this

section. Table 5.17 shows trains 2 and 3 and 5 start at time zero. Train 4 is scheduled

to be the fourth to start at time 86. Train 1 is the last to start and commences its run

at time 250.

Table 5.17: Start and finish times of train runs of MIP to optimise the total waiting time Train index

Run index

Siding number

Visit time(s)

Empty bins

Full bins

Total run time(s)

Run start time (s)

Run finish time (s)

1 3 6 1840 120 90 2414 250 2664 2 1 6 758 54 84 1582 0 1582 3 2 14 202 112 112 982 0 982 4 5 20 988 90 - 1890 86 1976

22 1175 10 110 5 4 20 854 - 90 1074 0 1074

22 273 120 20 Total 5 - - 506 506 9478 2664

* s: second

Figure 5.10 shows the segment index, which includes sections to be used during each

train run. The waiting time for each train is illustrated during each run. The two

0

4

8

12

16

20

24

28

32

36

40

44


CPU

in se

cond

Search techniques

SSS of standard MIP/100

DSS of standard MIP/100



n/a n/a n/a n/a

169

directions of the trains are implemented on the terminal segments without any

interruption.

Figure 5.10: MIP for scheduling 5 trains on 11 segments with the Total waiting time of 1984

Optimising the time table for the sugarcane rail transport system is introduced in

Section 5.5 and includes all variables to solve many problems.

5.2.3.3 Comparisons of CP and MIP Models of Total Waiting Time Criterion

The comparison of the CP and MIP model results are analysed in this section.

Generally, by using the total waiting time as an objective function, the standard CP

and standard MIP models gave the same results for the total waiting time value,

while the standard CP model is faster compared with the standard MIP. There was

no significant difference between CPU of SSS and DSS for the CP and MIP models.

Both strategies, with different search techniques, are implemented nearly in the same

CPU time as shown in Table 5.18.

Two search techniques are not applicable to the standard MIP, SBS and DDS, while

for the standard CP only the BFS search technique is not applicable. IDFS for the

standard MIP takes more time than other techniques.

170

Table 5.18: Standard MIP and CP results using SSS and DSS to optimise the total waiting time

After applying the CA algorithms, the CPU time of the MIP model reduced sharply

to be very close to the CPU time of CP. Additionally, all search techniques in MIP

work well, while BFS still does not work in the CP model. This is illustrated in

Table 5.19. Table 5.19: MIP and CP results using SSS and DSS for CA algorithms

Search technique

MIP CP

SSS DSS SSS DSS

Waiting time (s)

CPU Time

(s)

Waiting time (s)

CPU Time

(s)

Waiting time (s)

CPU Time

(s)

Waiting time (s)

CPU Time

(s) DFS 1984 2.72 1984 2.70 1984 1.17 1984 1.20 SBS 1984 3.45 1984 3.45 1984 1.19 1984 1.20 DDS 1984 3.66 1984 3.63 1984 1.22 1984 1.19 BFS 1984 2.44 1984 2.42 n/a n/a n/a n/a IDFS 1984 3.77 1984 3.77 1984 1.17 1984 1.19

Sear

ch

tech

niqu

e

MIP CP SSS DSS SSS DSS

Waiting time (s)

CPU Time (s)

Waiting time (s)

CPU Time

(s)

Waiting time (s)

CPU Time

(s)

Waiting time (s)

CPU Time (s)

DFS 1984 1928 1984 1918 1984 707 1984 699 SBS n/a n/a n/a n/a 1984 717 1984 700 DDS n/a n/a n/a n/a 1984 698 1984 699 BFS 1984 1740 1984 1744 n/a n/a n/a n/a IDFS 1984 4502 1984 4482 1984 715 1984 699

171

5.3 A large Scale Case Study for Testing Blocking Segment CP and MIP Models

To further test the model, the previous case study was extended to include a larger

part of the Kalamia Mill rail network. That is, the extended case study includes 51

sections, 42 sidings, 21 segments, 5 trains, 10 runs and 6217 tonnes as a total

allotment for all sidings. Figure 5.11 and Table 5.20 show the distances between

sidings, allotment of each siding, and siding capacities.

The previous results indicated that the CP model works well with small cases.

However, the MIP model has provided good results especially with the DFS search

technique in particular, to solve the makespan problem. The DFS search technique

has provided stable results with SSS and DSS. Integrating the MIP model and

different CP search techniques will obtain good solutions for this case. Computing

Acceleration (CA) Algorithms are applied directly to reduce the CPU time where the

total number of sections is reduced to 33; the total number of sidings has become 22,

and there are 10 segments. The two strategies SSS and DSS are tested with the

different search techniques to solve the current case study. ILOG OPL software is

used to solve the current case study of the sugarcane rail transport system.

172

Figure 5.11: Larger case study: bigger part of rail network of Kalamia mill Rail Data

Rail data in Table 5.20 has more details about this case that are related to distances

between sidings, sidings’ capacity and harvester data. Harvest Data will be shown in

Table 5.20.

0.27

Siding capacity

Distance between sidings

Allotment of siding

173

Table 5.20: The distance between sidings in the extension case study

Section numbe

r

From To Distance (km)

Sectional running time (s)

1 Mill JN - 35 0 0 2 JN - 35 JN - 40 .48 58 3 JN - 40 HEA .32 39 4 JN - 40 JN - 39 1.59 192 5 JN - 39 JN - 41 4.22 508 6 JN - 41 Gainsford 2 .27 33 7 JN - 41 Gainsford 4 2.03 245 8 JN - 39 Kal Plains .98 118 9 Kal Plains JN 38 1.82 219

10 JN 38 Brandon1 .19 23 11 JN 38 Brandon3 .94 113 12 Brandon3 Brandon4 1.6 193 13 JN - 35 Chiverston 2 1.68 202 14 Chiverston 2 Chiv terminus 2.4 289 15 Mill Town pts. J2 .71 86 16 Town Pts. J2 JN37 .66 80 17 JN37 Lillesmers .36 43 18 JN37 Town 3 1.53 184 19 Town 3 Town Terminus 3.13 377 20 Town PTS J2 Mainline 1A 1.11 134 21 Mainline 1 a Mainline 1 .44 53 22 Mainline 1 Mainline 3 2.19 264 23 Mainline 3 JN33 1.07 129 24 JN33 Mainline 4A .32 39 25 Mainline 4A Mainline 4B .01 1 26 JN33 Central PTS J8 .13 16 27 Central PTS J8 Central 1a .33 40 28 Central 1a Central 1 1.4 169 29 Central 1 Central 2 1.06 128 30 Central 2 Central 3 1.01 122 31 Central PTS J8 Creek points 2.00 201 32 Creek points Jarvisfield 2A .76 92 33 Jarvisfield 2A Jarvisfield 2B .14 17 34 Jarvisfield 2B Jarvisfield 3 1.75 211 35 Jarvisfield 3 Jarvisfield 6 1.39 167 36 Jarvisfield 6 JN_27 .73 88 37 JN_27 Jarvisfield 8A 1.46 176 38 Jarvisfield 8A JN-29 .15 18 39 JN-29 Jarvisfield 8B .39 47 40 JN-29 Jarvisfield 8C .37 45 41 JN_27 J/Field term B 1.5 181 42 J/Field term B J/Field term A .43 52 43 Creek points Ivanhoe points 1.9 229 44 Ivanhoe points Ivanhoe 2 1.25 151 45 Ivanhoe 2 Ivanhoe 3 1.24 149 46 Ivanhoe 3 Ivanhoe Terminus .99 119 47 Ivanho points Norham 3 .19 23 48 Norham 3 Norham _4 1.22 147 49 Norham 4 Norham Depot 1.94 234 50 Norham Depot Rita _ISLAND4 .68 82 51 Rita island 4 Rita island PTS .67 81

174

Harvest Data

The harvest data includes the siding capacity and the siding allotment. Other

parameters such as the harvester rates and the harvester start time are presented in

Section 5.5. Table 5.21 below details this data. Table 5.21: Sidings capacity and allotments in the extension case study

Siding section number

Siding Capacity (bins)

Allotment (tones)

Allotment (bins)

3 Hea 120 - - 6 Gainsford 2 140 1044 174 7 Gainsford 4 160 - - 8 Kal Plains 162 - - 10 Brandon1 136 - - 11 Brandon3 252 - - 12 Brandon4 110 - - 13 Chiverston 2 128 - - 14 Chiv terminus 114 676 114 17 Lillesmers 132 - - 18 Town 3 110 - - 19 Town Terminus 120 - - 20 Mainline 1 A 130 540 90 21 Main_Line 1 62 - - 22 Main_Line 3 148 780 130 24 Mainline 4A 100 - - 25 Mainline 4B 96 - - 26 Central 1A 68 - - 27 Central 1 150 - - 28 Central 2 136 - - 29 Central 3 154 585 98 30 Jarvisfield 2A 118 0 - 31 Jarvisfield 2B 118 0 - 32 Jarvisfield 3 146 0 - 33 Jarvisfield 6 124 0 - 34 Jarvisfield 8A 118 0 - 35 Jarvisfield 8B 124 300 50 36 Jarvisfield 8C 136 0 - 37 J/Field Term B 100 564 94 38 J/Field Term A 152 1050 175 39 Norham 3 126 0 - 40 Norham 4 124 0 - 41 Norham Depot 120 0 - 42 Rita Island 4 140 678 113

Total - - 6228 1038

175

Train Data

Train data includes the train index, name and speed. The train capacity includes

maximum number of empty bins and maximum number of full bins that can be

downloaded by the train. Table 5.22 details the train data.

Table 5.22: Train number, capacity and speed in the extension case study

5.3.1 Results of Makespan Minimisation Objective

Table 5.23 shows the results of the DFS search technique and the SSS search

strategy. The makespan of the current case is 25205, obtained after 870.54 seconds,

and no better solution was reached after 24 hours. The other search techniques did

not provide any solutions during the first 24 hours. The number of variables in the

current case study is high even though the algorithms were applied. This means that

the sugarcane system is an expansive system that includes a large number of

variables used in 829754 constraints of the MIP model.

Table 5.23: SSS results using makespan for the larger case study

Table 5.24 shows the results of the integration of DSS and the different search

techniques. An optimal solution was found in reasonable time with a makespan of

15414. DSS works efficiently with all search techniques even with the larger case

study. A comparison of the two search strategies identifies that DSS was more

efficient than SSS for this case. The number of choice points and failure points was

Train index

Train name Max empty (bins)

Max fulls (bins)

Average speed (km/h)

1 Rita-Island 120 124 30 2 Jarvisfield 120 124 30 3 Kilrie 120 124 30

Search technique

SSS

Nb. of variables

Nb. of constraints

Nb. of choice points

Nb. of failure points

Makespan (s)

CPU Time

(s)

Solver memory

(kb)

DFS 65161 829754 26335 4186 25205 870.54 364797

176

reduced by the DDS strategy to 22254 and 14 respectively for the different search

techniques.

Table 5.24: DSS results using makespan for the larger case study

Train Scheduling

In the larger case study, five trains performed 10 runs to deliver and collect 6228

tonnes of cane. Table 5.25 shows the scheduling of the train runs, the start and finish

times for each run and the number of bins delivered to and collected from each

siding.

Table 5.25: Start and finish times of train runs using the makespan objective function

Search technique Makespan CPU Time(s) Solver memory (kb) DFS 15414 75.70 311516 SBS 15414 75.84 312231 DDS 15414 77.98 312231 BFS 15414 75.86 312231 IDFS 15414 75.53 312231

Train index

Run index

Siding Number

Visit Time

Empty bins

Full bins

Run time (s)

Run start time (s)

Run finish time (s)

1 1 6 758 120 90 1582 0 1582 5 6 2340 54 84 1582 1582 3164

2 2 14 202 114 114 982 0 982 4 22 13405 10 26 2282 13132 15414

30 14151 98 98 3 7 41 5871 94 - 3382 4413 7795

42 6052 26 31 22 7258 - 93

8 22 8068 120 - 3741 7795 11536 41 9719 - 94

4 9 42 10475 120 - 3817 8401 12218 20 12132 - 90

10 22 12492 - 11 3196 12218 15414 50 13734 113 113

5 3 39 1652 49 50 3817 0 3817 42 2126 - 24

6 20 7414 90 - 4017 3817 7834 39 5469 1 - 42 5891 29 120

Total 10 - - 1038 28398

177

Note:

Visit time in table 5.25 includes the arrival time of each train at each siding.

Delivering and collecting visit time is the same (delivering and collecting operations

are conducted continuously) at the siding when the train returns from this siding to

the mill or when this siding is located at the end of the rail network. If the siding is at

the middle of the train pass-way, the delivering visit time “outbound direction” is

different from the collecting visit time “inbound direction”.

5.3.2 Results of Total Waiting Time Minimisation Objective

This case used minimising the total waiting time as an objective function to

investigate the MIP model. 5 trains were used to satisfy the sidings and mill

requirements. Table 5.26 shows the results for the DFS search technique and the

SSS search strategy. The total waiting time of 18495 was obtained after 3220

seconds.

Table 5.26: SSS results using total waiting time for the larger case study

Table 5.27 shows the results of the DSS and the search techniques integration. An

optimal solution was found in a reasonable time, with a total waiting time 18495.

The DSS works efficiently with DFS as a search technique even with the larger case

study. A comparison of the two search strategies identifies that DSS was more

efficient than SSS using the total waiting time objective. The number of choice and

failure points was reduced by the SSS and DDS strategies to 2881 and 2228

respectively during the search process.

Table 5.27: DSS results using total waiting time for the larger case study

Search technique

SSS Nb. of

variables Nb. of

constraints Nb. of choice points


Total waiting

time

CPU Time

(s)

Solver memory

(kb) DFS 54400 671106 2881 2228 18495 3220.72 958730

Search technique

DSS Nb. of

variables Nb. of

constraints Nb. of choice points


Total waiting

time

CPU Time

(s)

Solver memory (kb)

DFS 54400 671106 2881 2228 18495 3210 958742

178

Train scheduling In the larger case study, 5 trains performed 10 runs to deliver and collect 6217 tonnes

of cane. Table 5.28 shows the scheduling of the train runs, the start and finish times

for each run and the number of bins delivered to and collected from each siding.

Table 5.28: Start and finish times of train runs using the total waiting time objective

5.4 Sensitivity Analysis of Blocking Segment MIP Sugarcane Rail Model

This section shows how sensitivity analysis is used to investigate a new approach to

the sugarcane rail problem. The integration of the MIP model and CP search

techniques is used to solve different problems. Small and large rails are used in the

sensitivity study. Different trains and multi runs will be used to check the

relationship between the number of trains, number of runs and the CPU time of the

solution technique to the sugarcane rail problem. Makespan and total waiting time

objectives are used as objective functions. DFS and the two strategies provide the

rail problem solution.

Train index

Run index

Siding Number

Visit Time

Empty bins

Full bins

Run time (s)

Run start time (s)

Run finish time (s)

1 1 6 758 120 90 1582 0 1582 4 6 2340 54 84 1582 1582 3164

2 2 14 202 114 114 982 0 982 6 30 4997 98 98 3296 3468 6764

22 5723 22 26 3 3 22 273 30 - 3382 0 3382

20 3162 90 17 5 22 6227 44 104 3382 3382 6764

20 6678 - 20 4 7 42 7417 - 110 5554 3554 9108

41 7184 94 - 22 8571 26 -

8 20 9194 - 11 3196 9108 12304 50 10624 113 113

5 9 41 12063 - 94 4327 9194 13521 42 11778 120 30

10 39 15173 50 50 3817 13521 17338 42 15595 55 35 22 16801 8 - 20 13607 - 42

Total 10 - - 1038 1038 31100

179

5.4.1 Small Rail Systems Total Waiting Time(TWT)

This section investigates the effect the number of trains and runs has on minimising

the total waiting time as an objective function. In these cases, the size of the rail

network is limited to include 26 sections, 11 segments, and 15 sidings. SSS and DSS

strategies will be integrated with the DFS search technique of Computing

Acceleration (CA) algorithms to solve different tested cases. A Comparison of the

results of alternative cases for 40 runs is shown in Table 5.29.

180

Table 5.29: Results of sensitivity analysis of total waiting time (a small rail)

Test

ed C

ases

Trai

ns

Run

s /Tr

ain

SSS DSS

Var

iabl

es

Con

stra

ints

Cho

ice

poin

ts

Failu

re p

oint

s

Tota

l Wai

ting

Tim

e (s

)

CPU

(s)

Var

iabl

es

Con

stra

ints

Cho

ice

poin

ts

Failu

re p

oint

s

Tota

l Wai

ting

Tim

e (s

)

CPU

(s)

C1 10 4 18090 258664 26294 25080 111430 2818 18090 258664 26294 25080 111430 2832

C2 8 5 17736 253030 13318 12351 79746 1759 17736 253030 13318 12351 79746 1760

C3 5 8 17250 247534 1112 1112 71960 1206 17250 247534 1112 1112 71960 1220

C4 4 10 17100 236052 559 559 53748 661 17100 236052 559 559 53748 671

C5 2 20 16818 196180 198 198 21546 464 16818 196180 198 198 21546 464

181

SSS and DSS results are nearly similar and both of them are working well with

different problems as shown in Table 5.29 and Figure 5.12. There is a very strong

link between the total waiting time and the number of trains, where the total waiting

time increases by adding and reducing the number of runs. However, minimizing the

total waiting time can occur by increasing the number of runs and removing some

trains from the system. Table 5.29 shows that CPU time increases with any increase

in the number of trains in the system for both the SSS and DSS strategies.

Figure 5.12: SSS and DSS results of different cases using a small rail to optimise TWT

The Percentage Improvement of the Total Waiting Time (PITWT) is calculated for

the different cases as follows:

(PITWT)ij= {{total waiting time of case i - total waiting time of case j} / {total

waiting time of case i,}}*100. Where i and j are index of cases and i < j.

(PITWT)12 = ((111430-79746)/111430)*100 = 28.4%; (PITWT)12 improvement from C1 to C2.

(PITWT)23 = ((79746-71960)/ 79746)*100 = 9.7%.; (PITWT)23 improvement from C2 to C3.

(PITWT)34 = ((71960-53748)/ 71960)*100 = 25.3%. ; (PITWT)12 improvement from C3 to C4.

(PITWT)45 = ((53748-21546)/ 53748)*100 = 59.9%.; (PITWT)23 improvement from C4 to C5.

(PITWT)15 = ((111430-21546)/ 111430)*100 = 80.6%; (PITWT)15 improvement from C1 to C5.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

10000 11000 12000

C1 C2 C3 C4 C5

Tim

e in

seco

nds

Problems

SSS Solution/10

SSS CPU

DSS Solution/10

DSS CPU

182

Makespan

Table 5.30 shows the results of the makespan criterion applied as an objective

function to the MIP model. Increasing the number of trains and decreasing the

number of runs, means the makespan will also decrease. This means that using a

large number of trains at the same time to deliver or collect the bins affects the

makespan value. The CPU time is larger when 20 trains are used at the same time,

while CPU time with a small number of trains is reasonable.

The Percentage Improvement of the Makespan (PIM) is calculated for the different

cases using a small rail as follows:

(PIM)ij= {{makespan of case i – makespan of case j} / {makespan of case i}}*100,

where i and j are index of cases and i<j.

(PIM)12 = ((37524-26960)/ 37524)*100 = 28.1%; (PIM)12 improvement from C1 to C2.

(PIM)23 = ((26960-24812)/ 26960)*100 = 7.96%; (PIM)23 improvement from C2 to P3.




(PIM)16 = ((37524-21312)/ 37524)*100 = 43.2; (PIM)16 improvement from C1 to C6.

183

Table 5.30: Results of sensitivity study of makespan (a small rail)

Test

ed C

ases

Trai

ns

Run

s for

eac

h tra

in SSS DSS

Var

iabl

es

Con

stra

ints

Cho

ice

poin

ts

Failu

re p

oint

s

Mak

espa

n

(s)

CPU

(s)

Var

iabl

es

Con

stra

ints

Cho

ice

poin

ts

Failu

re p

oint

s

Mak

espa

n

(s)

CPU

(s)

C1 2 20 16819 1642664 437 438 37524 337 16819 1642664 438 437 37524 330

C2 4 10 17101 1581140 852 851 26960 302 17101 1581140 852 851 26960 301

C3 5 8 17251 1537592 628 628 24812 275 17251 1537592 628 628 24812 275

C4 8 5 17737 236738 1089 1089 22730 599 17737 236738 1089 1089 22730 603

C5 10 4 17191 1402920 0 0 21812 229 17191 1402920 0 0 21812 229

C6 20 2 20221 1025604 0 0 21312 2244 20221 1025604 0 0 21312 2240

184

SSS and DSS strategies are nearly the same when using a large number of runs and

trains, as shown in Figure 5.13.

Figure 5.13: SSS and DSS results of different problems using a small rail to optimise makespan

5.4.2 Large Rail Systems

Total Waiting Time

The previous case study used 5 trains and two runs for each train. In this section a

sensitivity analysis of the total waiting time is undertaken by changing the train

number from 5 to 3 and 2, and the runs from 2 to 4 and 5. Table 5.31 indicates that

decreasing the number of trains can improve the value of the total waiting time.

The Percentage Improvement of the Total Waiting Time (PITWT) for the different

cases using a larger rail is shown as follows:

(PITWT)12 = ((18495-13324)/18495)*100 = 27.9%; (PITWT)12 improvement from C1 to C2.

(PITWT)23 = ((13324-7024)/ 13324)*100 = 47.28%; (PITWT)23 improvement from C2 to C3.

(PITWT)13 = ((18495-7024)/ 18495)*100 = 62%; (PITWT)23 improvement from C1 to C3.

These total waiting time results show that using 2 trains and 5 runs in C1 is better

than using 5 trains and 2 runs in C3, where the improvement from C1 to C2 is 62%.

0

500

1000

1500

2000

2500

3000

3500

4000

C1 P2 C3 C4 C5 C6

Tim

e in

seco

nds

Problems

Solution of SSS/10

CPU of SSS

Solution of DSS/10

CPU of DSS

185

Table 5.31: Sensitivity analysis of changing some variables in the system using total waiting time

Test

ed C

ases

Trai

ns

Run

s /Tr

ain

SSS DSS

Var

iabl

es

Con

stra

ints

Cho

ice

poin

ts

Failu

re p

oint

s

Tota

l Wai

ting

Tim

e (s

)

CPU

(s)

Var

iabl

es

Con

stra

ints

Cho

ice

poin

ts

Failu

re p

oint

s

Tota

l Wai

ting

Tim

e (s

)

CPU

(s)

C1 5 2 54400 671106 2881 2228 18495 3220 54400 671106 2881 2228 18495 3210

C2 3 4 64800 829034 1392 1015 13324 4566 64800 829034 1392 1015 13324 4562

C3 2 5 53860 646490 222 0 7024 2867 53860 646490 222 0 7024 2844

186

Makespan

This section presents the sensitivity analysis applied to the makespan using different

problems. Table 5.32 shows the effects of changing the number of trains and runs on

the makespan value, where the makespan value is minimised, in spite of increasing

the number of trains and decreasing the number of runs.

Table 5.32: Sensitivity study for makespan criterion

The Percentage Improvement of the Makespan (PIM) in the case of increasing the

number trains and decreasing the number of runs is described as follows:

(PIM)12 = ((27839 - 21411)/ 27839)*100 = 23.08%; (PIM)12 improvement from C1 to C2.

(PIM)23 = ((21411 - 15414)/ 21411)*100 = 28%; (PIM)23 improvement from C2 to C3.

(PIM)13= ((27839 - 15414)/ 27839)*100 = 44.63%; (PIM)13 improvement from C1 to C3.

5.5 Blocking Section MIP and CP Results

DFS techniques are integrated with the DSS strategy to produce the solution for the

blocking section MIP and CP models for the case study in Figure 5.1. DFS is a stable

technique in most of the previous case studies and gave good results in reasonable

time in particular with the DSS strategy. Two objectives are investigated, minimising

the makespan and total waiting time (Masoud et al. 2010c). MIP and CP have nearly

the same results and CPU time. Table 5.33 shows the DSS results for blocking

section MIP using makespan and the total waiting objectives. Minimising makespan and the total waiting time are used as different objectives for

more investigation of the section blocking models. Table 5.33 describes that the

number of constraints and the CPU time of MIP are increased by changing the

objective function from makespan to the total waiting time.

Tested Cases

Trains number

Runs /Train

Variables Constraints Makespan (s)

Solver memory (kb)

C1 1 10 53701 607714 27839 321665 C2 2 5 53981 646966 21411 333370 C3 5 2 55901 670706 15414 342173

187

Table 5.33: Makespan and total waiting time results for the blocking section MIP model

The results of the blocking section CP model are obtained using makespan and the

total waiting time. Table 5.34 shows the number of constraints in CP using

makespan and the total waiting is lower than in MIP by:

((17802-13760)/17802)*100 =22.7%, where CP combines many constraints to work

as one constraint. Table 5.34: Makespan and total waiting time results for the blocking section CP model

5.5.1 The Comparisons between the Blocking Segment and Sections Models

The main aim of developing blocking section models is to reduce the value of the

makespan and the total waiting time objectives. Small and large cases in Figures 5.1

and 5.11 are used by applying blocking segment and section models. The Percentage

Improvement of the makespan and the total waiting time (PIM and PITWT) is

calculated from blocking segment to blocking section.

(PIM) segment, section = ((makespan of blocking segment – makespan of blocking section) / makespan of blocking segment)*100

Sear

ch

Tech

niqu

e

Obj

ectiv

e fu

nctio

n

DSS

Var

iabl

e N

umbe

r

Con

stra

ints

N

umbe

r

Cho

ice

Poin

ts

Failu

re

Poin

ts

CPU

Tim

e (s

)

Mem

ory

Solv

er(k

b)

Solu

tion

Val

ue (s

)

DFS

Makespan 1786 17802 491 491 1.17 6255 2432 Waiting time 1786 17852 510 510 2.36 7797 1610

Sear

ch

tech

niqu

e

Obj

ectiv

e fu

nctio

n

DSS

Var

iabl

e N

umbe

r

Con

stra

ints

nu

mbe

r

Cho

ice

poin

ts

Failu

re

poin

ts

CPU

Tim

e (s

)

Mem

ory

solv

er (k

b)

Solu

tion

Val

ue (s

)

DFS Makespan 1808 13760 1156 1156 0.91 6235 2432 Waiting time 1808 13760 1152 576 0.55 6786 1610

188

(PITWT) segment, section = ((TWT of blocking segment – TWT of blocking section) /

TWT of blocking segment)*100

Small case:

(PIM) segment, section = ((2664- 2432)/2664)*100 =8.7%

(PITWT) segment, section = ((1984- 1610)/1984)*100 =18.9%

Large case:

(PIM) segment, section = ((15414 - 15175)/15414)*100 =1.5%

(PITWT) segment, section = ((18495 - 16065)/18495)*100 =13.1%

Figure 5.14 shows the comparison between blocking section and segment models

using large and small cases, where the total waiting time improvement is more

significant than makespan.

Figure 5.14: Blocking segment and section results using makespan and total waiting time

0

500

1000

1500

2000

2500

3000

Makespan Waiting time

Tim

e in

seco

nd

Smal case study

Blocking sections

Blocking segemnts

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000

Makespan Waiting time

Tim

e in

seco

nd

Large case study

Blocking sections Blocking segemnts

189

5.6 The Results of Inclusion of the Delivery and Collection Time Constraints The delivery and collection time constraints are investigated in this section using

Kalamia Mill. This case includes 7 harvesters, 4 trains and delivery and collection of

875 bins under time constraints. The ACTSS schedule checker and simulator

(McWhinney and Penridge, 1991; Pinkney and Everitt, 1997) was used to examine

the schedule produced by the MIP model. An optimal (or near optimal) number of train is determined under the system’s

constraints (see Chapter 3 for detail). The best makespan solution is found as 24.63

hr and the total operating time is 18.46 hr with 4 trains and 12 runs (trips).

Figure 5.15 shows a graph of harvester utilisation. The bar graphs are shaded in the

period when a harvester is operating. Gaps in the graph indicate periods when the

harvester is waiting for bins. Although there are some small gaps in the harvest

shown in Figure 5.15, these gaps were caused by differences in allocating shunting

time between the model and the ACTSS simulator and not by a weakness in the

model. Hence the model achieves the objective of maintaining a continuous supply

of empty bins to the harvesters.

Figure 5.15: Harvester usage for seven harvester model

Note: In ACTSS, shunting time is 15min for the total number of delivery or

collection bins of each visit at each harvester, while in the model, shunting time

depends on the number of delivered or collected bins where each bin takes 15

seconds.

Maintaining a continuous supply of full bins to the mill is demonstrated in the mill

yard stock chart shown in Figure 5.16. Since the stock of full bins never reduces to

zero, a continuous supply of full bins to the mill is demonstrated. It is easy,

190

however, to achieve a continuous supply of full bins to the mill in a model simply by

increasing the number of full bins at the mill at the start of the simulation. The

challenge, however, is to minimise the size of the bin fleet and increasing the number

of full bins at the mill to overcome deficiencies in the model can be identified by

considering the required size of the bin fleet. In this model, a bin fleet of 753 bins is

required to achieve the objective of filling 875 bins, indicating that 16% of the bin

fleet is filled twice during the day and showing a reasonably small bin fleet for the

harvesting task.

Figure 5.16: Mill yard stock chart for the seven harvester model

Figure 5.17 shows the train utilisation and the automatically generated train run

details are presented in Table 5.35.

Figure 5.17: Train utilisation for the seven harvester model

191

Table 5.35: Train runs for seven harvester model Train Run Location Start time Activity

Delivered empty bins

Collected full bins

Jarvisfield 1 Kalamia 7:02:00 Start_run

Mainline_1a 7:05:39 43 0

Mainline_3 7:25:56 35 0

Central_3 7:50:56 42 0

Central_3 8:05:56 0 31

Mainline_1a 8:21:14 0 28

Kalamia 8:39:53 End_run

2 Kalamia 10:25:00 Start_run

Mainline_1a 10:28:39 33 0

Mainline_3 10:48:56 35 0

Central_3 11:13:56 10 0

Central_3 11:28:56 0 34

Mainline_3 11:38:57 0 30

Mainline_1a 11:59:14 0 50



Mainline_1a 23:04:39 14 0

Mainline_3 23:24:56 60 0

Central_3 23:49:56 46 0

Central_3 24:04:56 0 33

Mainline_3 24:14:57 0 10


Kilrie 1 Kalamia 1:40:00 Start_run

Gainsford_2 2:09:04 70 0

Gainsford_2 2:24:04 0 70



Gainsford_2 11:52:04 70 0

Gainsford_2 12:07:04 0 70



Gainsford_2 17:28:04 34 0

Gainsford_2 17:43:04 0 34


Norham 1 Kalamia 8:08:00 Start_run

J/Field_Term_B 8:35:54 47 0

J/ Field_Term _A 8:51:46 69 0

J/ Field_Term _A 9:06:46 0 48

J/ Field_Term _B 9:08:39 0 50

192

Mainline_3 9:42:37 0 26



J/ Field_Term _A 13:09:46 62 0

J/ Field_Term _A 13:24:46 0 76

J/ Field_Term _B 13:59:39 0 12

Mainline_3 14:33:37 0 36



J/ Field_Term _B 20:01:54 47 0

J/ Field_Term _A 20:17:46 44 0

J/ Field_Term _A 20:32:46 0 51

J/ Field_Term _B 20:33:39 0 32

Mainline_3 21:07:37 0 28

Mainline_1a 21:27:54 0 12


Rita Island 1 Kalamia 11:08:00 Start_run

Chiv_terminus 11:32:31 52 0











Total 875 875

193

5.7 Conclusion

The sugarcane railway operations are complicated and have a large number of

variables. As a result, the proposed scheduling model is complicated and needs to be

solved in a reasonable time, because of the dynamic nature of the system. The

combination of mixed integer programming and constraint programming search

techniques is used as an integrated approach to obtain appropriate solutions in a

reasonable time. Some heuristics are proposed to solve rail conflicts and simplify the

system by removing unused segments and sections from the model. A case study

tests many search techniques and evaluates the performance of each technique. The

dichotomic search strategy, combined with any of the search techniques, gives high

quality solutions. The comparison of the results of the blocking section models and

blocking segment models, and sensitivity analysis of the railway utilisation are

described in detail in this chapter.

Inclusion of the delivery and collection time constraints results are presented in real

life where the sidings visit time is optimised to remove any delays in delivering or

collecting bins throughout the rail network. Metaheuristic techniques are adapted in

Chapter 6 for a larger number of harvesters and trains throughout the rail network to

reduce the CPU time of executing the model codes.

194

Chapter 6

Metaheuristic Techniques

Chapter Outline

6.1 Introduction....................................................................................................................197

6.2 Neighbourhood Structure................................................................................................198

6.2.1 Adjacent Pairwise Interchange (API) …………………...............................200

6.2.2 Non-Adjacent Pairwise Interchange (NAPI).................................................201

6.2.3 Extraction and Forward Shifted Reinsertion (EFSR)……………...…….....201

6.2.4 Extraction and Backward Shifted Reinsertion (EBSR).................................202

6.3 Neighbourhood Structure in the Railway Scheduling Problem ....................................202

6.4 Simulated Annealing (SA) Technique............................................................................207

6.4.1 Simulated Annealing Technique for Solving Job Shop Scheduling

Problem…………………………………...………………………………...208

6.4.2 New Simulated Annealing Algorithms for Sugarcane Rail Cases……….....210

6.5 Tabu Search (TS) Technique..........................................................................................213

6.5.1 Tabu Search Technique for Solving Job Shop Scheduling Problem.……...213

6.5.2 A new Tabu Search Technique for Sugarcane Rail Cases...............................215

6.6 Metaheuristic Results of Sugarcane Rail Systems..........................................................216

6.6.1 TS and SA Results by Changing Number of Trains.........................................219

6.7 Hybrid Metaheuristic Techniques for Sugarcane Rail Systems.....................................221

6.7.1 Hybrid SA/TS Technique..................................................................................222

6.7.2 Hybrid TS/SA Technique ..................................................................................226

6.7.3 Hybrid Techniques Result for Sugarcane Rail Systems......................................229

6.7.4 Analysis of Hybrid Techniques...........................................................................232

6.8 Hyper Metaheuristic Techniques for Sugarcane Rail Transport Systems......................234

195

6.8.1 Hyper SA/TS Technique.......................................................................................235

6.8.2 Hyper TS/SA Technique.......................................................................................238

6.8.3 Hyper Techniques Result for Sugarcane Rail Systems.........................................241

6.8.4 Analysis of Hyper Techniques.............................................................................244

6.9 Hybrid and Hyper Metaheuristic Techniques Test Cases...............................................246

6.10 Hyper and Hybrid Metaheuristic Technique (TS/SA) and MIP...................................247

6.11 Study Analysis of Elements of Metaheuristic Techniques...........................................249

6.12 Inclusion of Delivery and Collection Time Constraints for a Hyper TS/SA

Results...........................................................................................................................252

6.13 Conclusion....................................................................................................................257

196


Masoud, M., Kozan, E., & Kent, G. (2011c). Hybrid/hyper metaheuristic techniques

for optimising sugarcane rail operations. Computer and Operations research

(submitted).

197

6.1 Introduction

Optimisation the sugarcane rail transport system is NP Hard problem and finding an

optimal solution in reasonable time is difficult. As a result, there is a need to develop

some techniques to find good solutions in reasonable time. This chapter describes

some metaheuristic techniques to optimise the sugarcane rail transport systems.

Metaheuristic techniques are a type of local search technique which usually starts by

determining a feasible solution which may be selected at random and then used to

obtain a better solution by manipulating the current solution. These techniques give

a near optimal or optimal solution, but not guaranteed optimality.

The main advantage of the metaheuristic techniques is obtaining near-optimal

solutions in a reasonable time for many large problems, especially in scheduling

problems where the solution will be represented by a complete schedule and then

obtaining a better schedule than the current schedule. This schedule can be specified

by a simple sequence of n jobs as in a non-preemptive single machine schedule, or a

sequence of k operations on a specific machine of m machines, as in a non-

preemptive job shop schedule. Start times and completion times are included in this

schedule. Design of the schedule representation is more complicated in the

preemption case.

Before the metaheuristic techniques are identified, a basic local search is described

first. According to Pinedo (2008), there are four main criteria to be investigated

when comparing local search techniques, namely:

- solution representation;

- neighbourhood structure;

- search method within neighbourhood; and

- acceptance-rejection criterion.

The basic local search is usually called iterative improvement, since each iteration or

step is only performed if the next solution is better than the current solution. The

stop criteria of the algorithm depend on obtaining the target value of the solution

“final solution”. The main procedures of local search techniques are shown in the

next generic algorithm.

A generic local search algorithm;

198

Begin

Select an initial solution s ЄS; where S is the search space of all solutions.

Set Best solution s*=s;

While Stop criteria is not achieved do

Determine a uor s'ЄN(s); N(s) is the space of all neighbourhoods of s.

Set s=s'

If C(s') <C(s*) then

Set s*=s'

End if

End while

End.

This research investigates two metaheuristic techniques, simulated annealing (SA)

and tabu search (TS). These techniques solve many cases particularly those of a large

scale. The classical SA and TS will be used to solve a job shop scheduling problem

as a small example to show how these techniques work. They are applied to the

sugarcane rail transport systems. The job shop scheduling problem is formulated in

this chapter using a disjunctive graph approach to obtain feasible solutions to use as

initial solutions in SA and TS techniques. The overall acceptance or rejection

criterion to stop the technique implementation is different from one technique to

another, and in SA, is based on a probabilistic process. The TS uses a deterministic

process.

6.2 Neighbourhood Structure

Neighbourhood structure is the core of many local search techniques such as SA and

TS. Neighbourhood structure is applied to obtain new feasible solutions by applying

small perturbations on a given feasible solution. In job shop scheduling, re-ordering

the sequence of operations on a critical path is used to produce small perturbations to

obtain a neighbourhood which produces a better makespan than the current

makespan. A critical path in a job shop schedule includes a set of operations in

which the first starts at time 0 and the last finishes at time t=Cmax . The completion

time of each operation on the critical path is equal to the starting time of the next

operation on that path. The sequence of operations on the critical path is changed to

199

analyse the effects on the makespan. These change for different job operations on

the same machine and are explained in the different types of neighbourhood

structure.

Definition1: if N: S →Z(S) is a neighbourhood structure, where S is the search space

of an optimisation or decision/feasibility problem, then:

- N is called connected if any solution s Є S can be obtained from any other

solution s' Є S by a specific number of steps according to N.

- In the optimisation problem, N is called opt-connected if from any solutions s

Є S, an optimal solution s* Є S (if existent) can be obtained after applying a

specific number of steps according to N.

- In a decision problem, N is called feasible-connected, if any solution s Є S, a

feasible solution sf Є S (if existent) can be obtained after applying a specific

number of steps according N.

For classical job shop scheduling, assume S and S' be two complete selections for a

given alternative graph G.

- If S and S' are consistent, P is a critical path in G(S) and Cmax(S') ≤ Cmax(S),

then at least one alternative arc of P does not belong to S'.

- If S is not consistent, and C is a positive cycle in G(s) then at least one arc of

C does not belong to a complete consistent selection S' .

The proof of this theory is very clear (Brucker, 2007).

Definition 2: a generic neighbourhood structure Assume P be a critical path of a given alternative graph G(S) where S is a complete

selection. Neighbourhood N1(S) is the set of all consistent selections that can be

obtained by replacing one alternative arc or more of an arbitrary critical path P by its

alternative. One alternative arc i → j of an arbitrary critical path P is replaced by its

alternative j → i while its alternative will be h → k if the swap is not allowed. If the

new selection is unfeasible or inconsistent, this selection can be repaired by

replacing alternative arcs.

The search technique within a neighbourhood can be conducted in many ways. One

of these ways is to select schedules in the neighbourhood randomly, and evaluate

200

these schedules using any performance criteria to decide which schedule is accepted

to produce new solutions. Another way is to select a schedule which is giving good

results, and to try to swap the jobs that have a significant effect on the objective

function.

Many techniques are used to design a neighbourhood structure. Some techniques are

random and others are specific such as pairwise interchange (PI) techniques. This

section shows some PI techniques.

6.2.1 Adjacent Pairwise Interchange (API)

Adjacent Pairwise Interchange (API) technique swaps any two adjacent jobs

according to the following scheme:

Initial parameters

The maximum number of iteration=max_iter

The index of iteration: iter

The schedule s is an initial schedule

Select i, j Єs, where i, j are two adjacent jobs.(Step 1)

Swap i and j

Store the new schedule s'

Evaluate the new schedule using Makespan criterion where,

If f(s') < f(s) then

Select s' as a new schedule

Else

Select s as a new schedule

End if

Set ietr=iter+1

If iter < max_iter then

go to step 1

Else

End if

End

201

This technique is best explained by: 10 jobs are sequenced as in a given sequence

1-2-3-4-5-6-7-8-9-10. Figure 6.1 shows jobs 5 and 6 are two adjacent jobs which

will be swapped to obtain a new sequence 1-2-3-4-6-5-7-8-9-10.,

Figure 6.1: API technique for 10 jobs

6.2.2 Non-Adjacent Pairwise Interchange (NAPI)

Non-Adjacent Pairwise Interchange (NAPI) swaps two non adjacent jobs. All

adjacent pairwise interchange steps will be applied in NAPI except step 1 where the

step 1 will be changed to be:

Select i, j Єs, where i and j are two non-adjacent jobs.

Suppose the sequence 1-2-3-4-5-6-7-8-9-10 is given. By using NAPI, jobs 4 and 8,

as two non-adjacent jobs, will be swapped to obtain a new sequence: 1-2-3-8-5-6-7-

4-9-10 as shown in Figure 6.2.

Figure 6.2: NAPI technique for 10 jobs

6.2.3 Extraction and Forward Shifted Reinsertion (EFSR)

Extraction and Forward Shifted Reinsertion (EFSR) technique follows the next steps:

Select i, j Єs, where i and j are two non-adjacent jobs and the position of i

before the position of j

Extract job i from its position and reinsert it direct after job j.

5 10 9 8 7 6 4 3 2

5 10 9 8 7 6 4 3 2

1

1

202

For example, in the sequence 1-2-3-4-5-6-7-8-9-10, job 4 is extracted from its

position and reinserted directly after job 8. The resulted sequence is:

1-2-3-5-6-7-8-4-9-10 as shown in Figure 6.3.

Figure 6.3: EFSR technique for 10 jobs

6.2.4 Extraction and Backward Shifted Reinsertion (EBSR)

Extraction and Backward Shifted Reinsertion (EBSR) includes two main steps:

Select i, j Єs, where i and j are two non-adjacent jobs and the position of i is

before the position of j.

Extract job j from its position and reinsert it directly before job j. For example, in the sequence 1-2-3-4-5-6-7-8-9-10, job 8 is extracted from its

position and reinserted directly before job 4. The out sequence is:

1-2-3-8-4-5-6-7-9-10 as shown in Figure 6.4.

Figure 6.4: EBSR technique for 10 jobs

6.3 Neighbourhood Structure in the Railway Scheduling Problem

Neighbourhood Structure technology is used in developing new metaheuristic

techniques to solve the railway scheduling problems. Neighbourhood can be

produced at any section particularly at the intersection points. A single cane rail

network with 22 sections is shown in Figure 6.5, where a and b are intersection

points. Point a includes three neighbourhoods (sR5R, sR6R and sR11R) and point b also

includes three neighbourhoods (sR13R, sR14R and sR16 R).

5 10 9 8 7 6 4 3 2

5 10 9 8 7 6 4 3 2

1

2

203

s22

s21

s20

Intersection Point b

s19

Intersection Point a s18

s17

s16

s1 s2 s3 s4 s5 s11 s12 s13 s14 s15

s6

s7

Figure 6.5: A single cane rail network with 22 sections

A transition matrix is used to represent any railway network. This matrix will

include all connections of the railway network between all sections. The main idea of

the transition matrix is to use the binary system, using 1 and 0, to representing any

transport network. This matrix uses 1 to mean there is a connection or a link between

two sections or a section and a passing loop or mill and section. 0 means there is no

connection between them. Figure 6.5 shows a small railway network and how it can

be represented using a transition matrix in Figure 6.6.

s8

s9

S10

204

Figure 6.6: Transition matrix for a single railway includes 22 sections

The Transition matrix is designed in Figure 6.6 to include all connections between

22 sections of the network of the single railway. This matrix helps to implement two

main types of neighbourhood structures: section neighbourhood and train

neighborhood. Analysis of the feasibility of the solutions is included in the transition

matrix, where 1 means there is a feasible solution while 0 means there is no feasible

solution. This technique means many infeasible solutions related to section

neighbourhood can be avoided. Two main neighbourhood structure techniques are

proposed in this research and explained in detail below.

205

Section Neighbourhood

A decision at each conjunction or intersection point has to be taken in section

neighbourhood technique. This decision relates to the selecting of the section to be

used. The direction of the arc represents the direction of the train. Figure 6.5 shows

two intersection points, a and b.

At point a

There are three sections (sR5R, sR6R, and sR11R) linked with point a. The neighbourhoods

that can be produced at point a as follows:

sR5R→sR11R outbound direction


sR6R→sR5R inbound direction


sR11R→sR6 R inbound direction


At point b

Three sections (sR13R, sR14R, sR16R) are linked to point b. All neighbourhoods that can be

produced at this point are shown as follows:







The makespan values are evaluated at each neighbourhood. All neighbourhoods are

permitted by considering the priorities of visiting the sidings to achieve the sidings

and mill requirements or other constraints on the models.

206

From the transition matrix, 1 means there is a link between one section and another.

This link represents a neighbourhood for any section that can be used to improve the

solution and 0 means a new neighbourhood cannot be constructed for the current

solution.

Train Neighbourhood

Some trains require a specific section for different operations. The sequence of

operations on these sections can produce new neighbourhoods by changing the

positions of these operations. Figure 6.7 presents three trains ka, kb, and kc where

ka requires section s to implement operations oa, kb requires section s to implement

operation ob and kc requires the same section to implement operation oc. As result,

many neighbourhoods of operations will be produced at section s as follows:

N1: oa→ob , where train ka precedes train kb The sequences of operations on s under N1 are oa→ob→oc or oc→ oa→ob.

oa →ob →oc means train ka precedes train kb and train kb precedes train kc,

oc→ oa→ob means train kc precedes train ka and train ka precedes train kb.

N2: oa→oc , where train ka precedes train kc The sequences of operations on s under N2 are oa→oc→ob or ob→ oa→oc.

oa→oc→ob means train ka precedes train kc and train kc precedes train kb,

ob→ oa→oc means train kb precedes train ka and train ka precedes train kc.

N3:ob→oc , where train kb precedes train kc The sequences of operations on s under N3 are ob →oc →oa or oa→ ob→oc.

ob →oc →oa means train kb precedes train kc and train kc precedes train ka,

oa→ ob→oc means train ka precedes train kb and train kb precedes train kc.

N4:ob→oa , where train kb precedes train ka

The sequences of operations on s under N4 are ob→oa→oc or oc→ ob→oa.

ob→oa→oc means train kb precedes train ka and train ka precedes train kc,

oc→ ob→oa means train kc precedes train kb and train kb precedes train ka.

207

N5:oc→oa , where train kc precedes train ka The sequences of operations on s under N5 are ob→oc→oa or oc→ oa→ob.

ob→oc→oa means train kb precedes train kc and train kc precedes train ka,

oc→ oa→ob means train kc precedes train ka and train ka precedes train kb.

N6:oc→ob , where train kc precedes train kb

The sequences of operations on s under N6 are oc→ob→oa or oa→ oc→ob.

oc→ob→oa means train kc precedes train kb and train kb precedes train ka, oa→ oc→ob means train ka precedes train kc and train kc precedes train kb .

kc

kRb RkRaR

s

Figure 6.7: Case of three trains require the same section

Neighbourhood structure techniques are used to develop new metatheuristic

techniques such as SA and TS. Sections 6.4 and 6.5 explain the application of SA

and TS techniques respectively to the job shop scheduling problem (refer to Example

3.1 for the detail) and to the sugarcane rail transport system.

6.4 Simulated Annealing (SA) Technique

SA was first introduced by Kirkpatrick et al. (1983) and Cerny (1985). The SA

technique is a metaheuristic technique using local search techniques to solve

combinatorial optimization problems. SA searches different possible solutions to

avoid being stuck in local optima. The SA procedure is as follows:

Junction point

208

SA starts with an initial solution as a current solution, currSolution,

and obtains a new solution, newSolution, using the neighbourhood of

the current solution. If the objective function value of the current

solution is f(currSolution) and the objective function value of the new

solution is f(newSolution), it can calculate the difference

∆= f(newSolution) - f(currSolution) to determine if the new solution is

acceptable. The new solution is acceptable (for minimization problems)

if ∆ is less than zero or exp(∆/T)> ε where ε is a small number and T is

the temperature; otherwise the new solution is rejected. Using the SA

technique, T will decrease through the iterations using the decreasing

stochastic parameter α.

6.4.1 Simulated Annealing Technique for Solving Job Shop Scheduling Problem

This section demonstrates how the SA technique applies to job shop scheduling

problems using makespan minimisation as an objective function. The SA procedure

is explained step by step below.

Select the initial simulated annealing parameters

Define a large value of the temperature T=T0

Select the value of the parameter α is between 0 and 1; 0<α<1

Set an initial feasible schedule using the disjunctive graph

Evaluate the initial value of objective function (makespan)

While (T is in cooling range)

Develop a heuristic technique to design a neighbourhood for the

current solution

Evaluate the new makespan; ∆=new Makespan – currMakespan

If ∆<0 then

Accept the new state

Else

If Pr(accepted)= exp(∆/T)> ε then


Else

Reject the solution

209

Return to the previous solution

End if

End if

Update T according to T= αT

End while

Stop criteria

∆ is not significant or there are no improvements in the makespan for

small change in T

The maximum number of iterations has been completed

End

An Example of a Simulated Annealing Application

Example 2.1 is used to explain how SA can solve a job shop problem. The final

solution is shown graphically in Figure 6.8, where the critical path (red arrows) is

obtained. All SA iterations are shown in Table 6.1. The best solution is obtained

after four iterations. The complete SA solution for the numerical example is given in

Appendix A.

Figure 6.8: Disjunctive graph of final solution for the 3 jobs and 3 machine example

Makespan=21

Table 6.1 shows the results of all iterations for solving Example 2.1.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7 M2 9 M3

3

6

4

6

2

3

6

4

4

M1

210

Table 6.1: Simulated annealing result for Example 2.1

Step Makespan (Cmax)

∆ Temperature (T)

Boltzmann probability

Decision Best value

0 - 100 - - 21 1 - 95 - accepted 2 90.25 0.988 accepted 3 - 85.73 - accepted 4 81.44 0.961 rejected

The Percentage Improvement of SA can be calculated as:

PISA= ((initial solution of CRmaxR -SA value)/initial solution)*100 of CRmax

PISA= ((29-21)/29)*100 = 8/29= 27.5%.

SA principles of JSS problem have been adapted to sugarcane rail transport systems

as identified in Section 6.4.2 using the SPT as an initial solution.

6.4.2 New Simulated Annealing Algorithms for Sugarcane Rail Cases

Two interactive SA algorithms are developed to solve sugarcane rail transport

systems. Firstly, initial feasible solutions are obtained and then the quality of the

solutions is improved. These algorithms depend on using two techniques of the

neighbourhood structure are shown in Section 6.3. Algorithm 1 uses the train

neighbourhood technique, while Algorithm 2 uses the section neighbourhood

technique. The integration of the two algorithms obtains SA results where the

section neighbourhood and train neighbourhood are used at the same time. The

procedures of the two algorithms are shown as follows:

Algorithm1 :


Define a large value for the temperature T=TR0

Select value of the parameter α; 0<α<1

Construct an initial solution using a disjunctive graph model

Define initial scheduling for all trains on all sections

Define initial sequencing of segments

Calculate initial makespan (currMakespan)


211

Construct a new neighbourhood

Apply train neighbourhood technique

Apply Adjacent pairwise techniques

Swap any two random trains

Require a specific segment

Require a specific section

Check the feasibility

Time window, section transition matrix, trains and sidings capacity

Keep the sequencing of sections without changing

Produce a new schedule and a new solution (new Makespan)

Evaluate the new makespan where ∆=new Makespan – currMakespan

If ∆<0 then


Else

If Pr(accepted)= exp(∆/t)> ε then


Else

Reject the solution and return to the previous solution

End if

End if


End while

Stop criteria

Until ∆ is not significant or there are no improvements in the

makespan for small change in T or,

Complete the maximum number of iterations, known in advance

End.

212

Algorithm 2

Produce an initial solution using a disjunctive graph model

Define initial scheduling for all trains on all sections

Define initial sequencing of sections

Calculate initial makespan (currMakespan)


Construct a new neighbourhood

Apply section neighbourhood technique

Apply adjacent pair wise techniques

Swap any two random sections or segments

Apply blocking section constraints or

Apply blocking segment constraints


Time window, section transition matrix, trains and sidings capacity

Keep the sequencing of trains without changing

Produce a new schedule and a new solution (new Makespan)

Evaluate the new makespan where ∆=new Makespan – currMakespan

If ∆<0 then


Else

If Pr(accepted)= exp(∆/t)> ε then


Else

Reject the solution and return to the previous solution.

End if

End if


End while

Stop criteria

Until ∆ is not significant or there are no improvements in the

makespan for small change in T or,

Complete the maximum number of iterations, known in advanced

End.

213

6.5 Tabu Search (TS) Technique

The TS technique was introduced by Glover (1989 and 1990) and starts with an

initial solution which it considers as the current seed. The objective function of the

current seed is then calculated. Using the neighbourhood structure, the

neighbourhoods of the current seed are determined. The objective function is

calculated for all neighbourhoods of the current seed. The best is selected and added

to the tabu list. If the new solution is better than the current solution, it is stored as

the new best solution and used as a new seed. These steps are repeated until the stop

criteria are satisfied.

6.5.1 Tabu Search Technique for Solving Job Shop Scheduling Problem

The TS Technique for the job shop scheduling problems follows the steps below:

Generate initial feasible scheduling as a current solution and the best solution

Set the tabu search parameters and assume that the tabu list is empty

Set the stop criterion

Define a specific number of iterations

Find no improvements of the solution

Obtain specific value of the objective function

Generate neighbourhoods of the current solution using the neighbourhood structure

Select a neighbourhood which is not tabu or satisfies a given aspiration criterion

Move this neighbourhood to a new solution

Update the tabu list

Store the new solution as the best solution

The objective function value is better for the new solution

Repeat the steps until a stop criterion is satisfied

Numerical example is solved to explain tabu search procedure in detail.

214

Tabu Search Numerical Example

The numerical Example 2.1 in Chapter 2 shows how TS works to solve scheduling

problems. An initial solution supposed for minimising the makespan as an objective

function. The neighbourhood structure is used to generate new solutions. Figure 6.9

shows after implementing 6 iterations, the best solution is makespan = 21.

Figure 6.9: Disjunctive graph of final solution for the 3 jobs and 3 machines example

Makespan=21

Table 6.2 shows the results of all iterations for the previous example. Appendix B

shows the complete solution for the numerical example using TS.

Table 6.2: TS technique result for example 2.1 Iteratio

n CRmax Tabu list Best

value

0 { } 21

1 { oR8R→o} 2 { oR8R→o, oR8R→oR1R} 3 { oR8R→oR4R, oR8R→oR1R, oR9R→oR3R } 4 { oR8R→oR4R, oR8R→oR1R, oR9R→oR3R , oR4R→oR1R} 5 { oR8R→oR4R, oR8R→oR1R, oR9R→oR3R , oR4R→oR1R, oR5R→oR3R} 6 { oR8R→oR4R, oR8R→oR1R, oR9R→oR3R , oR4R→oR1R, oR5R→oR3R, oR5R→oR9R}

* CRmax R: Makespan

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

215

The Percentage Improvement of Tabu Search(PITS) is calculated as follows:

PITS = ((initial solution of Cmax-TS value)/initial solution )*100) of Cmax

PITS = ((29-21)/29)*100 =27.58%.

A new TS algorithm for the sugarcane rail operations has been developed to solve

large-scale problems to improve the quality of the solutions as shown in Section

6.5.2. This algorithm has two main parts where the first obtains the initial solution

using the shortest processing time (SPT) heuristic and the second includes tabu

search steps to improve the current solution.

6.5.2 A New Tabu Search Technique for Sugarcane Rail Cases

Anew algorithm will be used to a solve sugarcane rail transport system problem.

This algorithm includes two main parts.

Algorithm notations k: index of train;k=1...K

s: index of sections;s=1...S

o: index of operations;o=1...O

rk: ready time of train k

pkso: processing time of train k on sections during operation o

dkso: start time of train k on section s during operation o

iter: index of iterations

max_iter: maximum number of iterations

Part1

Set s=1, o=1

For k=1 to K

Select min (rk +pkso)

Next k

For k=1 to K

For s=2 to S

For o=1 to O

Select min (dkso + pkso)

216

Next o

Next s

Next k

Obtain the schedule as initial solution using makespan (currMakespan)

Part2:

If (iter ≤ max_iter) then

Building the neighbourhood structure for the initial solution in part1


Time window, section transition matrix, trains and sidings

capacity

Let (k,k') are the adjacent of two trains which require same section s

by operations o and o' respectively

Swapping (k,k') →(k',k)

Obtain new schedules

Calculate new makespan for all schedules (new Makespan)

Select the best solution

If newMakespan ≤ currMakespan then

Store the new solution as the best solution

Update the tabu list

Else

End if

End if

End

6.6 Metaheuristic Results of Sugarcane Rail Systems

The metaheuristic techniques, SA and TS, were investigated by comparing their

results to the initial solution, SPT, to optimise the efficiency of the sugarcane rail

transport system. The investigation took the form of a numerical experiment where

solutions were obtained for railway networks where the number of sections was

varied (10, 15 and 20) and the number of trains was varied (from 3 to 13) in a 1×10

and 2×13 factorial experiment as shown in Table 6.3. The experiment was

conducted using a PC with a Core 2 CPU chip at 2.39GHz speed and 3.50 GB of

217

RAM. Two parameters were examined to assess the techniques: CPU time and the

Percentage Improvement (PI) of the makespan of the TS and SA compared to the

makespan of the initial solution. The yellow colour highlights the best CPU time of

the metaheuristic techniques, while the green colour represents the best solution

quality.

Table 6.3 also shows that the TS results are better than the SA results for most of the

tested cases and the CPU time when applying the TS technique is shorter than SA.

As a result, TS is faster than SA for obtaining makespan. The Percentage

Improvement of TS and SA (PITS and PISA) is calculated for all tested cases to

show exactly the metaheuristic effects on the initial solution. The best PITS and

PISA values are 13.61% and 13.33% respectively using 20 sections and 8 trains.

Table 6.3: TS and SA results for sugarcane rail transport system under section blocking constraint

Tested cases (sections/trains)

Initial solution

SPT

TS

PITS (%)

TS CPU(s)

SA

PISA (%)

SA CPU(s)

(10/3) 9112 9029 .91 9029 .91 (10/4) 10146 9830 3.11 9830 3.11 (10/5) 13356 3.96 (10/6) 14836 5.79 4.82 (10/7) 15806 8.48 7.32 (10/8) 16840 8.72 158 7.01 155 (10/9) 20050 6.05 169 4.41 200 (10/10) 21530 3.36 180 3.25 212 (15/3) 11683 11600 0.59 7 11600 0.59 75 (15/4) 12563 12154 3.25 33 12154 3.25 124 (15/5) 16464 14376 4.34 74 14376 3.88 159 (15/6) 17685 16230 8.22 143 16472 6.86 (15/7) 18377 13.32 163 11.18 226 (15/8) 19257 17021 11.61 188 10.62 239 (15/9) 23158 8.83 230 6.70 277 (15/10) 24379 7.59 290 6.885 319 (15/11) 25071 5.72 298 3.63 359 (15/12) 25951 24326 6.26 323 6.16 385 (15/13) 29852 27968 6.31 337 6.08 396 (20/3) 13579 0.49 12 0.49 125 (20/4) 14288 13926 2.53 42 2.53 152 (20/5) 18684 3.49 117 3.49 202 (20/6) 19720 8.67 177 7.58 230 (20/7) 20273 12.57 211 12.04 260 (20/8) 20982 18126 13.61 268 18185 13.33 345 (20/9) 25378 12.05 332 6.72 380 (20/10) 26414 7.29 363 7.29 418 (20/11) 26967 5.76 402 5.06 464 (20/12) 27676 25951 6.23 428 5.92 507 (20/13) 32072 8.19 499 7.71 543

218

The worst PITS and PISA values occurred while using a small number of trains and

a significant number of sections such as the 3 train and 20 section case. There is a

strong relationship between the size of the case and improvements to the solution

quality using metaheuristic techniques, where in the case of increasing the number of

trains, the improvement of the initial solution increases until it reaches 7 trains in a

15 section group, and 8 trains in 10 and 20 sections. Following this increase the

improvements then start to decrease. For example, in the 15 section group, PITS and

PISA reached 13.32% and 11.18 respectively with 7 trains, and then decreased.

Figure 6.10 summarises the makespan results for the SA and TS solution techniques

for the numerical experiment. The solution quality (compared to the initial solution)

was generally better for fewer sections and more trains.

.

Figure 6.10: Metaheuristic techniques using makespan for different cases

Figure 6.11 shows that the CPU time increased relatively consistently with the

number of sections and the number of trains. The CPU time of TS is shorter than the

CPU time of SA in all tested cases.

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000

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3)

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11)

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12)

(20/

13)

Mak

espa

n in

Sec

onds

Tested Cases Index

SPT

TS

SA

219

Figure 6.11: CPU time of TS and SA for different cases

6.6.1 TS and SA Results by Changing Number of Trains

The effects of changing the number of trains on a specific number of sections on TS

and SA are investigated in this section. The changing number of trains in the cases of

using 10 and 15 and 20 sections are shown in Figures 6.12, 6.13, and 6.14.

Generally, TS and SA work well with all section group cases and different trains,

particularly in the cases that include 7 and 8 trains. In the case of a small number of

trains, there is no significant difference in the results between the SPT technique and

TS and SA techniques. SPT works well with a small number of trains on the rail

network because the total number of conflict points decreased and then the total

waiting time at any section decreased as well.

Figure 6.12a shows the 10 section group results with a different number of trains. In

this case, TS and SA have the same solution in 2 out of 8 cases, while TS is better

than SA in 6 out of 8 cases. This means that TS and SA have the same solution in

25% of the total number of tested cases in this group. The TS is better than SA in

75% of the tested cases in the same group. The best PITS value is in case (10/8)

where 8 trains work with 10 sections.

0

50

100

150

200

250

300

350

400

450

500

550

600

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3)

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4)

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5)

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6)

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7)

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8)

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9)

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7)

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9)

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10)

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11)

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12)

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13)

CPU

tim

e( in

Sec

onds

)

Tested Cases

TS SA

220

The 15 section group results with a different number of trains are shown in Figure

6.12b. One case out of 11 has the same TS and SA results, while TS works better

than SA in 10 out of 11 cases. As a result, TS can improve around 90.8% of the

group tested cases better than SA, while SA and TS have the same results in solving

9.2% of these cases. The case (15/7) has the best PITS and PISA values, where the

number of trains is 7 with 15 sections.

Figure 6.12a: Makespan of metaheuristics on 10 sections Figure 6.12b: Makespan of metaheuristics on 15 sections

Results for the 20 sections group show in Figure 6.12c that three out of 11 cases

have the same TS and SA results while TS is better than SA in 8 out 11 cases. As a

result, TS is better than SA in 72% of the tested group, while around 28% of the

tested cases in this group have the same results. The best PITS and PISA values in

this group are in the solving case of (20/8), where the number of trains is 8 with 20

sections.

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000

3 4 5 6 7 8 9 10

Mak

espa

n (S

)

Total Number of Trains

SPT TS SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000

3 4 5 6 7 8 9 10 11 12 13

Mak

espa

n (S

)


SPT TS SA

221

Figure 6.12c: Makespan of metaheuristics on 20 sections

6.7 Hybrid Metaheuristic Techniques for Sugarcane Rail Systems

Hybrid techniques depend on integrating heuristic and metaheuristic techniques

consecutively to produce good solutions in a reasonable time. Generally, hybrid

metaheuristic techniques work better than metaheuristic techniques such as SA and

TS individually. The TS technique uses tabu lists, which includes recently visited

solutions to avoid the local optima. TS has a deterministic nature, and for that reason

cannot avoid cycling. On the other hand, SA is a stochastic search technique that

takes a long time, but can stop cycling. Therefore, TS and SA can be integrated to

improve the quality of the solutions and reduce the CPU time of implementing the

case studies.

Many researchers introduced hybrid techniques based on TS and SA to solve real

scheduling problems such as the group scheduling and machining speed selection

problems (Zolfaghari & Liang, 1999), modeling machine loading problems

(Swarnkar & Tiwari, 2004), packing circles into a larger container circle (Zhang &

Deng, 2005), and using the integration of TS and SA for minimisation problems.

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000

3 4 5 6 7 8 9 10 11 12 13

Mak

espa

n (S

)


SPT TS SA

222

This section presents the two types of hybrid techniques of TS/SA and SA/TS to

solve the rail system problem with the specific sugarcane system constraints. These

techniques depend on the integration of the two metaheuristic techniques SA and TS,

and SPT as a heuristic technique. The TS/SA and SA/TS sensitivity analysis is

introduced in this Chapter.

The main advantages of these techniques are easy implementation and good quality

solutions that are better than the individual metaheuristic techniques. The main

disadvantage of these techniques is that CPU time is larger than for the individual

metaheuristic techniques.

6.7.1 Hybrid SA/TS Technique The SA/TS hybrid technique uses the SA solution as an initial solution to apply the

TS technique. The SA solution is stored for comparison with any TS solution at each

iteration to obtain better solutions. The final SA solution is therefore integrated with

all TS individual iterations to improve the SA solution as shown in Figure 6.13. The

three main steps are explained as follows:

Step 1 is to obtain an initial solution using the SPT heuristic.

Step 2 is to obtain the SA solution using the initial solution in step 1.

Step 3 is to use the final solution in step 2 as an initial solution to implement TS as a

hybrid technique. The main aim of step 3 is to improve the quality of the SA

solution.

The stop point in this technique is based on the maximum number of iterations in TS,

however a good hyper technique solution can be achieved without completing all TS

iterations (terminating point) as shown in Figure 6.13.

223

Figure 6.13: Hybrid SA/TS technique

This SA/TS hybrid technique procedure is detailed as follows:


Define a large value of the temperature T=T0

Select the value of the decreasing parameter α; 0<α<1

Set an initial feasible scheduling using SPT technique


Apply simulated annealing technique

Obtain MakeaspanSA

Select initial tabu search parameters

Tabu list ={ }.

Maximum number of iterations, max_iter

Set an initial feasible scheduling simulated annealing technique

While (iter <=max_iter)

Generate a new neighbourhood for new solution

224


Time window, sections transition matrix, trains and sidings capacity

Evaluate the new Makespan SA/TS

∆=new Makespan SA/ TS – Makespan SA

If ∆ ≤ 0 then

Accept the new solution

Store the new solution

Else

Reject the new solution

Update tabu list

End if

iter =iter+1

End while

End .

Figure 6.14 details the main steps of the hybrid SA/TS technique

225

NO

YES

NO YES

Figure 6.14: The detailed hybrid SA/TS technique

Initial conditions A large value for the temperature T=T0. The value of the decreasing parameter α;

0<α<1. Tabu list ={ }

Set number of iterations as a maximum

Iter <=max_iter

∆<=0

Accept the new solution Reject the new solution

An initial solution using SPT technique

Applying simulated annealing technique

End



∆=new Makespan SA/TS– Makespan SA


SA

TS

Update tabu list iter =iter+1

Start

Obtaining MakeaspanSA

Check the feasibility; time window, transition matrix for sections, trains and sidings capacity

SPT

226

6.7.2 Hybrid TS/SA Technique

The TS/SA hybrid technique uses the TS solution as an initial solution and applies

SA to improve the quality of the final solution. Comparing the TS solution as the

initial solution with all SA iterations is the main idea of this technique as shown in

Figure 6.15. The three main steps of the TS/SA hybrid technique are as follows:

Step 1 is to obtain an initial solution using the SPT heuristic.

Step 2 is to obtain the solution of TS using the initial solution in step 1.

Step 3 is to use the final solution in step 2 as an initial solution to implement SA as a

hybrid technique. The main aim of step 3 is to improve the quality of the TS

solution.

The stop point in this technique is based on the maximum number of iterations in

SA; however a good hyper technique solution can be achieved without completing all

SA iterations (terminating point) as shown in Figure 6.15.

Figure 6.15: Hybrid TS/SA technique

227

The hybrid TS/SA technique procedure is detailed as follows:

Select initial tabu search parameters

Set Tabu list ={ }

Set number of iterations as a maximum number, max_iter

Set an initial feasible scheduling by using SPT technique


Apply tabu search technique

Obtain MakespanTS

Set an initial feasible scheduling tabu search technique

Select initial simulated annealing parameters

Define a large value for the temperature T=T0

Select the value of the decreasing parameter α; 0<α<1


Design a neighbourhood for the current solution

Evaluate the new Makespan TS/SA

∆=new Makespan TS/SA – MakespanTS

If ∆<0 then

Accept the hybrid TS/SA solution

Else

If Pr(accepted)=exp(∆/T)> ε then

Accept the hybrid TS/SA solution


Else

Reject the hybrid TS/SA solution

End if

End if


End while

End.

Figure 6.16 shows the main steps of the hybrid TS/SA technique. The main loops of

the SPT, TS and TS to produce the hybrid solution are shown below.

228

NO

YES

NO YES

YES

NO

Figure 6.16: The detailed hybrid TS /SA technique

Initial conditions A large value for the temperature T=T0. The value of the decreasing parameter α;

0<α<1. Tabu list ={ }

Set number of iterations as a maximum number

T≤Tmax

∆<=0



An initial solution using SPT technique

Applying Tabu Search

End



∆=new Makespan TS/SA – Makespan TS


TS

SA

T= αT

Start

Obtaining MakespanTS

Pr(accepted)=exp(∆/T)> ε


SPT

229

6.7.3 Hybrid Techniques Result for Sugarcane Rail Systems

The hybrid techniques, hybrid SA/TS and hybrid TS/SA, were investigated by

comparing their results to those using individual metaheuristic techniques, SA and

TS and also to the initial solution.

The investigation took the form of a numerical experiment where solutions were

obtained for railway networks where the number of sections was varied (15, 20, 25

and 30) and the number of trains was varied (4, 8 and 12) in a 4×3 factorial

experiment. The experiment was conducted using a PC with a Core 2 CPU chip at

2.39GHz speed and 3.50 GB of RAM. Two parameters were examined to assess the

techniques: CPU time and the Percentage Improvement of the makespan of the

hybrid SA/TS and hybrid TS/SA compared to the makespan of the initial solution

(PISA/TS and PITS/SA) The initial solution was obtained using the SPT and this

technique was used to obtain TS and SA results.

Table 6.4 shows that the hybrid techniques work better than the metaheuristic

technique, while the CPU time of all cases is higher than the individual metaheuristic

techniques (SA or TS). The hybrid PISA/TS and PITS/SA indicate the improvement

of makespan for each solution technique, and reveal which technique is more

effective in producing the results.

Table 6.4 shows that the hybrid SA/TS technique works better than hybrid TS/SA

with cases (15/8), (20/8), (20/12), (25/12), (30/8) and (30/12), while hybrid TS/SA

works well with only case (15/12). The different metaheuristic techniques have the

same PISA/TS and PITS/SA values using 4 trains in each group where the

improvement value reduced by increasing the number of sections. The best solution

improvement was in case (20/8) for all solution techniques. All section groups with 8

trains worked well and had better improvements than the case solution in the same

group. Generally, the improvements reduced by increasing the number of sections to

30. The solution quality of the TS technique is better than the SA technique and the

CUT time for TS is still shorter than the SA. The yellow colour is the best CPU time

for different metaheuristic techniques, while the green one is the best percentage

improvement of makespan for different metaheuristic solution techniques.

230

Table 6.4: Comparison of makespan of TS, SA and hybrid techniques

Test

ed c

ases

(s

ectio

ns/tr

ains

)

Initi

al so

lutio

n

SPT

TS SA Hybrid SA/TS Hybrid TS/SA

Mak

espa

n

PITS

(%)

CPU

Mak

espa

n

PISA

(%)

CPU

Mak

espa

n PI

SA/T

S (%

)

CPU

Mak

espa

n PI

TS/S

A (%

)

CPU

(15/4) 12563 12154 3.25 33 12154 3.25 124 12154 3.25 141 12154 3.25 139 (15/8) 19257 17021 11.61 188 10.62 239 16809 12.71 448 17097 11.215 447 (15/12) 25951 24326 6.26 323 6.16 385 7.61 692 23963 7.66 675 (20/4) 14288 13926 2.53 42 2.53 152 13926 2.53 228 13926 2.53 224 (20/8) 20982 18126 13.61 268 18185 13.33 345 17620 16.02 624 17717 15.56 614 (20/12) 27676 25951 6.23 448 5.92 507 25326 8.49 1010 25542 7.71 983 (25/4) 18234 17818 2.28 47 17818 2.28 196 17818 2.28 268 17818 2.28 261 (25/8) 24828 23879 3.82 347 23940 3.58 398 23393 5.78 736 23393 5.78 731 (25/12) 31522 30243 4.1 547 30383 3.61 647 29977 4.9 1240 30356 3.699 1221 (30/4) 23221 22812 1.76 63 22812 1.76 246 22812 1.76 320 22812 1.76 324 (30/8) 29915 28590 4.43 437 28590 4.43 496 28580 4.461 905 28589 4.43 927 (30/12) 36609 35524 2.96% 690 35546 2.9% 769 34929 4.59% 1451 35525 2.96% 1453

231

Figure 6.17 summarises the makespan results of the hybrid techniques (hybrid

SA/TS and hybrid TS/SA) compared to heuristic and metaheuristic techniques. The

solution quality (compared to the initial solution, SA and TS) was generally better

for the two hybrid techniques.

Figure 6.17: Comparison of TS, SA and hybrid techniques using makespan

As stated above, the main disadvantage of hybrid techniques TS/SA and SA/TS is

that they are time consuming when the CPU time is higher than the metaheuristic

techniques, TS and SA. Figure 6.18 shows the CPU time for all cases with different

techniques. Generally, the hybrid SA/TS is more time consuming than the hybrid

TS/SA with all groups except in section group 30 where the hybrid SA/TS CPU time

is less than the hybrid TS/SA CPU time. In all cases the CPU time of TS is shorter

than the CPU time of the SA.

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000

(15/4) (15/8) (15/12) (20/4) (20/8) (20/12) (25/4) (25/8) (25/12) (30/4) (30/8) (30/12)

Mak

espa

n(s)

Tested Cases

SPT TS SA Hybrid SA/TS Hybrid TS/SA

232

Figure 6.18: CPU time of TS, SA and hybrid techniques for different cases

6.7.4 Analysis of Hybrid Techniques

This section focuses on the results of each series of groups with a different number

of trains in order to show the sensitivity analysis of the hybrid techniques. This

analysis is to obtain the makespan with a different number of trains with the same

number of sections. Figure 6.19(a) shows that the hybrid technique results are the

same in 33.33% of the total number of the cases, particularly with a small number of

trains (4 trains). The hybrid SA/TS is better than hybrid TS/SA in 33.33% of the

total cases especially with 8 trains, and hybrid TS/SA is better than hybrid SA/TS in

33.33% of the total cases, particularly with 12 trains. As a result, the hybrid

techniques provide the same results for the 15 section group.

Figure 6.19(b) focuses on the hybrid technique results of the 20 section group with

different numbers of trains. Both hybrid techniques have the same PITS in 33.33%

of the total cases in this section group, while SA/TS is better than TS/SA in 66.66%

of the total cases.

0 100 200 300 400 500 600 700 800 900

1000 1100 1200 1300 1400 1500

(15/4) (15/8) (15/12) (20/4) (20/8) (20/12) (25/4) (25/8) (25/12) (30/4) (30/8) (30/12)

CPU

Tim

e in

Sec

onds

Tested Cases

TS SA Hybrid SA/TS Hybrid TS/SA

233

Figure 6.19a: Makespan of hybrid techniques nn15 sections Figure 6.19b: Makespan of hybrid techniques on 20 sections

The hybrid techniques in the 25 section group are close to each other, where 66.66 %

of the total cases have the same results as the two hybrid techniques, while the

hybrid SA/TS is better than the TS/SA in 33.33% of the total cases, as shown in

Figure 6.19(c).

Figure 6.19(d) shows the results of 30 section group cases. The hybrid SA/TS works

well and better than the TS/SA in 66.66% of the total cases in this group, while both

of them have the same result of 33.33% of the total cases in the same group,

especially with a small number of trains.

Figure 6.19c: Makespan of hybrid techniques on 25 sections Figure 6.19d: Makespan of hybrid techniques on30 sections

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000

4 trains 8 trains 12 trains

Mak

espa

n (s

)


SPT TS SA Hybrid SA/TS HybridTS/SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000


Mak

espa

n(s)


SPT TS SA Hybridr SA/TS HybridTS/SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000


Mak

espa

n (s

)


SPT TS SA Hybridr SA/TS HybridTS/SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000


Mak

espa

n (s

)


SPT TS SA Hybrid SA/TS Hybrid TS/SA

234

6.8 Hyper Metaheuristic Techniques for Sugarcane Rail Transport Systems

Hyper techniques are the complete integration between different metaheuristic

techniques, heuristic techniques, or both, to improve the quality of the solution and

the CPU time. In this research, SPT, as heuristic technique, and SA and TS as

metaheuristic techniques, are integrated interactively where the interaction occurs

between each iteration in TS and each iteration in SA, to improve good solutions in a

reasonable time.

The hybrid approach uses solution techniques consecutively, while the hyper

approaches use solution techniques interactively. The proposed hyper approach

therefore has a greater chance to find the global optimal solution in a reasonable time

than the hybrid approach. This is because at each stage of the hyper iterations, any

local or suboptimal solutions are removed and push the solution to global optimality.

Additionally hybrid approaches are more consuming of CPU time than hyper

approaches for solving large-scale problems.

Many researchers examined and applied hyper techniques with different views to

solve different scheduling problems: Burke et al. (2007) developed hyper techniques

to solve Timetabling Problems; Cuesta et al. (2005) used ant colony techniques to

develop a hyper approach to solve the 2D Bin Packing Problem; Cowling et al.

(2003) developed a Hyper-heuristic approach to schedule a sales summit. Cowling’s

approach depends on using local search techniques combined with some heuristics.

Hyper techniques have two main steps: heuristic selection and move acceptance.

Most hyper techniques are integrating deterministic and non deterministic heuristic

techniques.

This research uses the two hyper techniques of SA/TS and TS/SA which depend on

the TS technique which has a deterministic nature, and SA as a stochastic search

technique. Both techniques are used interactively and consecutively where there is a

complete interaction between them. The sequencing of the metaheuristic techniques

is considered in the two hyper techniques such as using TS and then SA or using SA

and then TS.

235

6.8.1 Hyper SA/TS Technique

The hyper SA/TS technique aims to improve the SA results by integrating TS

iterations with the SA solution at each iteration. The SA/TS technique checks each

solution by SA after each iteration by entering one iteration from the TS technique to

improve it. Figure 6.20 shows the integration of the first solution in SA (after the

first iteration) with the first solution in TS is used to produce the solution two in SA.

Solution two in SA is integrated with solution two in TS to produce solution three in

SA. The stop point of the hyper SA/TS algorithm depends on the number of SA

iterations (the temperature T and parameter α).

Figure 6.20: Hyper SA/TS technique

A good hyper SA/TS technique solution can be achieved without completing all

iterations, unlike the hybrid SA/TS technique where a good solution requires

finishing SA completely. As a result, a hyper SA/TS technique is less time

consuming than hybrid SA/TS technique. The hyper SA/TS algorithm is detailed as

follows:

Select initial simulated annealing parameters

A large value for the temperature T=T0

The value of the decreasing parameter α; 0<α<1

Tabu list = { }

236

Set number of iterations as a maximum number Set an initial feasible scheduling by using SPT technique


While (T is in cooling range or Iter <=max_iter)

Design a neighbourhood for the current solution


Time window, sections transition matrix, trains and sidings capacity

Evaluate the new Makespan SA

Evaluate the new makespan where ∆= Makespan SA –currMakespan

If ∆<0 then




∆1=new Makespan SA/TS– Makespan SA

If ∆1<0 then


Else


End if

Else



Else {

Reject the solution

}

End if

End if

Set iter =iter+1


Update tabu list-includes the best solution to prevent any repetition for that solution

End while End Figure 6.21 shows the main loops and the integration of SPT, SA, and TS techniques

to produce the hyper SA/TS solution.

237

No

YES

NO YES

NO YES

Yes

NO

Figure 6.21: The detailed SA/TS hyper Technique

Start

Initial conditions A large value for the temperature T=T0.

The value of the decreasing parameter α; 0<α<1, Tabu list ={ } Set a maximum number of iterations, Iter=1

T >=0 OR


Evaluate the new MakespanSA

∆= Makespan SA – currMakespan

∆<=0




T= αT; iter =iter+1; Update tabu list

An initial solution by using SPT technique

Evaluate the Makespan for an initial solution

End

Generate a new neighbourhood for the new solution

Evaluate the new MakespanSA/TS

∆1=new Makespan SA/TS– MakespanSA


TS

SA

∆1<=0


Generate a new neighbourhood for the current solution

SPT

238

6.8.2 Hyper TS/SA Technique

The hyper TS/SA technique integrates TS and SA, with the initial solution obtained

by SPT for TS. TS is used at the beginning of each iteration and the solution is

checked at the end of each iteration by applying SA iterations. The stop point of the

hyper TS/SA algorithm depends on the number of TS iterations. Figure 6.22 shows

the integration of the first solution in TS (after the first iteration) with the first

solution in SA used to produce solution two in TS. Solution two in TS is integrated

with solution two in SA to produce solution three in TS. The stop point of the hyper

TS/SA algorithm depends on the number of TS iterations. A good hyper TS/SA

technique solution can be achieved without completing all iterations, unlike the

hybrid TS/SA technique where a good solution requires finishing TS completely. As

a result, a hyper TS/SA technique is less time consuming than the hybrid TS/SA

technique.

Figure 6.22: Hyper TS/SA technique

The main steps of the hyper TS/SA algorithm are explained as follows:

239

Set the initial tabu search parameters

A large value for the temperature T=T0.

The value of the decreasing parameter α; 0<α<1.

Tabu list ={ }.

Maximum number of iterations, max_iter.

Set an initial feasible scheduling by using SPT technique.

Evaluate the initial value of objective function (makespan).

While (iter <=max_iter or T is in cooling range)

Design a neighbourhood for the current solution.


Time window, sections transition matrix, trains and sidings capacity.

Evaluate the new MakespanTS, where ∆= MakespanTS -currMakespan

If ∆<0 then

Accept the new state.

Else

{

Reject the solution.

Apply simulated annealing technique.


Accept the new state.

Else

Reject the solution

End if

}

End if

Update T according to T= αT.

Update tabu list

“includes the best solution to prevent any repeating for that solution”

End while

Stop criteria:

Until ∆ is not significant or there is no improvements in the makespan for

small change in T

Total number of iterations has completed.

End

240

Figure 6.23 shows the main loops and the integration of SPT, TS, and SA techniques

to produce the hyper TS/SA solution.

Yes No

NO YES

YES

Figure 6.23: The detailed hyper TS/SA Technique

Start

Iter <=max_iter

Generate a new neighbor for current solution

Evaluate the new MakespanTS

∆= new MakespanTS – currMakespan

∆<=0




An initial solution by using SPT technique

Evaluate the Makespan for an initial solution

End



∆1=new Makespan TS/SA– Makespan TS


SA

TS

∆1<=0

iter =iter+1; Update tabu list; T= αT

Initial conditions A large value for the temperature T=T0.

The value of the decreasing parameter α; 0<α<1, Tabu list = { } Set a maximum number of iterations, Iter=1


No

Yes

No

SPT

241

6.8.3 Hyper Techniques Result for Sugarcane Rail Systems

The hyper techniques were investigated by comparing their results to those using

individual metaheuristic techniques, SA and TS and also to the initial solution. The

same cases used to investigate the hybrid technique in section 6.7.3 are used to

investigate the hyper techniques in this section. The investigation took the form of a

numerical experiment where solutions were obtained for railway networks where the

number of sections was varied (15, 20, 25 and 30) and the number of trains was

varied (4, 8 and 12) in a 4×3 factorial experiment as shown in Table 6.5. The

experiment was conducted using a PC with a Core 2 CPU chip at 2.39GHz speed

and 3.50 GB of RAM. Two parameters were examined to assess the techniques:

CPU time and the Percentage Improvement (PI) of the makespan of the hyper SA/TS

and the hyper TS/SA compared to the makespan of the initial solution.

Generally, Table 6.5 shows that the hyper TS/SA works better than the hyper SA/TS

technique in most of the cases, where there is an improvement in the quality of the

makespan solutions. The results of cases (20/8), (20/12), (25/8) and (30/8) improved

more through hyper TS/SA than through hyper SA/TS, while the solution for case

(25/12) improved by using hyper SA/TS more than by using hyper TS/SA. Both

hyper techniques have the same results when solving cases (15/4), (15/8), (15/12),

(20/3), (25/4), (30/4), and (30/12). All section groups with 8 trains work well and

displayed greater improvements especially for group section 20. With 4 trains, the

results for both techniques were the same and there were no improvements using

metaheuristic techniques. The yellow colour shows the best CPU time for different

metaheuristic techniques, while the green one shows the best percentage

improvement of makespan for different metaheuristic solution techniques.

242

Table 6.5: Metaheuristic and hyper techniques results for different cases

Test

ed c

ases

(sec

tion/

train

) In

itial

solu

tion

SP

T

TS SA Hyper SA/TS Hyper TS/SA

Mak

espa

n

PITS

CPU

Mak

espa

n

PISA

CPU

Mak

espa

n

PISA

/TS

CPU

Mak

espa

n

PITS

/SA

CPU

(15/4) 12563 12154 3.25 33 12154 3.25 124 12154 3.25 66 12154 3.25 51 (15/8) 19257 17021 11.61 188 10.62 239 16917 12.15 170 16917 12.15 164 (15/12) 25951 24326 6.26 323 6.16 385 24326 6.26 292 24326 6.26 251 (20/4) 14288 13926 2.53 42 2.53 152 13927 2.53 100 13927 2.53 83 (20/8) 20982 18126 13.61 268 18185 13.33 345 17656 15.85 258 17652 15.87 231 (20/12) 27676 25951 6.23 448 5.92 507 25512 7.82 454 25326 8.49 402 (25/4) 18234 17818 2.28 47 17818 2.28 196 17818 2.28 118 17818 2.28 96 (25/8) 24828 23879 3.82 347 23940 3.58 398 23763 4.29 323 23592 4.98 282 (25/12) 31522 30243 4.1 547 30383 3.61 647 29637 5.98 538 30146 4.365 449 (30/4) 23221 22812 1.76 63 22812 1.76 246 22812 1.76 147 22812 1.76 122 (30/8) 29915 28590 4.43 437 28590 4.43 496 28590 4.43 384 28569 4.5 332 (30/12) 36609 35524 2.96% 690 35546 2.9% 769 35192 3.87 628 35192 3.87 544

243

Figure 6.24 summarises the makespan results for the two hyper techniques compared

with heuristic and metaheuristic techniques. The solution quality (compared to the

initial solution, SA and TS) was generally better for the two hybrid techniques.

Table 6.24: Metaheuristic and hyper techniques results of different cases

Figure 6.25 shows the CPU time for all cases using hyper techniques, where both

hyper techniques provide improvements for all metaheuristic CPU time. Hyper

TS/SA CPU time is better than hyper SA/TS and increasing the number of sections

or trains affects the CPU time.

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000

(15/4) (15/8) (15/12) (20/4) (20/8) (20/12) (25/4) (25/8) (25/12) (30/4) (30/8) (30/12)

Mak

espa

n (s

)

Tested Cases

SPT TS SA Hyper SA/TS Hyper TS/SA

244

Figure 6.25: Comparison of CPU time of TS and SA and hyper techniques

6.8.4 Analysis of Hyper Techniques

The sensitivity analysis of the two hyper techniques is shown in this section. The

sensitivity analysis depends on changing the number of trains with the same number

of sections. Figure 6.26a shows the group results for 15 sections with a different

number of trains. Both hyper techniques have the same results with all group cases

and show good improvement in the case of 8 trains.

Hyper TS/SA in Figure 6.26b performs better than hyper SA/TS in 66.66% of the

total cases in the 20 section group. Both TS/SA and SA/TS have the same results in

33.33% of the total cases, especially with a small number of trains - 4. 8 trains with

20 sections provided the best result for both hyper techniques between all cases in all

section groups, where the PISA/TS and PITS/SA results were the highest.

0

100

200

300

400

500

600

700

800

(15/4) (15/8) (15/12) (20/4) (20/8) (20/12) (25/4) (25/8) (25/12) (30/4) (30/8) (30/12)

CPU

Tim

e in

Sec

onds

Tested Cases

TS SA Hyper SA/TS HyperTS/SA

245

Figure 6.26a: Makespan of hyper techniques on 15 sections Figure 6.26b: Makespan hyper techniques on 20 sections

The 25 sections group in Figure 6.26c has the same percentage improvement of the

two hyper techniques in 33.33% of the total cases, especially in the case of 4 trains.

Hyper TS/SA had a better solution than hyper SA/TS in solving 8 trains, while the

hyper SA/TS was better at solving the 12 train case. The two hyper techniques have

nearly the same efficiency in solving the cases of the 30 sections group with the

same quality except in case (30/8) where the percentage improvement of the hyper

TS/SA technique increased by a small value of .07%, as shown in Figure 6.26d.

.

Figure 6.26c: Makespan of hyper techniques on25 sections Figure 6.26d: Makespan of hyper techniques on30 sections

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000


Mak

espa

n (s

)


SPT

TS

SA

Hyper SA/TS

Hyper TS/SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000


Mak

espa

n 10

00


SPT

TS

SA

Hyper SA/TS

Hyper TS/SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000


Mak

espa

n (s

)


SPT

TS

SA

Hyper SA/TS

Hyper TS/SA

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000


Mak

espa

n 10

00


SPT TS SA Hyper SA/TS Hyper TS/SA

246

6.9 Hybrid and Hyper Metaheuristic Techniques Test Cases

Hybrid and hyper techniques were used in this research to solve many cases in the

sugarcane rail transport system. Generally, the quality of the solutions from hybrid

techniques in tested cases such as (15/8), (15/12), (20/8), (20/12), (30/8) and (30/12)

is better than those from the hyper techniques, while the percentage improvement in

the case (25/12) solution using the hyper SA/TS technique is better than that using

hybrid techniques. Cases (15/4), (20/4), (25/4) and (30/4) have the same percentage

improvement value as shown in Figure 6.27.

Figure 6.27: Hybrid and hyper techniques results using makespan for some tested cases

While hybrid techniques have good results in many cases, these techniques are still

time consuming. However, hyper techniques can produce the same results in many

cases, and close to the good results produced by hybrid techniques in other cases, but

done in a reasonable time compared to hybrid techniques. Figure 6.28 shows the

CPU time of hybrid and hyper techniques for the tested cases. The CPU time results

indicate that the hyper techniques are shorter than hybrid techniques in producing the

solution.

0 2000 4000 6000 8000

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000

(15/4) (15/8) (15/12) (20/4) (20/8) (20/12) (25/4) (25/8) (25/12) (30/4) (30/8) (30/12)

Mak

espa

n (s

)

Tested Cases

Hyper SA/TS Hyper TS/SA Hybrid SA/TS Hybrid TS/SA

247

Figure 6.28: CPU time of some tested cases using hybrid and hyper techniques

6.10 Hyper and Hybrid Metaheuristic Technique (TS/SA) and MIP

The hybrid and hyper techniques were investigated by comparing their results to

those using individual metaheuristic techniques, SA and TS and also to the initial

solution, lower bound solution and the solution obtained using constraint

programming techniques.

The comparison of hybrid and hyper metaheuristic techniques and the optimal

solution of mixed integer programming using CPLEX software is shown in Table

6.6. Hybrid and Hyper techniques work with all problems while MIP works well

with small problems. MIP results using CPLEX is time consuming for large-scale

problems, where the CPU increased sharply from 0.66 to 774 seconds by increasing

the number of trains from 4 to 8 on 15 sections. MIP is not applicable for many cases

which include a large number of trains and sections such as (15/12), (20/8), (20/12),

(25/8), (25/12), (30/8) and (30/12), while hybrid and hyper techniques are applicable

for all cases. The best improvement is case (20/8), where the hybrid SA/TS equals

16.02% and Hyper TS/SA is 15.87%. Generally, the improvements reduced by

increasing the number of sections to 30. The yellow colour shows the best CPU time

for different metaheuristic techniques, while the green one shows the best percentage

improvement of makespan of different metaheuristic solution techniques.

0 100 200 300 400 500 600 700 800 900

1000 1100 1200 1300 1400 1500

(15/4) (15/8) (15/12) (20/4) (20/8) (20/12) (25/4) (25/8) (25/12) (30/4) (30/8) (30/12)

CPU

tim

e (s

)

Tested Cases

Hyper SA/TS Hyper TS/SA Hybrid SA/TS Hybrid TS/SA

248

Table 6.6: Metaheuristic, hybrid, hyper, and MIP results of different tested cases

Test

ed c

ases

(S

ectio

ns/T

rain

s)

V

aria

bles

C

onst

rain

ts

Initi

al so

lutio

n (c

onst

ruct

ive

tech

niqu

e SP

T)

MIP-CPLEX Optimal

TS SA Hyper SA/TS Hyper TS/SA Hybrid SA/TS Hybrid TS/SA

Perc

enta

ge

Impr

ovem

ent(

PI )

CPU

PITS

CPU

PISA

CPU

PISA

/TS

CPU

PITS

/SA

CPU

PISA

/TS

CPU

PITS

/SA

CPU

(15/4) 4625 54464 12563 9 0.66 3.25 33 3.25 124 3.25 66 3.25 51 3.25 141 3.25 139 (15/8) 10689 166528 19257 15.3 774 11.61 188 10.62 239 12.15 170 12.15 164 12.71 448 11.215 447 (15/12) 18193 344112 25951 n/a n/a 6.26 323 6.16 385 6.26 292 6.26 251 7.61 692 7.66 675 (20/4) 7765 96624 14288 14.1 1.67 2.53 42 2.53 152 2.53 100 2.53 83 2.53 220 2.53 224 (20/8) 17449 295648 20982 n/a n/a 13.61 268 13.33 345 15.85 258 15.87 231 16.02 624 15.56 614 (20/12) 29053 607632 27676 n/a n/a 6.23 448 5.92 507 7.82 454 8.49 402 8.49 1010 7.71 983 (25/4) 11705 150784 18234 5.99 2.11 2.28 47 2.28 196 2.82 118 2.82 96 2.28 268 2.28 261 (25/8) 25809 467168 24828 n/a n/a 3.82 347 3.58 398 4.29 323 4.98 282 5.78 736 5.78 731 (25/12) 42313 945552 31522 n/a n/a 4.1 547 3.61 647 5.98 538 4.365 449 4.9 1240 4.5 1221 (30/4) 16445 216944 23221 4.9 3.39 1.76 63 1.76 246 1.76 147 1.76 122 1.76 320 1.76 324 (30/8) 35769 671008 29915 n/a n/a 4.43 437 4.43 496 4.43 384 4.5 332 4.461 905 4.43 927 (30/12) 57973 1357872 36609 n/a n/a 2.96 690 2.9 769 3.87 628 3.87 544 4.59 1451 2.96 1453

249

6.11 Study Analysis of Elements of Metaheuristic Techniques

The stochastic element α value in SA and maximum number of iterations in TS have

a significant effect on the final results of any solution technique. Typical values for

cool parameter α in simulated annealing vary between 0.8 and 0.99 (Aarts et al.,

2005). Figure 6.29 shows the results for α values 0.88, 0.90, 0.92 and 0.95. The

makespan for hyper SA/TS using α = 0.90 is better than the makespan values using

other α values in most cases for solution quality.

Figure 6.29: Effect of α value on the hyper SA/TS solution

CPU time for all cases using different α values is described in Figure 6.30.

Figure 6.30: Effect of α value on the hyper SA/TS solution

0

2

4

6

8

10

12

14

16

18

α=0.88 α=0.90 α=0.92 α=0.95

Perc

enta

ge Im

prov

emen

t ( %

)

(15/4)

(15/8)

(15/12)

(20/4)

(20/8)

(20/12)

(25/4)

(25/8)

(25/12)

(30/4)

(30/8)

(30/12)

0

100

200

300

400

500

600

700

800

900

α=0.88 α=0.90 α=0.92 α=0.95

CPU

tim

e (s

econ

d)

(15/4)

(15/8)

(15/12)

(20/4)

(20/8)

(20/12)

(25/4)

(25/8)

(25/12)

(30/4)

(30/8)

(30/12)

250

Figure 6.31 shows no improvement in the hyper TS/SA value after 250 iterations and

it also shows the percentage improvements for different cases where the best

percentage improvement was 15.87% for case study 20/8. By increasing the number

of iterations, the CPU increases sharply for no benefit.

Figure 6.31: Effect of number of iterations on hyper TS/SA solution

Figure 6.32 shows the CPU time of all cases using hyper TS/SA technique using

different number of iterations. CPU time of all cases study is growing up sharply by

increasing number of iterations.

Figure 6.32: Effect of number of iterations on CPU time of hyper TS/SA solution

0

2

4

6

8

10

12

14

16

18 10

20

40

60

80

100

150

200

250

Perc

enta

ge Im

prov

emen

t (%

)

Number of iterations

(15/4)

(15/8)

(15/12)

(20/4)

(20/8)

(20/12)

(25/4)

(25/8)

(25/12)

(30/4)

(30/8)

(30/12)

0

100

200

300

400

500

600

700

10

20

40

60

80

100

150

200

250

CPU

tim

e (s

econ

d)

Number of iterations

(15/4)

(15/8)

(15/12)

(20/4)

(20/8)

(20/12)

(25/4)

(25/8)

(25/12)

(30/4)

(30/8)

(30/12)

251

Figure 6.33 describes the average of makespan using hyper TS/SA technique for

different runs in case study 30/12 as a largest problem in Table 6.6 and example for

explaining how makespan is calculated for the other cases.

Figure 6.33: The average of makespan of hyper TS/SA case study 30/12 with different runs

Minimum value of makespan is 34930 and maximum value is 36334 for 165 runs for

the 30/12. The average value of makespan, 35192, is calculated where the makespan

value determined after 145 runs from Figure 6.33.

35100

35150

35200

35250

35300

35350

35400

35450

35500

5 15 25 35 45 55 65 75 85 95 105 115 125 135 145 155 165

Ave

rage

of m

akes

pan

Number of Runs

Average of Makespan

252

6.12 Inclusion of Delivery and Collection Time Constraints for a Hyper TS/SA

Results

The hyper TS/SA technique is used to solve a complicated real life example to

produce high quality solution in reasonable time. This case includes 30 single

sections, 15 double sections, 15 harvesters, and delivery and collection of 1552 bins

under time constraints. Empty bins are delivered in the outbound direction while full

bins are collected in the inbound direction. The delivery and collection operations

can be executed at the same time when the siding is located at the end of each rail

branch. The delivery of the last visit is used as an inventory for the next day. In the

sugarcane rail system, delaying the last visit time for each harvester can increase the

mill stock of bins.

An optimal (or near optimal) number of train is determined under the system’s

constraints (see Chapter 3 for detail). The best makespan solution is found as 26.3

hr with seven trains and 21 runs (trips). A graphical presentation of the solution is

shown in Figure 6.34. We found infeasible solutions for 4, 5 and 6 trains because

some of the system constraints are not satisfied, for example, four trains only

delivered 1434 bins and six trains only delivered 1417 bins in 24 hours and these

cases could not satisfy the delivering and collecting 1552 bins constraints; and for

Eight trains with 22 runs satisfy the system’s constraints and we find a feasible

solution for this case, however the total operating time and makespan of the system

is found larger than 7 trains with 21 runs case. More than nine trains created

congestions at the network so the maximum number of trains is determined as 9

trains for this case study.

Train operations for delivering and collecting 1552 bins of the practical example are

shown in Table 6.7. As seen in this table, six runs are assigned for first train (T1),

five runs are assigned for the second train (T2), three runs are assigned for third train

(T3), three runs are assigned for fourth train (T4), two runs are assigned for fifth

train (T5), one run is assigned for sixth train (T6) and one run is assigned for seventh

train (T7).

253

Figure 6.34: A sample train schedule of seven trains, 45 sections and 15 harvesters

254

Table 6.7: Train operations for delivering and collecting empty and full bins

Train Run Siding Start Time

Activity

Delivered empty bins

Collected full bins

T1 1 Mill 05:10:18 Start run S2 05:11:38 49 0 S6 05:28:02 49 0 S8 05:44:24 22 0 S6 06:01:48 0 49 S4 06:18:10 0 24 S2 06:38:24 0 27

Mill 06:40:48 End run 2 Mill 06:40:48 Start run

S2 06:58:48 33 0 S4 07:19:12 49 0 S6 07:35:24 33 0 S8 07:51:36 5 0

S10 08:22:12 0 29 S8 08:37:48 0 48 S6 08:54:36 0 23


S4 09:56:24 16 0 S8 10:14:24 47 0

S10 10:30:12 49 0 S10 10:45:29 0 23 S8 11:00:36 0 9 S6 11:16:48 0 23 S4 11:33:45 0 45


S2 12:13:48 14 0 S6 12:35:24 14 0

S10 14:09:36 49 0 S10 14:24:36 0 45 S8 14:40:12 0 38 S6 14:56:24 0 1 S4 15:12:36 0 16


S4 16:13:48 31 0 S8 17:48:36 22 0

S10 18:04:12 27 0 S10 18:19:12 0 49 S8 18:34:48 0 1 S4 18:52:48 0 11 S2 19:12:36 0 39


S10 20:57:36 49 0 S10 21:12:36 0 28 S2 21:36:36 0 30

Mill 21:39:10 End run

255

T2 1 Mill 01:13:48 Start run S12 01:30:36 49 0 Mill 01:40:48 End run

2 Mill 01:50:24 Start run S20 02:15:20 51 0 S30 02:43:12 49 0 S30 03:01:12 0 32 S20 03:29:24 0 20 Mill 03:38:24 End run

3 Mill 04:59:24 Start run S16 05:16:12 49 0 S18 06:19:48 45 0 S20 06:40:48 26 0 S24 07:37:12 0 38 S22 07:59:24 0 24 S20 08:19:12 0 38 Mill 08:58:12 End run


5 Mill 20:06:00 Start run S16 20:17:45 33 0 S26 21:31:12 11 0 S28 21:49:48 9 0 S30 22:10:12 33 0 S30 22:28:12 0 15 S28 22:49:12 0 37 S26 23:07:48 0 48 Mill 23:35:24 End run

T3 1 Mill 8:27:25 Start run S18 08:59:24 20 0 S20 09:19:48 34 0 S22 09:39:36 45 0 S24 10:01:48 21 0 S24 10:16:48 0 4 S22 1038:24 0 24 S20 10:58:12 0 39 S18 11:19:12 0 33 Mill 11:37:12 End run


3 Mill 23:04:48 Start run S26 24:16:12 36 0 S28 24:34:48 36 0 S28 25:19:12 0 8 S26 25:37:48 0 42 S16 26:04:48 0 30 Mill 26:19:48 End run

256

T4 1 Mill 11:06:39 Start run S22 11:48:36 9 0 S24 12:10:48 38 0 S24 12:25:48 0 21 S22 12:48:15 0 27 S20 13:07:12 0 23 S18 13:28:12 0 29 Mill 13:46:48 End run

2 Mill 13:46:48 Start run S14 14:22:48 0 19 Mill 14:28:48 End run

3 Mill 17:16:12 Start run S14 17:52:48 34 0 S14 18:07:48 0 30 Mill 18:13:48 End run

T5 1 Mill 13:15:55 Start run S18 13:47:24 25 0 S20 14:08:24 19 0 S22 14:28:12 36 0 S24 14:49:48 0 30 S22 15:12:35 0 15 S20 15:31:48 0 10 S18 15:52:12 0 28 S16 16:09:36 0 17 Mill 16:10:48 End run

2 Mill 17:15:30 Start run S16 17:33:10 14 0 Mill 17:48:10 End run

T6 1 Mill 15:40:10 Start run S24 17:14:24 39 0 S24 17:29:24 0 5 S16 18:04:12 0 49 Mill 18:24:10 End run

T7 1 Mill 17:37:25 Start run S26 18:45:36 49 0 S28 19:04:12 45 0 S30 19:25:12 14 0 S30 19:43:12 0 49 S28 20:03:36 0 45 S26 20:22:12 0 6 Mill 20:50:24 End run

Total allotment

1552 1552

257

6.13 Conclusion

Metaheuristic techniques are adapted to the sugarcane rail transport system to obtain

near optimal solutions in a reasonable time. Hybrid and hyper techniques improve

the quality of the solution and decrease the CPU time. The two hybrid techniques

proposed are SA/TS and TS/SA both of which depend on integrating the

metaheuristic techniques. The hybrid techniques improved the quality of the solution

while the CPU time of the solution is high. As a result, the two hyper techniques of

SA/TS and TS/SA reduce the CPU time. The results indicated that the hyper TS/SA

produces better solutions than SA/TS with a shorter CPU time. The previous results

using the mixed integer programming technique identified that an optimal solution

can easily be produced for the small large-scale cases in the short time. Mixed

integer programming however is time consuming for large-scale cases. The hybrid

metaheuristic TS/SA on the other hand can solve large-scale cases in a reasonable

time.

The hyper TS/SA technique is applied to solve a more complicated real life example,

which includes Delivery and Collection Time constraints and high quality solution in

reasonable time is obtained.

258

Chapter 7

Conclusions and Future Work

Chapter Outline

7.1 Introduction.........................................................................................................259

7.2 Theoretical Contributions...................................................................................259

7.3 Practical Contributions.......................................................................................262

7.4 Future Work.......................................................................................................263

259

7.1 Introduction

Mathematical models, CP and MIP, have been developed to optimise the sugarcane

rail transport system using different solution techniques. Mathematical modelling

provides many benefits in terms of accuracy of results, prediction of urgent and

future problems and subsequent solutions which are faster than some other

techniques. The integration of the different solution techniques can improve the

efficiency of these techniques in producing high quality solutions and reducing the

CPU time. This research has investigated the impact of integrating CP search

techniques such as Best First Search Technique (BFS), Depth-First Search Strategy

(DFS), Slice Based Search (SBS), Limited Discrepancy Search (LDS), Depth-bound

Discrepancy Search (DDS), Interleaved Depth First Search (IDFS), Standard Search

Strategy (SSS) and Dichotomic Search Strategy (DSS) with MIP and CP models on

the quality of problem solutions. Linear relaxation is integrated with constraint

stratification techniques such as Constraint Propagation and search techniques to

solve the CP and MIP models. Exact solution techniques cannot solve large-scale

problems in a reasonable time. Therefore, metaheuristic techniques such as simulated

annealing (SA) and tabu search (TS) are used to reduce the CPU time of large-scale

problems. Hybrid and hyper metaheuristic techniques are proposed in this research to

improve the solutions of SA and TS. Real time constraints of the sugarcane rail

transport system were developed to optimise the efficiency of the system. The

theoretical and practical contributions of this thesis and future work are summarised

in this Chapter.

7.2 Theoretical Contributions Generic mathematical models and solution techniques were developed to optimise

the efficiency of the sugarcane rail transport system. The theoretical contributions

are:

A blocking parallel-machine job shop scheduling (BPMJSS) technique was

developed to solve the sugarcane rail transport system.

260

CP and MIP Models were developed to address different types of sugarcane

train scheduling problems. Each model includes rail operation scheduling

constraints and sugarcane system constraints.

• Blocking Segment MIP Model of the Sugarcane Rail Problem

• Blocking Segment CP Model of the Sugarcane Rail Problem

• Blocking Section CP Model of Sugarcane Rail System

• Blocking Section MIP Model of Sugarcane Rail System

• Inclusion of the Delivery and Collection Time Constraints to the Models.

The standard and CA algorithm results were examined in CP and MIP.

Different objective functions were developed and investigated in this

research - that is, minimising both makespan and the total waiting time.

These objective functions were applied to different sized sugarcane rail

problems so as to promote the efficiency of CP and MIP models.

Constraint Propagation and search techniques such as DFS, SBS, DDS, BFS

and IDFS were applied to CP and MIP models. The performance of each

technique was examined within the CP and MIP models. The integration of

these techniques as one solver, improved the quality of the CP and MIP

solutions and reduced the CPU time.

The integration of the different search techniques of DFS, SBS, DDS, BFS

and IDFS with the two search strategies SSS and DSS was used in

Optimisation Language Programming (OPL).

Different sizes of the sugarcane rail problem were tested and compared with

different solution techniques to investigate the new CP and MIP models and

the efficiency of solutions techniques.

Algorithms were developed to solve the rail network issues such as

delivering and collecting conflicts, and train conflict.

• Collecting and Delivering Conflict Elimination

• Terminal Segment Conflict Elimination

261

• Intermediate Segment Conflict Elimination

• Algorithms for Solving Train Conflict

• Segment Blocking Determination (SBD)

• Rail Conflicts Elimination (RCE)

• Section Conflict Elimination

• Computing Acceleration (CA) Algorithms

• Segment Elimination Algorithm

• Section Elimination Algorithm

The blocking constraints of the train scheduling were developed to the case

study to satisfy the safety conditions and solve train conflict using :

• Blocking Terminal Segments

• Blocking Intermediate Segments

New neighbourhood techniques were developed based on the section and

train neighbourhoods and were used in the metaheuristic techniques.

The new metaheuristic techniques of SA and TS were adapted to solve large-

scale size problems in the sugarcane rail transport system. These techniques

were applied to the sugarcane rail system and included all constraints of the

sugarcane rail system.

Hybrid metaheuristic techniques were developed to solve the large-scale size

problems. Hybrid SA/TS and hybrid TS/SA were developed to improve the

quality of the solutions when the CPU time is higher than the SA or TS. The

two hybrid techniques used the heuristic technique SPT to obtain an initial

solution. The final solution can be obtained in two ways by improving the

initial solution by:

• SA and then implementing the TS on SA.

• TS and then implementing the SA on TS.

Hyper metaheuristic techniques were developed to improve the quality of the

solution and the CPU time. That is, SPT as a heuristic technique, and SA and

262

TS as metaheuristic techniques were integrated interactively where the

interaction occurs between each iteration in TS and each iteration in SA. Two

hyper techniques developed in this research are:

• Hyper SA/TS technique to improve each solution of each iteration in

SA by integrating iteration of TS technique.

• Hyper TS/SA technique to improve each solution of each iteration in

TS by integrating iteration of the SA technique.

These techniques and the exact solutions were compared to investigate the

accuracy of these techniques. C# language was the design code for these

techniques.

The blocking segment and blocking section constraints were tested and

compared to investigate the sensitivity analyses of the rail utilisation using

different size problems.

7.3 Practical Contributions

The practical contributions are related to the real sugarcane transport system and the

real problems of train conflict, and the collecting and delivering times at the railway

sidings. The many practical contributions are:

The Kalamia Mill was used as a real case study of a complex sugarcane rail

transport system. The results show that the developed solutions can find near

optimal solution to the sugarcane rail transport problem in reasonable time.

ACTSS schedule checker and simulator is used to investigate the MIP model

in real situations (Kalamia mill)

A real time table was developed for the sugarcane rail transport system using

MIP model in Chapter 5. Real life constraints were developed to optimise the

visit times for each siding by each train. A specific scenario was established

to include the constraints of harvester start time and harvester rate at each

siding and satisfy the siding allotment.

Scheduling softwares was developed using graphic interface to optimise the

operation time of the sugarcane rail transport system based on the

263

metaheuristic techniques. Sugarcane rail transport system codes were

developed by:

• OPL and CPLEX codes to examine the CP and MIP models with small

cases.

• C# Language Programming including the section blocking constraints to

examine metaheuristic(SA and TS), hybrid(SA/TS and TS/SA) and

hyper (SA/TS and TS/SA) metaheuristic techniques.

• Practical example was solved including 15 harvesters throughout 45 rail

sections (30 single sections and 15 double sections) and 7 trains. Train

operations were optimised at each siding to satisfy the mill and harvester

requirements

7.4 Future Work

The proposed methodology has a great potential to be applied to other rail systems.

Railways play a vital role in transporting the coal from mines to ports, where the

majority of the coal is transported by rail. The trains used are the longest in the world

with a length of more than 2 kilometres. Each train can carry between 2100 and 8600

net tonnes of coal. The coal mining railway system collects the full wagons of coal

from mines and transports them to the port. The transport sector has a significant

impact on the overall cost of a coal mining production system.

All sections in a coal mining rail network are standard which means the length of

sections is sufficient for the length of trains so blocking constraints can be applied.

The blocking section cannot be useful if the length of the train is greater than the

length of the section. The blocking section MIP model in Chapter 4 can be applied to

a coal mining rail system application.

The blocking section constraints are important to ensure the passing of trains without

accidents or conflict. The mine’s requirements of the empty wagons must be

achieved without interruption in the production operations of coal. The main outputs

of the model are efficient schedules for coal rail transport systems that optimise the

performance of the system.

264

The coal mining rail problem will be a part of future work. A real case study will be

solved using real visit time constraints.

As a summary, this research has the potential to be developed further by:

Extending the metaheuristic techniques to solve dynamic train scheduling

problems using dynamic variables such as unexpected events or accidents in

real life train scheduling problems.

Investigating further techniques to manage the large-scale problems of rail

transport systems.

Developing cane road system models for transporting the empty and full bins

between harvester locations and sidings by trucks. The integration between

the rail models and road models will help in optimising the cane transport

system between harvesters and mill.

Applying a blocking section MIP model to solve the coal mining rail problem

as another application using real large-scale problem.

Testing the MIP model with the coal mining rail problem.

265

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Appendix A

A hyper Branch and Bound Technique for Job Shop

Scheduling Problems While exact techniques give an optimal solution for some problems, they are very

consuming time even in the small cases. The hyper branch and bound technique is

used to solve a job shop problem and investigate that a branch and bound techniques

as exact technique is time consuming to solve small problem such Example 3.1 in

Chapter 3. This example is a job shop problem 3/3/G/Cmax which is strongly

NP –hard problem. The solution steps of the hyper branch and bound technique to

solve Example 3.1 are described as follows:

Level 0 The problem is represented by conjunctive graph as shown in Figure A1.

Figure A1: Conjunctive graph for the 3/3/G/CRmax Rproblem

The initial parameters are calculated as follows:

α = {(1, 1), (1, 4), (2, 7)},

t (α) =min{0+4,0+4,0+3}=3,

i*=2,

α'={(2,7)}.

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3

3

3

4

4

6

6

6

2

4

277

Level 1

Operation (2, 7) scheduled first on machine M2, and there are two disjunctive arcs

will be drawn from this operation, (2, 7) to (2, 2) and (2, 7) to (2, 6), as shown in

Figure A2.


The lower bound is LRBR= L {0, (2, 7), (1, 8), (3, 9), 10} =13

The problem n/1/CRmaxR of the three machines MR1R, MR2R, and MR3 Ris solved in Table A1. To

find better lower bound, the data on each machine is concluded for all jobs as shown

in Table A1 and Figure A3.

Table A1: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 1

MR1 MR2 MR3

JR1 JR2 JR3 JR1 JR2 JR3 JR1 JR2 JR3


ΦR13R=1

3

ΦR13R=1

2

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=1

2

ΦR33R=4


Optimal solution for MR1 Optimal solution for MR2 Optimal solution for MR3


4 8 14 time 3 4 7 10 12time 4 10 14 20time

LR1R=8 LR2R=0 LR3R=8

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3 6

6

6

2

4

3

4

4

3

278

Figure A3: Optimal solutions of M1, M2, and M3 in level 1for n/1/Cmax problem

As a result, the new lower bound can be calculated: LB*=13+max {8, 0, 8} =21.

To find the next operation to be scheduled, operation (2, 7) is deleted from α and the

next operation (1, 8) in the same job 3 is added.

Update α to be:

α = {(1, 1), (1, 4), (1, 8)}

t(α)=min{0+4,0+4,3+6}=4

i*=1, and

α'={(1,1),(1,4),(1,8)} where si*j<t(α).

As a result, stage 1 is branched into three parts. One of branches is the operation (1,

1) scheduled first, another scheduled branch is operation (1, 4) first and the operation

(1, 8) scheduled first as well. Hence, the operation (1, 1) is denoted as level 2(a),

operation (1, 4) scheduled first as level 2(b), and operation (1, 8) scheduled first as

level 2(c) as shown in Figure A4:

Figure A4: The branching procedure at level 1

Level 2

Level 2a

Operation (1, 1) scheduled first on machine M1, therefore two disjunctive arcs are

drawn; (1, 1) to (1, 4) and (1, 1) to (1, 8) as shown in Figure A5.

1, 1 1, 4 1, 8

2, 7

279


The lower bound LRBR= L {0, (1, 1), (1, 4), (3, 5), (2, 6), 10} =16.

The problem n/1/CRmaxR is solved to obtain the better lower bound as shown in Table

A2 and Figure A6.

Table A2: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 2a

MR1 MR2 MR3



ΦR11R=1

6

ΦR12R=1

2

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=8 ΦR33R=4




4 8 14 time 3 4 7 14 16time 7 13 17 23time

LR1R=2 LR2R=0 LR3R=10 Figure A6: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3R in level 2a

The new lower bound is LRBRP

*P= LRB R+ max {LRiR} for i=1 to m.

LRBRP

*P= 16+max {2, 0, 10} =26

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3 6

6

6

4

4

4

2

3

3

280

To find the next operation to be scheduled after operation (1, 1), the operation (1, 1)

is deleted from α and then the operation (2, 2) is added to α where it is the second

operation in job1.

α = {(2, 2), (1, 4), (1, 8)}

t(α)=min{4+3,4+4,4+6}=7 , then i*=2 and α'={(2, 2)},

So in the next branching, the operation (2, 2) is scheduled on M2 at level 3(a) as

shown in Figure A7.

Level 1

2a 2 b 2 c

3a

Figure A7: The branching procedure at level 2a

Level 2b

In this level operation (1, 4) scheduled first on M1 after operation (2, 7), therefore

there are two disjunctive arcs will be drawn; (1, 4) to (1, 1) and (1, 4) to (1, 8) as

shown in Figure A8.

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3

1, 1 1, 4 1, 8

2, 7

2, 2

4

4

4

6

6

6

2

3

3

281

Figure A8: Disjunctive graph for the level 2b

LB = L {0, (1, 4), (1, 1), (2, 2), (3, 3), 10} =17.

Table A3 and Figure A9 are used to obtain the better lower bound.

Table A3: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 2b

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=4 S12=0 S13=4 S21=8 S22=10 S23=0 S31=11 S32=4 S33=10

Φ11=1

3

Φ12=1

7

Φ13=1

0

Φ21=9 Φ22=2 Φ23=1

3

Φ31=6 Φ32=8 Φ33=4

d11=8 d12=4 d13=13 d21=11 d22=17 d23=7 d31=17 d32=15 d33=17


J2 J1 J3 J3 J1 J2 J2 J3 J1

4 8 14 time 3 8 11 13 time 4 10 14 20time

L1=1 L2=0 L3=3

Figure A9: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 2b

The new lower bound is LB*=17+max {1, 0, 3} =20

To find the next operation to be scheduled after operation (1, 4), operation (1, 4) is

deleted from α and operation (3,5 ) is added to α.

α ={(1, 1), (3, 5), (1, 8)}, t(α)=min{4+4,4+6,4+6}=8, then i*=1 and α'={(1,1),(1,8)}.

So, level 2(b) is branched into two parts, One of branches is the operation (1, 1)

scheduled first on M1 and another scheduled branch is operation (1,8) will be

scheduled first on M1. Hence, the operation (1, 1) is denoted as level 3 bi and

operation (1,8) scheduled first as level 3 bii. As shown in Figure A10.

Level 1

2a 2 b 2c

3bi 3bii

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8

282

Figure A10: The branching procedure at level 2b

Level 2c

In this level, operation (1, 8) is scheduled first on M1 after operation (2, 7), therefore

there are two disjunctive arcs will be drawn. (1, 8) to (1, 1) and (1, 8) to (1, 4) as

shown in Figure A11.

Figure A11: Disjunctive graph for the level 2c

The lower bound is LRBR= L {0, (2, 7), (1, 8), (1, 1), (2, 2), (3, 3), 10} =22.

Table A4 and Figure A12 are used to obtain the better lower bound

Table A4: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 2c

MR1 MR2 MR3



ΦR11R=13 ΦR12R=1

2

ΦR13R=1

9

ΦR21R=9 ΦR22R=2 ΦR23R=2

2

ΦR31R=6 ΦR32R=8 ΦR33R=4




3 9 13 17 time 3 13 16 19 21 time 9 13 19 25time

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3

6

6

6

4

4

4

3

3

2

283

L1=3 L2=0 L3=3 Figure A12: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 2c

The new lower bound is LB

*=22+max {3, 0, 3} =25


deleted from α and operation (3, 9) is added to α.

As a results, α = {(1, 1), (1, 4), (3, 9)}, t(α)=min{9+4,9+4,9+4}=13,

then i*=1, 3 and α’= {(1, 1), (1, 4), (3, 9)}.

So, level 2(c) is branched into three parts, One of branches is the operation(1, 1)

scheduled first on M1, another scheduled branch is operation (1, 4) will be scheduled

first on M1 and the operation (3,9) will be scheduled first on M3.. Hence, the

operation (1, 1) is denoted as level 3 ci and operation (1,4) scheduled first as level

3cii and the operation (3,9) will be scheduled at level 3ciii as shown in Figure A13.

Level 1

2a 2b 2c

2ci 3cii 3ciii

Figure A13: The branching procedure at level 2c

To start level 3, Figure A14 shows the graph at level 2 and the all branches which are

evaluated in the level 3.

Level 1

2a 2 b 2 c

1,1 1,4 1,8

2,7

1, 4 1, 1 3, 9

1, 1 1, 4 1, 8

2, 7

284

3a 3bi 3bii 3ci 3cii 3ciii

Figure A14: The complete branching procedure at level 2

Level 3

Level 3a

Operation (2, 2) scheduled on M2 after operation (1, 1), therefore a disjunctive arc is

drawn from (2, 2) to (2, 6) as shown in Figure A15.

Figure A15: Disjunctive graph for the level 3a

The lower bound is LRBR= L {0, (1, 1), (1, 4), (3, 5), (2, 6), 10} =16.


Table A5: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 3a

MR1 MR2 MR3

JR1 JR2 JR3 JR1 JR2 JR3 JR1 JR2 JobR3


ΦR11R=16 ΦR12R=1

2

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=8 ΦR33R=4


1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3 6

6

6

4

4

4

3

3

2

285


J1 J2 J3 J3 J1 J2 J3 J2 J1

4 8 14 time 3 4 7 14 16time 7 13 17 23time

L1=2 L2=0 L3=10 Figure A16: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 3a

The new lower bound LB*=16+max {2,0,10}=26.

To find the next operation to be scheduled, operation (2, 2) will be deleted from α

and then the operation (3,3) will be added to α.

As a result, α will be updated as follows: α ={(3, 3), (1, 4), (1, 8)} , t(α)=min{7+6,

4+4, 4+6}=8, then i*=1, and α'= {(1, 4),(1, 8)},

So, level 3(a) will be branched into two parts, One of branches is the operation (1, 4)

scheduled first on M1, and another scheduled branch is the operation (1, 8) is

scheduled first on M1. Hence, the operation (1, 4) is denoted as level 4ai and

operation (1, 8) scheduled first as level 4aii as shown in Figure A17.

Level 1

2a 2 b 2 c


4ai 4aii Figure A17: The branching procedure at level 3a

Level 3bi In this level, operation (1, 1) scheduled on M2 after operation (1, 4), therefore there

is one disjunctive arc from (1, 1) to (1, 8) as shown in Figure A18.

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

1, 8 1, 4

0 10

J1

J2

4 5 6

M1 M3 M2

1 2 3

M1

M2 M3 3

6

6

4

2

4

286

Figure A18: Disjunctive graph for the level 3bi

The lower bound LRBR= L {0, (1, 4), (1, 1), (1, 8), (3, 9), 10} =18

Table A6 and Figure A19 are used to obtain the better lower bound. Table A6: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 3bi

MR1 MR2 MR3



ΦR11R=14 ΦR12R=1

8

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=8 ΦR33R=4




4 8 14 time 3 8 11 13 time 4 10 11 17 21time


Figure A19: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3 Rin level 3bi

The new lower bound LRBRP

*P=18+max {0, 0, 3} =21



As a results, α = {(2, 2),(3, 5), (1, 8)}, t(α)=min{8+3,4+6,8+6}=10,

then i P

*P=1 and α'={(3,5)}.

So in the next branching, operation (3, 5) is scheduled at level 4bi after operation (1,

1) as shown in Figure A20.

Level 1

7 8

J3

M2 M1 9

M3

2, 7

3

6

4

287

2a 2 b 2 c


4bi

Figure A20: The branching procedure at level 3bi

Level 3bii

Operation (1, 8) scheduled first on M1 after operation (1, 4), therefore one

disjunctive arc (1, 8) to (1, 4) as shown in Figure A21.

Figure A21: Disjunctive graph for the level 3bii

The lower bound = {0, (1, 4), (1, 8), (1, 1), (2, 2), (3, 3), 10}=23


Table A7: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 3bii

MR1 MR2 MR3



ΦR11R=13 ΦR12R=2

3

ΦR13R=1

9

ΦR21R=9 ΦR22R=2 ΦR23R=2

2

ΦR31R=6 ΦR32R=8 ΦR33R=4

1, 1 1, 4 1, 8

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3 6

6

4

4

3

3

2

6

4

288

d11=14 d12=4 d13=10 d21=17 d22=23 d23=4 d31=23 d32=21 d33=23


J2 J3 J1 J3 J1 J2 J2 J3 J1

4 10 14 time 4 14 17 19 time 4 10 14 17 23time

L1=0 L2=0 L3=0 Figure A22: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 3bii

The new lower bound LB*=23+max {0, 0, 0} =23,

To find the next operation to be scheduled after operation (1, 8), operation (1, 8) will

be deleted from α and operation (3, 9) will be added to α.

As a results, α = {(1, 1), (3, 5), (3, 9)} and t (α) =min {10+4,4+6,9+4}=10,

Then i*=3 and α'= {(3, 5), (3, 9)}.

So, level 3bii will be branched into two parts, One of branches is the operation (3, 5)

scheduled first on M3, and another scheduled branch is the operation (3, 9) will be

scheduled first on M3. Hence, the operation (3, 5) is denoted as level 4(bii)a and

operation (1,8) are scheduled first as level (4bii)b as shown in Figure A23.

Level 1

2a 2 b 2 c


(4bii)a (4bii)b

Figure A23: The branching procedure at level 3bii

Level 3ci

Operation (1, 1) scheduled first on M1 after (1, 8), therefore there is one disjunctive

arc; (1, 1) to (1, 4) as shown in Figure A24.

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9

1 2 3

M1

M2 M3 6

3

4

289

Figure A24: Disjunctive graph for the level 3ci

The lower bound LRBR= L {0, (2, 7),(1, 8), (1, 1), (1, 4),(3, 5), (2, 6),10}=25.


Table A8: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 3ci

MR1 MR2 MR3



ΦR11R=16 ΦR12R=1

2

ΦR13R=2

2

ΦR21R=9 ΦR22R=2 ΦR23R=2

5

ΦR31R=6 ΦR32R=8 ΦR33R=4




3 9 13 17 time 4 13 16 23 25 time 9 13 17 23 29time


Figure A25: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3 Rin level 3ci

The new lower bound is LRBRP

*P= 25+max {0, 0, 4} =29.

To find the next operation to be scheduled after operation (1, 1), operation (1, 1) will

be deleted from α and operation (2, 2) will be added to α.

As a results,

α = {(2, 2), (1, 4), (3,9)},

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

6

6

4

4

3

2

290

t(α)=min{13+3,13+4,9+4}=13, then

i*=3,

α'={(3, 9)}, So in the next branching, the operation (3, 9) is scheduled on M3 at level

4(ci) as shown in Figure A26.

Level 1

2a 2 b 2 c


4ci

Figure A26: The branching procedure at level 3ci

Level 3cii

Opration (1, 4) scheduled first on M1 after (1, 8), therefore there is one disjunctive

arc; (1, 4) to (1, 1) as shown in Figure A27.

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3,9

0 10

J1

J2

J3

4 5 6

M1 M3 M2

1 2 3

M1

M2 M3 6

6

4

4

4

3

3

2

291

Figure A27: Disjunctive graph for the level 3cii

LB= L {0,(2, 7), (1, 8),(1, 4), (1, 1), (2, 2), (3, 3), 10}=26.


Table A9: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 3cii

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=13 S12=9 S13=3 S21=17 S22=19 S23=0 S31=20 S32=13 S33=9

Φ11=13 Φ12=1

7

Φ13=2

3

Φ21=9 Φ22=2 Φ23=2

6

Φ31=6 Φ32=8 Φ33=4

d11=17 d12=13 d13=9 d21=20 d22=26 d23=3 d31=26 d32=24 d33=26


J3 J2 J1 J3 J1 J2 J3 J2 J1

3 9 13 17 time 3 17 20 22 time 9 13 19 20 26time

L1=0 L2=0 L3=0


The new lower bound is LB*=26+max {0, 0, 0} =26,



As a results, α = {(1, 1), (3, 5), (3, 9)} and t(α) =min{13+4,13+6,9+4}=13,

then i*=3 and α'={(3,5)}.

So in the next branching, the operation (3, 5) will be scheduled on M3 at level 4(cii)

as shown in Figure A29.

7 8

M2 M1 9

M3

6

292

Level 1

2a 2 b 2 c


4cii

Figure A29: The branching procedure at level 3cii

Level 3ciii

In this level, operation (3, 9) scheduled first after operation (1, 8), therefore two

disjunctive arcs will be drawn as shown in Figure A30.

Figure A30: Disjunctive graph for the level 3ciii

LRBR= L {0, (2, 7), (1, 8), (1, 1), (2, 2), (3, 3), 10} =22.


Table A10: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 3ciii

MR1 MR2 MR3



1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5

0

7 8

10

J1

J2

J3

4 5 6

M1 M3 M2

M2 M1 9

M3

1 2 3

M1

M2 M3 6

6

6

4

4

4

3

3

2

293

Φ11=13 Φ12=1

2

Φ13=1

9

Φ21=9 Φ22=2 Φ23=2

2

Φ31=6 Φ32=8 Φ33=2

2

d11=13 d12=14 d13=9 d21=16 d22=22 d23=3 d31=22 d32=20 d33=3


J3 J2 J1 J3 J1 J2 J3 J2 J1

3 9 13 17 time 3 13 16 19 21time 9 13 19 25time

L1=3 L2=0 L3=3 Figure A31: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 3ciii


By the same way level 4 is completed, Figure A32 shows the graph at level 4 and the

all branches which will be branched from level 3.

Level 0

Level 1

2a 2 b 2 c


4ai 4aii 4bi (4bii)a (4bii)b 4ci 4cii

Figure A32: The complete branching procedure at level4

Level 4

Level 4ai

Operation (1, 4) scheduled on machine 1 after operation (2, 2). Therefore it fixes a

adjunctive arc (1, 4) to (1, 8) as shown in Figure A33.

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

24

24 21 23 24 29

29

1 2 3

M1

M2 M3 3

6

4

294

Figure A33: Disjunctive graph for the level 4ai

The lower bound LRBR= L {0, (1, 1), (1, 4), (1, 8), (3, 9), 10}=18

Table A11 and Figure A34 are used to obtain the better lower bound. Table A11: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 4ai

MR1 MR2 MR3



ΦR11R=18 ΦR12R=1

4

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=8 ΦR33R=4




4 8 14 time 3 4 7 14 16 time 8 14 18 24time


Figure A34: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3 Rin level 4ai


*P=18+max {0, 0, 6} =24.

To find the next operation to be scheduled, operation (1, 4) is deleted and operation

(3, 5) is added to α.

α ={(3, 3), (3, 5), (1, 8)} and t(α)=min{7+6,8+6,8+6}=13,

then i P

*P=3 and α'={(3,3),(3,5)}.

So in the next branching, the operation (3, 3) and (3, 5) is scheduled on MR3R at level

5aia and 5aib. The branching at level 4ai is shown in Figure A35.

0

8

10

J1

J2

J3

4 5 6

M1 M3 M2

7

M2

9 M3

4

6

2

3

6

4

M1

295

Level 0

Level 1

2a 2 b 2 c



Figure A35: The complete branching procedure at level4ai

Level 4aii Operation (1, 8) scheduled on machine 1 after operation (2, 2), therefore it fixes a

disjunctive arc: (1, 8) to (1, 4) as shown in Figure A36.

Figure A36: Disjunctive graph for the level 4aii

The lower bound LRBR= L {0, (1, 1), (1, 8), (1, 4), (3, 5), (2, 6), 10} =22


Table A12: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 4aii

MR1 MR2 MR3


1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

24

3, 3 3, 5

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

296

S11=0 S12=10 S13=4 S21=4 S22=20 S23=0 S31=7 S32=14 S33=10

Φ11=22 Φ12=1

2

Φ13=1

8

Φ21=9 Φ22=2 Φ23=2

1

Φ31=6 Φ32=8 Φ33=4

d11=4 d12=14 d13=10 d21=16 d22=22 d23=4 d31=22 d32=20 d33=22


J1 J3 J2 J3 J1 J2 J1 J2 J3

4 10 14 time 3 4 7 20 22 time 7 13 14 20 24time

L1=0 L2=0 L3=2 Figure A37: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 4aii

The new lower bound LB

*=22+max {0, 0, 2} =24.



α ={(3, 3), (1, 4), (3, 9)} and t(α)=min{7+6, 9+12, 10+4}=13, then i*=3 and

α'={(3, 3), (3, 9)}.

So in the next branching, the operation (3, 3) and (3, 9) will be scheduled on M3 at

level 5aiia and 5aiib. The branching at level 4aii can be shown in Figure A38.

Level 0

Level 1

2a 2 b 2 c



Figure A38: The complete branching procedure at level4aii

Level 4bi

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

24

3, 3 3, 9

297

In this level, operation (3, 5) is scheduled on the machines after operation (1, 1),

therefore it fixes two arcs : (3,5) to (3,3) and (3,5) to (3,9) as shown in Figure A39.

Figure A39: Disjunctive graph for the level 4bi

The lower bound is LRBR= L {0, (1, 4), (1, 1), (1, 8), (3, 9), 10} =18.

Table A13 and Figure A40 are used to obtain the better lower bound. Table A13: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 4bi

MR1 MR2 MR3



ΦR11R=14 ΦR12R=1

8

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=1

2

ΦR33R=4




4 8 14 time 3 8 11 13 time 4 10 11 17 21time


Figure A40: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3 Rin level 4bi


*P=18+max {0, 0, 3} =21.

To find the next operation to be scheduled, we delete operation (3, 5) and add (2, 6)

to α.

α = {(2, 2), (2, 6), (1, 8)} and t(α)=min{8+3,10+2,8+6}=11,

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

298

then i*=2 and α'={(2, 2),(2, 6)}.

So in the next branching, the operation (2, 2) and (2, 6) will be scheduled on M2 at

level 5bia and 5bib. The branching at level 4bii can be shown in Figure A41.

Level 0

Level 1

2a 2 b 2 c



Figure A41: The complete branching procedure at level 4bi

Level 4 biia

In this level, operation (3, 5) is scheduled first after operation (1, 8) and two arcs are

fixed as shown in Figure A42:

Figure A42: Disjunctive graph for the level 4biia

The lower bound LRBR= L {0, (1, 4), (1, 8), (1, 1), (2, 2),(3, 3), 10}=23.

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

24

2, 2 2,6

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

299


Table A14: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 4biia

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=10 S12=0 S13=4 S21=14 S22=10 S23=0 S31=17 S32=4 S33=10

Φ11=13 Φ12=2

3

Φ13=1

9

Φ21=9 Φ22=2 Φ23=2

2

Φ31=6 Φ32=1

2

Φ33=4

d11=14 d12=4 d13=10 d21=17 d22=23 d23=4 d31=23 d32=17 d33=23


J2 J3 J1 J3 J1 J2 J2 J3 J1

4 10 14 time 3 14 17 19 time 4 10 14 17 23time

L1=0 L2=0 L3=0

Figure A43: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 4biia




α = {(1, 1), (2, 6), (3, 9)}, t(α)=min{0+4, 10+2, 10+4}=12, then i*=2,

α'={(2, 6)}, So in the next branching, the operation (2, 6) will be scheduled on M2 at

level 5biia after operation (3, 5). The branching at level 4biia is shown in Figure

A44.

Level 0

Level 1

2a 2 b 2 c



1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

23 26 29

300

Figure A44: The complete branching procedure at level 4biia

Level 4biib

In this level, operation (3, 9) is scheduled after operation (1, 8), and two arcs are

fixed as shown in Figure A45, (3, 9) to (3, 5) and (3, 9) to (3, 3),

Figure A45: Disjunctive graph for the level 4biib

The lower bound LRBR = L {0, (1, 4), (1, 8), (1, 1), (2, 2), (3, 3), 10} =23


Table A15: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 4biib

MR1 MR2 MR3



ΦR11R=13 ΦR12R=2

3

ΦR13R=1

9

ΦR21R=9 ΦR22R=2 ΦR23R=2

2

ΦR31R=6 ΦR32R=8 ΦR33R=1

2




4 10 14 time 3 14 17 19 time 10 14 20 26time

LR1R=0 LR2R=0 LR3R=3 Figure A46: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3 Rin level 4biib

2,6

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

301


The branching terminated no operation after (3, 9).

Level 4ci In this level operation (3, 9) is scheduled after operation (1, 1), and two arcs are

fixed: (3, 9) to (3, 5) and (3, 9) to (3, 3) as shown in Figure A47.

Figure A47: Disjunctive graph for the level 4ci

The lower bound LRBR= L {0, (2,7),(1,8),(1,1),(1,4),(3,5),(2,6),10}=25

Table A16 and Figure A48 are used to obtain the better makespan.

Table A16: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 4ci

MR1 MR2 MR3



ΦR11R=16 ΦR12R=1

2

ΦR13R=1

9

ΦR21R=9 ΦR22R=2 ΦR23R=2

5

ΦR31R=6 ΦR32R=8 ΦR33R=1

2




3 9 13 17 time 3 13 16 23 25 time 9 13 17 23 29time


0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

302

Figure A48: Solving n/1/Cmax problem for the three machines M1,M2, and M3 in level 4ci


The branching terminated no operation after (3,9).

Level 4cii

The operation (3, 5) will be scheduled after (1, 4). It fixes two arcs: (3, 5) to (3, 3)

and (3, 5) to (3, 9) as shown in Figure A49.

Figure A49: Disjunctive graph for the level 4cii

The lower bound is LRBR=L {0, (2, 7), (1, 8), (1, 4), (1, 1), (2, 2), (3, 3), 10}=26


Table A17: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 4cii

MR1 MR2 MR3



ΦR11R=13 ΦR12R=1

7

ΦR13R=2

3

ΦR21R=9 ΦR22R=2 ΦR23R=2

6

ΦR31R=6 ΦR32R=1

2

ΦR33R=4




3 9 13 17 time 3 17 20 22 time 13 19 23 29time

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

303

L1=0 L2=0 L3=3



*=26+max {0, 0, 3} =29.

To obtain the next operation after operation (3, 5), operation (3, 5) is deleted from α

and added the next one (2, 6) where:

α = {(1, 1),(2, 6), (3, 9)} and t(α)=min{13+4, 19+2, 19+4}=17,

then i*=1 and α'={(1, 1)}.

So in the next branching, the operation (1, 1) has to be scheduled on M1 at level

5(cii) but as seen in Figure A35, this operation has not any disjunctive arc to fix .

Hence, the next operation (2, 6) in t(α) is selected because its value is less than (3, 9).

So, operation (2, 6) is scheduled after operation (3, 5) and α'= {(2, 6)} at level 5(cii)

as shown in Figure A51.

Level 0

Level 1

2a 2 b 2 c



1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

23

2,6

26 29

24

3, 3 3, 5

24

3, 3 3,9 2,6

24

2, 2 2,6

29

304

Figure A51: The complete branching procedure at level 4cii

Level 5 Level 5aia

In Figure A52, Operation (3, 3) is scheduled on M3 after operation (1, 4). It fixes two

arcs:

(3, 3) to (3, 5) and (3, 3) to (3, 9)

Figure A52: Disjunctive graph for the level 5aia

LRBR= L {0, (1, 1), (2, 2), (3, 3), (3, 5), (2, 6), 10} =21


Table A18: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 5aia

MR1 MR2 MR3



ΦR11R=21 ΦR12R=1

4

ΦR13R=1

0

ΦR21R=1

7

ΦR22R=2 ΦR23R=2

0

ΦR31R=1

4

ΦR32R=8 ΦR33R=4



0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

305

J1 J2 J3 J3 J1 J2 J2 J3 J1

4 8 14 time 3 4 7 19 21 time 7 13 19 23time

L1=0 L2=0 L3=2

Figure A53: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5aia


Stop where no branching.

Level 5aib Operation (3, 5) scheduled on M3 after operation (1, 4). It fixes two disjunctive arcs:

(3, 5) to (3, 3) and (3, 5) to (3, 9) as shown in Figure A54.

Figure A54: Disjunctive graph for the level 5aib

LRBR= L {0, (1, 1), (1, 4), (3, 5), (3, 3), 10} =20

Table A19 and Figure A55 are used to find the better lower bound.

Table A19: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 5aib

MR1 MR2 MR3



ΦR11R=20 ΦR12R=1

6

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=1

2

ΦR33R=4

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

306

d11=4 d12=8 d13=16 d21=14 d22=20 d23=10 d31=20 d32=14 d33=20


J1 J2 J3 J3 J1 J2 J2 J3 J1

4 8 14 time 3 4 7 14 16 time 8 14 18 24time

L1=0 L2=0 L3=4 Figure A55: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5aib


To obtain the next operation after operation (3,5), we delete (3,5) from α and add

(the next one (2,6) where : α ={(3, 3), (2, 6) ,(1, 8)} and

t(α)=min{10+6,10+2,4+6}=10,

Then i*=1, but Figure A36 shows the operations (1,8) and (2,6) haven’t any new

disjunctive arcs to branch. Then operation (3, 3) is selected to schedule after (3,5)

and α'={(3,3)}. So in the next branching, the operation (3,3) will be scheduled on

M1 after (3,5) at level 6(aii).

Level 5aiia

The operation (3, 3) will be scheduled first after operation (1, 8). It fixes two

disjunctive arcs: (3, 3) to (3, 5) and (3, 3) to (3, 9) as shown in Figure A56.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

307

Figure A56: Disjunctive graph for the level 5aiia

The lower bound LB= L {0, (1, 1), (1, 8), (1, 4), (3, 5), (2, 6), 10} =22

Table A20 and Figure A57 are used to obtain better lower bound. Table A20: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5aiia

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=0 S12=10 S13=4 S21=4 S22=20 S23=0 S31=7 S32=14 S33=13

Φ11=22 Φ12=1

2

Φ13=1

8

Φ21=1

7

Φ22=2 Φ23=2

1

Φ31=1

4

Φ32=8 Φ33=4

d11=4 d12=14 d13=10 d21=8 d22=22 d23=4 d31=14 d32=20 d33=22


J1 J3 J2 J3 J1 J2 J1 J2 J3

4 10 14 time 3 4 7 20 22 time 7 13 14 20 24time

L1=0 L2=0 L3=2 Figure A57: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5aiia


Stop at node (3, 3)

Level 5aiib

The operation (3, 9) will be scheduled after operation (1, 8). It fixes two disjunctive

arcs: (3, 9) to (3, 3) and (3, 9) to (3, 5) as shown in Figure A58.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

308

Figure A58: Disjunctive graph for the level 5aiib

The lower bound LB= L {0, (1, 1), (1, 8), (3, 9), (3, 5), (2, 6), 10} =22


Table A21: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5aiib

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=0 S12=10 S13=4 S21=4 S22=20 S23=0 S31=14 S32=14 S33=10

Φ11=22 Φ12=1

2

Φ13=1

8

Φ21=9 Φ22=2 Φ23=2

1

Φ31=6 Φ32=8 Φ33=1

2

d11=4 d12=14 d13=10 d21=16 d22=22 d23=4 d31=20 d32=20 d33=14


J1 J3 J2 J3 J1 J2 J3 J2 J1

4 10 14 time 3 4 7 20 22 time 10 14 20 26time

L1=0 L2=0 L3=6 Figure A59: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5aiib


Stop at node (3, 9).

The level 5bia

The operation (2, 2) will be scheduled first on machine M2 after operation (3, 5).

Therefore it fixes one disjunctive arc (2, 2) to (2, 6) as shown in Figure A60:

0

1 2 3

10

J1

J2

M1

4 5 6

M2 M3

M1 M3 M2

3

6

4

6

2

4

309

Figure A60: Disjunctive graph for the level 5bia

The lower bound LRBR= L {0, (1, 4), (1, 1), (1, 8), (3, 9), 10} =18


Table A22: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 5bia

MR1 MR2 MR3



ΦR11R=14 ΦR12R=1

8

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

3

ΦR31R=6 ΦR32R=1

2

ΦR33R=4




4 8 14 time 3 8 11 13 time 4 10 11 17 21time


Figure A61: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3 Rin level 5bia


*P=18+max {0, 0, 3} =21.

To obtain the next operation after operation (2, 2), (2, 2) is deleted from α and added

the next one (3, 3) where,

α = {(3, 3), (2, 6), (1, 8)},

t(α)=min{11+6, 11+2, 8+6}=13, then i P

*P=2,

α'={(2, 6)}, So in the next branching, the operation (2, 6) has to schedule on MR2R

after operation(2, 2) at level 6(bia).

8

J3

7

M2

9 M3

3

6

4

M1

310

But as we see, the operations (1, 8) and (2, 6) already they finished all branches. This

means no new branching or disjunctive arcs, as a result, operation (3, 3) will be

scheduled on machine 3 after operation(2, 2) at level 6bia instead of (2, 6).

i*=3,

α'={(3, 3)}.

Level 5bib

The operation (2, 6) will be scheduled first after operation (3, 5), there it fixes one

adjunctive arc: (2, 6) to (2, 2) as shown in Figure A62.

Figure A62: Disjunctive graph for the level 5bib



Table A23: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 5bib

MR1 MR2 MR3



ΦR11R=14 ΦR12R=1

8

ΦR13R=1

0

ΦR21R=9 ΦR22R=2 ΦR23R=1

4

ΦR31R=6 ΦR32R=1

7

ΦR33R=4


0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

311


J2 J1 J3 J3 J1 J2 J2 J3 J1

4 8 14 time 3 12 15 17 time 4 10 14 18 24time

L1=0 L2=0 L3=3

Figure A63: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5bib


Stop the branching here.

Level 5biia

The operation (2, 6) will be scheduled after operation (3, 5), it fixes one disjunctive

arc: (2, 6) to (2, 2) as shown in Figure A64.

Figure A64: Disjunctive graph for the level 5biia

The lower bound LRBR = L {0, (1, 4), (1, 8), (1, 1), (2, 2), (3, 3), 10} =23.

Table A24 and Figure A65 are used to obtain the better lower bound. Table A24: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 5biia

MR1 MR2 MR3



ΦR11R=13 ΦR12R=2

3

ΦR13R=1

9

ΦR21R=9 ΦR22R=1

1

ΦR23R=2

2

ΦR31R=6 ΦR32R=1

7

ΦR33R=4

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

312

d11=14 d12=4 d13=10 d21=17 d22=14 d23=4 d31=23 d32=12 d33=23


J2 J3 J1 J3 J2 J1 J2 J3 J1

4 10 14 time 3 10 12 14 17 time 4 10 14 17 23time

L1=0 L2=0 L3=0

Figure A65: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 5biia


The branching terminated here, no branching.

Level 5cii The operation (2, 6) will be scheduled after operation (3, 5).

There is one disjunctive arc: (2, 6) to (2, 2) as shown in Figure A66

Figure A66. Disjunctive graph for the level 5cii

The lower bound is LRBR= L {0,(2, 7), (1, 8), (1, 4), (3, 5), (2, 6), (2, 2), (3, 3), 10}=30.


Table A25: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 5cii

MR1 MR2 MR3


0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

313

S11=13 S12=9 S13=3 S21=21 S22=19 S23=0 S31=24 S32=13 S33=19

Φ11=13 Φ12=2

1

Φ13=2

7

Φ21=9 Φ22=2 Φ23=3

0

Φ31=6 Φ32=1

7

Φ33=4

d11=21 d12=13 d13=9 d21=24 d22=30 d23=3 d31=30 d32=19 d33=30


J3 J1 J2 J3 J1 J2 J2 J3 J1

3 9 13 17 time 3 21 24 26 time 13 19 23 24 30time

L1=0 L2=0 L3=0



After operation (2, 6) the branching terminated.

Level 6aii

The operation (3, 3) is scheduled after operation (3, 5). It fixes one arc: (3, 3) to (3,

9) as shown in Figure A68.

Figure A68. Disjunctive graph for the level 6aii



Table A26: Solving n/1/CRmaxR problem for the three machines MR1R, MR2R, and MR3RR Rin level 6aii

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

314

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=0 S12=4 S13=4 S21=4 S22=14 S23=0 S31=14 S32=8 S33=20

Φ11=24 Φ12=2

0

Φ13=1

0

Φ21=1

3

Φ22=2 Φ23=1

6

Φ31=1

0

Φ32=1

6

Φ33=4

d11=4 d12=8 d13=20 d21=14 d22=24 d23=11 d31=20 d32=14 d33=24


J1 J2 J3 J3 J1 J2 J2 J1 J3

4 8 14 time 3 4 7 14 16 time 8 14 20 24time

L1=0 L2=0 L3=0 Figure A69: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 6aii


*=24+max {0, 0, 0} =24.

Stop the branching here where no disjunctive arcs more to fix.

Level 6bia

The operation (3, 3) will be scheduled after operation (2, 2). It fixes one a disjunctive

arc (3, 3) to (3, 9) as shown in Figure A70.

Figure A70: Disjunctive graph for the level 6bia

The lower bound LRBR=L {0, (1, 4), (1, 1), (2, 2), (3, 3), (3, 9), 10}=21


0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

315

Table A27: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 6bia

M1 M2 M3

J1 J2 J3 J1 J2 J3 J1 J2 J3

S11=4 S12=0 S13=8 S21=8 S22=11 S23=0 S31=11 S32=4 S33=17

Φ11=17 Φ12=2

1

Φ13=1

0

Φ21=1

3

Φ22=2 Φ23=1

6

Φ31=1

0

Φ32=1

6

Φ33=4

d11=8 d12=4 d13=14 d21=11 d22=21 d23=8 d31=17 d32=11 d33=21


J2 J1 J3 J3 J1 J2 J2 J1 J3

4 8 14 time 3 8 11 13 time 4 10 11 17 21time

L1=0 L2=0 L3=0

Figure A71: Solving n/1/Cmax problem for the three machines M1, M2, and M3 in level 6bia


The branching terminated at node (3, 3).

After building the complete search tree, Figure A72, the nodes at very bottom of the

tree correspond to all the active schedules, where the complete selection at the very

low level at the tree.

316

Level 0

Level 1

2a 2 b 2 c



5aia 5aib 5aiia 5aiib 5bia 5bib 5biia 5cii

The optimal solution Figure A72: The complete branching procedure at level 6bia

From the previous levels of the solution of Example 3.1, the optimal solution is

makespan=21 under critical path: {0, (1, 4), (1, 1), (2, 2), (3, 3), (3, 9), 10}.

1, 1 1, 4 1, 8

2, 7

1, 1 1, 8 1, 1 1, 4 3, 9 2, 2

3, 5 3, 9 3, 5 3, 9 3, 9 3, 5 3, 5 1, 8 1, 4

21

26 20 25

26 21 23 26 25 25

23

2, 6

26 29

24

3, 3 3, 5

24

3, 3 3, 9 2, 6

24

2, 2 2, 6

29

23 24 24 28 21 24 23 30

3,3 3,3 21 21

316

Simulated Annealing for Job Shop Scheduling Problems

Example 2.1 is solved using simulated annealing to explain how this technique can

be applied to a job shop problem. The main steps are described as follows:

Step 0

Figure B1 shows that the initial schedule is {0→7→8→4→5→9→6→10} with the

initial makespan 29. Initial conditions of simulated annealing algorithms are: T=100,

α=.95 and ε=.98. The solution by simulated annealing is shown by the next steps:

Figure B1: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example

Makespan=29

CRmaxR(0)=29 and T=100

STEP 1

In this step, the new solution is generated using the neighbourhood structure. The

disjunctive graph is changed from (1, 8)→(1, 4) to (1, 4) →(1, 8) in the critical path.

The new disjunctive graph is shown in Figure B2. The new temperature, new

makespan and ∆ are calculated as following:

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

317

Figure B2. Disjunctive graph of an initial solution for the 3 jobs and 3 machines example

Makespan=23 T=αT = .95*100= 95, CRmaxR(1)=23

= CRmaxR(1)- CRmaxR(0)=23-29=-6<0 then the new schedule is accepted.

STEP 2

The new solution is generated using the neighbourhood structure. The disjunctive

graph is changed from (1, 8)→(1, 1) to (1, 1) →(1, 8) in the critical path. The new

disjunctive graph is shown in Figure B3. The new temperature, new makespan and ∆

are calculated as following:


Makespan=24

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

M1

318

T=αT = .95*95= 90.25

Cmax(2)=24

Δ= Cmax(2)- Cmax(1)=24-23=1>0 then

Pr=exp(-∆∕T)= exp(-1∕90.25)=0.988>ε

The new schedule is accepted.

Step3

The new solution is generated in Step 3 as well by changing the neighbourhood on the critical path as shown in Figure B4.

(3, 9)→(3, 3) to (3, 3) →(3, 9)


Makespan=21

The new temperature, new makespan and ∆ are calculated as follows:

T=αT

.95*90.25= 85.73

CRmaxR(3)=24

= CRmaxR(3)- CRmaxR(2)=21-24=-3then

The new schedule is accepted

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

319

Step 4


Makespan=24

The new solution is generated by changing the neighbourhood on the critical path as shown in Figure B5.

(1, 4)→(1,1) to (1, 1) →(1, 4)

.95*85.73= 81.44

CRmaxR(4)=24

= CRmaxR(4)- CRmaxR(3)= 3 then Pr=exp(-∆∕T)= exp(-3∕81.44)=0.961<ε

The new schedule will be rejected. The solution at step three is used again to produce new solution Figure B5 shows that all disjunctive arcs on the critical path are used before in the previous steps. As a result, we stop at this stage because no more changes can be implemented. Table B1 is deduced using the previous steps.

Table B1: simulated annealing result for Example 2.1

step Makespan CRmax

∆ Temperature T

Boltzmann probability

Decision Best value found

0 - 100 - - 21 1 - 95 - accepted 2 90.25 0.988 accepted 3 - 85.73 - accepted 4 81.44 0.961 rejected

The Percentage Improvement of SA can be calculated as:

PISA= (initial solution of CRmaxR -SA value)/initial solution of CRmax

PISA= ((29-21)/29)*100 = 8/29= 27.5%

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

320

Appendix Ｃ

Tabu Search for Job Shop Scheduling Problems

The numerical Example 2.1 in Chapter 2 is solved by tabu search to describe how

tabu search can be applied to a job shop scheduling problem.

An initial solution will be supposed for minimising the makespan as an objective

function. Neighbourhood structure will be used to generate new solutions.

Iteration 0 According to the feasible solution in Figure C1, the sequence of operations on the

machines as following:

Figure C1: Disjunctive graph of an initial solution for the 3 jobs and 3 machines example

Makespan=29

MR1R OR8R→OR4R→ OR1


MR3R OR5R→ OR9R→OR3

Where the makespan CRmax R=29 and the critical path for this solution is:

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

321

O0→O7→O8→O4→O5→O9→O3→O10, where O0 and O10 are dummies operations.

Tabu list={ } Iteration1 Three neighbourhoods can be constructed: (O4→O8), (O9→O5), (O3→O9)

Where

At neighbourhood (O4→O8), the sequence of operations on machines is:

M1 O4→O8→ O1

M2 O7→O2→ O6

M3 O5→ O9→O3

Figure C2 shows that makespan 23.


Makespan=23

At neighbourhood (OR9R→OR5R):




Figure C3 shows that makespan=26.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

322


Makespan=26

At neighbourhood (OR8R→ OR1R):




Figure C4 shows that makespan 30.


Makespan=30

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

M1

323

By comparison the three previous values of the makespan and by considering the

minimisation the makespan, we select Makspan = 23 as a seed for the new solutions.

The critical path for this solution is: O0→O4→O8→O1→O2→ O3→O10

As a result tabu list will be update to be:

Tabu list = {O8→O4} Iteration 2 The critical path is:

O0→O4→O8→O1→O2→ O3→O10

Neighbourhoods

O8→O4, O1→O8

At O8→O4

M1 O8→O4→ O1

M2 O7→O2→ O6

M3 O5→ O9→O3

Figure C5 shows that makespan =29.


Makespan=29

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

324

At O1→O8

M1 O4→O1→ O8

M2 O7→O2→ O6

M3 O5→ O9→O3


Makespan=24

Figure C6 shows that makespan=24.

By comparison the two previous values of the makespan and by considering the

minimisation the makespan, we select makespan = 24 with critical path

OR0R→OR4R→OR1R→OR8R→OR9R→ OR3R →OR10R as a seed for the new solutions.

As a result tabu list will be update to be:

Tabu list ={ OR8R→OR4R, OR8R→OR1R} Iteration3 The critical path is OR0R→OR4R→OR1R→OR8R→OR9R→ OR3R →OR10

The neighbourhoods: OR1R→OR4R, OR3R→OR9

At neighbourhood: OR1R→OR4R:




0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

325


Makespan=24

The makespan=24 as shown in Figure C7.

At OR3R→OR9




Figure C8 shows that makespan = 21.


Makespan=21

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

M1

326

By comparison the two previous values of the makespan and by considering the

minimisation the makespan, makespan=21 with critical path,

O0→O4→O1→O2→O3→ O9→O10 as a seed for the new solutions.

Tabu list = {O8→O4, O8→O1, O9→O3} From the critical path, the new neighbourhoods will be as: O1→O4 and O9→ O3.

The neighbourhood O9→O3 at the tabu list, so will be ignored and one

neighbourhood, O1→O4, will be used.

Iteration4

O1→O4

M1 O1→O4→ O8

M2 O7→O2→ O6

M3 O5→ O3→O9

The new makespan will be 24 with critical path: O0→O1→O4→O5→O3→

O9→O10 as shown in Figure C9.


Makespan=24

The new Tabu list = {OR8R→OR4R, OR8R→OR1R, OR9R→OR3R, OR4R→OR1R}

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

327

The new neghibour are: O4→O1, O3→O5, O9→O3.

The neighbourhoods, O4→O1, O9→O3, will be ignored because they are at the tabu

list and O3→O5 will be used to generate a new solution.

Iteration5

At: O3→O5

M1 O1→O4→ O8

M2 O7→O2→ O6

M3 O3→ O5→O9

The new makespan = 23 under critical path O0→O1→O2→O3→O5→ O9→O10 as

shown in Figure C10.


Makespan=23

The new tabu list= {OR8R→OR4R, OR8R→OR1R, OR9R→OR3R, OR4R→OR1R, OR5R→OR3R}

The new neighbourhoods for the critical path: OR0R→OR1R→OR2R→OR3R→OR5R→ OR9R→OR10R

are: OR5R→OR3R, OR9R→ OR5

The OR5R→OR3 Rwill be ignored and OR9R→ OR5 Rwill be used in the next iteration.

0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

328

Iteration 6

At O9→ O5

M1 o1→o4→ o8

M2 o7→o2→ o6

M3 O3→ O9→O5


Makespan=26

The new makespan is 26 under critical path: OR0R→OR1R→OR4R→OR8R→OR9R→

OR5R→OR6R→OR10R as shown in Figure C11.

The new tabu list = {OR8R→OR4R, OR8R→OR1R, OR9R→OR3R, OR4R→OR1R, OR5R→OR3R, OR5R→ OR9R}

The new neighbourhoods are: OR8R→OR4R, OR4R→OR1R, OR5R→OR9

We note that all neighbourhoods are at tabu list so, we can stop.

The best solution found until now is makespan=21 in Figure C8 and the sequence of

the operations on machines:


MR2 R OR7R→OR2R→ OR6


0

1 2 3

8

10

J1

J2

J3

M1

4 5 6

M2 M3

M1 M3 M2

7

M2

9 M3

3

6

4

6

2

3

6

4

4

M1

329

Table C1 can be deduced from the previous results:

Table C1: Tabu search technique result for example 2.1

Iteration Makespan Cmax

Tabu list Best value

0 { } 21 1 { OR8R→OR4R} 2 { OR8R→OR4R, OR8R→OR1R} 3 { OR8R→OR4R, OR8R→OR1R, OR9R→OR3R } 4 { OR8R→OR4R, OR8R→OR1R, OR9R→OR3R , OR4R→OR1R} 5 { OR8R→OR4R, OR8R→OR1R, OR9R→OR3R , OR4R→OR1R,

OR5R→OR3R}

6 { OR8R→OR4R, OR8R→OR1R, OR9R→OR3R , OR4R→OR1R, OR5R→OR3R, OR5R→OR9R}

The Percentage Improvement of Tabu Search(PITS) is calculated as follows:

PITS = ((initial solution of CRmaxR-TS value)/initial solution )*100) of CRmax

PITS = ((29-21)/29)*100 =27.58%.