AN ANT COLONY OPTIMIZATION-BASED ALGORITHM FOR MINIMIZINGTHE MAKESPAN OF A JOB SHOP PROBLEM

DIYANA ABD MAHAD

UNIVERSITI TEKNOLOGI MALAYSIA

AN ANT COLONY OPTIMIZATION-BASED ALGORITHM FOR MINIMIZINGTHE MAKESPAN OF A JOB SHOP PROBLEM

DIYANA ABD MAHAD

A thesis submitted in fulfilment of therequirements for the award of the degree of

Master of Science (Mathematics)

Faculty of ScienceUniversiti Teknologi Malaysia

MAY 2011

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To

My Caring family,

My Supportive abah (Abdul Mahad @ Abdul Hamid bin Abdullah),

My lovely emak (Zainon bt Babjee),

My Dearest (Muhamad Mohzan b Hj. Jaafar)

Lecturers and friends.

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ACKNOWLEDGEMENT

In the name of the Almighty, Allah s.w.t, who has created mankind withknowledge, wisdom and power. Alhamdulillah, praise be to Allah for his uncountableblessings, strengths and guidance to face the hard times with patience. Without Hisblessing, it would be hard for me to accomplish finishing this study, successfully.

Next, I would like to thank my supervisor, Prof. Dr. Zuhaimy b Hj. Ismailfor his motivations, encouragement, guidance and continuous support. His wideknowledge and his logical way of thinking have been a great inspiration to me.

I also want to thank Pn. Shazirawati bt Mohd Puzi for helping with Latex, myexaminers Assoc. Prof. Dr. Hjh. Rohanin bt Ahmad and Assoc. Prof. Dr. Leong WahJune for devoting their time to read my thesis and also for their valuable and beneficialcomments. I really appreciate it. I should also dedicate my appreciation to Dr. SutinahSalim and Teoh Lay Eng for the set of data that I took from their Master’ thesis.

I am also thankful to my beloved brother, Mohd Faizal bin Abdul Mahad, forgiving me such a great idol that I admired since I was born. From deep into my heart Iam so proud to have a brother like him. As well I appreciate my father, Abdul Mahad@ Abdul Hamid bin Abdullah, my mother, Pn. Zainon bt. Babjee and siblings whonever stop to inspire me to work harder and for the mental and financial supports.Thanks for providing me a loving and adoring environment.

Last but not least, I appreciate the effort of Muhamad Mohzan bin Jaafar for hisendless support for me and Siti Nurul Huda binti Atan who was always be my friendin need and my appreciation also goes to those who were involved in completing thisproject.

Thank you.

Diyana Abd Mahad,

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ABSTRACT

Combinatorial optimization is a branch of optimization in applied mathematicsand computer science, related to operations research, algorithm theory andcomputational complexity theory that sit at the intersection of several fields, includingartificial intelligence, mathematics and software engineering. The quest for a solutionto NP-hard problem has brought many researchers into developing approximationmethod, an algorithm which attempts to find solutions to hard optimization problemsbut gives no guarantee that the solution is the best possible solution. In this researchAnt Colony Optimization (ACO) heuristics algorithm is proposed to solve Job ShopScheduling Problem (JSP). An appropriate ACO algorithm based on the job shopproblem is developed and implemented on a case study of JSP with the aims ofimproving the performance of the algorithm in term of computational effort and time.It is about minimizing the total completion time, such as the makespan of a selectedn-jobs and m-machines problem. Results from the case study have shown that theproposed ACO algorithm has a competitive advantage over the best given solution.

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ABSTRAK

Pengoptimuman berkombinatorik adalah salah satu cabang pengoptimumandi dalam aplikasi matematik dan sains pengkomputeran, ianya berkaitan dengankajian operasi, teori algoritma dan teori pengkomputeran kompleks yang terletak padapenggabungan beberapa bidang; termasuklah pengaturcaraan pintar, matematik danperisian kejuruteraan. Penekaan ini ialah satu penyelidikan kepada masalah NP-rumitdan telah membawa ramai penyelidik kepada pembentukan kaedah penghampiraniaitu suatu algoritma yang mana direka untuk mencari penyelesaian kepada jawapanmasalah optima rumit, akan tetapi ianya tidak memberi jaminan bahawa satupenyelesaian itu adalah jawapan yang paling tepat. Dalam kajian pengoptimumankoloni semut ini, kaedah heuristik telah dicadangkan untuk menyelesaikan masalahaliran kerja. Satu Algoritma Koloni Semut berdasarkan masalah perkilangan untukmasalah aliran kerja, dengan tujuan memperbaiki satu prestasi algoritma dalam skopkesungguhan pengkomputeran dan masa. Ini adalah untuk meminimumkan jumlahmasa penyudahan, iaitu jangka buatan untuk masalah n kerja dan m mesin yangterpilih. Keputusan daripada masalah sebenar ada menunjukkan tentang algoritmakoloni semut adalah di antara saingan sihat yang memberikan jawapan yang palingbaik.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION iiDEDICATION iiiACKNOWLEDGEMENT ivABSTRACT vABSTRAK viTABLE OF CONTENTS viiLIST OF TABLES xLIST OF FIGURES xiLIST OF ABBREVIATIONS xiiiLIST OF SYMBOLS xivLIST OF APPENDICES xvi

1 INTRODUCTION 11.1 Introduction 11.2 Research background 2

1.2.1 Job Shop Problem (JSP) 21.2.2 Ant Colony Optimization (ACO) 3

1.3 Problem Statement 51.4 Objective of the Study 51.5 Scope of the Study 61.6 Significance of the Study 61.7 Thesis Organization 7

2 LITERATURE REVIEW 82.1 Introduction 82.2 Behaviour of Real Ants 82.3 Classical Job Shop Problem 112.4 Computational Complexity 132.5 Shifting Bottleneck 13

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2.6 Review Relevant Research 142.7 Research Findings on ACO 172.8 Summary 18

3 RESEARCH METHODOLOGY 193.1 Introduction 193.2 Job Shop Problem 19

3.2.1 Introduction 193.2.2 Definition for Job Shop 203.2.3 JSP Mathematical Formulation 203.2.4 Constraint for the JSP 223.2.5 Assumptions for the JSP 233.2.6 The JSP Representation 243.2.7 Problem Formulation 263.2.8 Graph Representation for Job Shop Problem 27

3.3 The Solution Approaches 303.3.1 Methodology of Ant Colony Optimization 323.3.2 The new ACO-based Algorithm 34

3.3.2.1 Initialization 363.3.2.2 The Ant Architecture 363.3.2.3 Daemon Action 383.3.2.4 Pheromone Update 383.3.2.5 Terminating Condition 40

3.4 Summary 40

4 INDUSTRIAL PROBLEM EXPLORATION 414.1 Introduction 414.2 Introduction to Manufacturing Company 414.3 The Manufacturing Framework 424.4 Company Problem Description 474.5 The Case Study 514.6 Sensitivity Analysis 594.7 Summary 67

5 IMPLEMENTATION OF ANT COLONY OPTIMIZATIONFOR JOB SHOP PROBLEM 685.1 Introduction 685.2 Development of ACO Algorithm 68

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5.3 Algorithm for ACO 705.4 Data Diagram 845.5 Summary 89

6 SYSTEM DEVELOPMENT FOR ACO SOFTWARE 906.1 Introduction 906.2 Program Fundamentals 906.3 Programming with Microsoft Visual Studio 916.4 The Visual Studio Application 926.5 Program Visualization 936.6 Summary 104

7 ANALYSIS OF RESULTS, CONCLUSION AND RECOM-MENDATION 1057.1 Introduction 1057.2 Results 1057.3 Analyze of The Results 1097.4 Conclusion 1137.5 Contribution 1147.6 Recommendation for Future Research 114

REFERENCES 115Appendices A – 7 ?? – ??

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LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 The Instance data with 3 machines and 3 jobs. 122.2 Relevant Research on Ant Colony Optimization . 152.3 Relevant Research on Job Shop Problem. 162.4 Relevant Research on Job Shop Scheduling Problem in Ant

Colony Optimization. 17

3.1 The Processing Time for 2/3/G/Cmax job shop problem. 29

4.1 The Processing Time for 2/3/G/Cmax job shop problem. 574.2 Random values of Cmax and makespan depending on the

parameter values of α, β and ρ. 604.3 Random values of Cmax and makespan depend on the values of

β parameters, when ρ = 0 . 614.4 Random value of Cmax and makespan depending on the values

of β parameters, when ρ = 1 and α = 0. 624.5 Random values of Cmax and makespan depend on the values of

β parameters, when ρ = 2. 634.6 Random values of Cmax and makespan depend on the values of

β parameters, when ρ = 1 and α = 1. 644.7 Random values of Cmax and makespan depend on the values of

ρ parameters, when β = 0. 654.8 Random values of Cmax and makespan depend on the values of

ρ parameters, when β = 1. 66

5.1 Short caption 85

7.1 The Stimulation Results. 1107.2 Results for makespan with 6 machines. 1107.3 Comparison the results between Shifting Bottleneck Heuristic

Method [1] with the new ACO-based Algorithm. 112

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LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Ant experimental for the bridge experiment. 92.2 The ants are moving on the straight line. 92.3 An unexpected obstacle has interrupted the initial path. 102.4 The same situation on the other side of the obstacle. 102.5 Visual of ants choosing the shorter path. 11

3.1 The definition of a 2/3/G/Cmax job shop problem into graph. 283.2 The result for a 2/3/G/Cmax job shop problem into graph. 303.3 The flow chart for work explanation. 323.4 The problem solving step. 33

4.1 The framework for the warehouse process. 434.2 The typical job shop layout problem that forms the case study. 444.3 The data for the product customized E07 rack. 484.4 The processing time for the product. 494.5 The data input with 6 machine. 504.6 The graph for input data Power Press proses. 524.7 The graph for input data Tapping process. 534.8 The graph for input data Clinching Process. 544.9 The graph for input data Bending Process. 554.10 The graph for input data Machining Process. 564.11 The total number of edges versus total number of jobs run on 6

machines. 574.12 The total number of edges versus total number of nodes and

operations. 584.13 The graph on sensitivity analysis for Table 3. 614.14 The graph for Sensitivity analysis for Table 4.4. 624.15 The graph for sensitivity analysis for Table 5.5. 634.16 The graph for sensitivity analysis for Table 6. 644.17 The graph for sensitivity analysis for Table 7. 65

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4.18 The graph for sensitivity analysis for Table 4.8. 66

5.1 The Pseudocode of ACO algorithm. 685.2 The framework for the proposed ACO algorithm. 705.3 The Gantt Diagram for the Power Press Process. 865.4 The Gantt Diagram for the Tapping Process. 865.5 The Gantt Diagram for the Clinching Process. 875.6 The Gantt Diagram for the Bending Process. 885.7 The Gantt Diagram for the Machining Process. 88

6.1 The Welcome GUI for the program. 936.2 The Project Properties for the Case Study. 946.3 The Input Data for the CNC Machining. 956.4 The Input Data for the Bending process. 966.5 The Input Data for the Clinching process. 976.6 The Input Data for the Tapping process. 986.7 The Input Data for the Power Press process. 996.8 The Results for the case study. 1006.9 The Debug for the case study. 1016.10 The flow run for the software development. 103

7.1 The consequence for Power Press Process. 1067.2 The Consequence for Tapping Process. 1067.3 The Consequence for Clinching Process. 1077.4 The consequence for Bending Process. 1087.5 The consequence for CNC Machining Process. 109

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LIST OF ABBREVIATIONS

ACO – Ant Colony Optimization

JSP – Job Shop Problem

Np-hard – Non-deterministic Polynomial Time Hard

TSP – Traveling Salesman Problem

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LIST OF SYMBOLS

Cmax – Makespan / total completion time

m – Total of machine

n – Total of job

i – node i

j – node j

Mm – m Machine

Jn – n Job

Jj – Jobs at node j

Oij – Operation at node i and node j

Pij – Processing Time

s – Source Node

t – Sink Node

λ – Wavelength

α – Real positive parameters

β – Real positive parameters

ρ – Evaporation rate

γ – Decision variable, generates a sequence

|O| – Operations

ηij – Heuristic value associates with the common Cij

τij – Pheromone value at node i and j

dij – Heuristic distance between node i and j

G – A representative of a graph

Seij – Empty Solution

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Sg – Good solution

Lk – Tour length of the k-th ant

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LIST OF APPENDICES

APPENDIX TITLE PAGE

CHAPTER 1

INTRODUCTION

1.1 Introduction

This chapter presents the introduction to this thesis. It begins with thedescription of the overall background followed by a brief history of Scheduling, JobShop Problem (JSP) and Ant Colony Optimization(ACO). Research objectives, scopeof the study and discussion on the research contribution are also given. This researchare comparing the results for the same case study taken from [1] and solves the jobshop problem by using the different methods.

Job Shop problem is a combinatorial problem classified as an NP-hard problem[2]. It has been widely studied for the last two decades and this problem can be foundin many of the real life problems. Many heuristic approaches have been introduced tosolve this problem.

Ant Colony Optimization is one of the heuristic methods. It emulates thebehaviour of ants colony in their search for food where they search for the shortestdistance due to the strength of the pheromone trail. This method was first introducedby Dorigo on 1992 and there is a range of application for optimization problems. Inthis study the exploration on the architecture of ACO in solving JSP has been done.

This study involves the modeling of JSP and solving problem using Ant ColonyOptimization techniques. The study considers a general problem of JSP with multimachines, which has no limitation. The Algorithm of Ant System was used to solvethis JSP.

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1.2 Research background

Scheduling has been defined as ”the art of assigning resources to tasks in orderto ensure the termination of these tasks in a reasonable amount of time” [3]. Thepurpose of job scheduling is to balance the entire system load while completing allthe jobs at hand as soon as possible according to the environment status [4]. Job shopscheduling problems are hard to solve due to their complexities in theory and practice.

There are three common shop models for scheduling; namely Open Shops,Flow Shops, and Job Shops. An open shop is ”build to order” and no inventory isstocked.

Production schedules therefore are usually generated by using heuristics inpractice. However, it is very difficult to evaluate the quality of these schedules, and theconsistency of performance may also be an issue. Manufacturing operations may bedivided into Product-focused and Process focused. Flow Shop is the Product-focused;since it has the deterministic processing times, static arrivals and also famed as easiestscheduling problem.

The assembling of telephones is one of the examples for Product-focused.A metal shop making parts for Proton (assembly line) and the General Hospitalis a Process-focused and well-known as the Job Shop Problem (JSP). JSP has thestochastic processing times and dynamic arrivals. Hence, the Process-focused is themost difficult scheduling problem. This study is focusing on Job Shop. The results forany scheduling problem are known as an approximation scheme.

The key factor for manufacturing productivity is scheduling. Effectivescheduling can improve on-time delivery, reduce inventory, cut lead times and improvethe utilization of bottleneck resources. Because of the combinatorial nature ofscheduling problems, it is often difficult to find optimal schedules, especially within alimited amount of computation time.

1.2.1 Job Shop Problem (JSP)

Job Shop Problem is a discrete problem of a combinatorial problem. It isgeneralized from the Travelling Salesman Problem (TSP). It has also been illustrated

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as a problem in computational complexity theory which is hard to solve in practice.Job Shop is an NP-hard problem. The general objective of scheduling is to decide inwhat rules, methods or order is used to perform jobs in the sequence so that certaindesired criteria are met.

Job-shop scheduling is a complicated and widely studied productionscheduling problem. In a job-shop, jobs must be processed on machines in a specifiedorder. The Scheduling must have both optimal and flexible solutions to respond tofluctuations in the demand and operations of the plants while minimizing costs andtimes of operations.

Mathematical programming has been applied extensively to job shopscheduling problems. Problems have been formulated using integer programming,mixed-integer programming and dynamic programming. The job shop schedulingproblem (JSP) has a finite set of jobs processed on a finite set of machines. Eachjob is characterized by a fixed order of operations; each of the jobs is to be processedon a specific machine and specified duration. Each machine can process at most onejob at a time [5]. Once a job initiates processing on a given machine it must completeprocessing and uninterrupted. In this research, near-optimal solution methodologiesfor job shop scheduling are examined.

1.2.2 Ant Colony Optimization (ACO)

The ant colony optimization algorithm (ACO) was first introduced by MarcoDorigo in 1992 [2], a probabilistic technique for solving computational problemswhich can be reduced to finding good paths through graphs. This approach wasinspired by the behaviour of ants in finding paths from the colony to food (the source).Unfortunately, it is difficult to analyze ACO algorithms theoretically, the main reasonbeing that they are based on sequences of random decisions (taken by a colony ofartificial ants) that are usually not independent and whose probability distributionchanges from iteration to iteration.

ACO has been applied successfully to a large number of difficult combinatorialoptimization problems including travelling salesman problems, quadratic assignmentproblems and scheduling problems. ACO is an algorithm approach, inspired bythe foraging behavior of real ants. ACO can also be applied to the solution of

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Combinatorial Optimization Problem (COP), which will be discussed later. ACOis a population-based metaheuristic used to find approximate solutions to difficultoptimization problems. The algorithm involves a set of ’agents’ called artificial antssearch for good solutions to a given optimization problem.

In this study, the proposed ACO is used by transforming the optimizationproblem into a problem of finding the best path on a weighted graph. Artificial ants(hereafter ants) incrementally build solutions by moving on the graph. The solutionconstruction process is stochastic and is biased by a pheromone model, that is a setof parameters associated with graph components (either nodes or edges) whose valuesare modified at runtime by the ants.

The basic mechanism of ACO metaheuristic is that a colony of artificial antscooperates in finding good solutions to combinatorial optimization problems. Animportant and interesting behaviour of ant colonies appears in their foraging behaviour.In particular, ants are capable of finding the shortest paths between food sources andtheir nest without using visual cues. While walking from food sources to the nest andvice versa, ants exploit on their ground a substance called pheromone. Ants can smellpheromone substance and when choosing their way they tend to choose in probabilitypaths marked by strong pheromone concentrations.

The following description is a very brief summary about real ants and how theyforage from real to artificial ants from Dorigo and Stutzle’s book [2]:

”An ant colony represents a highly structured insect society that can

accomplish complex tasks that in some cases far exceed the individual capabilities of

a single ant. Ants coordinate their activities via indirect communication; for example,

a foraging ant that has found food will deposit a chemical on the ground (pheromones)

that increases the probability that other ants in the colony will follow its same path.”

Many researchers use ACO to solve NP-hard problem such as travellingsalesman problem [6], graph colouring [7], vehicle routing problem [8] and so on.This study applies the ACO algorithm to Job Shop Problem. Assumes each job is anant and the algorithm sends the ants to search for resources.

The job problems, consists of scheduling given jobs with same order at allmachines. The job can be processed at most on one machine; meanwhile one machine

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can process at most one job. The most common objectives and problem in this study,is the makespan. Hence, it is a decision making process with the goal of minimizingmakespan. This research studies job shop scheduling with performance measure. Thisstudy focuses on representations and formulations of job shop with the makespanobjective.

In production scheduling, the main problem is to minimize the makespan. Thespecific problem is assigning the operation:

• no one job in the job shop problem is pre-empted.

• the precedences given by the chain relations are respected.

• no two or more jobs are processed at the same time on the same machine.

• the maximum of the makespan, Cmax of all operations is minimized.

This research is using the case study taken from [1] and compare the resultsby using a different method. The research in [1] had been done by taking the set ofdata from ’XYZ Company’, and solve the JSP by using Shifting Bottleneck HeuristicMethod, whereas to this research takes the set of data from [1] and solve the JSP byusing a new ACO-based algorithm.

1.3 Problem Statement

The research aims to develop an algorithm based on ACO specially forminimizing the makespan of a JSP.

1.4 Objective of the Study

The objectives of this research are as follows:

(i) To explore Heuristic Method of Ant Colony Optimization for Job ShopProblem and its application in real-life phenomenon.

(ii) To develop an algorithm for solving JSP based on ACO.

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(iii) To compare the efficiency of the new ACO-based algorithm with existingShifting Bottleneck.

1.5 Scope of the Study

The objective of the research mostly focuses on minimizing the totalcompletion time, such as the makespan. In this study, we will describe on how tominimize the makespan for the n-jobs and m-machines problem, which is classifiedas combinatorial optimization problem. Therefore, it is an NP-hard problem class andnear optimum solution technique is preferred. Also considered on how to improvethe performance of the modified algorithm; such as the quality of the results, thecomputational effort, the time to it takes to arrive at the solution. For the sake onimproving this algorithm the best parameters values are chosen.

1.6 Significance of the Study

Scheduling has been widely studied on production and our real life problems.Since the key of manufacturer productivity is scheduling, it minimizes the cost andthe times of operations. The industrial relocalize trying to maintain profit throughthe development of effective production schedules, better utilization of resources andoverall, better planning. The significance of effective planning methods has increasedand will likely continue to do so.

Moreover, the main idea of ACO is to use parameterized probabilistic modelto construct solutions which is then used to update the model parameters valueswith the aim of increasing the probability of constructing high quality solutions byprobabilistically making a number of local decisions.

In this research, a procedure to achieve the desired makespan with near optimalnumber of machines of each machine type was developed. The needs for efficientmethods to obtain approximate solutions of high quality in a reasonable amount oftime has also been justified. The problem in this study is to minimize the makespanwhile ensuring that not more than one job can be processed at the same time on thesame machine and when a jobs starts, it must be completed and cannot be interrupted.

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1.7 Thesis Organization

The thesis is divided into seven chapters. The first chapter introduces theresearch. This chapter gives explanation on the ACO and JSP, problem statement,objective of the study, scope of the study, significance of the study and researchproblem.

Chapter 2 is the Literature Review. It reviews the behaviour of real ants, whileexplaining the Classical Job Shop Problem. The literature also reviews the relevantresearch, research findings on ACO and motivations for extended research.

Chapter 3 introduces definition of Job Shop Problem genuinely. Next, theconstraint of JSP is reviewed and some assumptions for JSP will be elaborated.This chapter also explained how to represent the problem into JSP, using the graphillustration.

Chapter 4 is the development of the algorithm. This chapter explains the ACOalgorithm step by step. There are five main steps in the algorithm that is; initialize,Construct the Ant Solutions, Daemon Action, Pheromone Update and TerminatingCondition.

Chapter 5 is where the implementation of ACO for JSP is discussed. The ideasof this research are also examined and considered. The system development for ACOsoftware and the main point from the programming with Microsoft Visual Studio andC # language are explain.

Chapter 6 describes the case study and the procedures used to solve theproblem. In this chapter the case study, framework and the layout from the DEN MetalCompany are clarified. The mathematical calculation as also discussed.

Chapter 7 summarizes the thesis, determines and computes the results, then theconclusion is clarified. Some recommendations for future research related to the topicsare also highlighted.

REFERENCES

1. Teoh Lay Eng, S. S. Minimizing Makespan and Total Weighted Tardiness

for Job Shop Scheduling Problem using Shifting Bottleneck Heuristic. Masterthesis. Faculty of Science (Mathematics), University Technology of Malaysia,Johor Bahru, Malaysia. 2005.

2. Marco Dorigo, T. S. Ant Colony Optimization. A Bradford Book, The MITPress, Cambridge, Massachusetts, London, England. 2004.

3. M. Dempster, J. L. and Kan, R. Deterministic and stochastic scheduling:Introduction, 1981.

4. Chang, R.-S., Chang, J.-S. and Lin, P.-S. An ant algorithm for balanced jobscheduling in grids. Future Generation Computer Systems, 2009. URL http://www.elsevier.com/locate/fgcs.

5. S. Binato, W. H. A greedy randomized adaptive search procedure for job shopscheduling, 2000.

6. E. Salari, K. E. An ACO algorithm for graph coloring problem. Congress on

computational intelligence methods an applications, 2005.

7. Xiaoxia Zhang, L. T. CT-ACO- hybridizing ant colony optimization with cycletransfer search for the vehicle routing problem. Congress on computational

intelligence methods an applications, 2005.

8. Goss. S., D. J., Aron. S. and Pasteels, J. Self-organized shortcuts in theArgentine Ant. Naturwissenschaften, 1989. 76: 579–581.

9. S. Goss, J.-L. D., S. Aron and Pasteels, J. M. Self-organized shortcuts in theArgentine ant. Naturwissenschaften, 1989. 76: 579581.

10. Beckers R., D. J. and Goss, S. Trails and U-turns in the selection of the shortestpath by the ant Lasius niger. Journal of theoritical biology, 1992. 159: 397–415.

11. B., H. and Wilson, E. The ants. Springer-Verlag, Berlin, 1990.

12. 2008. URL http://iridia.ulb.ac.be/.

13. Fisher, H. and Thompson, G. Probabilistic Learning Combinations of Local

116

Job-Shop Scheduling Rules in: Industrial Scheduling. Englewood Cliffs, NJUSA: Prentice-Hall. 1963.

14. Kuo-Ling Huang, C.-J. L. Ant colony Optimization combined with taboo

search for the job shop scheduling problem. Taipei, Taiwan. 2006.

15. Arora, B., Sanjeev; Barak. Computational Complexity: A Modern Approach.Cambridge. 2009.

16. Colorni A., M. V. T. M., Dorigo M. Ant system for job-shop scheduling.Belgian Journal of Operations Research (JORBEL), Statistics and Computer

Science, 1994. 34: 39–53.

17. Dorigo M., D. C. G. The Ant Colony Optimization Meta-heuristic’. McGraw-Hill. 1999.

18. Dorigo M., G. L. Ant colonies for the traveling salesman problem. vol. 43.Biosystem. 1997.

19. G.H., S. A hierarchical control architecture for job-shop manufacturing

systems. Master’s Thesis. Eindhoven University of Technology, Eindhoven.1992.

20. Blum, C. and Sampels, M. An Ant Colony Optimization Algorithm for ShopScheduling Problems. Journal of Mathematical Modelling and Algorithms.,2004. 3: 285–308.

21. Dell’ Amico, M. and Trubian, M., eds. Applying tabu search to the job-shop

scheduling problem. 1993.

22. Adams, B. E., J. and Zawack, D. The Shifting Bottleneck procedure for jobshop scheduling. Management Science, 1998. 34(3): 391–401.

23. Pezzela, F. Genetic Algorithm for the Flexible JSP (FJSP). 2007.

24. J. L. Bruno, J. R. L. G. W. H. K. R. S. K. S., E. G. Coffman and Ullman., J. D.Computer and Job-Shop Scheduling Theory. Pennsylvania State University:A Wiley-Interscience Publication. 1976.

25. URL http://en.wikipedia.org/wiki/Job_shop.

26. Neapolitan, N. 1996.

27. M. Dorigo, A. C., V. Maniezzo. The ant system : optimization by a colony ofcooperating agents. IEEE Trans., 1996. 26: 29–41.

28. van der Zwaan, S. Ant Colony Optimization for Job Shop Scheduling.

29. Jun Zhang, X. H. Implementation of an Ant Colony Optimization techinuquefor job shop scheduling problem. Transactions of the Institute of Measurement

117

and Control, 2006. 28: 93.

30. M. Dorigo, V. M. Positive feedback as a search strategy. Technical Report,

Dipartimento di Eettronica, Politecnico do Milano, Milan, Italy., 1991.

31. T.M.Willems, J. R. Neural Networks for job-shop scheduling. vol. 2.Pergamon Press Ltd. 1994.

32. Betul Yagmahan, M. M. Y. Ant Optimization for multi-objective Flow Shop

Scheduling Problem. Turkey. 2007.

33. Simon French. B.A., D., M.A. Sequencing and scheduling : An introduction

to the mathematics of the Job-Shop. Department of decision theory, Universityof Manchester: Ellis Horwood Limited. 1982.

34. Johnston, B. JAVA programming today. Upper Saddle River, New Jersey,Columbus, Ohio: Pe. 2004.

35. Weiss, M. A. Data Structures and Algorithm Analysis in Java. 2nd ed. FloridaInternational University: Greg Tobin. 2007.

36. URL http://www.ecma-international.org.

37. URL http://www.cplusplus.com/doc/tutorial/

namespaces/.