This article was downloaded by: [New York University]On: 25 November 2013, At: 07:11Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Production ResearchPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tprs20

A MILP model for the n-job, M-stage flowshop withsequence dependent set-up timesBELLUR N. SRIKAR a & SOUMEN GHOSH ba Department of Management , Michigan State University , East Lansing, Michigan, 48824,U.S.A.b American Airlines , Dallas, Texas, U.S.A.Published online: 22 Oct 2007.

To cite this article: BELLUR N. SRIKAR & SOUMEN GHOSH (1986) A MILP model for the n-job, M-stage flowshop with sequencedependent set-up times, International Journal of Production Research, 24:6, 1459-1474, DOI: 10.1080/00207548608919815

To link to this article: http://dx.doi.org/10.1080/00207548608919815

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

A MILP model for the n-job, M-stage flowshop with sequence dependent set-up times

BELLUR N. S R I K A R t and SOUMEN GHOSHf

The problem of an n-job, M-stage flowshop with sequence dependent set-up times is discussed. A mixed integer linear program (MILP) is formulated with considerably reduced number of integer binary variables, than if the same problem were formulated with integer variables defined as X i j = 1 if job i is sequenced immediately before job j, and 0 otherwise. The inclusion of set-UQ times, sequence dependency and M stages makes this formulation different from most other flowshop formulations in the literature, since such problems have not been given due consideration. This paper contributes a unique formu- lation to handle flowshop problems in the process industry where the set-up times have a definite dependence on the sequence in which jobs are processed. Computational results have been compared for problems up to six jobs and stages where the objective has been to minimize the makespan. The MILP has been solved using the SCICONIC-VIM mixed integer solution package on the Prime-550 minicomputer.

Introduction

The problem of scheduling n jobs on M machines is one of the classical prob- lems in flowshop scheduling tha t has interested researchers for many years. Though an enormous amount of research exists on this flowshop problem, over the last three decades very little work has ever attempted t o look into the 'n jobs on M machines' flowshop problem where the set-up times are sequence depen- dent. Indeed, even when ignoring sequence dependent set-up times, the general flowshop problem is highly combinatorial and one for which no efficient approach is known other than Johnson's results (Johnson 1954) for minimizing makespan in a two and three stage flowshop. The objective of this paper is to present a mixed integer linear program (MILP) of a flowshop where set-up times and their sequence dependencies are explicitly considered.

There seems t o be a surge of interest and awakening in the recent past as to the practical importance of such problems. Though extensive research on flow- shop problems exist in the literature, most of them either totally ignore set-up times, or consider them independent of the job sequence. However, increasing number of researchers have realized tha t sequence dependent set-up times are encountered very frequently in practice, and need t o be addressed adequately. Such problems occur, especially in process industry operations, whenever pro- cessing on a facility has to be shutdown after a particular job is completed, in order t o bring the facility to a desired state t o process the next job a t hand. This changeover delay is typically called 'set-up' time, and its magnitude depends on

Revision received April 1985. f American Airlines, Dallas, Texas, U.S.A. f Michigan State University, Department of Management, East Lansing, Michigan

48824, U.S.A.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

1460 B. N . Srikhr and S . Ghosh

the similarity in technological processing requirements for the two jobs. Typi- cally, similar technological requirements of two consecutive jobs would require lesser changeover or set-up time and conversely, greater the dissimilarities, larger is the changeover times (White and Wilson 1977).

The completion time for all the jobs through all the processors, besides being a function of the processing, set-up and idle times, will also depend on the sequence or ordering of the jobs through the processors. Some of the many examples of such problems found in real life include: (1) the printing industry, where the presses have to be cleaned and settings changed depending on the colour of ink, size of paper and types used, (2) container manufacture and bottling, where set- tings are changed when the containers or bottle sizes change, (3) textile industry, where weaving and dyeing set-up operations depend on the jobs, (4) food pro- cessing industry, (5) oilwell workover scheduling, and (6) bulk production and packaging industries (Gupta 1982, Barnes and Vanston 1981, White and Wilson 1977). In all these situations, sequence dependent set-up times play a very major role; however, research has typically been confined to prohlems with only one stage.

Background Though sequence dependent set-up times are encountered in a variety of pro-

duction situations, we will limit our discussions hereonly t o flowshop problems. Flowshop problems can typically be classified into two main classes: (1) single stage problems, and (2) multi-stage problems. While single stage problems corre- spond to a very 'special' case of flowshop problems, multi-stage problems are the real flowshop problems. It should also be pointed out that parallel processing can occur in the single stage flowshop, as well as within different stages in the multi- stage flowshop. In all flowshops where set-up times are sequence dependent, one of the objectives must attempt, either implicitly or explicitly, to minimize the total set-up time also. Minimization of set-up times makes the sequence depen- dent set-up time scheduling problems resemble the classical 'travelling salesman problems' t o varying extents. The one machine flowshop problem is analogous to the 'single-travelling salesman' problem (TSP) and the single stage parallel pro- cessing machines flowshop is similar t o the M-travelling salesman problem (MTSP). There is extensive literature on TSP, MTSP and their variations (Dantzig and Ramser 1959, Bellman 1962, Clark and Wright 1964, Little et al. 1963, Bellmore and Nemheuser 1968, Svestka and Huckfeldt 1970, Benton and Srikar 1982). These methodologies are directly applicable t o the single stage flow- shop problems by suitably modifying them t o process a dummy job a t the s t a r t and end of the sequence.

The single stage single processor sequence dependent flowshop problems have been researched extensively. Heuristic approaches have been proposed by Gavett (1965), Lockett and Muhlemann (1972), and whi te and Wilson (1977). Moore (1975), Driscoll and Emmons (1977), and Barnes and Vanston (1981) use branch and bound and dynamic programming methods. Taha (1971) and Sielken (1976) have formulated the problem as integer programming models considering holding costs and delay penalties. Sawicki (1973) studies the tardiness in such flowshops by simulation.

The single stage parallel processor sequence dependent flowshop problems,

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

MZLP male1 for the n-job, M-stage fiwshop 1461

where the processors are identical, have also received considerable attention. This could be due t o the fact t h a t direct application from the solution techniques of the M-travelling salesman problem is possible. These problems have been studied by Prabhaker (1974), Dean and White (1975), Geoffrion and Graves (1976), Bulfin and Parker (1976)and Parker et al. (1977).

Very little work has been done t o date with the n-job, M-machine sequence dependent flowshop problems. The only work addressing a two-stage flowshop is by Uskup and Smith (1975). Gupta (1982) proposes a branch and bound pro- cedure t o address an M-stage generalized jobshop problem with sequence depen- dent set-up times. He also points ou t that his procedure is limited t o very small problems because of excessive computational times. He further points out the lack of attention given t o sequence dependent set-up time problems in the literature.

We evidence from this brief review tha t while there is some work done on single stage flowshop problems with'sequence dependency, there is certainly a void in the literature for the multistage sequence dependent set-up time flowshop problems. We present a MILP formulation of the n-job, M-stage sequence depen- dent flowshop problem in this paper. Though the sequencing aspect of the problem has similarities with the TSP, the binary sequencing variables are uniquely defined, resulting in a considerable reduction in the number of integer variables than if we had used binary variables z i j , as defined by conventional travelling salesman formulations (Srikar et al. 1983). Also, this definition obviates the need for subtour breaking constraints in the MILP formulation.

MILP model formulation

I n general, the flowshop problem has been defined as follows:

'Given n nonidentical jobs t o be processed on M machines, the processing time of job i on machine j is tij (i'= 1, 2, . . . , n ; j = 1, 2, . . . , M), the problem is t o find that ordering of jobs which minimizes total process time or the makespan' (Gupta 1969).

However, in such a definition, set-up times are independent of job orderings and hence the processing and set-up times for a job i on machine j can be included in tij .

We will first s ta te some of the assumptions considered t o formulate the MILP model of the sequence dependent set-up time (SDST) flowshop problem. The first set are standard assumptions considered in the literature t o describe general flow- shop problems (Gupta 1977, Bansal 1977).

(1) All n jobs are processed on all M machines in the same order; i.e. each job has a prescribed technological order which is the same for all jobs and fixed.

(2) All jobs and machines are available at the beginning of the scheduling process; i.e. at zero time.

(3) Each job requires a known and finite processing time on various machines and is independent of the order in which the jobs are processed.

(4) Each job is processed only once on each machine.

(5) Splitting of any particular job is not allowed.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

1462 R. N . Srikar and S . Ghosh

(6) Jobs are independent of each other.

(7) Once a job starts on a machine i t has t o be completed before another job can be scheduled on the same machine (i.e. no pre-emption).

Additional assumptions we make in this paper t o formulate the SDST flow- shop are as follows:

(8) Set-up times for jobs on each machine are dependent on the order in which jobs are processed.

(9) Triangular inequality is not violated in the set-up times. Hence the sum of times to 'changeover from job i to j and then from job j to k is always greater than the time from job i t o k. This is a reasonable assumption since set-up times are based on the technological similarities of the two jobs being sequenced together. It is also evident from the literature t h a t this is a reasonable and frequently made assumption (Sawicki 1973, White and Wilson 1977, Driscoll and Emmons 1977).

(10) Sequencing decisions are handled independently of capacity decisions.

This MILP formulation of the SDST flowshop problem is unique in tha t i t does not use the traditional integer binary sequencing variable, Xij; where the binary variable is 1 if job i immediately precedes job j, and zero otherwise. Instead, we use the aij binary variable, a s defined later, for sequencing jobs. The MILP formulation is based on completion times of jobs on each machine. For every job pair to be scheduled on a given machine, mutually exclusive 'either-or' constraint pairs are developed, t o force completion times of the latter job t o exceed the completion time of the former job, by at least the sum of set-up and processing times of the latter job. This will be discussed in greater detail with the model formulation.

The notations used in the model formulation are described below. The model parameters are defined as

n number of jobs to be processed M number of processing stages (machines)

aij processing time of job j on stage i sijk sequence dependent set-up time of changing over from job j t o

k on stage i n a very large positive real number

The decision variables are defined as

cij latest completion time of job j on stage i Ei earliest completion time of the last job on stage i

1 if job j is scheduled anytime before job k 6 . = { ' 0 otherwise

where ajk is defined only for k > j B 1.

The djk variable definition is the key t o our formulation, since this requires only (n2 - n)/2 binary variables and also does away with subtour breaking constraints.

It should be pointed out t h a t assumption 1 restricts the formulation to provide optimal permutation schedules only. With the same processing order on each stage (i.e. permutation schedules), the solution space is restricted t o n ! pos-

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

M I L P model for the n-job, M-stage jlowshop i463

sible sequences; whereas without this restriction, the solution space would consist of (n! )M sequences. While i t is well known (see Baker, 1974, p. 141) tha t consider- ation of permutation schedules is enough to derive the global optimum for flow- shops without SDST only in the following two cases: (a) regular measure of performance, and M = 2 ; (b) makespan objeciive, and M = 2 or M = 3, i t may still be difficult to locate the optima efficiently. Consequently, trying to locate the optima for M > 3 by considering all (n!)M possible schedules is a n impossible task !

Also in the formulation t o be presented here, assumption 1 restricts variable a,, to be independent of the stage. For an exact formulation, hjk should depend on stage i and needs to be redefined as Sij, . This would give rise t o M ( n 2 - n)/2 binary variables, increasing i t by factor M . Although introduction of hi,, can make the formulation more general and also applicable t o the n-job, M-machine jobshop problem, we choose t o consider the simplified version for lesser computa- tional burden, but a t the cost of obtaining a restricted (permutation schedules) optima.

It is also pertinent t o point out tha t the use of variables dij is original only in the way they are used in this SDST formulation. Such variables have been used in classical job-shop formulations using graph-theoretic models based on disjunctive constraints (for more details and references on this, see Baker, 1974, p. 206-212). Similar uses of these variables have also been made by Hill (1980) in formulating the travelling salesman and vehicle scheduling problems.

Objective function

The objective function of the MILP could either be to minimize mean flow time or to minimize the makespan time. Both will now be discussed. T o minimize the mean flow time we can minimize the sum of all the completion times, resulting in eqn. (1)

This objective function includes all the completion times t o ensure tha t all jobs are finished at their earliest possible times on all stages.

The alternative objective function criterion would be to minimize the make- span time, which means we have t o ensure t h a t the completion time of the final stage is minimized. This is illustrated in eqn. (2).

Minimize 2, = EM (2)

Either eqns. (1) or (2) can be used in the MILP as the objective function criterion for which optimal schedules need t o be generated. Our remaining discussion con- siders the second criterion.

When considering minimizing makespan as the optimizing criterion, it is important t o note tha t individual job completion times need not necessarily be minimized. For a given minimum makespan time, the individual job completions can be delayed, depending on the slack available in the system, resulting in the job's latest completion times. It is this time t h a t we refer to in the paper as the

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

1464 R. N . Srikar and S . Ghosh

latest completion time on the job. I f no slack or idle time for a job exists in the system, then the earliest and the latest completion times will be the same for tha t job. The notation for the completion time used in the MILP formulation, cij, is the' latest completion time for job j on stage i which can still result in the minimum makespan time..

Constraint sets

The constrain? fall into four categories, namely: ( a ) completion time forcing constraints; ( b ) stage-link constraints; ( c ) 'either-or' sequencing constraints; and ( d ) sequence completion time constraints.

Completion time forcing eonslraint

Constraint set ( 3 ) is needed t o ensure t h a t all jobs are scheduled and tha t the completion time of any job k on stage 1 is at least a s great as the processing time required for tha t job on tha t stage.

Stage link constraints

constraint se t ( 4 ) links the completion time of a particular job in one stage t o its own completion time in the previous stage.

c,>c( , - , , , + a , i = 2 , 3 , . . . , M , and k = 1 , 2 , . . . , n ( 4 )

This constraint set ensures t h a t the completion time of job k in stage i must be greater than its completion time in stage ( i - 1 ) by a t least the amount of pro- cessing required in stage i . The set-up time is not explicitly considered here because of the fact tha t stage i may have been waiting for job k to be completed in stage ( i - 1 ) . Hence the set-up could have been done during this idle period and be of no consequence. If job k has to wait in stage ( i - 1 ) for completion of the set-up of job k in stage i , then the set-up time is accounted for in the 'either- or' constraints described later. However, constraint sets ( 3 ) and ( 4 ) together ensure tha t all jobs are scheduled on all stages and do not have trivial zero com- pletion times.

Eith,er-or sequencing constraints

Constraint sets (5) and (6) are either-or type constraints which determine the sequencing of the jobs on all the M stages.

cij - cik + nsjk B silk + a i j (5) where

i = . . M , = , 2 , n k = 1 , 2 , . . . , n , and k > j > l

cik - ci j + n(l - d jk ) >, sijk + a , ( 6 ) where

i = 1 , 2 , . . . , M , j = 1 , 2 , . . . , n,. k = 1 , 2 , . . . , n and k > j > l

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

MILP model for the n-job, M-stagejlowshop 1465

cil = max {c( i - l ) lr cil + s i31) + a i l (14)

If c ( ~ - , , , > cil + si3, then constraint (7) is binding and becomes a strict equal- ity, and the set-up from job 3 to job 1 is performed during idle time and does not affect the completion time cil . I n this case constraint (9) becomes a strict inequal- ity. Conversely, if c(,-,,, < cil + s i3 , , then constraint (9) is a strict equality and constraint (7) a strict inequality. Hence the completion of job 1 on stage i is affected by the completion of job 3 and the set-up from 3 to 1. For the given sequence, we see from eqn. (14) tha t one of constraints (7) and (9) is a strict equality and the other is a strict inequality.

The completion time of job 1 has now been determined from eqns. (7) or (9), and since job 2 does not follow job 1, constraint (8) is automatically satisfied as long as the triangular inequality assumption holds valid. This is established by the following two properties.

The either-or sequencing constraints work in a manner that , for any given sequence of the n jobs, only one of constraint sets (5) and (6) will be binding. If job k is scheduled before job j , (where k > j ) , then by definition bjk becomes zero, and constraint (5) forces the completion of job j to be at least as great as the completion time of job k, plus the sum of set-up time from job k to job j and its processing time. Notice, however, k may or may not be the immediate predecessor of j. Constraint set (6) in this situation will not be binding since n is a large number. Conversely, if k is not scheduled before job j then bj, = 1, and constraint set (6) will be binding, while constraint set (5) will not. Thus ajk the restricted binary variable, partitions the solution set into those where job j is scheduled anywhere before job k and those where job j is scheduled anywhere after job k.

To further illustrate the operations of constraint sets (5) and (6), consider the following illustration for any stage i of a three job sequencing problem which follows the triangular inequality, sij, + siPk 2 sijk. Consider the job sequence 2-3-1 for the sake of discussion. Without loss of generality, the discussion would hold true for any sequence. By earlier definition, the values of bjkwill be

b l l = 0, b13 = 0, and = 1.

Consider the sequencing of job 1 in relation to jobs 2 and 3. Given below are the set of relevant constraints from eqns. (4), (5) and (6).

C i l 3 C ( ~ - I ) I + ail (from constraint set (4)) (7)

cil - ci2 + n . 0 2 siz1 + ail

cil - ci3 + a . 0 3 si31 + ail

ci2 - ci3 + n . 1 2 s~~~ + ai2

(8)

> (from constraint set (5))

ci2 - cil + n(l - 0) 3 si12

I (::; + ai2

ci3 - cil + n(l - 0) 2 si l , + a,, > (from constraint set (6)) I :::; ci3 - ci2 + n ( l - 1) 1, 8i23 + ai3 1 ~ 3 )

Since we are minimizing the makespan, which is a function of completion times, cil will take the smallest possible values to satisfy inequalities (7), (8) and (9). Also, since in the sequence, job 3 precedes job 1, we can express the values of ci, from inequalities (7) and (9) as follows

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

1466 B. N . Srikar and S . Ghosh

Properly 1 . On any stage i of an SDST flowshop problem the relation cij - cik 2 smj + aij is always true if the triangular inequality holds true.

Proof. Consider the Gantt chart in Fig. 1 for any stage i :

set-up times

processing times

cij - cik 2 sikq + siqj + aij

Figure 1.

By triangular inequality assumption, sikq 3 sikp + sipq is true; or cij - cik 2 sikj + a i j , for same reason. The same logic holds true for constraint (6), except that

we are dealing with situations where j precedes k, so ajk = 1 by definition.

Property 2 . For the sequence 2-3-1, constraint (8) is a strict 'greater than' inequality as long as triangular inequality holds true.

Proof. For the given sequence 2-3-1, by definition a, , = 0 , so constraint (15) is binding. Adding constraints (9) and (13)

By triangular inequality assumption, we have

Substituting eqn. (16) in eqn. (15) we get

Similarly, for any two jobs that are not sequenced immediately, the correspond. ing constraint will be a strict 'greater than' relation.

Sequence complelion lime constraints Finally, the sequence completion time constraint set (18) selects the com-

pletion time of the last job in the sequence to be the sequence completion time.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

MILP model for the n-job, M-stageJEowshap 1467

This is useful in determining the makespan time for the flowshop problem.

E M a c M j , j = 1 , 2 , . . . , n (18)

Given the triangular inequality assumptions, constraint sets (3) and (4) link job completion times between and within stages with implicit consideration to either wait times in stage (i - I ) , or idle times in stage i. Constraint sets (5) and (6) ensure tha t for any stage i, the completion times of jobs hold the right relationship as the jobs are sequenced, since only one of them is a strict equality.

The solution to the above MILP, under optimality conditions, will generate a sequence chain consisting of all jobs. This is similar to the sequence in a TSP where a disconnected tour is generated visiting all the nodes once, and without any subtours. Our formulation avoids all subtour breaking constraints tha t would be necessary if sequencing variables are defined as in the classical TSP-type sequencing problem (Miller et al. 1960). The number of subtour breaking con- straints in a conventional MILP formulation for n jobs could be anywhere between 2" - 2n - 2 and (n - l ) (n - 2) constraints, depending on the type of constraints used, as shown by Rellmore and Nemheuser (1968). The above formu- lation also has fewer integer variables than the classical TSP problem. For an n-job problem the TSP type of definition requires (n2 - n) binary variables. This formulat~on requires only half the number of binary variables. The following section illustrates the size of the formulation in more detail, and the computa- tional requirements.

Numerical example and computational results

The MILP for~nulation of the SDST flowshop problem described in this paper was set up with a matrix generator programmed in Fortran 77. Thk model was solved on the Prime 550 minicomputer using the SCICONIC/VM mixed integer programming software. The basic method of the algorithm is to solve a sequence of relaxed problems with upper and lower bounds on integer variable vectors. Initially, bounds on the vectors are placed far enough to ensure tha t the optimal solution is covered. As the algorithm proceeds, nodes of the tree to be searched are generated. A list of outstanding nodes, called the eligibility list, is maintained, together with the value of the current best integer feasible solution, which is used as the value of the upper bound. The bounds aid in determination of which nodes are to be fathomed from the list as the searching proceeds.

Consider a problem where four jobs are to be sequenced through two stages and we need to minimize the makespan time. Table 1 gives the processing time and the set-up time matrices for this problem. The optimal solution for this problem, given the assumptions made in earlier sections, is illustrated by a Gantt chart in Fig. 2. The figure also illustrates the values of the integer variables tha t define the sequence. The solution to the problem does not depend on any assump- tions about the initial state of machines, except tha t any job can be scheduled right away.

Sample problems were run to demonstrate the feasibility of the MILP model, and to investigate the computing requirements for obtaining an optimal solution. A total of llOproblems were run with jobs and stages ranging from three to six in varying combinations. The da ta for these problems were generated from a uniform distribution of random numbers ranging between 1 and 99. The set-up

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

B. N . Srikar and S. Ghosh

Jobs

Stages 1 2 3 4

1 28 93 69 22 2 53 41 54 85

(a)

To job

From job 1 2 3 4

To job

From job 1 2 3 4

Table 1 . 4 x 2 flowshop scheduling problem data. (a ) Input processing time uniformly distributed between 1 and 99.

( 6 ) Input setup times by stages uniformly distributed between 1 and 99.

time matrices generated have their diagonal elements equal t o zero, and also ensure tha t the triangular inequality is enforced. .

Given the values of parameters n and M, the problem dimension can be expressed in terms of the number of continuous variables ( C V ) , integer variables ( I V ) , and constraints ( N C ) , as follows

Relative problem sizes and computational results are summarized in Table 2 . The objective function used is to minimize the makespan time (eqn. ( 2 ) ) . The same MILP formulation can be altered easily t o consider minimization of mean flow time by replacing the objective function with eqn. ( 1 ) . The relationship between computation times, and number of stages is illustrated in Fig. 3. The relationship between computation times and number of constraints is illustrated in Fig. 4. Figure 5 illustrates the relationship between computation times and number of jobs.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

MILP model for the n-job, M-stageJlows?wp

Stage 2 4

Job completion times

Idle time

Set-up time

n = 4

Number of binary variables (n2 - n ) / 2 = 6

Value of binary variables b , , = 1, b, , = 1 , S , , = 0, b , , = 1 , S , , = 0 , b , , = 0

Sequence = 4 - 1 - 2 - 3

M = 2

Number of constraints = 2n + ( M - I ) ( n ) + M ( n ) ( n - 1 )

= 36 Figure 2. Illustration of the example problem.

Discussion

The MILP model presented in this paper is a new formulation tha t has not been attempted before in flowshop literature. This formulation has far fewer number of variables and constraints as opposed t o other TSP-like sequencing problems, though like all MILP formulations we do have certain limitations on the size of the problem tha t can be handled. This formulation should, however, add t o the foundation work tha t is being done in sequence dependent n x M flowshop problems. Sample problems indicate that one can solve smaller size problems in reasonable time. It can be noted from Table 2 and Fig. 5 that solu- tion times start getting prohibitive beyond the (6 x 6) combination on the given computer system and solution package. It is also evident tha t moderately sized problems with few stages, or few jobs with many stages can be solved optimally in reasonable t ime.The solution times are affected significantly by both the number of binary variables and the number of constraints. Up to the (5 x 5) combination, the solution times increase almost linearly with the number of constraints, as seen in Figs. 3 and 4. On the other hand, Fig. 5 indicates tha t up to five jobs, solution times though increasing rapidly with increase in jobs, are not drastic across stages. However, Figs. 3, 4 and 5 all show tha t after five jobs, solution times are extremely sensitive with increase in stages. Though these behaviours are specifi- cally observed on a slower minicomputer, on other bigger and faster computer systems one could expect the same behaviour on larger sized problems.

Although the model is successful insofar as formulating the SDST flowshop problem, i t has demanding computational requirements for larger problems.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

1470 B. N . Srikar and S . Ghosh

Number of

Jobs Stages CV

2 8 3 12

3 4 16 5 20 6 24

Mean Mean Number Mean nodes till total

of timet optimal nodes I V NC CPU sec solution evaluated

3 2 1 12.2 8.6 11.8 3 30 12.0 6.0 18.8 3 39 14.6 9.6 10.8 3 48 15.8 9.4 11.4 3 57 18.0 11.0 11.4

2 14 15 78 328.0 433.0 713.0 3 2 1 15 114 561.4 529.0 810.0

6 4 28 15 150 730.2 308.0 721.0 5 35 15 186 854.4 268.8 641.0 6 42 15 222 1312,0§ 463.0 802.0

t Mean times are calculated generally, for five data sets of each size. 1 The means are out of ten data sets. 8 The means are out of eight data sets.

Table 2. Problems and computational statistics.

However, the optimal solution is reached in most cases, quite early in the search of the branch and bound tree, and a vast amount of time is spent in fathoming nodes to prove optiniality. This encourages the belief tha t one could either get very good feasible solutions for the model, or even the optimal solution, within a reasonable amount of time. The search for the integer optimum can therefore be terminated when once we are satisfied tha t the best integer solution obtained so far is within the acceptable limits.

The MILP formulation presented, though no t practical enough to handle very large problems by itself, can be very useful t o obtain good or even the optimal solution in reasonable computational time when used in combination with methods that generate good upper bounds. Though the problem size gets large with moderate sized combinations of jobs and stages, the formulation proposed could be used if the effective number of jobs were decreased by aggregating the jobs into fewer families which have similar set-up and processing requirements (Sawicki 1973). The formulation can then give optimal sequencing of batches or families of jobs, and this solution could subsequently be used as a good starting bound for the disaggregated problem. Computational requirements should decrease drastically when the effective number of jobs is decreased. Since no heu- ristic procedures exist to date to solve multi-stage sequence dependent flowshop

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

MILP model for the n-job, M-stage flowshop

1400

1300

1200

1100

1000

BOO

Number of stages

Figure 3. Time versus stages for each job

- VI

Y eoo VI

Number of constraints

Figure 4. Time versus constraints for each job.

200

100

5 jobs

/'

A'- . ' . 4 jobs _.a'- - .- 3 jobs

20 40 80 BO 100 120 140 160 180 200 220 240

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

B. N . Srikar and S. Ghosh

Number of jobs

Figure 5. Time versus jobs for each stage.

problems, this formulation can serve as an effective benchmark t o test the devel- opment of other solution procedures.

Conclusion

The process industry flowshop problem of n jobs with M stages and sequence dependent set-up times has been modelled as a mixed integer linear program. The formulation has been successfully solved for combinations of up t o six jobs and six stages on the Prime 550 minicomputer using the SCICONIC/VM package. Though optimal solutions have been found in all cases, i t is difficult to predict the behaviour of one problem based on another problem, especially using a branch and bound algorithm. As with any N P hard problem, the computational require- ments for this problem.increase rapidly with increase in problem size. However, this formulation can be useful to get good approximation of optimal solutions to larger problems after reducing their size by aggregation of jobs into families. This formulation should also serve as a foundation t o launch future work on sequence dependent flowshop problems.

It is hoped tha t this work will help in filling the gap t h a t exists in the liter- ature on multi-stage sequence dependent set-up time flowshops. There seems to be a need to focus future research interests into such problems tha t would enhance process industry operations.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

MILP model for the n-job, M-stage flowshop

Acknowledgment

W e would like to t ake the oppor tuni ty t o t hank t he referees for their valuable comments.

Le probldme d'un travail-n, dans un mouvement d'atelier phase-M avec temps de preparation dependant de sequence est debattu. Un programme lin- eaire a nombre entier mixte (PLNEM) est formule avec un nombre considbra- blement r6duit de variables binaires a nombre entier, comme si le meme probldme etait formule avec des variables nombre entier definies comme X i j = 1 si le travail i est mis en sequence immediatement avant le travail j, et 0 autrement. L'inclusion du temps de preparation, de la dependance de sg- quence et des phases M rend cette formulation differente de la majorith des autres formulations de mouvement d'atelier existantes dans la litbirature spb- cialiske, car il n'avait pas 6th donni: la consid6ration voulue a de tels prob- limes. Cet article contribue a donner une formulation unique pour resoudre les probl6mes de mouvement d'atelier dans I'industrie de traitement oli les temps de preparation ont une dipendance difinie sur la sequence selon laquelle les travaux sont traiths. Les ksultats calcules ont eth compares pour des prob- ldmes allant jusqu'a 6 travaux et phases e t dont I'objectif Btait de minimiser la plage. IR PLNEM a eth resolu en utilisant un systime de solution a nombre entier mixte SCICONIC-V/M sur un mini-ordinateur Prime-550.

Es wird das Problem eines DurchfluBbetriebs mit n Auftragen und M Stufen besprochen, dessen Riistzeiten von der Arbeitsreihenfolge abhangen, Ein gcmischtes. ganzzahliges Linearprogramm namens MILP wird vorgestellt, zu dessr.11 Formulierung eine bedeutend kleinere Anzahl von ganzzahligen Bin- arvariablen verwendet wir als wenn die Losung des gleichen Problems mit ganzzahligen Variablen erfolgen wiirde, die definiert sind als X i j = 1, wenn Auftrag i unmittelbar vor Auftrag j angeordnet ist, oder als 0, wenn das nicht der Fall ist. Durch die Einbeziehung der Riistzeiten, der Reihenfolgen- Abhangigkeit und der M stufen unterscheidet sich diese Formulierung von den meisten anderen in der Literatur fiir DurchfluBbetriebe genannten Ansatzen, in denen solche Probleme nicht gebiihrend beriicksichtigt wurden. Diese Arbeit bietet eine einzigartige Formulierung fur die Behandlung von DurchfluBproblemen in der verfahrenstechnischen Industrie, wo die Riistzeiten in einem ganz bestimmten Abhangigkeitsverhaltnis mit der Reihenfolge stehen, in der die Auftrage ausgefuhrt werden. Die Rechenergebnisse werden fur Probleme mit bis sechs Auftragen und Stufen verglichen, bei denen es darum ging. die Fertigungsdauer zu minirnieren. Die Losung des MILP- Programms e'rfolgte auf dem Prime-550-Minirechner mit Hilfe des Programm- pakets SCICONIC-V/M fur gemischte ganzzahlige Losungen.

References BAKER, K. R., 1974, Introduction to Sequencing and Scheduling (New York: John Wiley

and Sons). BANSAL, S. P. , 1977, Minimizing the sum of completion times of n jobs over m machines in

a flowshop-a branch and bound approach. A.I . I .E . Transactions, 9,306. BARNES, J . W., and VANSTON, L. K., 1981, Scheduling jobs with linear delay penalties and

sequence dependent setup costs. Operations Research, 29, 146. BELLMORE, M., and NEMHGUSER, G. I,., 1968, The travelling salesman problem: a survey.

Operalions Research, 16, 538. BELLMAN, R., 1962, Dynamic programming treatment of the travelling salesman problem.

Journnl of the Association of Computing Machinery, 9, 61. . .

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013

1474 MILP &el for the n j o b , M-stage flowshap

BENTON. W. C., and SRIKAR, B., 1982, Evaluating an efficient heuristic for the multiple vehicle problem, Working Paper Series 82-72, C.A.S., Ohio State University, U.S.A.

BULFIN, R. L., and PARKER, R. G., 1976, On a two facility scheduling problem with sequence dependent processing time. A.I.I.E. Transactions, 8,202.

CLARK, G., and WRIGHT, J., 1964, Scheduling vehicles from a central depot to a number of delivery points. Operations Research, 12, 568.

DANIELS, V.. 1983, Lot sizing and sequencing problems in processing industries. Unpub- lished dissertation, The Ohio State University, U.S.A.

DANTZIO, G., and RAMSER, J., 1959, The truck dispatching problem. Management Science, 6, 80.

DEANE, R. H., and WHITE, E. R., 1975, Balancing workloads and minimizing setup costs in the parallel processing shop. Operalional Research Quarterly, 26, 45.

DRISCOLL, M7. C., and EMMONS, H., 1977, Scheduling production on one machine with changeover costs. A.I.I.E. Transactions, 9, 388.

GAVETT. J . W., 1965, Three heuristic rules for sequencing jobs to a single production facility. Management Science, 11, B-166.

GEOFFRION, A. M. , and GRAVES, G. W., 1976, Scheduling parallel production lines with changeover costs: practical application of a quadratic assignment/LP approach. Operations Research. 24, 595.

GUPTA, J. N. D., 1969, A general algorithm for t h e n x M scheduling problem. Inlenur(wn- a1 J o u m l of Production Research, 7, 241.

GUPTA, J. N. D., 1977, A review of flowshop scheduling research in LP. Ritzman el al. (editors), Disaggregation: Problems in Manufacturing and Seruice Organizalion (The Hague: Martinus Nijhof).

GUPTA, S. K. , 1982, n jobs and m machines job-shop problems with sequence-dependent set-up times. International Journal of Production Research, 20, 643.

HILL, A. V., 1980, Mixed-integer programming formulations for the travelling salesman and vehicle scheduling problems. Working paper #52, Graduate School of Business Administration, University of Minnesota, U.S.A.

JOHNSON, S. M., 1954, Optimal two- and three-stage production schedules with setup times included. Naval Rfisearch Logisties Quarterly, 1, 61.

IJlTTL~, J. D. C., MURTHY, V. G., SWEEIY. D. W., and KAREL, C., 1963, An algorithm for the travelling salesman problem, Operations Research, 11, 972.

LOCKETT, A . G., and MUHLEMANI, A. P., 1972, A scheduling problem involving sequence dependent changeover times. Operations Research, 20, 895.

MILLER, C. E., TUCKER, A. W., and ZERNLII, R. A,, 1960, Integer programming formula- tion of travelling salesman problem. Journal of the Association of Computing Machin- ery, 7, 326.

MOORE, J. E., 1975, An algorithm for a single machine scheduling problem with sequence dependent setup times and scheduling windows. A.I.I.E. Transacfions. 7, 35.

PARKER, R. G., DEANE, R. H., and HOLMES, R. A,, 1977, On the use of a vehicle routing algorithm for the parallel processor problem with sequence. dependent changeover costs. A.I.I .E. Transactions, 9, 155.

PRABHAKAR, T., 1974, A production scheduling problem with sequencing considerations. Management Science, 21, 34.

SAWICKI, J. D., 1973, The problems of tardiness and saturation in a multi-class queue with sequence-dependent setups. A.I . I .E. Transactions, 5, 250.

SCICONIC/VM (1983) Version 1.2, Scicon Computer Services Limited, London. STELKEN, R. L., 1976, Sequencing with setup costs by zero-one mixed integer linear pro-

gramming. A.I.I.E. Transactions, 8, 369. SRIKAR, B. N., DANIELS, V., and RITZMAX, L. P., 1983. Multistage-multicycle flowshop

sequencing. Working paper, Series 83-81, C.A.S., Ohio State University, U.S.A. SVESTKA. J., and HUCKFELDT, V. E., 1970, Computational experience with an M-salesman

problem. Management Science, 19, 790. TAHA, H . A,, 1971, Sequencing by implicit ranking and zero-one polynomial programming.

A.I.I.E. Trasaclions, 3, 299. USKUP, G., and SMITH, S. B., 1975, A branch and bound for two-stage production sequenc-

ing problems. Opemtionr Research, 23, 11 8. WHITE, C. H., and WILSON, R. C., 1977, Sequence dependent set-up times and job sequenc-

ing. Internationnl Journal of Production Research, 16, 191.

Dow

nloa

ded

by [

New

Yor

k U

nive

rsity

] at

07:

11 2

5 N

ovem

ber

2013