Comparing Mixed Integer and Constraint Programming for the No-Wait ...

Comparing Mixed Integer and Constraint Programming for the No-Wait Flow Shop Problem with Due Date Constraints Hamed Samarghandi

Farzad Firouzi Jahantigh

Department of Finance and Management Science, Edwards School of Business, University of Saskatchewan Saskatoon, Saskatchewan, Canada, S7N 5A7 [email protected]

Department of Industrial Engineering University of Sistan and Baluchestan Zahedan, Iran [email protected]

Abstract— This paper deals with the no-wait flow shop scheduling problem with due date constraints. In the nowait flow shop problem, waiting time is not allowed between successive operations of jobs. Moreover, the jobs should be completed before their respective due dates. The considered performance criterionis makespan. This problem is strongly NP-Hard. This paper develops two distinct mathematical modelsfor the problem; namely, a mixed integer programmingand a constraint programming model are developed. To investigate the performance of these formulations, a number of test problems are solved and the results are reported. Keywords- Flow Shop Scheduling; No-wait; Due Date Constraints; Makespan; Mixed Integer Programming; Constraint Programming

I.

INTRODUCTION

In the no-wait flow shop problem, a special case of the classical flow shop problem, no waiting time is allowed between successive operations of jobs. In other words, once processing is started, no interruption is permitted between operations of the same job. In this paper, completion of each job is associated with a due date, i.e., jobs must be completed before their due dates. In this paper it is assumed that all the jobs are ready at time zero (all release dates are zero) and the considered performance measure is makespan.It can be proved that Hard[1].

F | no  wait, di | Cmax

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is strongly NP-

Industrial applications mentioned in the literature

F | no  wait, d | C

i max for include chemical industries [2], food industries [3], steel production [4], pharmaceutical industries [5], and production of concrete products [6].Due date constraints have rarely been studied as hard constraints. This is mainly due to the fact that generating a feasible solution for the problem, or proving that a feasible solution does not exist, turns into a very challenging task, especially when due dates become tighter. Interested readersare referred to [7-21]for a review of the methods that relax the due date constraints to solve the no-wait scheduling problem.

Mathematical programming techniques have long been employed to solve sequencing and scheduling problems. [22] developed a mixed integer programming for a flow shop system with more than one machine. [23] developed a mixed integer linear programming (MILP) based on the all-integer model of[24]. [25] performed an empirical study to evaluate the performance of the different mixed integer programming (MIP) models for permutation flow shop problems; results of this study were in line with the results of [26]for the case of regular job shop and flow shop problems. [27] developed a mixed binary integer programming (MBIP) model for reentrant job shop scheduling problem. [28] developed seven MBIP formulations for the flow shop sequencing problem. [29]developed a linear programming model for the nowait flow shop problem with fuzzy objectivefunctions. [30] developed a mathematical programming model to

minimize the earliness and tardiness costs in a flow shop context,where processing times can be zero.

Brackets are used to indicate consecutive jobs, i.e., refers to the starting time of the job planned to be

S[ i ]

This study develops two mathematical programming formulations for F | no  wait, di | Cmax . More specifically, an MIPand a constraint programming (CP) model are developed. Due date constraints are dealt with as hard constraints. Computational experiments in this paper reveal that the number of jobs in F | no  wait, di | Cmax instances should be very small so that the problem instances can be solved to optimality. The limitation on the number of jobs in a problem instance is more prominent when the due dates become tighter.

II.

processed after i th job in a given sequence. III.

THE DEVELOPED MODELS

A. MIP Model The first model employs the decision variable defined by (1).

1 if J j is placed immediatelyafter J i in thesequenc xij   Otherwise  0 i, j  1,2,..., n (1) The model,which is a mixed integer programming, is as follows:

PROBLEM STATEMENT

In the considered F | no  wait , di | Cmax it is assumed that: 1) all jobs follow the same predefined order of operations; 2) no preemption/interruption is allowed; 3) no job can be processed by more than one machine at the same time,and no machine can process more than one operation at the same time; and 4) there should be no waiting time between consecutive operations of a job.The following notation is used throughout the rest of this paper.

Min Cmax Cmax  Soim  pim ; i  1,2,..., n Sok 1  M (1  xik )  Soi1  pi1; i  1,2,..., n; k  1,2,..., n (4)

Soi [ j ]  Soij  pij ; i  1,2,..., n; j  1,2,..., m  1 (5)

M

m|M | N

n|N |

Soim  pim  di ; i  1,2,..., n

Set of machines Number of machines

n

x

Set of jobs

i 1

ij

Number of jobs n

Ji oij

Job

x

i

j 1

j th operation of J i Processing time of the

pij

operation of

Ji

n

n

i 1 j 1

Starting time of

Ji

Soij

Starting time of

oij

di

Due date of

ij

 n 1

Soij  0; i  1,2,..., n; j  1,2,..., m xij {0,1}; i  1,2,..., n; j  1,2,..., n

Ji

In this model, the objective function is to minimize the makespan; M is a sufficiently large number. (3) defines that makespan equals the finish time of the last operation of the last job. (4)assures that the operations do not overlap; this constraint is binding if J k is

l

Makespan of

 1; i  1,2,..., n

 x

on its respective

Si

Sequence

ij

xij  x ji  1; i  1,2,..., n; j  1,2,..., n

j th

machine

l Cmax

 1; j  1,2,..., n

l

scheduled immediately after J i in the sequence.(5) imposes the no-wait constraints. (6)represents the due

2

date constraint;according to (6), the last operation of each job should finish before its associated due date. Constraints (7), (8), (9), and (10) guarantee that all the jobs will appear exactly once in the sequence.

factor of the due dates. For each test problem, four different tightness factor settings were considered; this results in a total of 32 test problems for F | no  wait, di | Cmax and eight test problems for

F | no  wait | Cmax . Test problems with due date

B. CP Model

constraints will be called Car+DD.Maximum solution time for each test problem was set to 300 seconds throughout the experiment.Error! Reference source not found. summarizes the numerical results of the developed models. In this table NFS stands for no feasible solution found in the given CPU time limit. According to this table, MIP Modeldominates the other formulation. Concluding remarks will be presented in the next sections. (13)

Unlike previous models, CP Modelis formulated based on the special characteristics and properties of constraint programming (CP). The decision variable that will be used for CP Model is defined as xi  j if

Jj

is placed in location i . Accordingly, the CP

model will be as follows:



min Sox ,m  pxn ,m n



V.

All Different( x1, x2 ,..., xn )

Sox

( i 1) ,k

 Sox ,k  pxi ,k ; k  1,2,..., m; i  1,2,..., n  1 i

(15)

Sox ,[ j ]  Sox , j  pxi , j ; j  1,2,..., m  1; i  1,2,..., n i

i

(16)

Sox ,m  pxi ,m  d xi ; i  1,2,..., n i

Soij  0; i  1,2,..., n; j  1,2,..., m xi 1,2,..., n; i  1,2,..., n In this model, (15) means that the jobs should not overlap. (16)represents the no-wait constraints and (17) belongs to the due date constraints. Numerical results will be presented in the next section. IV.

COMPUTATIONAL EXPERIMENTS

Conducting numerical experiments is an effective approach to compare the performance of the developed models. IBM ILOG CPLEX V12.5 was used to solve the developed mathematical models. All the numerical experiments were performed on a PC equipped with a 2GHz Intel Pentium IV CPU and 2 GB of RAM. To perform the computational analysis, eight test problems were selected from the literature available for F | no  wait | Cmax : Car problems that were introduced by[31]. These test problems are available from the OR-Library1. The optimal solution of the selected test problems for F | no  wait | Cmax is known from the literature. Due dates data for the test problems were taken from [32]. TF is the tightness 1

Beasley, J.E. OR-Library: distributing test problems by electronic mail. July 2009 [cited 2014 March]; Available from: http://people.brunel.ac.uk/~mastjjb/jeb/info.html.

3

CONCLUSION

(14) The no-wait flow shop problem with due date constraints and makespan criterion was considered in this paper. The problem is strongly NP-Hard. Two mathematical modelswere developed for the problem; namely, a mixed integer programming modeland a constraint programming model.A thorough computational experiment was conducted to compare the performance of the developed models. Computational results illustrate that finding a feasible (17) solution for F | no  wait , di | Cmax is not an easy task when dealing with tight due dates. Numerical (18) results revealed that the mixed integer programming model outperforms the other formulation. Developing tight lower and upper bounds (19) for is a useful future research F | no  wait, di | Cmax direction. Also, finding a feasible solution for problems with tight due dates is challenging. Therefore, developing an approach that is able to efficiently generate feasible solutions is very promising.

TABLE 1

COMPUTATIONAL RESULT MIP Model

Problem

Size n*m

Car01+DD

11*5

Car02+DD

13*4

Car03+DD

12*5

Car04+DD

14*4

Car05+DD

10*6

Car06+DD

8*9

Car07+DD

7*7

Car08+DD

8*8

Due Date Tightness Factor TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4 TF=1 TF=2 TF=3 TF=4

Best Feasible OFV Found

Optimality Proved

8,152 8,168 NFS NFS 8,646 9,139 NFS NFS 9,170 9,148 NFS NFS 9,674 NFS NFS NFS 9,159 9,454 11,537 NFS 9,690 9,690 9,690 NFS 7,705 7,705 7,705 NFS 9,372 9,372 9,573 NFS

No No No Yes No No No Yes No No No Yes No No No Yes No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes [6]

[2]

[3]

[4] [5]

OFV

Optimality Proved

CPU Time

300 8,152 No 300 300 8,164 No 300 300 NFS No 300 4 NFS Yes 23 300 8,465 No 300 300 9,002 No 300 300 NFS No 300 298 NFS No 300 300 9,091 No 300 300 9,120 No 300 300 NFS No 300 305 NFS Yes 37 300 9,798 No 300 300 NFS No 300 300 NFS No 300 4 NFS No 300 300 9,159 No 300 300 9,454 No 300 174 11,537 No 300 25 NFS Yes 6 10 9,690 Yes 24 10 9,690 Yes 28 10 9,690 Yes 25 290 NFS Yes 9 2 7,705 Yes 1 2 7,705 Yes 1 2 7,705 Yes 1 14 NFS Yes 0 11 9,372 Yes 49 11 9,372 Yes 56 11 9,573 Yes 69 12 NFS Yes 3 Grabowski, J. and Pempera, J., 2000, Sequencing of jobs in some production systems. European Journal of Operational Research, 125: p. 535-550. [7] Rajasekera, J., Murr, M., and So, K., 1991, A due-date assignment model for a flow shop with application in a lightguide cable shop. Journal of Manufacturing Systems, 10(1): p. 1-7. [8] Hunsucker, J. and Shah, J., 1992, Performance of Priority Rules in a Due Date Flow Shop. Omega, 20(1): p. 73-89. [9] Sarper, H., 1995, Minimizing the sum of absolute deviations about a common due date for the two-machine flow shop problem. Applied mathematical modelling, 19(3): p. 153161. [10] Brah, S., 1996, A comparative analysis of due date based job sequencing rules in a flow shop with multiple processors. Production Planning & Control, 7(4): p. 362-373. [11] Gupta, J.N., Lauff, V., and Werner, F. 2000, On the solution of 2-machine flow shop problems with a common due date. in Operations Research Proceedings 1999. Springer. [12] Gowrishankar, K., Rajendran, C., and Srinivasan, G., 2001, Flow shop scheduling algorithms for minimizing the

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