A new heuristic for m-machine flowshop scheduling problem with bicriteria of makespan and maximum tardiness

doi:10.1016/S0305-0548(02)00143-0

Computers & Operations Research

Volume 31, Issue 2, February 2004, Pages 157-180

https://doi.org/10.1016/S0305-0548(02)00143-0 Get rights and content

Abstract

This paper addresses the m-machine flowshop problem with the objective of minimizing a weighted sum of makespan and maximum tardiness. Two types of the problem are addressed. The first type is to minimize the objective function subject to the constraint that the maximum tardiness should be less than a given value. The second type is to minimize the objective without the constraint. A new heuristic is proposed and compared to two existing heuristics. Computational experiments indicate that the proposed heuristic is much better than the existing ones. Moreover, a dominance relation and a lower bound are developed for a three-machine problem. The dominance relation is shown to be quite efficient.

Scope and purpose

The majority of research on scheduling problems addresses only a single criterion while the majority of real-life problems require the decision maker to consider more than a single criterion before arriving at a decision. The scheduling literature reveals that the research on multi-criteria is mainly focused on the single-machine problem as a result of the difficulty of the multiple machines problem. This paper addresses an m-machine flowshop scheduling problem with a multi-criteria objective function.

Introduction

Makespan and maximum tardiness are among the most commonly used criteria in the flowshop scheduling research. Makespan is a measure of system utilization while maximum tardiness is a measure of performance in meeting customer due dates. The makespan criterion for m-machine flowshop has been addressed in the literature by many researchers including Nawaz et al. [1], Chu et al. [2], Zegordi et al. [3], and Riane et al. [4]. The tardiness criterion has also been addressed in the literature by many researchers, e.g., Townsend [5], Kim [6], [7], Srinivasaraghavan and Rajendran [8], and Armentano and Ronconi [9].

The research mentioned so far addressed only a single criterion while the majority of real life problems require the decision maker to consider more than a single criterion before arriving at a decision. The research on multiple criteria is mainly focused on the single machine scheduling problem, see Nagar et al. [10]. The reason for this is that the scheduling problem with multiple-machine is difficult even with a single criterion. Therefore, considering more than a single criterion makes the multiple-machine problem even more difficult to solve.

This paper considers both makespan and maximum tardiness as the performance measures where the objective is to minimize a weighted sum of makespan and maximum tardiness. A survey of the literature has revealed that the m-machine problem with this objective function is addressed only by Daniels and Chambers [11] and Chakravarthy and Rajendran [12]. Both papers addressed the problem in order to minimize the objective function value subject to the constraint that maximum tardiness is not greater than a given value. Daniels and Chambers [11] provided two dominance relations and developed a lower bound for the two-machine problem, and presented two heuristic algorithms; one for the two-machine and the other for the m-machine problem. Chakravarthy and Rajendran [12] presented a simulated annealing algorithm for the m-machine problem and compared their algorithm with that of Daniels and Chambers [11].

This paper considers the same problem addressed by Daniels and Chambers [11] and Chakravarthy and Rajendran [12]. The paper also considers the problem without the constraint on the maximum tardiness. A new heuristic is proposed and shown by extensive computational experiments that the proposed heuristic perform much better than those of Daniels and Chambers [11] and Chakravarthy and Rajendran [12]. Furthermore, an efficient dominance relation and a lower bound are established for the three-machine problem.

Section snippets

Problem formulation

Let n be the number of jobs, m the number of machines, C_max the makespan, T_max the maximum tardiness, t_i,j the processing time of job i on machine j, t_[i,j] the processing time of the job in position i on machine $j, d_{i}$ the due date of job $i, d_{[i]}$ the due date of the job in position $i, T_{[i]}$ the tardiness of the job in position i and C_[i,j] the completion time of the job in position i on machine j. Then, C_max=C_[n,m] and T_max=max{max(C_[i,m]−d_[i],0) for i=1,2,…,n}.

The objective function value (OFV) is

A dominance relation and a lower bound

Dominance relations are known to improve the efficiency of implicit enumeration techniques such as branch-and-bound or dynamic programming. Daniels and Chambers [11] established two dominance relations and a lower bound for a two-machine flowshop. In this section, a dominance relation and a lower bound are developed for a three-machine flowshop. The efficiency of the dominance relation is shown in Section 5.

Let TP_j,k represent the total processing time of the jobs in positions 1,2,…,j on

Heuristics

A survey of literature has revealed that the problem considered in this paper has been addressed by Daniels and Chambers [11] and Chakravarthy and Rajendran [12], each providing a heuristic algorithm. In this section, their heuristics are briefly defined and a new heuristic is proposed.

Comparison of DCH, SAH, MNEH and PH

An extensive evaluation of the existing heuristics DCH, SAH, MNEH and the proposed heuristic with two levels of $L (PH (L=5)$ and PH(L=10)) is conducted. The existing and proposed heuristics were coded in FORTRAN language and run on a SUN SPARC Station 20. The processing times on each machine were randomly generated from a discrete uniform distribution in the range [1, 100]. This is a common range used in the literature, e.g., Chakravarthy and Rajendran [12], Rajendran and Ziegler [13] and Woo and

Summary and future research

The m-machine flowshop problem with the objective of minimizing the weighted sum of makespan and maximum tardiness is addressed. The problem with and without an upper bound on the maximum tardiness is considered. A new heuristic is proposed. The existing two heuristics and the new proposed heuristic (with two levels) are compared. The computational results show that the proposed heuristic with both levels perform much better than both of the existing heuristics. Also a dominance relation is

Acknowledgements

I would like to thank unanimous referees for their valuable suggestions and comments, which considerably improved the paper.

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Ali Allahverdi is an Associate Professor in the Department of Industrial and Management Systems Engineering of Kuwait University. He received his B.S. in Petroleum Engineering from Istanbul Technical University and his M.Sc. and Ph.D. in Industrial Engineering, from Rensselaer Polytechnic Institute, New York. His current research interests include scheduling and modular production. He has published numerous articles in Computers and Operations Research, European Journal of Operational Research, Journal of the Operational Research Society, International Transactions in Operational Research, International Journal of Industrial Engineering, and Computers and Industrial Engineering. He has also published articles in Naval Research Logistics, OMEGA The International Journal of Management Science, International Journal of Production Economics, Mathematical and Computer Modeling, Journal of Intelligent Manufacturing, International Journal of Parallel and Distributed Systems, and Communications in Statistics: Simulation and Computation.

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A new heuristic for m-machine flowshop scheduling problem with bicriteria of makespan and maximum tardiness

Abstract

Introduction

Section snippets

Problem formulation

A dominance relation and a lower bound

Heuristics

Comparison of DCH, SAH, MNEH and PH

Summary and future research

Acknowledgements

OMEGA The International Journal of Management Sciences

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

Computers and Operations Research

European Journal of Operational Research

European Journal of Operational Research

Computers and Operations Research