Short Communication
A note on the two-stage assembly flow shop scheduling problem with uniform parallel machines

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Abstract

We study the problem of minimizing the makespan in a two-stage assembly flow shop scheduling problem with uniform parallel machines. This problem is a generalization of the assembly flow shop problem with concurrent operations in the first stage and a single assembly operation in the second stage. We propose a heuristic with an absolute performance bound which becomes asymptotically optimal as the number of jobs becomes very large. We show that our results slightly improve earlier results for the simpler assembly flow shop problem (without uniform machines) and for the two-stage hybrid flow shop problem with uniform machines.

Introduction

The concept of concurrency in a flow shop environment has been introduced by Lee et al., 1993, Potts et al., 1995 who considered the two-stage flow shop problem AmCmax with m concurrent operations in the first stage and a single assembly operation in the second stage. An extension of the AmCmax problem to three stages was studied by Koulamas and Kyparisis (2001). The two-stage assembly flow shop with uniform parallel machines (AFQCmax) problem studied in this paper generalizes the NP-hard AmCmax problem and is described as follows. Each job Jj, j = 1, 2,  , n, consists of a set of m operations {O1,j,O2,j,  , Om,j} followed by an assembly operation OA,j. The ith concurrent operation at stage 1, Oi,j, requires pi,j units of processing, is processed on some machine vi (from the set of li uniform parallel machines) with speed si,vi1, vi = 1,  , li, i = 1, 2,  , m, and it requires pi,j/si,vi time units. The assembly operation OA,j at stage 2, requires pA,j units of processing, is processed on some machine vA (from the set of lA uniform parallel machines) with speed sA,vA1, vA = 1,  , lA, and it requires pA,j/sA,vA time units. Each machine can process at most one job at a time. For each job Jj the assembly operation OA,j may start only after all concurrent operations O1,j, O2,j,   , Om,j have been completed. We assume that all jobs are simultaneously available at time zero. Preemption is not allowed, i.e. any commenced operation has to be completed without interruptions. Let Cj be the completion time of Jj at the assembly stage. The scheduling objective is to minimize the maximum job completion time (makespan) Cmax = max1⩽jnCj. We are interested in determining absolute performance bounds for heuristics for the AFQCmax problem, that is bounds on the difference between the makespans of a heuristic and an optimal solution respectively.

When there is only one concurrent operation at stage 1, the AFQCmax problem reduces to the two-stage flexible or hybrid flow shop problem with uniform parallel machines, denoted as HF2QCmax. Sevastianov, 2002, Kyparisis and Koulamas, 2006 developed heuristic algorithms with absolute performance bounds for the HF2QCmax problem. Another heuristic with absolute performance bounds was developed by Kyparisis and Koulamas (2002) for the related assembly-line scheduling problem with identical parallel machines per stage and with concurrent operations.

In this paper, we propose a heuristic for the AFQCmax problem with an absolute performance bound which becomes asymptotically optimal as the number of jobs becomes very large. To our knowledge, this is the first paper in the literature which studies the AFQCmax problem.

We close this section by reviewing the literature for the related AmCmax problem. Hariri and Potts (1997) developed a branch and bound algorithm for the AmCmax problem and established several dominance theorems which were incorporated into the branch and bound algorithm. Haouari and Daouas (1999) proposed a branch and bound algorithm based on recursive enumeration of potential inputs and outputs of the machines for the A2∥Cmax problem with two concurrent operations in the first stage. Sun et al. (2003) proposed a number of heuristic algorithms for the A2∥Cmax problem and showed through numerical experiments that the proposed heuristics can solve all of the worst cases which cannot be solved by the existing heuristic algorithms. Tozkapan et al. (2003) considered the AmwiCi problem with the objective of minimizing the total weighted flowtime and developed a branch and bound algorithm for the AmwiCi problem which incorporates a lower bounding procedure and a dominance criterion. Allahverdi and Al-Anzi (2006) consider the A2∥Lmax problem with the objective of minimizing the maximum lateness (in the presence of job due dates) and they propose three heuristics for the problem, namely, the earliest due date (EDD) heuristic, a Tabu search heuristic and the particle swarm optimization (PSO) heuristic. Finally, Cheng and Wang (1999) considered a related problem of scheduling the fabrication and assembly of components in a two-machine flowshop so as to minimize the makespan. Each jobs consists of a component unique to that job processed individually on the first machine and a component common to all jobs processed in batches on the first machine with a setup needed to form each batch. The assembly operation of a job is performed on the second machine. Cheng and Wang (1999) show that this problem is NP-complete with either batch availability or item availability for the common components. They also identify several properties of an optimal solution and some polynomially solvable cases.

The rest of the paper is organized as follows. In Section 2 we present an asymptotically optimal heuristic with an absolute performance bound. Some discussion on how our results compare with earlier results for the AmCmax and HF2QCmax problems are presented in Section 3.

Section snippets

An asymptotically optimal heuristic for the AFQCmax problem

Consider any concurrent operation in stage 1. Given a permutation π = (1,  , n), the Modified List Scheduling (LS′) rule creates a schedule for that operation by assigning the job to be scheduled next (in the order π) to the uniform machine on which it will finish the earliest. Define the quantitiesSi=vi=1lisi,vi,ti,j=pi,jSi,Tik1:k2=j=k1k2ti,j,pmaxi=maxj=1,,n{pi,j},where Si denotes the total speed of all the uniform machines dedicated to the ith concurrent operation, ti,j denotes the shop

Discussion

The primary objective of this note was to develop an asymptotically optimal heuristic for the AFQCmax problem. The development of heuristic H0 in conjunction with Theorem 1 accomplished this objective. However, Theorem 1 can also be used for slightly improving earlier results for the AmCmax problem and/or for either matching or slightly improving earlier results for the HF2QCmax problem as well.

Corollary 1

For the AmCmax problem, an asymptotically optimal schedule S0 can be constructed using heuristic H

Acknowledgment

We would like to thank two anonymous referees for their constructive comments which helped us improve an earlier version of this paper.

References (17)

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