Elsevier

Operations Research Letters

Volume 33, Issue 6, November 2005, Pages 609-614
Operations Research Letters

No-wait flexible flowshop scheduling with no-idle machines

https://doi.org/10.1016/j.orl.2004.10.004Get rights and content

Abstract

This paper considers a two-stage flexible flowshop scheduling problem with no waiting time between two sequential operations of a job and no idle time between two consecutive processed jobs on machines of the second stage. We show its complexity and present a heuristic algorithm with asymptotically tight error bounds.

Introduction

Melting and casting are two key operations in iron and steel manufacturing, which correspond to the two stages of a flowshop. In the production process, jobs are converted from molten steel into billets. The molten steel is produced in the first stage and transported into the second stage to be cast. Except of the transportation time, no extra waiting time is allowed between the two stages to keep the molten steel in a high temperature. Furthermore, no idle time should be allowed between any two consecutive processed jobs on each machine in the second stage to keep the billets cast continuously. In practice, all items almost need the same processing times in the melting stage, and items’ processing times in the casting stage may be different according to the different product requests. And the processing time of each job in the second stage is no smaller than that in the first stage. Due to the fact that processing time spent in the second stage is more than that in the first stage, a factory in real case provides a melting and casting flexible system with three and four machines in the two stages, respectively.

In this paper, we present the system with m and m+1 machines in the two stages, respectively. The objective is to minimize the maximum completion time Cmax. We describe the scheduling problem as F2(Pm,Pm+1)(no-wait,no-idle).

In the problem, the two-stage operations are processed in two machine centers {Z1,Z2} with m and m+1 parallel machines, respectively. Denote the machines by M1,M2,,Mm in center Z1 and Mm+1,Mm+2,,M2m+1 in center Z2. Jobs will be processed in centers Z1 and Z2 in turn without any delay. The processing time of job j (1jn) is pj1=1 in center Z1 and pj21 in center Z2. Each machine in center Z2 has no idle time between any two sequential processed jobs.

A no-wait flexible flowshop represents a generalization of the no-wait flowshop and the identical parallel machine shop, which have been extensively studied. The efficient algorithms for minimizing Cmax in a no-wait flexible flowshop are not likely to exist. More details can be found in Hall and Sriskandarajah [4] and Mokotoff [6]. For no-wait and no-idle shop, Giaro [2] presents that even many trivial questions about the existence of schedule are NP-hard.

In this paper, we discuss the complexity of the problem in Section 2. In Section 3, we present a heuristic algorithm with worst case error bounding analysis. Conclusions are given in Section 4.

Section snippets

Complexity

In this section, we make use of the results that the well-known Partition Problem (PP) is NP-complete [5] and 3-Partition is strongly NP-hard [1].

Partition Problem (PP). Given a sequence {c1,c2,,cn} of numbers summing up to Q, there exists a subsequence summing up to Q/2?

3-Partition. Given a number B and a 3n-element set {c1,c2,,c3n} of numbers such that cj=nB, B/4<cj<B/2, does there exist a partition of it into n 3-element subsets each with sum B?

A problem named General 3-Partition is

Algorithm and error bounding analysis

The following algorithm named MLPT is a modification of LPT [3]. As the processing times of all jobs in the first stage are the same, processing times of the second stage are mainly considered in the algorithm. In the algorithm, a machine is said active at time t if the machine works well following the no-idle constraint before time t. Otherwise, the machine is said inactive at time t. A machine in a state of inactive at time t will never work after this time.

Conclusions

There are some remaining problems for further research: Is the problem strongly NP-hard when m>1? Is there an algorithm which can competitive with MLPT in the worst case version?

References (6)

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Partially supported by Grant 9732003-12-124 and NSFC 70471008.

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