Discrete Optimization
Approximation algorithms for the parallel flow shop problem

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Abstract

We consider the NP-hard problem of scheduling n jobs in m two-stage parallel flow shops so as to minimize the makespan. This problem decomposes into two subproblems: assigning the jobs to parallel flow shops; and scheduling the jobs assigned to the same flow shop by use of Johnson’s rule. For m = 2, we present a 32-approximation algorithm, and for m = 3, we present a 127-approximation algorithm. Both these algorithms run in O(n log n) time. These are the first approximation algorithms with fixed worst-case performance guarantees for the parallel flow shop problem.

Highlights

► We consider the problem of scheduling n jobs in m two-stage parallel flow shops. ► For m = 2, we present a 3/2-approximation algorithm so as to minimize the makespan. ► For m = 3, we present a 12/7-approximation algorithm. ► Both these algorithms run in O(n log n) time. ► These are the first approximation algorithms with fixed worst-case guarantees.

Introduction

Consider the problem of scheduling a set of n independent jobs J={J1,,Jn}, in which each job Jj consists of a chain of two operations (O1j, O2j) (j = 1,  , n), in a hybrid flow shop, also called a flexible flow shop, so as to minimize the length of the schedule, that is, the makespan. A hybrid flow shop is an extension of the classical flow shop, where there are m1 identical machines Mi1 (i = 1,  , m1) in stage 1 and m2 identical machines Mi2 (i = 1,  , m2) in stage 2. The first operation O1j of any job Jj needs first be processed on one of the machines in stage 1 during an uninterrupted processing time p1j  0, and then the second operation O2j needs to be processed on one of the machines in stage 2 during an uninterrupted processing time p2j  0.

The hybrid flow shop problem of minimizing makespan has been well studied (Ruiz and Vazquez-Rodriguez, 2010, Ribas et al., 2010, Naderi et al., 2010). Obviously, if m1 = m2 = 1, then the problem is polynomially solvable in O(n log n) time by Johnson’s rule (Johnson, 1954). However, if m1  2, or by symmetry m2  2, the problem becomes strongly NP-hard (Hoogeveen et al., 1996). Many researchers have focused on the special case with a single machine in one stage (Chen, 1995, Gupta, 1988, Gupta and Tunc, 1991, Gupta et al., 1997). For a review of the literature for the hybrid flow shop problem with a single machine in one stage, see Linn and Zhang, 1999, Wang, 2005). For the general case, Chen, 1994, Lee and Vairaktarakis, 1994 present O(n log n)-time heuristics with worst-case performance guarantee ratio 2  1/max{m1, m2}. If, for any instance of the problem, the makespan of the schedule generated by some heuristic does not exceed ρ times the optimal makespan, where ρ is a constant that is as small as possible, then ρ is the worst-case performance ratio of the heuristic. A heuristic with a worst-case performance ratio of ρ is called referred to as a ρ-approximation algorithm.

A hybrid flow shop is a manufacturing system that offers much flexibility, but as Vairaktarakis and Elhafsi (2000) point out, this superior performance comes at the expense of sophisticated material handling systems, like automated guided vehicles and automated transfer lines. As an alternative to the hybrid flow shop, Vairaktarakis and Elhafsi (2000) introduced the parallel flowline design, which is a flexible manufacturing environment with m identical parallel two-stage flow shops F1,  , Fm, each consisting of a series of two machines M1i and M2i (i = 1,  , m). Each job needs first to be assigned to one of the flow shops, and once assigned, it will stay there for both operations. See Fig. 1 for a hybrid two-stage flow shop, where the arrows indicate the routes that the different jobs may follow, and Fig. 2 for a parallel two-stage flow shop. In the remainder, we will refer to a parallel flowline design as a parallel flow shop.

The makespan parallel flow shop problem breaks down into two consecutive subproblems; first assigning each job to one of the m flow shops, and then scheduling the jobs in each flow shop so as to minimize the makespan. Whereas this second problem can obviously be solved in polynomial time by Johsnon’s rule (Johnson, 1954), the first subproblem makes the problem NP-hard, as proved by Vairaktarakis and Elhafsi (2000), who also presented an Onj=1n(p1j+p2j)3 time dynamic programming algorithm for its solution. Qi (2008) gave a faster algorithm, running in Onj=1n(p1j+p2j)2 time.

Vairaktarakis and Elhafsi (2000) concluded empirically, on the basis of computational experiments with several heuristics for both problems, that the parallel flow shop entails only a minor loss in throughput performance in comparison with the hybrid flow shop; accordingly, it is an attractive alternative to the hybrid flow shop, with its complicated routings. Other heuristics for the parallel flow shop problem have been presented by Cao and Chen, 2003, Al-Salem, 2004.

In contrast to the makespan hybrid flow shop problem, no approximation results for the makespan parallel flow shop are known. In this paper, we present a 32-approximation algorithm for the parallel flow shop problem with m = 2 in Section 2. For m = 3, we present a 127-approximation algorithm in Section 3. These results are the first polynomial-time algorithms with fixed worst-case ratios for the parallel flow shop problem.

Section 4 ends the paper with some conclusions, where we point out that our algorithms and their worst-case performance guarantees also apply to the parallel flow shop problem where each job Jj after the completion of its first operation may be transferred to another flow shop for the processing of its second operation and where such a transfer requires a transportation time τj  0. This transportation time effectively introduces a minimum time lag between the completion time of the first operation and the start time of the second operation of a job. Note that if τj = 0 for each Jj, then the parallel flow shop problem with transportation times boils down to the hybrid flow shop problem. For the hybrid flow shop problem with m1 = m2 = 2, our approximation algorithm has the same worst-case performance ratio as the one by Chen, 1994, Lee and Vairaktarakis, 1994. At the other extreme, if τj = ∞ for each Jj, then transfer between flow shops is effectively prohibited, and we have the original parallel flow shop problem.

Section snippets

A 32-approximation algorithm for m = 2

In the remainder of the paper, we assume that the job set J={J1,,Jn} has been re-indexed according to Johnson’s rule; that is, for any pair of jobs (Ji, Jj) we have that i < j if and only ifmin{p1i,p2j}min{p1j,p2i}.

For any instance of the m parallel two-stage flow shop problem, we refer to the Johnsonian schedule σ as the schedule that is obtained by assigning all the jobs to the first flow shop F1 and processing them in order of Johnson’s rule. Cmax(J) denotes the makespan of the Johnsonian

A 127-approximation algorithm for m = 3

For m = 3, we essentially design a similar approach as for Algorithm SPLT1; we start by cutting the Johnsonian schedule σ into two parts. We will do this in such a way that the makespan of the first part is bounded from above by 47Cmax(J)127Cmax and the makespan of the second part is bounded from above by 1621Cmax(J)167Cmax; remember from Lemma 1 that Cmax(J)3Cmax if m = 3. We then use algorithm SPLT1 to cut the second part into two further parts and guarantee that both these further parts

Conclusions

We have developed approximation algorithms with worst-case performance guarantees for scheduling jobs in a flexible manufacturing environment with two and three two-stage parallel flow shops. The key idea is to judiciously cut the Johnsonian schedule in two and three parts, respectively, and schedule each part in a different flow shop.

Our results apply also to the makespan parallel flow shop problem with transportation times, in which the operations of the same job can be performed in different

Acknowledgements

The authors thank the anonymous reviewers and the editor for their valuable comments on improving an earlier version. This research was supported in part by projects of National Natural Science Foundation of China (Nos. 70832002 and 10971034).

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