Theory and Methodology
Two-stage open shop scheduling with a bottleneck machine

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Abstract

It is known that for the open shop scheduling problem to minimize the makespan there exists no polynomial-time heuristic algorithm that guarantees a worst-case performance ratio better than 5/4, unless P≠NP. However, this result holds only if the instance of the problem contains jobs consisting of at least three operations. This paper considers the open shop scheduling problem, provided that each job consists of at most two operations, one of which is to be processed on one of the m⩾2 machines, while the other operation must be performed on the bottleneck machine, the same for all jobs. For this NP-hard problem we present a heuristic algorithm and show that its worst-case performance ratio is 5/4.

Introduction

In the general open shop scheduling model, each job consists of several operations to be processed on all or some of the given machines. For each job, the order of its operations (the processing route) is not restricted and must be chosen, different jobs being allowed to be assigned different routes. In all problems considered in this paper the objective function to be minimized is the makespan, i.e., the maximum completion time of all jobs on all machines. This problem has been introduced by Gonzalez and Sahni [6].

In this paper we consider the open shop scheduling problem in which the processing route for each job consists of two operations at most. Moreover, in our model the number of machines is arbitrary, and one of the operations of each job has to be processed on a particular machine, the same for all jobs. We call such a machine the bottleneck machine.

A classical example of the open shop is an automotive garage with specialized shops, as given by Gonzalez and Sahni [6]. To adapt the example to our model, we may assume that each of the coming cars requires exhaust pipes to be replaced. Besides that each car requires one additional piece of work, different for various cars: wheel aligning, or engine tuning, or tire replacing, etc.

The model under consideration is closely related to some other scheduling problems known in the literature, such as the two-stage open shop problem with parallel machines in one stage [4] and the two-stage open shop problem with a non-bottleneck machine [10]. For these and similar shop scheduling problems, research has recently been focused on developing and analyzing approximation algorithms. Because of their relative simplicity, these models have appeared to be helpful for testing various scheduling techniques. On the other hand, even the short route problems are not straightforward to solve. It is known that for the open shop with three operations per job finding a schedule close to an optimal one is no easier than to solve the problem to optimality, see [12].

This paper offers a polynomial-time heuristic which generates a schedule for the two-stage open shop problem with a bottleneck machine. The makespan of the found schedule is at most 5/4 times the optimal value.

The remainder of the paper is organized as follows. Section 2 gives a formal description of the problem. Section 3 briefly overviews the relevant results in the area of open shop approximation. In Section 4 we study properties of greedy open shop scheduling algorithms. 5 Fixing operations of, 6 Fixing operations of, 7 Fixing operations of describe three phases of our heuristic and analyse the worst-case performance of the corresponding algorithms. The full description of our 5/4-heuristic and discussion of the tightness of the obtained performance bound are given in Section 8. The concluding remarks are contained in Section 9.

Section snippets

Problem description

We start with a formal description of the open shop. We are given m machines and the set of n jobs where job j consists of at most m operations. Each operation has to be processed on a specified machine. The processing times of all operations are known. For each job, the order in which its operations must be processed is not known in advance, and can be different for different jobs. There is no preemption in processing an operation. No two jobs can be processed on a machine at a time, and each

Review of open shop approximation

The quality of a heuristic schedule S is usually measured by the ratio Cmax(S)/Cmax(S*) where S* is an optimal schedule for the given problem instance.

An algorithm H is called a ρ-approximation algorithm if it creates a schedule SH such that the inequality Cmax(SH)/Cmax(S*)⩽ρ holds for all problem instances. The smallest possible value of ρ is called the worst-case performance ratio guaranteed by algorithm H.

As shown in [9], the Om||Cmax problem admits a polynomial approximation scheme, i.e.,

Preliminaries

In this section, we introduce additional notation, recall basic properties of greedy open shop algorithms and discuss the general strategy of our heuristic algorithm for the O(m+1)|mj⩽2|Cmax problem.

Given an instance of the problem, for a non-empty set Q of jobs denotea(Q)=∑j∈Qa(j),b(Q)=∑j∈Qb(j),and define a(∅)=b(∅)=0.

For the O(m+1)|mj⩽2|Cmax problem, the makespan of any schedule S is no less than the total workload on a machine as well as the total processing time of a job, so thatCmax(S)⩾max

Fixing operations of u-jobs and v-jobs

The purpose of this section is to describe an algorithm which finds a heuristic schedule for the O(m+1)|mj⩽2|Cmax problem, provided that exactly one operation of each u-job and of each v-job is fixed.

Define the sets of fixed and reserved operationsFk={OAk,vk},k=1,2,…,m;F0={OB,uk|k=1,2,…,m},Rk={OAk,uk},k=1,2,…,m;R0={OB,vk|k=1,2,…,m}andXk={OAk,j|j∈Nk⧹{uk,vk}},k=1,2,…,m;X0={OB,j|j∈Nk⧹{uk,vk},k=1,2,…,m}.

In the preliminary stage of finding schedule S1 we find a schedule SF1 for processing the fixed

Fixing operations of u-jobs

Let S1 be the schedule found by Algorithm S1. Consider the case ruled out by Lemma 5 that some machine AfM0 terminates schedule S1 and job uf is critical; see Fig. 5.

In this section we describe how to find an alternative schedule S2 such thatminCmax(S1),Cmax(S2)54Cmax(S*).

Recall that Algorithm S1X assigns the u-jobs to be processed on B in the order of their numbering given by (6). Thus, in the case under consideration,Cmax(S1)=∑j=1fb(uj)+a(uf).

Temporarily disregard all jobs other than the u

Fixing operations of v-jobs

Let S1 be the schedule found by Algorithm S1. Consider the case ruled out by Lemma 5 that machine B terminates schedule S1 and a job vf such that AfM0 is critical; see Fig. 6.

In this section we describe how to find an alternative schedule S3 such thatCmax(S3)⩽54Cmax(S*),provided that (15) holds.

The structure of S1 suggests thatCmax(S1)⩽a(vf)+b(V0).

If (15) holds, then it follows from (1) and (3) thatb(V)⩾b(V0)>34Cmax(S*).

In order to describe the set of fixed operations for the new schedule S3,

The algorithm

For the sake of completeness, in this section, we put together all previously described algorithms, thereby obtaining the main heuristic. We also show that the worst-case ratio of our algorithm is 5/4.

Algorithm H

  • Input: An instance of the O(m+1)|mj⩽2|Cmax problem.

  • Output: A heuristic schedule SH.

  • 1.

    Find a schedule S1 by running Algorithms S1X and Algorithms S1.

  • 2.

    If some machine AhM0 terminates schedule S1 and job uh is critical, go to Step 3; otherwise, go to Step 4.

  • 3.

    Find schedule S2 by running Algorithms S2X

Conclusion

The paper addresses the simplest multi-machine open shop scheduling problem. For this model, a reasonably good worst-case ratio of 5/4 has been reached with small computational effort. This complements a recent open shop non-approximability result by Williamson et al. [12]. Our algorithm combines the greedy approach with special scheduling decisions regarding jobs with long operations. We think this technique can further be explored. The immediate candidate to look at is the m-machine open shop

Acknowledgements

This research was partly supported by the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union, INTAS-93-257-Ext.

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