An improved shifting bottleneck procedure for the job shop scheduling problem

Abstract

In this paper, the job shop scheduling problem with the objective to minimize the makespan is discussed. A theorem on the shifting bottleneck procedure (SB) for solving this problem is proven, which guarantees an application of the procedure slightly modified from SB to obtain feasible solution of the problem.

Based on this theorem, an improved shifting bottleneck procedure (ISB) for the job shop scheduling problem has been proposed. Besides ISB is implemented straightly, a refined version that combines ISB with the strategy of back tracking is presented. These two procedures have been tested on many benchmarks with various sizes and hardness levels. The computational experiment shows that ISB is more efficient and effective than SB. Also, some encouraging results about the refined version are obtained.

Introduction

The job shop scheduling problem with which we are concerned can be described as follows: There are a set of jobs and a set of machines. Each job consists of a series of operations, and each operation with fixed processing time is processed by a certain machine. The problem consists in scheduling the jobs on the machines with the objective to minimize the makespan—the time needed to finish all the jobs. Any schedule is subjected to two constraints: (i) the precedence of the operations on each job must be respected; (ii) once a machine starts processing an operation it cannot be interrupted, and each machine can process at most one operation at a time.

Fig. 1 illustrates an instance of the problem with three jobs, four machines and 12 operations. Each block stands for an operation, and the blocks with the same stripe are processed by the same machine. The length of a block implies its processing time.

The job shop scheduling problem, which is among the hardest combinatorial optimization problems, is strongly NP-complete [1]. During the past decades, many researchers have been focusing on the problem and proposed several effective algorithms for it. These algorithms can be classified as optimization and approximation algorithms. The optimization algorithms are based on the branch and bound scheme [2], [3], [4], [5]. These algorithms have made considerable achievement, however, their implementation needs too much computational cost. On the other hand, approximation algorithms, which are more effective for large size problem instances, use several approaches classified as: (1) local search [6], (2) dispatching priority rules [7], (3) shifting bottleneck approach [8], [2], (4) stimulated annealing [9], [10], and (5) tabu search [11], [12], [13], [14], [23].

The research results of the problem show that it is hard to obtain a good-enough solution only by single search scheme. So, the hybrid heuristic methods that combine several heuristic schemes have been proposed. It is well known that shifting bottleneck procedure (SB) and tabu search are widely used in hybrid heuristic methods [15], [16], [17]. Adams et al. [8] proposed SB, but they also declared that it is unavailable sometimes, i.e., dead lock, a sort of cycle, will occur for some instances, and it was confirmed by Storer et al. [18]. This phenomenon is interesting and attractive, thus we study it carefully. The progress made by us are as follows: the difficulties of SB is overcome partly, in other words, SB is modified slightly to negate the statement of Adams et al. [8]. Also, a procedure named an improved shifting bottleneck procedure (ISB) was proposed. We proved that using ISB could obtain feasible solution for any instance of the problem.

To improve the quality of the solutions obtained by ISB, a refined version—ISBB that embeds ISB into the strategy of back tracking—is proposed. Many instances with various sizes and hardness levels are used to test our procedures, and these two algorithms are compared with other typical algorithms. The computational experiments show that the performance of ISB is better than that of SB, and that ISBB is one of the effective algorithms with single local search scheme. Although ISB and ISBB are not the best ones at present, they are promising algorithms. This paper is arranged as following: Section 2 introduces SB briefly, Section 3 gives the proof of our theorem, 4 The improved shifting bottleneck procedure, 5 Partial back tracking describe ISB and ISBB, respectively, Section 6 shows the results of our computations and conclusions.