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Job Shop Scheduling with Beam Search View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bilkent University Institutional Repository European Journal of Operational Research 118 (1999) 390±412 www.elsevier.com/locate/orms Theory and Methodology Job shop scheduling with beam search I. Sabuncuoglu *, M. Bayiz Department of Industrial Engineering, Bilkent University, 06533 Ankara, Turkey Received 1 July 1997; accepted 1 August 1998 Abstract Beam Search is a heuristic method for solving optimization problems. It is an adaptation of the branch and bound method in which only some nodes are evaluated in the search tree. At any level, only the promising nodes are kept for further branching and remaining nodes are pruned o permanently. In this paper, we develop a beam search based scheduling algorithm for the job shop problem. Both the makespan and mean tardiness are used as the performance measures. The proposed algorithm is also compared with other well known search methods and dispatching rules for a wide variety of problems. The results indicate that the beam search technique is a very competitive and promising tool which deserves further research in the scheduling literature. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Beam search; Job shop 1. Introduction (Lowerre, 1976). There have been a number of applications reported in the literature since then. Beam search is a heuristic method for solving Fox (1983) used beam search for solving complex optimization problems. It is an adaptation of the scheduling problems by a system called ISIS. La- branch and bound method in which only some ter, Ow and Morton (1988) studied the eects of nodes are evaluated. In this search method, at any using dierent evaluation functions to guide the level only the promising nodes are kept for further search and compare the performance of beam branching and the remaining nodes are pruned o search with other heuristics for the single machine permanently. Since a large part of the search tree is early/tardy problem and the ¯ow shop problem. pruned o aggressively to obtain a solution, its They also proposed a variation of this technique running time is polynomial in the size of the called the ®ltered beam search and found optimal problems. settings of the search parameters. This search technique was ®rst used in arti®cial In another study, Chang et al. (1989) used beam intelligence for the speech recognition problem search as a part of their FMS scheduling algorithm called bottleneck-based beam search (BBBS). Re- * Corresponding author. Tel.: 90 312 266 4126; fax: 90 312 sults indicate that BBBS outperforms widely used 266 4126; e-mail: [email protected] dispatching rules for the makespan criterion. 0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 1 9 - 1 I. Sabuncuoglu, M. Bayiz / European Journal of Operational Research 118 (1999) 390±412 391 Another beam search application to FMSs is re- mance of beam search (with respect to optimum ported by De and Lee (1990) who showed that the solutions) and compare it with other well-known solution quality of ®ltered beam search algorithm algorithms. In addition, we investigate the eec- is better than depth-®rst type heuristics in terms of tiveness of various rules as local and global func- the average maximum lateness and average ¯ow- tions of the beam search applications. The time measures. The authors also showed that beam previous research indicates that the values of ®lter search was better than breadth ®rst type heuristics and beam width aect the performance of the in terms of number of nodes created during the beam search. Hence, we also examine the perfor- search. In another study, Hatzikonstantis and mance of beam search for various values of ®lter Besant (1992) proposed a heuristic called A* for and beam width and ®nd their proper values. the job shop problem with the makespan criterion. Furthermore, we test two well-known schedule A* algorithm is very similar to the beam search generation schemes (active and nondelay schedule method. The only dierence is that A* algorithm is generation schemes) in the context of the beam a best-®rst search based heuristic and aims to ®nd search applications to the job shop problems. minimum-cost paths in search trees. Computa- The rest of the paper is organized as follows. tional tests indicate that this heuristic search al- Section 2 gives de®nitions of the job shop prob- gorithm performs better than dispatching rules. lem. Then a beam search based algorithm is de- Finally, Sabuncuoglu and Karabuk (1998) pro- veloped for the problem. This is followed by a posed a ®ltered beam search algorithm for more discussion on test problems and computational complex FMS environment in which AGVs are experience with the proposed algorithm. The paper explicitly modeled in addition to the routing and ends with concluding remarks and suggestions for sequence ¯exibilities. Their computational experi- further research. ments show that the beam search performs better than the machine and AGV scheduling rules under all experimental conditions for the makespan, 2. Problem de®nition mean ¯ow time and mean tardiness criteria. Their results also indicate that the beam search based The job shop problem is to determine the start scheduling algorithm exploits ¯exibilities inherent and completion time of operations of a set of jobs in FMS more eectively than other methods. An on a set of machines, subject to the constraint that overview of the beam search and its applications to each machine can handle at most one job at a time optimization problems can be found in Morton (capacity constraints) and each job has a speci®ed and Pentico (1993). processing order through the machines (prece- Even though beam search has been used to dence constraints). Explaining the problem more solve a wide variety of optimization problems, its speci®cally, there are a ®nite set J of jobs and a performance is not generally known for scheduling ®nite set M of machines. For each job j 2 J, a j j problems. Because, in the existing research work, permutation (r1; . ; rm) of the machines (where beam search is primarily applied to the FMS m jMj) represents the processing order of job j scheduling problem with additional considerations through the machines. Thus, j must be processed j j on MHS ®nite buer capacities and ¯exibilities ®rst on r1, then on r2, etc. Also, for each job j and and compared with only some dispatching rules. the machine i, there is a nonnegative integer pji, the Hence, its relative performances with respect to the processing time of job j on machine i. known optimum solution and other recently de- Since, this problem is NP-Hard (Garey and veloped heuristics are not known. Besides, it has Johnson, 1979) and very dicult to solve, early not been thoroughly studied as a problem solving studies on this problem were directed at develop- strategy with certain evaluation functions and ment of eective priority dispatching rules. But search parameters. later, due to the general de®ciencies exhibited by This paper attempts to achieve some of these priority dispatching rules, researchers concentrat- objectives. First of all, we measure the perfor- ed on more complex techniques. Tabu search 392 I. Sabuncuoglu, M. Bayiz / European Journal of Operational Research 118 (1999) 390±412 (Glover, 1989, 1990), large step optimization method. The former method is called one-step (Martin et al., 1989), simulated annealing (Matsua priority evaluation function, and the latter case is et al., 1988; Aarts et al., 1991), neural networks called total cost evaluation function. The one-step (Sabuncuoglu and Gurgun, 1996) and genetic al- priority evaluation function has a local view, gorithms (Nakano and Yamada, 1991) are exam- whereas, the total cost evaluation employs a pro- ples of the formalized applications of such jecting mechanism to estimate costs from the cur- scheduling techniques to the job shop problem. A rent partial solution. Therefore, evaluation is comprehensive bibliography of these studies for based on a global view of the solution. Unfortu- the job shop problem is given by Jain and Meeran nately, there is a trade-o between these two ap- (1996). In this paper, we measure the performance proaches: one-step (local) evaluation is quick but of beam search for the makespan and mean tar- may discard good solutions. On the other hand, diness criteria. Makespan, Cmax is the duration in more thorough evaluation by the global function is which all operations for all jobs are completed. more accurate but computationally more expen- Tardiness is the positive dierence between the sive. completion time and due date of a job. The ob- A ®ltering mechanism is also proposed in the jective is to determine starting times for each op- literature to reduce the computational burden of eration in order to minimize the makespan or beam search. During ®ltering some nodes are dis- mean tardiness while satisfying all the capacity and carded permanently based on their local evalua- precedence constraints: tion function values. Only the remaining nodes are subject to global evaluation. The number of nodes à Cmax min Cmax retained for further evaluation is called the ®lter width (a). min max Ci : 8i 2 J; feasible schedules As shown in Fig. 1, we determine the promising ! nodes (beam nodes) by performing local and X à global evaluations and proceed with the search T 1=jJj min max 0; Cidi ; feasible schedules through these selected nodes. After determining i2J the ®rst beam nodes at level 1, we apply the al- where Ci and di are the completion time and due gorithm to these nodes independently and generate date of job i, respectively.
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