An optimal method for the preemptive job shop scheduling problem

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Abstract

In this study, a powerful solution methodology is developed for minimizing makespan in the preemptive Job Shop Scheduling Problem (pJSSP). Some new properties of the problem are stated and proved via theorems on the basis of which a new dominant set is introduced for the problem. These properties give rise to a dramatic decrease in the search space and provide the potential for exact methods to be successfully used in the solution of this notoriously NP-hard problem. The exact method presented here is a branch and bound algorithm developed on the basis of a new disjunctive graph. Its efficiency is enhanced by the effective use of such techniques as dominance rules or lower bounds. The capability of the approach is investigated by using it to solve the well-known benchmark problems and comparing the results obtained with those from the best methods in common use. The results indicate that the proposed method is capable of optimally solving 24 open benchmark problems including the famous 10×10 problems. Additionally, it is the first optimal method ever developed to find optimal solutions to some large-scale problems of the size 30×10 and 50×10.

Introduction

Job shop environments, as a generalization of single machine and flow shop environments, are associated with products of a wide variety, each of which has a unique production process. Scheduling in these environments can, therefore, have a considerable impact on the system’s efficiency that can be further enhanced by preemption forming one of the most important scheduling assumptions. Preemption may be allowed in such processes as machining operations in which the amount of processing a job receives is not lost after preemption. Among other applications of preemption is included the assignment of human resources to different project activities. As illustrative examples, reference may be made to Guo et al. [1] who used a special case of this problem in the apparel industry, or to Anderson et al. [2] who, taking the hardware CPU and I/O for machines, used the 2-machine version of the problem to schedule computer systems.

Throughout this paper, a scheduling problem is designated by the triple notation α|β|γ [3]. It must be mentioned that preemption may decrease problem complexity in certain cases but it is not certainly the case with Job Shop Scheduling Problems (JSSP). For example, the two machine pJSSP with only three jobs (J2|n=3,prmp|Cmax) is an NP-hard problem; however, the nonpreemptive version with any arbitrary number of jobs (J2|n=k|Cmax) is solvable in a polynomial time [4]. The high complexity of the pJSSP has led researchers to use heuristic or approximation algorithms to solve the problem. This is verified by the literature review below.

A number of researchers have addressed a special case of the pJSSP with two machines. For example, Sevastianov and Woeginger [5] proposed a polynomial time approximation algorithm with the worst case ratio of 1.5 for the problem J2|prmp|Cmax. Kimbrel and Saia [6] considered this problem in online and offline environments and developed randomized heuristics with the worst case ratio of 2. Anderson et al. [2] studied the same problem and developed an algorithm that yields a schedule of length at most Pmax/2 greater than the optimal schedule length, where Pmax is the length of the longest job.

Others have developed approxim\ation algorithms for the problem in its general form. For instance, Goldberg et al. [7] proposed approximation algorithms with the ratio O(logmμ/loglogmμ) for the problem Jm|prmp|Cmax, where m is the number of machines and μ is the maximum number of operations per job. Bansal et al. [8] developed approximation algorithms with the ratio O(logm/loglogm) for the problems Jm|prmp|Cmax or Jm|pij=1|Cmax. They presented a (2+ε) approximation, in which 0<ε<1, for a constant number of machines and an algorithm with the ratio 1.45 for the problem J2|prmp|Cmax. Jansen et al. [9] presented a polynomial time approximation scheme for JSSP and pJSSP with a fixed number of machines and a fixed number of operations per job. Leighton et al. [10], [11] developed approximation algorithms with a fixed ratio for the problem Jm|pij=1|Cmax.

Another group of authors have addressed pJSSP with some additional assumptions. Jansen et al. [12] studied JSSP and pJSSP with controllable processing times, in the sense that processing time may be decreased at an additional cost. They developed polynomial time algorithms to minimize both makespan and cost. Akcora et al. [13] studied a case of pJSSP in which a reward is given to each operation based on its completion time. Considering the maximization of the total rewards as the objective function, they developed a heuristics using a mathematical model.

Exact solution methods for solving the Jm|prmp|Cmax problem have been rarely reported in the literature if ever. The most important attempt in this field is due to Le Pape and Baptiste in the form of scientific reports [14], conferences presentations [15], and papers [16], [17] during the period from 1994 to 1999. Unlike other researchers, they developed optimal methods in addition to heuristics to solve the problem. Common in all their work is the use of constraint programming and its related techniques. In 1998, they proposed different constraint propagation techniques based on time tables, disjunctive constraints, and edge finding to optimally solve the problem Jm|prmp|Cmax [16]. To the best of our knowledge, this study includes the best records in optimally solving the benchmark pJSSPs. In their following study in1999, they combined different heuristic search strategies with various constraint propagation techniques [17].

More recently, Baptiste et al. [18] have shown that there exists an optimal schedule for preemptive job shop scheduling problems with integral data in which all interruptions and all starting and completion times occur at integral dates. This implies that we can break down each operation into a chain of unit length components. They have also introduced new upper bounds on the minimal number of interruptions.

The desirability of optimal solutions and the unavailability of exact methods reported in the literature for the problem Jm|prmp|Cmax encouraged us to conduct this study aimed at developing efficient, exact methods for the optimal solution of such problems. We recently developed novel mathematical models for the preemptive shop scheduling problems using a commercial software for their solution [19]. The dimension in these models, unlike those of the previously reported ones, depends solely on the number of jobs and machines irrespective of processing times. Here, we develop a branch and bound solution that is capable of solving tougher instances.

In Section 2, the problem Jm|prmp|Cmax is represented by a disjunctive graph which serves as the basis of a branch and bound method presented in Section 3. Dominance rules, lower bounds, and a preference rule are used to improve its efficiency. In Section 4, the method is used to solve the famous benchmark problems [20]. The results are then compared with those obtained from the best methods available. The last Section provides conclusions and suggestions for future studies.

Section snippets

Disjunctive graph

In the pJSSP, there are n jobs and m machines. Each job has its own sequence of operations, and each operation should be processed on a particular machine. The objective is to schedule operations on machines so that the maximum completion time is minimized. Problem assumptions can be stated as follows. Processing times are deterministic and sequence independent. All jobs are ready to be processed at time zero. Only one job can be processed on each machine at a given period of time. Each job

A branch and bound method based on the disjunctive graph

A branch and bound method is introduced here to obtain the optimal solution. As already proved in Section 2, set BSAS constitutes the dominant set for the problem Jm|prmp|Cmax. Hence, the branching scheme is based on the generation of set AS where, by using dominance rules, the generation of schedules belonging to set (AS\BS) is avoided. In this process, scheduling starts from time 0 and jobs are gradually scheduled on machines in a depth-first search until feasible schedules are finally

Computational experiments

There are famous benchmark instances for the families of the problem Jm||Cmax that have been used in different studies to evaluate the efficiency of the proposed methods [24], [25], [26], [27], [28]. These benchmark problems are available in the OR benchmark library [20]. Although these problems are old, a large number of them still await optimal solutions.

As reported in the literature, the rather high complexity of the problem Jm|prmp|Cmax has led most researchers to using heuristic or

Conclusion

In this study, the problem Jm|prmp|Cmax was represented by a disjunctive graph. The graph and problem properties were then used to develop a branch and bound method to find optimal solutions. Dominance rules, lower bounds, and other techniques were also used to increase the efficiency of the method proposed. Finally, the method was used to find optimal solutions for some of the well-known benchmark problems. The results were then compared with those obtained from the best methods so far known.

References (38)

  • CY Zhang et al.

    A very fast TS/SA algorithm for the job shop scheduling problem

    Computers & Operations Research

    (2008)
  • EJ Anderson et al.

    Tighter bounds on preemptive job shop scheduling with two machines

    Computing (Vienna/New York)

    (2001)
  • ML. Pinedo

    Scheduling: theory, algorithms, and systems

    3rd ed.

    (2008)
  • P Brucker et al.

    Preemptive job-shop scheduling problems with a fixed number of jobs

    Mathematical Methods of Operations Research

    (1999)
  • SV Sevastianov et al.

    Makespan minimization in preemptive two machine job shops

    Computing (Vienna/New York)

    (1998)
  • T Kimbrel et al.

    Online and offline preemptive two-machine job shop scheduling

    Journal of Scheduling

    (2000)
  • Goldberg L, Paterson M, Sirinivasan A, Sweedyk E. Better approximation guarantees for job-shop scheduling. In:...
  • N Bansal et al.

    Job shop scheduling with unit processing times

    Mathematics of Operations Research

    (2006)
  • K Jansen et al.

    Makespan minimization in job shops: a linear time approximation scheme

    SIAM Journal on Discrete Mathematics

    (2003)
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