Shifting bottleneck scheduling for total weighted tardiness minimization—A computational evaluation of subproblem and re-optimization heuristics
Introduction
The job shop scheduling problem [1] in its standard form is specified by a set of jobs to be processed on a shop floor. The shop floor contains a set of machines , each of which has to be passed exactly once by each job. Accordingly, jobs are split into separate operations with a fixed machine assignment: oik denotes the operation of job i to be processed on machine k and pik its positive integer processing time. The technological order of machines, also called the routing, is predefined and may differ from job to job. Each machine can only process one job at a time and no preemption is allowed.
A feasible schedule is represented by a set of integral operation starting times such that precedence and capacity constraints are satisfied. Given a feasible schedule, let Ci denote the completion time of job i, that is, the completion time of the last operation of the job. Then the tardiness Ti of job i with respect to its due date di is computed as . Given job weights wi, the total weighted tardiness (TWT) is then simply equal to . Using the common three-field notation of [2], the total weighted tardiness job shop scheduling problem is written as .
Minimizing total weighted tardiness on a single machine is known to be NP-hard in the strong sense [3], hence as a generalization of the latter is also classified as strongly NP-hard.
The range of available scientific literature on total weighted tardiness job shops is quite limited compared to the classic makespan (Cmax) problem. Singer and Pinedo [4] proposed a dedicated branch-and-bound algorithm, which remained the only exact approach in this area until now. Later, the same authors presented a shifting bottleneck procedure (SBP), incorporating problem-specific concepts [5]. The same method also serves as a sub-component of a time-window based decomposition approach for large TWT job shops [6]. The shifting bottleneck paradigm has also been applied for more specialized job shop models, as arising from semiconductor manufacturing scenarios: Mason et al. [7] develop a modified shifting bottleneck procedure for complex job shops involving additional characteristics like sequence-dependent setup times and parallel machines. Mönch and Driessel [8] and Mönch et al. [9] investigate a similar kind of problem in this context. Mönch and Zimmermann [10] study the performance of a shifting bottleneck heuristic under stochastic settings in a semiconductor manufacturing environment.
All of the above approaches specifically address the total weighted tardiness performance measure. However, they considerably differ as far as subproblem solving is concerned. A simple priority rule based approach is adopted by Mason et al. [7], while Mönch et al. [9] apply a genetic algorithm based subproblem solver. Scholz-Reiter et al. [11] as well as Bilyk and Mönch [12] propose a variable neighborhood search algorithm for that purpose. Pinedo and Singer [5] rely on an enumeration algorithm of branch-and-bound type for solving single machine subproblems arising from . Bülbül [13] recently proposed a relaxation-based technique for this purpose and integrated it into a shifting bottleneck procedure hybridized with a tabu search heuristic. Braune et al. [14] present a new exact subproblem solver built upon sophisticated problem-specific concepts, particularly dominance rules and lower bounds.
Local search based methods are the second important group of approaches for total weighted tardiness job shops. Singer [15] introduced a cluster-based neighborhood search method. An iterated local search (ILS) algorithm based on the reversal of critical arcs in the graph representation of solutions has been developed by Kreipl [16]. He considers composite moves consisting of at most three related swaps of adjacent operations according to the scheme of Suh [17]. De Bontridder [18] presents a tabu search algorithm for a generalized job shop based on a maximum cost flow model. Relevant work of the recent past includes the genetic local search algorithm of Essafi et al. [19] and the more general local search framework proposed by Mati et al. [20]. A further hybrid genetic algorithm is described in [21]. Kuhpfahl and Bierwirth [22] perform a comprehensive computational comparison of neighborhood structures in the total weighted tardiness context. A simulated annealing approach has been presented by Zhang and Wu [23], while Braune et al. [24] apply an iterated local search algorithm based on advanced approximate move evaluation concepts.
In this paper, several new shifting bottleneck procedures for are presented. All of them rely on a newly proposed heuristic subproblem solving approach which incorporates a considerable amount of problem-specific knowledge. A dominance rule known from the recently published exact subproblem solver [14] is embedded into a systematic improvement scheme for (partial) operation sequences. The sequence improver is then “plugged” into a list scheduling algorithm based on priority dispatching rules. Hence, the resulting procedure can be considered a dominance-based heuristic [25], [26].
While the proposed heuristic is able to deliver near-optimal solutions, it is still advisable to use the exact subproblem solver in some situations. The impact on solution quality and runtime behavior of the superordinate bottleneck scheduler is quantified in an experimental way, based on benchmark instances of job shop type.
The different variants of the proposed bottleneck approach emerge from alternative re-optimization schemes. Besides the conventional, single-machine oriented re-optimization, an iterated local search (ILS) algorithm is employed for improving partial solutions at the job shop level. It is actually used in two configurations: (1) As a standalone re-optimizer and (2) in combination with conventional re-optimization.
The paper is organized as follows: Section 2 describes the disjunctive graph model which serves as the basis for bottleneck scheduling and all related activities. Section 3 gives a brief overview of the shifting bottleneck procedure itself, followed by a detailed coverage of the two main topics of the paper, subproblem solving (cf. Section 4) and re-optimization (cf. Section 5). Comprehensive computational experiments and their discussion are the contents of Section 6. The final Section 7 provides concluding remarks and outlooks on future research.
Section snippets
Disjunctive graph model
In the job shop case, a disjunctive graph [27] can be defined as a triple . V denotes the set of all nodes (vertices), which encompasses not only the ones corresponding to operations but also a source node X and, depending on the objective function, one or more sink nodes. The total weighted tardiness as a so called “min-sum” objective in fact requires the incorporation of n sink nodes , one for each job (cf. Fig. 1). The precedence constraints between operations of the same job
Shifting bottleneck framework
The main steps of a shifting bottleneck procedure (SBP) as initially proposed by Adams et al. [28] for the makespan objective can be considered in an abstract way and thus independent of a particular objective. Fig. 2 provides a flow chart representation of the basic framework of a bottleneck scheduling algorithm. Note that M refers to the set of all machines while M0 denotes the set of already scheduled machines which is continuously updated during the run of the algorithm.
Pinedo [5] and
Heuristic subproblem solving
The subproblems encountered during bottleneck scheduling are of type [5], where symbol refers to the delayed precedence constraints between operations on the same machine and Tzi denotes the tardiness of an operation with respect to a local due date associated with sink Yz in the graph representation. From the job shop point of view, release time and local due date related symbols have to be extended by a machine index k to be unique: The release date of operation oik
Re-optimization
Re-optimizing already scheduled machines is vital for effective shifting bottleneck scheduling, since unfavorable sequencing decisions taken in earlier stages of the scheduling process can still be fixed. Following the notation of [28], let M0 be the set of already scheduled machines and , the order in which those machines have been added to M0. For each , a new subproblem is set up by re-introducing disjunctive arcs between all operations on k(i), while keeping the
Computational results
The computational study presented in this section is split into two stages. First, a performance analysis of the heuristics introduced in Section 4 is presented. The experiments have been carried out at the subproblem level, using single machine instances occurring at different phases of a shifting bottleneck based optimization run. The final integration of the heuristic subproblem solver into a shifting bottleneck procedure and the combination with different re-optimization schemes is the
Conclusion and perspectives
The subject of this paper has been the presentation and a computational study of enhanced shifting bottleneck heuristics in the context of total weighted tardiness minimization in job shop scheduling. The proposed enhancements are related to the following two major aspects:
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Subproblem solving: Various approaches for improving subproblem sequences have been devised. These approaches are based on dedicated dominance principles and can either be embedded in a list scheduling algorithm or used for
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