Scheduling in an assembly-type production chain with batch transfer☆
Introduction
In this paper we study an assembly-type production scheduling problem, which can be used to model the coordination of production scheduling between cooperative parties in a supply chain. Consider a set of jobs (or products) to be processed from time zero onwards in a two-stage flowshop with three machines (or party firms in a supply chain). In the machine configuration, the first stage has two parallel machines whose outputs, i.e., components or parts, will be transferred to the second-stage machine, which is dedicated to assembly operations. Each job has three specific operations to be performed on the three machines, respectively. Each machine can process at most one operation at a time. No preemption is allowed. Operations on the second-stage machine are processed in batches and a constant setup time is needed whenever a batch is formed. The setup is non-anticipatory, i.e., a setup can commence only when all the parts of the jobs in the same batch are transferred to and available on the assembly machine. Batch availability is assumed for the batch process, i.e., a job is finished when the batch it belongs to is completed as a whole. The objective is to sequence, as well as to group, the jobs to minimize the makespan, i.e., the maximum completion time of all the jobs. Notice that centralized decision making is assumed. That is, the sequencing and batching policies are determined for the assembly machine and applied to the other two machines.
The problem under consideration is related to two well-studied scheduling problems, namely the hybrid flowshop scheduling problem and the batch scheduling problem. Johnson [1] first introduced the flowshop scheduling model and proposed a solution algorithm to minimize the makespan in a two-machine environment. In the past few decades, this seminal work has inspired numerous research endeavors in the scheduling literature [2], [3]. As a generalization of Johnson's two-machine flowshop, the three-machine assembly flowshop problem motivated by the manufacture of fire engines was studied by Lee et al. [4]. Like the three-machine flowshop problem [5], the problem to minimize the makespan in an assembly flowshop becomes computationally intractable. This kind of production setting is not only common in individual manufacturing organizations but also prevalent in supply chains where a manufacturer receives parts or materials from its upstream suppliers for final assembly or packaging. The incorporation of batch considerations into the scheduling model is motivated by the observation that components or parts are usually delivered to a downstream party in batches, such as in full truck loads (FTL). In supply chain management, either centralized or decentralized decision making can be assumed to reflect real situations [6]. In the problem under consideration, we assume that the assembly organization is dominant in the industry and therefore decides the production policies for optimizing the objectives. Also, we assume that the two-part suppliers begin their processing at the same time and their production processes dedicated to the n jobs (orders) must be continuous and cannot be interrupted by any other orders or requests. For heuristic algorithms for the three-machine assembly-type production scheduling, the reader is referred to Sun et al. [7].
Batching is one of the major characteristics of the studied problem. The major advantage of batching is the achievement of gains in operational efficiency that results from setup reductions. Over the past few decades, combining scheduling with batching has received significant research attention. Interest in batch scheduling is due to its relevance to real- world manufacturing and its theoretical challenges. Potts and Van Wassenhove [8], and Webster and Baker [9] have reviewed different batching models. In three recent survey papers by Allahverdi et al. [10], Cheng et al. [11], and Potts and Kovalyov [12], concise and comprehensive reviews on scheduling problems with batching and setup times/costs were presented. Amongst the different models, Lee et al. [13] studied the so-called “burn-in” operations in the semi-conductor industry. In the burn-in model, the processing time of a batch is defined as the longest processing time of the jobs contained in the batch. Ahmadi et al. [14] considered a batch-scheduling problem in a two-machine flowshop, where the processing time of a batch is constant regardless of the number and type of jobs it contains. The batching model considered in this study was previously studied by Albers and Brucker [15], Coffman et al. [16] and Santos and Magazine [17]. In this model the jobs assigned in the same batch require a common setup and their processing is continuous on the machine. Therefore, the processing time of a batch is the setup time plus the total processing times of the jobs belonging to the batch. Following the continuous batching model, Cheng and Wang [18] studied a two-machine flowshop in which the operations on the first machine are processed individually while the operations on the second machine are processed in batches. They showed that the problem is NP-hard and identified some polynomially solvable cases. Cheng et al. [19] considered the same configuration except that both machines process the jobs in batches. They presented strong NP-hardness proofs and developed efficient algorithms for several special cases. Glass et al. [20] studied a scheduling model similar to that of Cheng et al. [19] with anticipatory machine-dependent setup times. Theoretically, the problem we study in this paper concerns a combination of the models presented by Lee et al. [4] and Cheng and Wang [18]. The general case with multiple machines in stage one has been studied by Kovalyov et al. [21]. They proposed a lower bound and a heuristic algorithm, and presented a performance ratio analysis of the heuristic.
This paper is organized into six sections. In Section 2 we present the notation used in this paper and give an example to illustrate the problem definition. In Section 3 we show the strong NP-hardness of the problem. Section 4 is dedicated to studying a special case that is polynomially solvable. In Section 5 we investigate several heuristics for finding approximate solutions. The results of the computational experiments conducted to evaluate the performance of proposed algorithms are discussed. Finally, we give some concluding remarks in Section 6.
Section snippets
Notation and example
In this section we introduce the notation that will be used in this paper. Also, we give a numerical example to illustrate the problem definition. Notation: job set to be processed , : two first-stage machines : second-stage machine : processing time of job i on machine : processing time of job i on machine : processing time of job i on machine : batch setup time : schedule for the job set : makespan of schedule : optimal makespan for job set
The problem
NP-hardness
The problem, where a discrete processor and a batch processor are arranged into a two-machine flowshop, is known to be NP-hard [18]. Therefore, the problem, as a generalization of , is naturally NP-hard, even if either machine or machine is ignored. The special case where all the jobs have the same processing time on the second-stage machine is polynomially solvable for both the problem [18] and the problem [4]. In this
Polynomially solvable case
In this section we deal with a property concerning the polynomial solvability of the problem. Based on the derived results, we further explore a property that is useful for computing a lower bound for optimal solutions. As discussed before, to optimally compose a solution to the problem we need to take batching and sequencing issues into account simultaneously. In this section we assume that for an input instance, a job sequence is given and fixed. Then, we show that an
Heuristic algorithms and computational experiments
The strong NP-hardness result presented in Section 2 hints that it is very unlikely to design efficient algorithms to optimally solve the problem. Furthermore, it is difficult to devise a branching tree for branch-and- bound algorithms due to the fact that the scheduling decisions consist of deciding jointly on how to batch the jobs and how to sequence the batches. Similarly, the design of heuristic and meta-heuristic algorithms may be based upon the theme of considering the
Concluding remarks
This paper addressed the three-machine assembly-type flowshop scheduling problem with batching considerations to minimize the makespan. We first showed that the problem remains NP-hard in the strong sense even when all the jobs have the same processing time on the second-stage machine. We developed an algorithm for optimally grouping jobs in a fixed sequence into batches. A lower bound was established through the use of a data transformation scheme and the above algorithm. To find
Acknowledgements
We are grateful to the anonymous referees for their constructive comments on earlier versions of this paper. We specially thank one of the referees who identified a derivation error in the original proof of Theorem 1.
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The first author was partially supported by the National Science Council under Grant NSC-93-2416-H-009-026. The second author was supported in part by The Hong Kong Polytechnic University under a grant from the Area of Strategic Development in China Business Services.