Minimizing the mean weighted absolute deviation from due dates in lot-streaming flow shop scheduling
Introduction
Traditional batch production, often with the support of an enterprise resource planning (ERP) system, transfers each lot (batch) from one machine (resource) to the next only when the operation is done. Thus, an idle downstream machine has to wait for a lot to be transferred when the lot on the upstream machine is not finished. Some solutions to this waiting problem (e.g., just-in-time, group technology) require a major reorganization of the way business is done. The use of sublots (or transfer batches) is a good alternative solution because changes in scheduling can be implemented in less time and at a lower cost than a reorganization [1]. The process of splitting a lot into a number of smaller sublots to allow the overlapping of successive operations in multi-stage production systems is called lot-streaming [2], [3]. This process is illustrated by the single job (lot), three equal-sublot, three machine lot-streaming flow shop with job processing times of 6, 3 and 9 time units shown in Fig. 1. The due date is in 12 time units. If the job is not split into sublots, the job completion time will be 18 time units and the job will be 6 units tardy (schedule 1). As Fig. 1(b) shows, when the lot is split into three equal-size sublots, the completion time is reduced to 12 units and the job can be delivered on time (schedule 2).
This paper presents a solution methodology for the n-job, m-machine lot-streaming flow shop scheduling problem which objective is to minimize the mean weighted absolute deviation from due dates. This objective emphasizes timely delivery and is consistent with the just-in-time philosophy. For a given sequence, the insertion of idle times between sublots and between jobs may improve the objective value in some cases. Thus, the solution methodology has two stages: the sequencing of jobs and the insertion of idle times between sublots and between jobs. Following the literature review in Section 2, linear programming formulations of four different lot-streaming flow shop problems are presented in Section 3. A linear programming formulation for the equal-size sublot problem with infinite capacity buffers between successive machines to find the optimal sublot completion times for a given job sequence is presented first. Since buffers between successive machines are usually limited in real flow shops, a linear programming formulation for the limited capacity buffer problem is presented next. In many practical situations, frequently encountered in the metal processing industry (particularly where metal is rolled while it is hot), delays between operations are prohibited [4]. A flow shop without delays between successive operations is called no-wait flow shop [5]. A linear programming formulation for the no-wait flow shop problem is also presented. A sublot is called consistent if the sublot size is the same at each machine [6], [1]. A linear programming formulation for the consistent sublot problem is finally presented. Pairwise interchange methods based on several rules to generate different initial sequences and neighborhood search mechanisms used to find the best sequences are presented in Section 4. In Section 5, computational results are provided for the cases of equal-size sublots, consistent sublots, infinite capacity buffers, and the no-wait flow shop. Finally, a summary of main results and conclusions are presented in Section 6.
Section snippets
Literature review
Much progress has been made in the small size lot-streaming scheduling area and most researches have focused on minimizing the makespan [7], [8]. The importance of adopting sublots in batch manufacturing was pointed out by Jacobs [9], and Graves and Kostreva [10].
Baker and Pyke [11] derived optimal consistent sublot sizes for the single-job, multiple-machine, two sublot lot-streaming flow shop problem. They also presented heuristic approaches using a Campbell et al. [12] for problems with more
Linear programming formulations for a given sequence
Schedules are assumed to be semi-active [24], [25], [26]. A semi-active schedule is the one in which starting sublots earlier or later does not improve the objective of the schedule. For job , let sj be the number of sublots, dj the due date, αj the earliness penalty, βj the tardiness penalty, pi,j the processing time on machine i, and pi,j,k the processing time of sublot k on machine i. Let ej and tj represent the earliness and tardiness of job j, respectively. If the completion time
Pairwise interchange methods
In scheduling problems, the computational effort required to solve a problem grows remarkably fast as the number of jobs increases. Thus, if the number of jobs is large, it might not be practical to solve the problem exactly when computing capability is expensive or when scheduling decisions need to be resolved in minutes or hours. In such cases, it is reasonable to consider heuristic procedures, such as pairwise interchange (PI) methods which can provide a near-optimal solution in reasonable
Computational results
The linear programming formulations and heuristic algorithms described in 3 Linear programming formulations for a given sequence, 4 Pairwise interchange methods, respectively, were coded in Visual FORTRAN with the IMSL library and run on a Pentium III PC. Since no sample problems were found in the literature that could be used as a benchmark for testing the heuristics, the test problems were generated randomly as follows. The size of each test problem was represented by the number of
Summary and conclusions
This paper has addressed the n-job, m-machine lot-streaming scheduling problem considering various types of buffers and sublots. Linear programming formulations for a given job sequence have been presented for this problem. The no-wait lot-streaming flow shop has also been considered. It is not generally possible to solve medium and large size problems of this type optimally due to the large solution space. Thus, only a subset of this space can be searched to obtain a good solution in a
Suk-Hun Yoon is a Ph.D. student in the Department of Industrial and Manufacturing Engineering at The Pennsylvania State University. He received a B.S. degree in Industrial Engineering from Seoul National University, Korea and a M.S. degree in Industrial Engineering from North Carolina A&T State University, Greensboro. His research interests include mathematical programming, production scheduling, metaheuristic algorithms, and image recognition.
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Suk-Hun Yoon is a Ph.D. student in the Department of Industrial and Manufacturing Engineering at The Pennsylvania State University. He received a B.S. degree in Industrial Engineering from Seoul National University, Korea and a M.S. degree in Industrial Engineering from North Carolina A&T State University, Greensboro. His research interests include mathematical programming, production scheduling, metaheuristic algorithms, and image recognition.
Jose A. Ventura received his Ph.D. in Industrial and Systems Engineering from the University of Florida. He is a Professor of Industrial and Manufacturing Engineering at The Pennsylvania State University. His research and teaching interests include mathematical programming, production scheduling, facility layout and location, and machine vision. He is a member of IIE, INFORMS and CIC-MHE.