No-wait flexible flowshop scheduling with no-idle machines☆
Introduction
Melting and casting are two key operations in iron and steel manufacturing, which correspond to the two stages of a flowshop. In the production process, jobs are converted from molten steel into billets. The molten steel is produced in the first stage and transported into the second stage to be cast. Except of the transportation time, no extra waiting time is allowed between the two stages to keep the molten steel in a high temperature. Furthermore, no idle time should be allowed between any two consecutive processed jobs on each machine in the second stage to keep the billets cast continuously. In practice, all items almost need the same processing times in the melting stage, and items’ processing times in the casting stage may be different according to the different product requests. And the processing time of each job in the second stage is no smaller than that in the first stage. Due to the fact that processing time spent in the second stage is more than that in the first stage, a factory in real case provides a melting and casting flexible system with three and four machines in the two stages, respectively.
In this paper, we present the system with m and machines in the two stages, respectively. The objective is to minimize the maximum completion time . We describe the scheduling problem as .
In the problem, the two-stage operations are processed in two machine centers with m and parallel machines, respectively. Denote the machines by in center and in center . Jobs will be processed in centers and in turn without any delay. The processing time of job j () is in center and in center . Each machine in center has no idle time between any two sequential processed jobs.
A no-wait flexible flowshop represents a generalization of the no-wait flowshop and the identical parallel machine shop, which have been extensively studied. The efficient algorithms for minimizing in a no-wait flexible flowshop are not likely to exist. More details can be found in Hall and Sriskandarajah [4] and Mokotoff [6]. For no-wait and no-idle shop, Giaro [2] presents that even many trivial questions about the existence of schedule are NP-hard.
In this paper, we discuss the complexity of the problem in Section 2. In Section 3, we present a heuristic algorithm with worst case error bounding analysis. Conclusions are given in Section 4.
Section snippets
Complexity
In this section, we make use of the results that the well-known Partition Problem (PP) is NP-complete [5] and 3-Partition is strongly NP-hard [1].
Partition Problem (PP). Given a sequence of numbers summing up to Q, there exists a subsequence summing up to ?
3-Partition. Given a number B and a -element set of numbers such that , , does there exist a partition of it into n 3-element subsets each with sum B?
A problem named General 3-Partition is
Algorithm and error bounding analysis
The following algorithm named MLPT is a modification of LPT [3]. As the processing times of all jobs in the first stage are the same, processing times of the second stage are mainly considered in the algorithm. In the algorithm, a machine is said active at time t if the machine works well following the no-idle constraint before time t. Otherwise, the machine is said inactive at time t. A machine in a state of inactive at time t will never work after this time.
Conclusions
There are some remaining problems for further research: Is the problem strongly NP-hard when ? Is there an algorithm which can competitive with MLPT in the worst case version?
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A discrete Water Wave Optimization algorithm for no-wait flow shop scheduling problem
2018, Expert Systems with ApplicationsCitation Excerpt :In other words, in order to satisfy the no-wait constraints, the starting time of a job on a certain machine may have to be postponed. The main reason for the research of NWFSP is the technology requirement in some manufacturing environment, such as in conventional industries including steel rolling (Aldowaisan & Allahverdi, 2012), food processing (Hall & Sriskandarajah, 1996), chemical industry (Rajendran, 1994) and pharmaceutical industry (Raaymakersa, 2000) and in some advanced manufacturing systems including just-in-time production systems (Shabtay, 2012), flexible manufacturing systems(Z. Wang, Xing, & Bai, 2005) and robotic cells (Agnetis, 2000). The previous research (Röck, 1984) has proved that the NWFSP is a NP-hard problem with minimizing the makespan when the number of machines is more than two.
A survey of scheduling problems with no-wait in process
2016, European Journal of Operational ResearchCitation Excerpt :Furthermore, Kubzin and Strusevich (2005) addressed the F2/no-wait/Cmax problem, where one of the machines is subject to maintenance, and provided a polynomial time approximation scheme. Wang, Xing, and Bai (2005) considered the FF2/no-wait/Cmax problem such that there is no idle time between two consecutive processed jobs on the machines at the second stage. They investigated the complexity of the problem and presented a heuristic algorithm with asymptotically tight error bounds.
An improved iterated greedy algorithm with a Tabu-based reconstruction strategy for the no-wait flowshop scheduling problem
2015, Applied Soft Computing JournalCitation Excerpt :Application domains of the NWFSP include chemical processing [2], plastic molding [3], food processing [3] and steel rolling [4]. Apart from these conventional industries, the NWFSP is also important for some advanced manufacturing systems such as just-in-time production systems [5], flexible manufacturing systems [6], and robotic cells [7] where jobs are continuously processed with no in-process waiting time. In this study, we consider the NWFSP with the objective of minimizing the makespan.
A discrete time exact solution approach for a complex hybrid flow-shop scheduling problem with limited-wait constraints
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2011, Expert Systems with ApplicationsCitation Excerpt :For the problem of parallel machine job shops, most of the earlier and much of the current work focuses on presenting heuristics. The problem of scheduling a two-stage no-wait/no-idle flexible flow shop was considered by Wang, Xing, and Bai (2005). Their system had m and m + 1 machines in the two stages, respectively.
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Partially supported by Grant 9732003-12-124 and NSFC 70471008.