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...FPl||[C.sub.max] problem considered in this note) are frequently encountered in Flexible Manufacturing Systems (FMSs) (see Maleki (1991) for detailed description of FMSs) in which each production stage might be either a flexible machine or a flexible manufacturing cell.
In the case of a flexible machine, it is well documented (Maleki, 1991), that its tool magazine can accommodate in excess of 100 different types of tools. Consequently, in each stage, the operation of a job can be processed by any of the machines having the required combination of tools in its tool magazine resulting in processing flexibility. Since the cost of operating an FMS is high, the objective of minimizing the maximum completion time and thus maximizing the production throughput is of utmost importance.
In the case of flexible manufacturing cells, (see Luggen (1991) for more information on flexible manufacturing cells), each cell consists of a set of flexible machines with multiprocessing capabilities. Each of these cells constitutes a multistation line and the flexible flowshop problem amounts to allocating jobs to multistation lines so that the system throughput is maximized.
Another application of flexible flowshops is in the electronics industry and more specifically in the fabrication of Printed Circuit Boards (PCBs). In that manufacturing process, automated pick-and-place machines are used to place components on the surface of the PCBs. All machines in a stage are equipped with identical component feeders, justifying the identical parallel machine assumption of the flexible flowshop. Since PCBs are usually produced in a batch format, the makespan minimization criterion is suitable for the PCB fabrication problem as well.
A final application of the flexible flowshop problem is in the chemical processing industry. Chemical processes are serial in nature and their equipment is costly to operate, therefore it is important to sequence their operations in fine detail as done in flexible flowshop models.
In summary, in all of the above applications, a multistage production process utilizes flexible machines or specialized machines with shared tool magazines in each stage. As a result, the production system possesses processing flexibility and the flexible flowshop makespan minimization methodology can be implemented to maximize system throughput.
The objective of this note is to analyze an effective easy-to-implement algorithm for the FP3||[C.sub.max] problem in order to address the issue of throughput maximization in the above stated real-life production environments. The algorithm under study was originally proposed by Soewandi and Elmaghraby (2001), to be called S-E from now on. S-E proposed three heuristics for the FP3||[C.sub.max] problem, provided worst-case ratio bounds for two of these heuristics and stated that they were unable to derive a worst-case ratio bound for the third one, the Three-Stage (TS) heuristic. The TS heuristic is based on the idea of reducing the FP3||[C.sub.max] problem to the embedded auxiliary FP3||[C.sub.max] problem (the standard three-stage flowshop...
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