Minimizing Makespan for Hybrid Flowshops with Batch, Discrete Processing Machines and Arbitrary Job Sizes

摘要

This research aims at a study of the hybrid flow shop problem which has parallel batch-processing machines in one stage and discrete-processing machines in other stages to process jobs of arbitrary sizes. The objective is to minimize the makespan for a set of jobs. The problem is denoted as: FF|batch1, sj|Cmax.The problem is formulated as a mixed-integer linear program. The commercial solver, AMPL/CPLEX, is used to solve problem instances to their optimality. Experimental results show that AMPL/CPLEX requires considerable time to find the optimal solution for even a small size problem, i.e., a 6-job instance requires 2 hours in average.A bottleneck-first-decomposition heuristic (BFD) is proposed in this study to overcome the computational (time) problem encountered while using the commercial solver. The proposed BFD heuristic is inspired by the shifting bottleneck heuristic. It decomposes the entire problem into three sub-problems, and schedules the sub-problems one by one. The proposed BFD heuristic consists of four major steps: formulating sub-problems, prioritizing sub-problems, solving sub-problems and re-scheduling. For solving the sub-problems, two heuristic algorithms are proposed; one for scheduling a hybrid flow shop with discrete processing machines, and the other for scheduling parallel batching machines (single stage). Both consider job arrival and delivery times. An experiment design is conducted to evaluate the effectiveness of the proposed BFD, which is further evaluated against a set of common heuristics including a randomized greedy heuristic and five dispatching rules. The results show that the proposed BFD heuristic outperforms all these algorithms.To evaluate the quality of the heuristic solution, a procedure is developed to calculate a lower bound of makespan for the problem under study. The lower bound obtained is tighter than other bounds developed for related problems in literature.A meta-search approach based on the Genetic Algorithm concept is developed to evaluate the significance of further improving the solution obtained from the proposed BFD heuristic. The experiment indicates that it reduces the makespan by 1.93% in average within a negligible time when problem size is less than 50 jobs.