Submodular batch scheduling on parallel machines

Abstract

This article studies a submodular batch scheduling problem motivated by the vacuum heat treatment. The batch processing time is formulated by a monotone nondecreasing submodular function characterized by decreasing marginal gain property. The objective is to minimize the makespan. We show the NP‐hardness of the problem on a single machine and of finding a polynomial‐time approximation algorithm with the worst‐case performance ratio strictly less than for the problem on parallel machines. We introduce a bounded interval to model the batch processing time using two parameters, that is, the total curvature and the quantization indicator. Based on the decreasing marginal gain property and the two parameters, we make a systematic analysis of the full batch longest processing time algorithm and the longest processing time greedy algorithm, and propose the instances with the bound of batch capacity for these two algorithms for the submodular batch scheduling problem. Moreover, we prove the submodularity of batch processing time function of the existing batch models including the parallel batch, serial batch, and mixed batch models. We compare the worst‐case performance ratios in the existing batch models with those deduced from our work in the submodular batch model. In most situations, the worst‐case performance ratios deduced from our work are comparable to the best‐known worst‐case performance ratios with tailored examinations.