Multiple identical serial-batch machines scheduling with release dates and submodular rejection penalties

Abstract

In this paper we study a scheduling problem with release dates and submodular rejection penalties on multiple (parallel) identical serial-batch machines. For this problem, each machine processes jobs in batches, jobs in a common batch start and finish simultaneously, and the duration of a batch is equal to the sum of a setup time and the total processing time of jobs in it. Some jobs are accepted and processed on the machines, while the left jobs are rejected with penalty and the total rejection penalty is determined by a submodular function. The objective function to be minimized is the sum of the makespan of accepted jobs and the total rejection penalty. For the scenario of an arbitrary number of machines and submodular rejection penalties, we give a 2-approximation algorithm by which the objective function value of the obtained schedule is no more than twice that of the optimal schedule. For the scenario of a fixed number of machines and linear rejection penalties, we give a pseudo-polynomial-time dynamic programming exact algorithm and a fully polynomial-time approximation scheme.