Scheduling problems with rejection to minimize the k-th power of the makespan plus the total rejection cost

Abstract

In this paper, we consider several scheduling problems with rejection on $m\ge 1$ identical machines. Each job is either accepted and processed on the machines, or it is rejected by paying a certain rejection cost. The objective is to minimize the sum of the k-th power of the makespan of accepted jobs and the total rejection cost of rejected jobs, where $k>0$ is a given constant. We also introduce the conception of “job splitting" in our problems. First, we consider the single machine scheduling problem, i.e., $m=1$. When job splitting is allowed, we propose an $O(n\log n)$-time optimal algorithm for the problem. When job splitting is not allowed, we show that this problem is polynomially solvable when $k\in (0,1]$ and it becomes binary NP-hard when $k>1$. Furthermore, for the NP-hard problem, we propose a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS). Finally, we also extend our problems and some results to $m\ge 2$ identical parallel machines.