Two-Agent Single Machine Scheduling with Deteriorating Jobs and Rejection

Abstract

We consider two two-agent scheduling problems with deteriorating jobs and rejection. Two agents $A$ and $B$ compete for the usage of a single machine. The actual processing time of job $J_j^X$ is $p_j^X=b_j^X\left(a+bt\right)$ , $X\in \left\{A,B\right\}$ , where $b_j^X$ is the normal processing time of $J_j^X$ , $a\ge 0$ , $b\ge 0$ and $t$ denotes the starting time of $J_j^X$ . A job is either rejected by paying a rejection penalty, or accepted and processed on the machine. The objective is to minimize the sum of the completion time of the accepted $A$ -jobs and total rejection penalty of the rejected $A$ -jobs subject to an upper bound on the sum of the given objective function $f^B$ of the accepted $B$ -jobs and total rejection penalty of the rejected $B$ -jobs, where $f^B\in \left\{C_{\max }^B,\sum C_j^B\right\}$ . We give dynamic programming algorithms for them, respectively. When $f^B=C_{\max }^B$ , we present a fully polynomial-time approximation scheme (FPTAS) for the case $a=0$ and $b=1$ . When $f^B=\sum C_j^B$ , a fully polynomial-time approximation scheme is also presented.