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 a + b t , X ∈ A , B , where b j X is the normal processing time of J j X , a ≥ 0 , b ≥ 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 ∈ C max B , ∑ C j B . 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 = ∑ C j B , a fully polynomial-time approximation scheme is also presented.