A note on two-agent single-machine scheduling problem with deteriorating jobs

This paper considers the two-agent scheduling problems with decreasing linear deterioration on a single machine. Two agents compete to perform their respective jobs on a common single machine, and each agent has his own criterion to optimize. The processing times of jobs are defined as a decreasing linear function of their starting times. The goal is to minimize the maximum cost of the first agent with the restriction that the makespan of the second agent cannot exceed a given upper bound. We propose the optimal properties and present the optimal polynomial time algorithm to solve the scheduling problem.