Pareto optimization for the two-agent scheduling problems with linear non-increasing deterioration

In this paper, we investigate two single-machine scheduling problems for two competing with linear non-increasing deterioration. Each agent wants to optimize its own objective which depends on the completion times of its jobs only and the restriction of linear non-increasing deterioration means that the actual processing time of a job will decrease linearly with the starting time. The objective functions we consider in this paper are the maximum earliness cost and total earliness cost. When each of the two agents has the maximum earliness cost as the objective function, we design a strongly polynomial-time algorithm to get all pareto-optimal pairs for the two-agent scheduling problem. When one agent has the maximum earliness cost as the objective function and the other agent has the total earliness cost as the objective function, a strongly polynomial-time algorithm is also presented to find all of the pareto-optimal pairs for the corresponding two-agent scheduling problem.