Two-agent single-machine scheduling to minimize resource consumption with a constraint on the makespan

In this paper, we introduce a new scheduling model in which both two agents and resource consumption exist simultaneously. Two agents compete to perform their respective jobs on a common single machine and each agent has his own criterion to optimize. The starting time of each A-agent's job is related to the amount of resource consumed. The objective is to minimize the total amount of resource consumption of agent A with the restriction that the makespan of agent B cannot exceed a given bound U. The optimal properties and the optimal polynomial time algorithm are proposed to solve the scheduling problem