Some new problems on two-agent scheduling to minimize the earliness costs

We consider several two-agent scheduling problems, where agents A and B have to share a single machine while processing their jobs. The objective is to minimize a certain objective function which depends on the completion time of all the jobs, while keeping the objective of agent B (with regard to its jobs only) below or at a fixed level Q. Specifically, we focus on minimizing certain objective functions of all the jobs such as maximum earliness cost, total weighted earliness cost, and total weighted earliness and tardiness cost, subject to an upper bound on a certain objective function of agent B. We introduce polynomial time solution for the maximum earliness cost problem, and prove NP-hardness for the total weighted earliness cost and total weighted earliness and tardiness cost cases. We also discuss some polynomially solvable cases for the total weighted earliness cost problem.