Multi-agent Scheduling With Release Date

Scheduling is to assign some tasks to time resources under some constraints, such that one- or multi-criteria attain to the optimum. In recent years, management problems in which multiple agents compete on the usage of a common processing resource are receiving increasing attention in different application enviomments and different methodological fields. The agents have to schedule their jobs on a common processing resource, and each agent wishes to minimize its objective function.In this paper, we consider the scheduling problems arising when two agents, each with a set of nonpreemptive jobs, compete to perform their respective jobs on a common processing resource. Each agent wants to minimize a certain objective function, which depends on the completion times of its jobs only. The problem is how to compute schedules which account for each agents cost function, and that can be used to support the negotiation between the two agents. There are two competing agents, called agent A and agent B. The agent A has to excute the job set J^A={J₁^A, J₂^A,…,J_nA^A}, whereas the agent B has to excute the job set J^B = {J₁^B, J₂^B,…,J_nB^B}. We call A-jobs and B-jobs the jobs of the two sets. B-jobs have nonnegative release dates, and A-jobs have zero release dates. The objective function is to minimize the maximum of completion time. Therefore, the scheduling problem which we consider can be written as (?), in which C_max^A and C_max^B are the objective function of agent A and agent B, repectively. Letσindicate a feasible schedule, in which, C_max^A(σ) is minimum, and agent B will acceptσof cost up toQ.In Agnetis et al.(2004, scheduling problems with two competing Agents), the results are established under the assumption that all the jobs have a common release date 0. Hence, the research in this paper is an extention of the literature, without considering the release dates . The main results of this paper are as follows:(1) For the model (?), we prove that it is binary NP-hard.(2) For the model (?), we provide a pseudo-polynomial time algorithm. (3) For the model (?), we provede a full polynomial-time approximation time scheme.(4) For the model (?), we prove that it is binary NP-hard.