Proportionate Flow Shop Scheduling with Multi-agents to Maximize Total Gains of JIT Jobs

Abstract

Different variants of the multi-agent scheduling have been studied in the literature due to its wide applications in artificial intelligence, decision theory, operations research, etc. Most of previous research focused on the single-machine environment and two-agent scheduling. In this paper, we address a multi-agent scheduling problem on a set of m machines in a proportionate flow shop system, where the job processing times are machine independent. Each agent desires to maximize its total gains of JIT jobs which are completed exactly at the due dates. The goal is to find a feasible schedule in which each agent’s cost function value does not less than a given lower bound. When the number of agents is part of the input, we use the reduction method to show that the general problem is strongly $$\mathcal {NP}$$ -complete even if all jobs have unit processing times. When the number of agents is fixed, we first develop a dynamic programming algorithm that runs in pseudo-polynomial time, then we design a fully polynomial time approximation scheme by exploiting the technique of trimming-the-state-space. The results presented in this paper imply that by relaxing each agent’s desired cost function value a small fraction, we can obtain an efficient approximate schedule for the problem with fixed number of agents in polynomial time, while when the number of agents is part of the input, the problem become much intractable, and it needs more sophisticated methods to solve in future research.