Two competitive agents to minimize the weighted total late work and the total completion time

Abstract

This paper studies deterministic constraint optimization problem with two competitive agents in which the following objective functions on a single machine: the total weighted late work and the total completion time. We show that the constraint optimization problem is the binary NP-hard by Knapsack problem reduction. Furthermore, we present a pseudo-polynomial time algorithm by early due date maximum not-late sequence, and an approximation Pareto curve by dynamic programming algorithm and two eliminated states, which time complexity of the two approximation algorithms are O(nA2nBQ∑(pjA+pjB)) and O(n4θ2logUBAlogUBB), where pj,θare processing time of job Jj, a given positive constant, and UBx an upper bound of the objective function of agent x,x∈{A,B}. Finally, we present a simple approximation algorithm by the earliest due date (EDD) rule, which jobs of agent B are assigned an dummy due date.