The Basic Core of a Parallel Machines Scheduling Game

Abstract

Problem definition: We consider the parallel machine scheduling (PMS) under job-splitting game defined by a set of manufacturers where each holds uniform parallel machines and each is committed to produce some jobs submitted to her by her clients while bearing the cost of the sum of completion times of her jobs on her machines. An efficient algorithm for this scheduling problem is well known. We consider the corresponding cooperative game, where the manufacturers are players that want to join forces. We show that collaboration is profitable. Yet, the stability of the cooperation depends on the cost allocation scheme; we focus on the core of the game. Methodology/results: We prove that the PMS game is totally balanced and its core is infinitely large, by developing a sophisticated methodology of linear complexity that finds a line segment in its symmetric core. We call this segment the basic core of the game. Managerial implications: This PMS game has the potential for various applications both in traditional industry and in distributed computing systems in the hi-tech industry. The formation of a partnership among entrepreneurs, companies, or manufacturers necessitates not only a plan for joining forces toward the achievement of the ultimate goals, but also an acceptable agreement regarding the cost allocation among the partners. Core allocations guarantee the stability of the partnership as no subset of players can gain by defecting from the grand coalition. Funding: This work was supported by the Henry Crown Israeli Institute for Business Research, the Coller Foundation, and the Israel Science Foundation [Grant 1489/19].