Proportionate flow shop games

In a proportionate flow shop problem several jobs have to be processed through a fixed sequence of machines and the processing time of each job is equal on all machines. By identifying jobs with agents whose costs linearly depend on the completion time of their jobs and assuming an initial processing order on the jobs, we face two problems: the first is how to obtain an optimal order that minimizes the total processing cost, the second is how to allocate the cost savings obtained by ordering the jobs optimally. In this paper we focus on the allocation problem. PFS games are defined as cooperative games associated to proportionate flow shop problems. It is seen that PFS games have a nonempty core. Moreover, it is shown that PFS games are convex if the jobs are initially ordered in decreasing urgency. For this case an explicit game independent expression for the Shapley value is provided. The authors thank two referees for their valuable suggestions for improvement. M.A. Mosquera acknowledges the financial support of Ministerio de Ciencia y Tecnología, FEDER, Xunta de Galicia (projects SEJ2005-07637-C02-02 and PGIDIT06PXIC207038PN). A. Estévez-Fernández ( ) Department of Econometrics and Operations Research, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands e-mail: [email protected] M.A. Mosquera Department of Statistics and Operations Research, Faculty of Business Administration and Tourism, University of Vigo, 32004 Ourense, Spain P. Borm · H. Hamers CentER and Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands