Family Sequencing and Cooperation

Abstract

This paper analyzes a single-machine scheduling problem with family setup times both from an optimization and a cost allocation perspective. In a so-called family sequencing situation jobs are processed on a single machine, there is an initial processing order on the jobs, and every job within a family has an identical cost function that depends linearly on its completion time. Moreover, a job does not require a setup when preceded by another job from the same family while a family specific setup time is required when a job follows a member of some other family. Explicitly taking into account admissibility restrictions due to the presence of the initial order, we show that for any subgroup of jobs there is an optimal order, such that all jobs of the same family are processed consecutively. To analyze the allocation problem of the maximal cost savings of the whole group of jobs, we define and analyze a so-called corresponding cooperative family sequencing game which explicitly takes into account the maximal cost savings for any coalition of jobs. Using nonstandard techniques we prove that each family sequencing game has a non-empty core by showing that a particular marginal vector belongs to the core. Finally, we specifically analyze the case in which the initial order is family ordered.

Multiple-Machine Scheduling with Learning Effects and Cooperative Games

“…Sequencing games have been extended by many researchers. The extensions focus on the allocation rules [20,21], the number of jobs [22,23], the admissible rearrangements of jobs [24,25], the initial order of jobs [26,27], the family or batch sequencing situations [28,29], the ready times [30], and the due-dates [31]. Convexity and balancedness of the corresponding sequencing games are the main goals of these studies as convex games and balanced games have nice properties.…”