Family Sequencing and Cooperation
CentER Discussion Paper Series No. 2012-040
28 Pages Posted: 17 May 2012
Date Written: May 16, 2012
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.
Keywords: Single-machine scheduling, Family scheduling model, Setup times, Cooperative Game, Core, Marginal Vector
JEL Classification: C71
Suggested Citation: Suggested Citation