Comparative evaluation of MILP flowshop models

Abstract

This paper investigates the performance of two families of mixed-integer linear programing (MILP) models for solving the regular permutation flowshop problem to minimize makespan. The three models of the Wagner family incorporate the assignment problem while the five members of the Manne family use pairs of dichotomous constraints, or their mathematical equivalents, to assign jobs to sequence positions. For both families, the problem size complexity and computational time required to optimally solve a common set of problems are investigated. In so doing, this paper extends the application of MILP approaches to larger problem sizes than those found in the existing literature. The Wagner models require more than twice the binary variables and more real variables than do the Manne models, while Manne models require more constraints for the same sized problems. All Wagner models require much less computational time than any of the Manne models for solving the common set of problems, and these differences increase dramatically with increasing number of jobs and machines. Wagner models can solve problems containing larger numbers of machines and jobs than the Manne models, and hence are preferable for finding optimal solutions to the permutation flowshop problem with makespan objective.

Keywords:

• flowshop scheduling
• integer programming
• model complexity
• empirical evaluation
• Wagner models
• Manne models

Acknowledgements

Constructive comments from two anonymous reviewers and the Co-Editor, John Wilson, significantly improved the results and presentation of this paper.