A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

Authors:
Yongpei Guan;Shabbir Ahmed;George L. Nemhauser;Andrew J. Miller
Affiliations:
School of Industrial & Systems Engineering, Georgia Institute of Technology, 30332, Georgia, GA, USA;School of Industrial & Systems Engineering, Georgia Institute of Technology, 30332, Georgia, GA, USA;School of Industrial & Systems Engineering, Georgia Institute of Technology, 30332, Georgia, GA, USA;Department of Industrial Engineering, University of Wisconsin, 53706, Madison, WI, USA
Venue:
Mathematical Programming: Series A and B
Year:
2006

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Abstract

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (ℓ,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (ℓ,S) inequalities to a general class of valid inequalities, called the ** inequalities, and we establish necessary and sufficient conditions which guarantee that the ** inequalities are facet-defining. A separation heuristic for ** inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the ** inequalities as cuts.