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The single-item lot-sizing polytope with continuous start-up costs and uniform production capacity

The single-item lot-sizing polytope with continuous start-up costs and uniform production capacity In this work we consider the uniform capacitated single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. In this work we explore a natural integer programming formulation for this problem. We consider the polytope obtained as convex hull of the feasible points in this problem. We state some general properties, study whether the model constraints define facets, and present an exponentially-sized family of valid inequalities for it. We analyze the structure of the extreme points of this convex hull, their adjacency and bounds for the polytope diameter. Finally, we study the particular case when the demands are high enough in order to require production in all the periods. We provide a complete description of the convex hull of feasible solutions in this case and show that all the inequalities in this description are separable in polynomial time, thus proving its polynomial time solvability. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Operations Research Springer Journals

The single-item lot-sizing polytope with continuous start-up costs and uniform production capacity

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References (9)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Business and Management; Operation Research/Decision Theory; Combinatorics; Theory of Computation
ISSN
0254-5330
eISSN
1572-9338
DOI
10.1007/s10479-015-1915-4
Publisher site
See Article on Publisher Site

Abstract

In this work we consider the uniform capacitated single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. In this work we explore a natural integer programming formulation for this problem. We consider the polytope obtained as convex hull of the feasible points in this problem. We state some general properties, study whether the model constraints define facets, and present an exponentially-sized family of valid inequalities for it. We analyze the structure of the extreme points of this convex hull, their adjacency and bounds for the polytope diameter. Finally, we study the particular case when the demands are high enough in order to require production in all the periods. We provide a complete description of the convex hull of feasible solutions in this case and show that all the inequalities in this description are separable in polynomial time, thus proving its polynomial time solvability.

Journal

Annals of Operations Research – Springer Journals

Published: Jun 7, 2015

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