OR Spectrum

, Volume 30, Issue 4, pp 773–785

# Equivalence of the LP relaxations of two strong formulations for the capacitated lot-sizing problem with setup times

• Meltem Denizel
• F. Tevhide Altekin
• Haldun Süral
Regular Article

## Abstract

The multi-item Capacitated Lot-Sizing Problem (CLSP) has been widely studied in the literature due to its relevance to practice, such as its application in constructing a master production schedule. The problem becomes more realistic with the incorporation of setup times since they may use up significant amounts of the available resource capacity. In this paper, we present a proof to show the linear equivalence of the Shortest Path (SP) formulation and the Transportation Problem (TP) formulation for CLSP with setup costs and times. Our proof is based on a linear transformation from TP to SP and vice versa. In our proof, we explicitly consider the case when there is no demand for an item in a period, a case that is frequently observed in the real world and in test problems in the literature. The equivalence result in this paper has an impact on the choice of model formulation and the development of solution procedures.

### Keywords

Production Capacitated lot-sizing Strong formulation Linear relaxation

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### References

1. Alfieri A, Brandimarte P, D’Orazio S (2002) LP-based heuristics for the capacitated lot-sizing problem: the interaction of model formulation and solution algorithm. Int J Prod Res 40(2):441–458
2. Chen WH, Thizy JM (1990) Analysis of relaxations for the multi-item capacitated lot-sizing problem. Ann Oper Res 26:29–72
3. Denizel M, Süral H (2006) On alternative mixed integer programming formulations and LP based heuristics for lot-sizing with setup times. J Oper Res Soc 57(4):389–399
4. Denizel M, Altekin FT, Süral H (2005) Equivalence of the LP relaxation solutions of two MIP formulations for the capacitated lot-sizing problem with setups. Working paper no: SUGSM-05-09, Faculty of Management, Sabanci University, IstanbulGoogle Scholar
5. Eppen GD, Martin RK (1987) Solving multi-item capacitated lot-sizing problems using variable redefinition. Oper Res 35:832–848
6. Jans R, Degraeve Z (2004) Improved lower bounds for the capacitated lot sizing problem with setup times. Oper Res Lett 32(2):185–195
7. Karimi B, Fatemi Ghomi SMT, Wilson JM (2003) The capacitated lot sizing problem: a review of models and algorithms. Omega 31:365–378
8. Krarup J, Bilde O (1977) Plant location, set covering and economic lot size: An O(mn)-algorithm for structured problems. In: Numerische methoden bei optimierungsaufgaben, band 3: Optimierung bei graphentheoritischen ganzzahligen problemen, Birkhauser, pp 155–186Google Scholar
9. Maes J, McClain JO, Van Wassenhove LN (1991) Multilevel capacitated lot sizing complexity and LP-based heuristics. Eur J Oper Res 53(2):131–148
10. Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New YorkGoogle Scholar
11. Stadtler H (1996a) Mixed integer programming model formulations for dynamic multi-item multi-level capacitated lotsizing. Eur J Oper Res 94:561–581
12. Stadtler H (1996b) On the equivalence of LP bounds provided by the shortest route and the simple plant location model formulation for dynamic multi-item multi-level capacitated lot-sizing, Schriften zur Quantitativen Betriebswirtschaftslehre, Technische Hochschule Darmstadt, 5/96, DarmstadtGoogle Scholar
13. Stadtler H (1997) Reformulations of the shortest route model for dynamic multi-item multi-level lotsizing. OR Spektrum 19:87–96Google Scholar
14. Süral H, Denizel M, Van Wassenhove LN (2006) Lagrangean relaxation based heuristics for lot sizing with setup times. Working paper no: SUGSM-06-01, Faculty of Management, Sabanci University, IstanbulGoogle Scholar

## Authors and Affiliations

• Meltem Denizel
• 1
• F. Tevhide Altekin
• 1
• Haldun Süral
• 2