# A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand

## Abstract

We study the single-item single-resource capacitated lot-sizing problem with stochastic demand. We propose to formulate this stochastic optimization problem as a joint chance-constrained program in which the probability that an inventory shortage occurs during the planning horizon is limited to a maximum acceptable risk level. We investigate the development of a new approximate solution method which can be seen as an extension of the previously published sample approximation approach. The proposed method relies on a Monte Carlo sampling of the random variables representing the demand in all planning periods except the first one. Provided there is no dependence between the demand in the first period and the demand in the later periods, this partial sampling results in the formulation of a chance-constrained program featuring a series of joint chance constraints. Each of these constraints involves a single random variable and defines a feasible set for which a conservative convex approximation can be quite easily built. Contrary to the sample approximation approach, the partial sample approximation leads to the formulation of a deterministic mixed-integer linear problem having the same number of binary variables as the original stochastic problem. Our computational results show that the proposed method is more efficient at finding feasible solutions of the original stochastic problem than the sample approximation method and that these solutions are less costly than the ones provided by the Bonferroni conservative approximation. Moreover, the computation time is significantly shorter than the one needed for the sample approximation method.

## Keywords

Stochastic lot-sizing Chance-constrained programming Joint probabilistic constraint Sample approximation approach Mixed-integer linear programming## Notes

### Acknowledgements

The work described in the present paper was partly funded by the French National Research Agency through its research funding program for young researchers (Project ANR-11-JS0002-01 LotRelax). The authors would also like to thank two anonymous referees for their detailed reviews that helped improving an initial version of this paper.

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