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

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## 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.

### References

- Beraldi, P., & Ruszczyǹnski, A. (2002). A branch and bound method for stochastic integer problems under probabilistic constraints.
*Optimization Methods and Software*,*17*(4), 359–382.CrossRefGoogle Scholar - Bonferroni, C. E. (1936). Teoria statistica delle classi e calcolo delle probabilita.
*Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze*,*8*, 3–62.Google Scholar - Bookbinder, J. H., & Tan, J. Y. (1988). Strategies for the probabilistic lot-sizing problem with service-level constraints.
*Management Science*,*34*(9), 1096–1108.CrossRefGoogle Scholar - Buschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. (2010). Dynamic capacitated lot-sizing problems: Classification and review of solution approaches.
*OR Spectrum*,*32*, 231–261.CrossRefGoogle Scholar - Calafiore, G., & Campi, M. C. (2005). Uncertain convex programs: Randomized solutions and confidence levels.
*Mathematical Programming*,*102*, 25–46.CrossRefGoogle Scholar - Chen, H. (2007). A Lagrangian relaxation approach for production planning with demand uncertainty.
*European Journal of Industrial Engineering*,*1*(4), 370–390.CrossRefGoogle Scholar - Cheng, J., Lisser, A., & Gicquel, C. (2014).
*A new partial sample average approximation method for chance-constrained problems*. Working paper. http://www.optimization-online.org/DB_HTML/2014/11/4622.html. - Di Summa, M., & Wolsey, L. A. (2008). Lot-sizing on a tree.
*Operations Research Letters*,*36*, 7–13.CrossRefGoogle Scholar - Guan, Y., Ahmed, S., Miller, A. J., & Nemhauser, G. L. (2006). On formulations of the stochastic uncapacitated lot-sizing problem.
*Operations Research Letters*,*34*, 241–250.CrossRefGoogle Scholar - Haugen, K. K., Lokketangen, A., & Woodruff, D. L. (2001). Progressive hedging as a metaheuristic applied to stochastic lot-sizing.
*European Journal of Operational Research*,*132*, 116–122.CrossRefGoogle Scholar - Jans, R., & Degraeve, Z. (2008). Modeling industrial lot sizing problems: A review.
*Industrial Journal of Production Research*,*46*(6), 1619–1643.CrossRefGoogle Scholar - Küçükyavuz, S. (2012). On mixing sets arising in chance-constrained programming.
*Mathematical Programming*,*132*, 31–56.CrossRefGoogle Scholar - Leung, J., Magnanti, T. L., & Vachani, R. (1989). Facets and algorithms for capacitated lot sizing.
*Mathematical Programming*,*45*, 331–359.CrossRefGoogle Scholar - Luedtke, J., & Ahmed, S. (2008). A sample approximation approach for optimization with probabilistic constraints.
*SIAM Journal of Optimization*,*19*(2), 674–699.CrossRefGoogle Scholar - Luedtke, J., Ahmed, S., & Nemhauser, G. L. (2010). An integer programming approach for linear programs with probabilistic constraints.
*Mathematical Programming*,*122*, 247–272.CrossRefGoogle Scholar - Pochet, Y., & Wolsey, L. A. (2006).
*Production planning by mixed integer programming*. Berlin: Springer.Google Scholar - Rockafellar, R. T., & Uryasev, S. P. (2000). Optimization of conditional value-at-risk.
*Journal of Risk*,*2*, 21–41.CrossRefGoogle Scholar - Nemirovski, A., & Shapiro, A. (2005). Scenario approximations of chance constraints. In G. Calafiore & F. Dabbene (Eds.),
*Probabilistic and randomized methods for design under uncertainty*(pp. 3–48). London: Springer.Google Scholar - Nemirovski, A., & Shapiro, A. (2006). Convex approximations of chance constrained programs.
*SIAM Journal of Optimization*,*17*, 969–996.CrossRefGoogle Scholar - Piperagkas, G. S., Konstantaras, I., Skouri, K., & Parsopoulo, K. E. (2012). Solving the stochastic dynamic lot-sizing problem through nature-inspired heuristics.
*Computers and Operations Research*,*39*, 1555–1565.CrossRefGoogle Scholar - Tarim, S. A., & Kingsman, B. G. (2004). The stochastic dynamic production/inventory lot-sizing problem with service-level constraints.
*International Journal of Production Economics*,*88*, 105–119.CrossRefGoogle Scholar - Tempelmeier, H. (2007). On the stochastic uncapacitated dynamic single-item lotsizing problem with service level constraints.
*European Journal of Operational Research*,*181*, 184–194.CrossRefGoogle Scholar - Tempelmeier, H., & Herpers, S. (2011). Dynamic uncapacitated lot sizing with random demand under a fillrate constraint.
*European Journal of Operational Research*,*212*, 497–507.CrossRefGoogle Scholar - Vargas, V. (2009). An optimal solution for the stochastic version of the Wagner–Whitin dynamic lot-size model.
*European Journal of Operational Research*,*198*, 447–451.CrossRefGoogle Scholar - Wagner, H. M., & Whitin, T. M. (1958). Dynamic version of the economic lot size model.
*Management Science*,*5*(1), 89–96.CrossRefGoogle Scholar - Zhang, M., Küçükyavuz, S., & Goel, S. (2014). A branch and cut method for dynamic decision making under joint chance constraints.
*Management Science*,*60*(5), 1317–1333.CrossRefGoogle Scholar