Regularized Methods for a Two-Stage Robust Production Planning Problem and its Sample Average Approximation

doi:10.1007/s40305-021-00373-6

Journal of the Operations Research Society of China ›› 2023, Vol. 11 ›› Issue (3): 595-625.doi: 10.1007/s40305-021-00373-6

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Regularized Methods for a Two-Stage Robust Production Planning Problem and its Sample Average Approximation

Jie Jiang¹, Zhi-Ping Chen²

1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China;
2. School of Mathematics and Statistics, Xi'an Jiaotong University; Center for Optimization Technique and Quantitative Finance, Xi'an International Academy for Mathematics and Mathematical Technology, Xi'an 710049, Shaanxi, China

Received:2020-08-23 Revised:2021-09-23 Online:2023-09-30 Published:2023-09-07
Contact: Zhi-Ping Chen, Jie Jiang E-mail:zchen@mail.xjtu.edu.cn;jiangjiecq@163.com
Supported by:
This work was supported by China Postdoctoral Science Foundation (No. 2020M673117), the National Natural Science Foundation of China (Nos. 11991023, 11735011 and 11571270), the World-Class Universities (Disciplines) and the Characteristic Development Guidance Funds for the Central Universities (No. PY3A058).

Abstract

Abstract: In this paper, we consider a two-stage robust production planning model where the first stage problem determines the optimal production quantity upon considering the worst-case revenue generated by the uncertain future demand, and the second stage problem determines the possible demand of consumers by using a utility-based model given the production quantity and a realization of the random variable. We derive an equivalent single-stage reformulation of the two-stage problem. However, it fails the convergence analysis of the sample average approximation (SAA) approach for the reformulation directly. Thus we develop a regularized approximation of the second stage problem and derive its closed-form solution. We then present conditions under which the optimal value and the optimal solution set of the proposed SAA regularized approximation problem converge to those of the single-stage reformulation problem as the regularization parameter shrinks to zero and the sample size tends to infinity. Finally, some preliminary numerical examples are presented to illustrate our theoretical results.

Key words: Production planning, Utility function, SAA, Complementarity problem, Regularized method

CLC Number:

Jie Jiang, Zhi-Ping Chen. Regularized Methods for a Two-Stage Robust Production Planning Problem and its Sample Average Approximation[J]. Journal of the Operations Research Society of China, 2023, 11(3): 595-625.

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References

[1] Rahmani, D., Ramezanian, R., Fattahi, P., Heydari, M.: A robust optimization model for multi-product two-stage capacitated production planning under uncertainty. Appl. Math. Model. 37(20–21), 8957–8971(2013)
[2] Pochet, Y., Wolsey, L. A.: Production Planning by Mixed Integer Programming. Springer, New York (2006)
[3] Aghezzaf, E. H., Sitompul, C., Najid, N. M.: Models for robust tactical planning in multi-stage production systems with uncertain demands. Comput. Oper. Res. 37(5), 880–889(2010)
[4] Alem, D. J., Morabito, R.: Production planning in furniture settings via robust optimization. Comput. Oper. Res. 39(2), 139–150(2012)
[5] Leung, S. C. H., Tsang, S. O. S., Ng, W. L., Wu, Y.: A robust optimization model for multi-site production planning problem in an uncertain environment. Eur. J. Oper. Res. 181(1), 224–238(2007)
[6] Mula, J.: Mathematical programming models for supply chain production and transport planning. Eur. J. Oper. Res. 204(3), 377–390(2010)
[7] Mula, J., Poler, R., García-Sabater, J. P., Lario, F. C.: Models for production planning under uncertainty: a review. Int. J. Prod. Econ. 103(1), 271–285(2006)
[8] Morgenstern, O., Von Neumann, J.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1953)
[9] Dentcheva, D., Ruszczyński, A.: Inverse stochastic dominance constraints and rank dependent expected utility theory. Math. Program. 108(2–3), 297–311(2006)
[10] Dentcheva, D., Ruszczynski, A.: Common mathematical foundations of expected utility and dual utility theories. SIAM J. Optim. 23(1), 381–405(2013)
[11] Fishburn, P. C.: Utility theory. Encyclopedia of Statistical Sciences 14(2004)
[12] Berry, S., Pakes, A.: The pure characteristics demand model. Int. Econ. Rev. 48(4), 1193–1225(2007)
[13] Pang, J. S., Su, C. L., Lee, Y. C.: A constructive approach to estimating pure characteristics demand models with pricing. Oper. Res. 63(3), 639–659(2015)
[14] Chen, X., Sun, H., Wets, R. J. B.: Regularized mathematical programs with stochastic equilibrium constraints: estimating structural demand models. SIAM J. Optim. 25(1), 53–75(2015)
[15] Sun, H., Su, C. L., Chen, X.: SAA-regularized methods for multiproduct price optimization under the pure characteristics demand model. Math. Program. 165(1), 361–389(2017)
[16] Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization, vol. 28. Princeton University Press, Princeton (2009)
[17] Chen, X., Sun, H., Xu, H.: Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems. Math. Program. 177(1–2), 255–289(2019)
[18] Ling, A., Sun, J., Xiu, N., Yang, X.: Robust two-stage stochastic linear optimization with risk aversion. Eur. J. Oper. Res. 256(1), 215–229(2017)
[19] Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation. Optimization 57(3), 395–418(2008)
[20] Shapiro, A.: Monte Carlo sampling methods. Handbooks Oper. Res. Manag. Sci. 10, 353–425(2003)
[21] Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2014)
[22] Cottle, R. W., Pang, J. S., Stone, R. E.: The Linear Complementarity Problem. SIAM, Philadelphia (1992)
[23] Rockafellar, R. T., Wets, R. J. B.: Variational Analysis. Springer, New York (2009)
[24] Burke, J. V., Lewis, A. S., Overton, M. L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15(3), 751–779(2005)
[25] Lewis, A. S., Overton, M. L.: Nonsmooth optimization via quasi-Newton methods. Math. Program. 141(1–2), 135–163(2013)

Regularized Methods for a Two-Stage Robust Production Planning Problem and its Sample Average Approximation

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