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

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  • 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 date: 2020-08-23

  Revised date: 2021-09-23

  Online published: 2023-09-07

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

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.

Cite this article

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 . DOI: 10.1007/s40305-021-00373-6

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