(761g) A Data-Driven Multistage Adaptive Robust Optimization Framework for Planning and Scheduling Under Uncertainty

Robust optimization has gained increasing popularity because it provides a good balance between solution quality, feasibility and computational tractability [1]. Robust optimization methods can be roughly classified into three categories, namely static robust optimization, two-stage adaptive robust optimization (ARO), and multistage ARO. Two-stage ARO grants more flexibility than static robust optimization by introducing recourse decisions, and typically generates less conservative solutions [2-5]. To overcome the limitation of two-stage structures, multistage ARO is recently proposed for non-anticipative sequential decision making processes [6, 7]. Notably, most of the existing robust optimization approaches assume that uncertainty sets are defined or given a priori and fall short of fully utilizing uncertainty data. Consequently, these approaches hedge against the uncertainty realization regardless of its occurrence probability, and could generate conservative solutions. The abundance of uncertainty data and recent advances in machine learning hold the promise to address the conservatism issue, and call for research efforts on data-driven robust optimization [8, 9].

To the best of our knowledge, data-driven approach for multistage ARO has not been considered in the existing literature. Therefore, this paper aims to fill the knowledge gap and take the first step of developing a data-driven multistage ARO framework. In this work, we propose a novel data-driven multistage ARO framework that leverages robust statistics and machine learning techniques for multistage ARO methods. To extract an accurate probability distribution from uncertainty data that could be possibly large-scale and messy, we introduce a nonparametric robust kernel density estimation (RKDE) [10], which is obtained via a kernelized iteratively re-weighted least squares (KIRWLS) algorithm. Based on the information extracted from uncertainty data, we further propose a data-driven uncertainty set for multistage ARO using quantile functions and uncertainty budgets to organically integrate uncertainty data information with the multistage ARO framework. The quantile functions provide the bounds of uncertain parameters based on cumulative distribution functions. Moreover, uncertainty budgets are employed to adjust the level of conservatism. We apply affine decision rules for recourse variables [11], and further develop a data-driven multistage adaptive robust counterpart in a general form to facilitate the solution of the aforementioned ARO problem. Moreover, we introduce an approach to capture correlations between uncertain parameters in the data-driven uncertainty set using covariance matrix [12]. We further develop a corresponding data-driven multistage adaptive robust counterpart that addresses not only the correlations between different uncertain parameters, but also the temporal correlations between different stages.

To illustrate the applicability of the proposed framework, two typical applications in process operations are presented. One application is the multi-period strategic planning of process networks [13], and the other one is the multipurpose batch process scheduling [14, 15]. In a case study on process planning, our proposed approach generates 23.9% more net present value (NPV) than that of the multistage ARO with a box uncertainty set. Compared with multistage ARO using KDE, the proposed method returns 22.6% more NPV. In a case study on batch process scheduling, the proposed approach returns 31.5% and 7.6% more profits than those of multistage ARO with a box uncertainty set and a KDE based uncertainty set, respectively.

References

[1] D. Bertsimas, D. B. Brown, and C. Caramanis, "Theory and applications of robust optimization," SIAM Review, vol. 53, pp. 464-501, 2011.

[2] H. Shi and F. You, "A computational framework and solution algorithms for two-stage adaptive robust scheduling of batch manufacturing processes under uncertainty," AIChE Journal, vol. 62, pp. 687-703, 2016.

[3] J. Gong, D. J. Garcia, and F. You, "Unraveling Optimal Biomass Processing Routes from Bioconversion Product and Process Networks under Uncertainty: An Adaptive Robust Optimization Approach," ACS Sustainable Chemistry & Engineering, vol. 4, pp. 3160-3173, 2016.

[4] J. Gong and F. You, "Optimal processing network design under uncertainty for producing fuels and value-added bioproducts from microalgae: Two-stage adaptive robust mixed integer fractional programming model and computationally efficient solution algorithm," AIChE Journal, vol. 63, pp. 582-600, 2017.

[5] D. Yue and F. You, "Optimal supply chain design and operations under multi-scale uncertainties: Nested stochastic robust optimization modeling framework and solution algorithm," AIChE Journal, vol. 62, pp. 3041-3055, 2016.

[6] E. Delage and D. A. Iancu, "Robust multistage decision making," in The Oper Res Rev. , ed, 2015, pp. 20-46.

[7] N. H. Lappas and C. E. Gounaris, "Multi-stage adjustable robust optimization for process scheduling under uncertainty," AIChE Journal, vol. 62, pp. 1646-1667, 2016.

[8] D. Bertsimas, V. Gupta, and N. Kallus, "Data-driven robust optimization," Mathematical Programming, 2017. DOI:10.1007/s10107-017-1125-8

[9] C. Ning and F. You, "Data-driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty," AIChE Journal, 2017. DOI: 10.1002/aic.15717

[10] J. Kim and C. D. Scott, "Robust kernel density estimation," Journal of Machine Learning Research, vol. 13, pp. 2529-2565, 2012.

[11] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, "Adjustable robust solutions of uncertain linear programs," Mathematical Programming, vol. 99, pp. 351-376, 2004.

[12] Y. Yuan, Z. Li, and B. Huang, "Robust optimization under correlated uncertainty: Formulations and computational study," Computers & Chemical Engineering, vol. 85, pp. 58-71, 2016.

[13] F. You and I. E. Grossmann, "Stochastic inventory management for tactical process planning under uncertainties: MINLP models and algorithms," AIChE Journal, vol. 57, pp. 1250-1277, 2011.

[14] Y. Chu, J. M. Wassick, and F. You, "Efficient scheduling method of complex batch processes with general network structure via agent-based modeling," AIChE Journal, vol. 59, pp. 2884-2906, 2013.

[15] J. M. Wassick, A. Agarwal, N. Akiya, J. Ferrio, S. Bury, and F. You, "Addressing the operational challenges in the development, manufacture, and supply of advanced materials and performance products," Computers & Chemical Engineering, vol. 47, pp. 157-169, 2012.