473269 Data-Driven Adaptive Nested Robust Optimization: Modeling Framework and Solution Algorithm for Process Design and Operations Under Uncertainty

Thursday, November 17, 2016: 2:43 PM
Monterey I (Hotel Nikko San Francisco)
Chao Ning, Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY and Fengqi You, Cornell University, Ithaca, NY

Optimization of process systems under uncertainty has attracted wide attention from both academia and industry [1, 2]. A plethora of optimization models and algorithms have been proposed to support decision making in the presence of uncertainty. Recently, adaptive robust optimization (ARO) offers a new paradigm for optimization under uncertainty by incorporating recourse decisions through a sequential decision-making framework [3-6]. However, the traditional ARO approach fails to take full advantage of the information embedded in uncertainty data. It does not take into account the correlation between uncertainty parameters or skewness of uncertainty data, which turn out to have substantial impacts on the ARO solutions. Besides, the traditional ARO is sensitive to outliers, yielding an optimized solution of low quality.

The aforementioned issues are relevant to process industries, in which a deluge of data has been collected and archived [7, 8]. In this work, we propose a data-driven adaptive nested robust optimization (DDANRO) approach for better decision-making from big data. A Bayesian nonparametric model – the Dirichlet process mixture model (DPMM) [9] – is adopted to learn an uncertainty set via a variational inference algorithm [10]. As a nonparametric model, DPMM is able to adjust its complexity to that of data. To be more specific, it utilizes potentially infinite mixtures of Gaussian distributions to characterize data. In this way, DPMM efficiently extracts valuable information from uncertainty data, including correlation and skewness. Following this DPMM approach, we propose a novel data-driven uncertainty set for ARO based on l1 and lnorms. We then integrate the statistical model and adaptive optimization model seamlessly in a four-level optimization framework (a min-max-max-min problem). Additionally, this DDANRO framework, as its name suggests, has two layers of robustness. The outer layer of the DDANRO framework is robust to outliers in uncertainty data. By using the weights of Gaussian components to hedge against outliers, components with smaller weights are “filtered out” as outliers by the outer layer. The inner layer of the DDANRO framework is robust to variations in the remaining – or “clean” – uncertainty data. The resulting DDANRO problem cannot be solved directly by any off-the-shelf optimization solvers due to its multi-level optimization structure. To address this computational challenge, we also propose a tailored column-and-constraint generation (C&CG [11]) algorithm. In this algorithm, each subproblem corresponds to a component of Gaussian mixtures. The algorithm iteratively solves a sequence of master problems and subproblems until the optimality gap reduces to a predefined tolerance.

The proposed modeling framework and solution algorithm are demonstrated using two applications on batch process scheduling [12] and on petrochemical complex planning [13], respectively. In the short-term batch process scheduling problem, processing time uncertainty and demand uncertainty are considered. The real batch processing time data are corrupted with outliers. The traditional ARO treats these outliers in processing time data as worst cases. Consequently, the scheduling strategy obtained from the traditional ARO performs less tasks over the same time horizon. In contrast, our method yields less conservative solutions, which means increased profits from batch processes. In the planning problem, the process network involves 38 processes and 28 chemicals. Both supply and demand are subject to uncertainty. The optimal process designs and process operations produced by our proposed method and the traditional ARO are quite different. In a case, the processes 1, 3, 8, 28 and 32 are planned to be expanded in the traditional ARO solution, while the processes 1, 3, 8, 14, 17, 28 and 32 are chosen to be expanded in the solution of our proposed model. Our proposed approach demonstrates superior performance in terms of net present value (NPV) over the traditional ARO. For example, in a case where product demands are correlated, the NPV of our proposed method is 7.5% higher than that of the traditional ARO.


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[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.

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