Robust counterpart optimization techniques for linear optimization and mixed integer linear optimization problems are studied in this paper. Different uncertainty sets, including those studied in ...literature (i.e., interval set; combined interval and ellipsoidal set; combined interval and polyhedral set) and new ones (i.e., adjustable box; pure ellipsoidal; pure polyhedral; combined interval, ellipsoidal, and polyhedral set) are studied in this work and their geometric relationship is discussed. For uncertainty in the left-hand side, right-hand side, and objective function of the optimization problems, robust counterpart optimization formulations induced by those different uncertainty sets are derived. Numerical studies are performed to compare the solutions of the robust counterpart optimization models and applications in refinery production planning and batch process scheduling problem are presented.
In this paper, we study uncertainty set construction for robust optimization using various polyhedral norms. We first introduce the classical symmetric polyhedral-norms induced uncertainty sets and ...the corresponding robust counterparts of a linear uncertain constraint. Then, we introduce a novel method for asymmetric uncertainty set construction based on the distributional information of the uncertain parameters. Deterministic robust counterpart formulations for both types of uncertainty sets are derived for a general linear uncertain constraint. We further derive the robust counterpart of a linear uncertain constraint where the uncertain parameters belong to (i) an intersection of two symmetric uncertainty sets and (ii) an intersection of asymmetric and symmetric uncertainty sets. Using a numerical example and a reactor design problem, we demonstrate that appropriate uncertainty set construction reduces solution conservativeness. We also highlight the significance of integrating the data and distributional information of uncertain parameters in terms of safeguarding feasibility alongside improving the solution.
•Proposed a novel scenario reduction algorithm considering both input and output space probabilistic distance and output performance of decision making.•Minimizes the differences on the best, worst ...and expected performance of the output measure of the original and the reduced scenario distributions.•Developed a novel mixed integer linear optimization based problem formulation.•A number of case studies demonstrating and comparing the performance of the proposed method with existing tools.
Realistic decision making involves consideration of uncertainty in various parameters. While large number of scenarios brings significant challenge to computations, the scenario reduction aims at selecting a small number of representative scenarios that can capture the wide range of possible scenarios. A novel scenario reduction algorithm is proposed in this paper to incorporate the consideration of both input data and output performance of decision making. The proposed optimal scenario reduction algorithm, OSCAR, is formulated as a mixed integer linear optimization problem. It minimizes not only the probabilistic distance between the original and reduced input scenario distribution, but also minimizes the differences between the best, worst and expected performances of the output measure of the original and the reduced scenario distributions. The proposed method leads to reduced distribution not only closer to the original distribution in terms of the transportation distance, but also captures the performance of the output.
Enterprise-wide optimization for the petroleum refining industry involves optimization of the supply chain involving manufacturing and distribution with emphasis on integration of the different ...decision making levels. The key manufacturing operations include crude oil loading and unloading, mixing of crude oil, production unit operations of conversion and separation, operations of blending, and distribution of products. Other components of the petroleum supply chain network include oil explorations, crude oil procurement, and sales and distribution of products. The main issues present in the petroleum industry across various decision levels (strategic, tactical, and operational) and within oil refinery operations are discussed. This paper presents an extensive literature review of methodologies for addressing scheduling, planning, and supply chain management of oil refinery operations. An attempt is also made to identify the future challenges in efficiently solving these problems.
To improve the quality of decision making in the process operations, it is essential to implement integrated planning and scheduling optimization. Major challenge for the integration lies in that the ...corresponding optimization problem is generally hard to solve because of the intractable model size. In this paper, augmented Lagrangian method is applied to solve the full-space integration problem which takes a block angular structure. To resolve the non-separability issue in the augmented Lagrangian relaxation, we study the traditional method which approximates the cross-product term through linearization and also propose a new decomposition strategy based on two-level optimization. The results from case study show that the augmented Lagrangian method is effective in solving the large integration problem and generating a feasible solution. Furthermore, the proposed decomposition strategy based on two-level optimization can get better feasible solution than the traditional linearization method.
Probabilistic guarantees on constraint satisfaction for robust counterpart optimization are studied in this paper. The robust counterpart optimization formulations studied are derived from box, ...ellipsoidal, polyhedral, “interval+ellipsoidal”, and “interval+polyhedral” uncertainty sets ( Li Z. ; Ding R. ; Floudas C.A. A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: I. Robust Linear and Robust Mixed Integer Linear Optimization. Ind. Eng. Chem. Res. 2011, 50, 10567 ). For those robust counterpart optimization formulations, their corresponding probability bounds on constraint satisfaction are derived for different types of uncertainty characteristic (i.e., bounded or unbounded uncertainty, with or without detailed probability distribution information). The findings of this work extend the results in the literature and provide greater flexibility for robust optimization practitioners in choosing tighter probability bounds so as to find less conservative robust solutions. Extensive numerical studies are performed to compare the tightness of the different probability bounds and the conservatism of different robust counterpart optimization formulations. Guiding rules for the selection of robust counterpart optimization models and for the determination of the size of the uncertainty set are discussed. Applications in production planning and process scheduling problems are presented.
The rolling horizon method has been proposed to address the integrated production planning and scheduling optimization problem. Since the method can generally result in small-scale optimization model ...and fast solution, it has been used in a number of applications in realistic industrial planning and scheduling problems. In this paper, it is first pointed out that the incorporation of valid production capacity information into the planning model can improve the solution quality in the rolling horizon solution framework. A novel method is then proposed to derive the production capacity information representing the detail scheduling model based on parametric programming technique. A heuristic process network decomposition strategy is further applied to reduce the computational effort needed for larger and more complex process networks. Several case studies have been studied, which illustrate the efficiency of the proposed methodology in improving the solution quality of rolling horizon method for integrated planning and scheduling optimization.
Distributionally robust optimization (DRO) is an increasingly popular approach for optimization under uncertainty when the probability distribution of the uncertain parameter is unknown. ...Well-explored DRO approaches in literature, such as Wasserstein DRO, do not make any specific assumptions on the nature of the candidate distributions considered in the ambiguity set. However, in many practical applications, the uncertain parameter may be sourced from a distribution that can be well modeled as a Gaussian Mixture Model (GMM) whose components represent the different subpopulations the uncertain parameter may belong to. In this work, we propose a new DRO method based on an ambiguity set constructed around a GMM. The proposed DRO approach is illustrated on a numerical example as well as a portfolio optimization case study for uncertainty sourced from various distributions. The results obtained from the proposed DRO approach are compared with those from Wasserstein DRO, and are shown to be superior in quality with respect to out-of-sample performance.
This paper addresses the uncertainty problem in process scheduling using robust optimization. Compared to the traditional-scenario-based stochastic programming method, robust counterpart optimization ...method has a unique advantage, in that the scale of the corresponding optimization problem does not increase exponentially with the number of the uncertain parameters. Three robust counterpart optimization formulations―including Soyster’s worst-case scenario formulation, Ben-Tal and Nemirovski’s formulation, and a formulation proposed by Bertsimas and Sim―are studied and applied to uncertain scheduling problems in this paper. The results show that the formulation proposed by Bertsimas and Sim is the most appropriate model for uncertain scheduling problems, because it has the following advantages: (i) the model has the same size as the other formulations, (ii) it preserves its linearity, and (iii) it has the ability to control the degree of conservatism for every constraint and guarantees feasibility for the robust optimization problem.
Two methods for multistage adaptive robust binary optimization are investigated in this work. These methods referred to as binary decision rule and finite adaptability inherently share similarities ...in dividing the uncertainty set into subsets. In the binary decision rule method, the uncertainty is lifted using indicator functions which result in a nonconvex lifted uncertainty set. The linear decision rule is further applied to a convexified version of the lifted uncertainty set. In the finite adaptability method, the uncertainty set is divided into partitions and a constant decision is applied for each partition. In both methods, breakpoints are utilized either to define the indicator functions in the lifting method or to partition the uncertainty set in the finite adaptability method. In this work, we propose variable breakpoint location optimization for both methods. Extensive computational study on an illustrating example and a larger size case study is conducted. The performance of binary decision rule and finite adaptability methods under fixed and variable breakpoint approaches is compared.
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