Sensitivity Analysis and Robust Optimization: A Geometric Approach for the Special Case of Linear Optimization

In this thesis, we study the special case of linear optimization to show what may affect the sensitivity of the optimal value function under data uncertainty. In this special case, we show that the robust optimization problem with a locally smaller feasible region yields a more conservative robust optimal value than the one with a locally bigger feasible region. To achieve that goal, we use a geometric approach to analyze the sensitivity of the optimal value function for linear programming (LP) under data uncertainty. We construct a family of proper cones where the strict containment holds for any pair of cones in the family. We then form a family of LP problems using this family of cones constructed above; the feasible regions of each pair of LPs in the family holds strict containment, every LP in the family has the unique optimal solution at the vertex of the cone and has the same objective function, i.e., every LP in the family shares the same optimal solution and the same optimal value. We rewrite he LPs so that they reflect the given data uncertainty and perform local analysis near the optimal solutions where the local strict containment holds. Finally, we illustrate that an LP with a locally smaller feasible region is more sensitive than an LP with a locally bigger feasible region

Robust Multi-Objective Sustainable Reverse Supply Chain Planning: An Application in the Steel Industry

In the design of the supply chain, the use of the returned products and their recycling in the production and consumption network is called reverse logistics. The proposed model aims to optimize the flow of materials in the supply chain network (SCN), and determine the amount and location of facilities and the planning of transportation in conditions of demand uncertainty. Thus, maximizing the total profit of operation, minimizing adverse environmental effects, and maximizing customer and supplier service levels have been considered as the main objectives. Accordingly, finding symmetry (balance) among the profit of operation, the environmental effects and customer and supplier service levels is considered in this research. To deal with the uncertainty of the model, scenario-based robust planning is employed alongside a meta-heuristic algorithm (NSGA-II) to solve the model with actual data from a case study of the steel industry in Iran. The results obtained from the model, solving and validating, compared with actual data indicated that the model could optimize the objectives seamlessly and determine the amount and location of the necessary facilities for the steel industry more appropriately.This article belongs to the Special Issue Uncertain Multi-Criteria Optimization Problem

Convex Relaxations and Approximations of Chance-Constrained AC-OPF Problems

This paper deals with the impact of linear approximations for the unknown nonconvex confidence region of chance-constrained AC optimal power flow problems. Such approximations are required for the formulation of tractable chance constraints. In this context, we introduce the first formulation of a chance-constrained second-order cone (SOC) OPF. The proposed formulation provides convergence guarantees due to its convexity, while it demonstrates high computational efficiency. Combined with an AC feasibility recovery, it is able to identify better solutions than chance-constrained nonconvex AC-OPF formulations. To the best of our knowledge, this paper is the first to perform a rigorous analysis of the AC feasibility recovery procedures for robust SOC-OPF problems. We identify the issues that arise from the linear approximations, and by using a reformulation of the quadratic chance constraints, we introduce new parameters able to reshape the approximation of the confidence region. We demonstrate our method on the IEEE 118-bus system

Development of an experiment-based robust design paradigm for multiple quality characteristics using physical programming

The well-known quality improvement methodology, robust design, is a powerful and cost-effective technique for building quality into the design of products and processes. Although several approaches to robust design have been proposed in the literature, little attention has been given to the development of a flexible robust design model. Specifically, flexibility is needed in order to consider multiple quality characteristics simultaneously, just as customers do when judging products, and to capture design preferences with a reasonable degree of accuracy. Physical programming, a relatively new optimization technique, is an effective tool that can be used to transform design preferences into specific weighted objectives. In this paper, we extend the basic concept of physical programming to robust design by establishing the links of experimental design and response surface methodology to address designers’ preferences in a multiresponse robust design paradigm. A numerical example is used to show the proposed procedure and the results obtained are validated through a sensitivity study