Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Multi-objective optimization of environmentally conscious chemical supply chains under demand uncertainty

In this work, we analyze the effect of demand uncertainty on the multi-objective optimization of chemical supply chains (SC) considering simultaneously their economic and environmental performance. To this end, we present a stochastic multi-scenario mixed-integer linear program (MILP) with the unique feature of incorporating explicitly the demand uncertainty using scenarios with given probability of occurrence. The environmental performance is quantified following life cycle assessment (LCA) principles, which are represented in the model formulation through standard algebraic equations. The capabilities of our approach are illustrated through a case study. We show that the stochastic solution improves the economic performance of the SC in comparison with the deterministic one at any level of the environmental impact.

Handling of Uncertainty in Life Cycle Inventory by Correlated Multivariate Lognormal Distributions: Application to the Design of Supply Chain Networks

Poster presented in the 24th European Symposium on Computer Aided Process Engineering (ESCAPE 24), Budapest, Hungary, June 15-18, 2014.

Robust Solutions of MultiObjective Linear Semi-Infinite Programs under Constraint Data Uncertainty

The multiobjective optimization model studied in this paper deals with simultaneous minimization of finitely many linear functions subject to an arbitrary number of uncertain linear constraints. We first provide a radius of robust feasibility guaranteeing the feasibility of the robust counterpart under affine data parametrization. We then establish dual characterizations of robust solutions of our model that are immunized against data uncertainty by way of characterizing corresponding solutions of robust counterpart of the model. Consequently, we present robust duality theorems relating the value of the robust model with the corresponding value of its dual problem.

Handling of Uncertainty in Life Cycle Inventory by Correlated Multivariate Lognormal Distributions: Application to the Design of Supply Chain Networks

In this work, we analyze the effect of incorporating life cycle inventory (LCI) uncertainty on the multi-objective optimization of chemical supply chains (SC) considering simultaneously their economic and environmental performance. To this end, we present a stochastic multi-scenario mixed-integer linear programming (MILP) coupled with a two-step transformation scenario generation algorithm with the unique feature of providing scenarios where the LCI random variables are correlated and each one of them has the desired lognormal marginal distribution. The environmental performance is quantified following life cycle assessment (LCA) principles, which are represented in the model formulation through standard algebraic equations. The capabilities of our approach are illustrated through a case study of a petrochemical supply chain. We show that the stochastic solution improves the economic performance of the SC in comparison with the deterministic one at any level of the environmental impact, and moreover the correlation among environmental burdens provides more realistic scenarios for the decision making process.