Multi‐parametric linear programming under global uncertainty

Vassilis M Charitopoulos,Lazaros G Papageorgiou,Vivek Dua

doi:10.1002/aic.15755

Vassilis M Charitopoulos, Lazaros G Papageorgiou + Show 1 more

Open Access

https://doi.org/10.1002/aic.15755

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Abstract

Multi‐parametric programming has proven to be an invaluable tool for optimisation under uncertainty. Despite the theoretical developments in this area, the ability to handle uncertain parameters on the left‐hand side remains limited and as a result, hybrid, or approximate solution strategies have been proposed in the literature. In this work, a new algorithm is introduced for the exact solution of multi‐parametric linear programming problems with simultaneous variations in the objective function's coefficients, the right‐hand side and the left‐hand side of the constraints. The proposed methodology is based on the analytical solution of the system of equations derived from the first order Karush–Kuhn–Tucker conditions for general linear programming problems using symbolic manipulation. Emphasis is given on the ability of the proposed methodology to handle efficiently the LHS uncertainty by computing exactly the corresponding nonconvex critical regions while numerical studies underline further the advantages of the proposed methodology, when compared to existing algorithms. © 2017 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 63: 3871–3895, 2017

Highlights

Despite the constantly growing computational power decision makers have at hand, the need for systematic treatment of uncertainty has always been of paramount importance, especially under the ever-changing market conditions that the industries have to face
The main assumption in stochastic programming is the availability of statistical data about the uncertain parameters providing either discrete or continuous probability distributions which the uncertain parameters follow
In this work, motivated by the continuously increasing demand for efficient and effective decision making we propose an algorithm for the explicit solution of general mp-LPs when global uncertainty is considered

Summary

Introduction

Despite the constantly growing computational power decision makers have at hand, the need for systematic treatment of uncertainty has always been of paramount importance, especially under the ever-changing market conditions that the industries have to face. To deal with the presence of uncertainty, several methodologies such as stochastic programming, robust optimisation, fuzzy mathematical programming, chance constrained programming and so forth have been proposed throughout the years.[1] Within the stochastic programming framework,[2] the decision maker typically takes some actions before the realisation of uncertainty (here and ) while at subsequent steps (after the uncertainty has been revealed) takes corrective actions Bertsimas and Sim[6] proposed a more flexible formulation, while preserving the linear form of the problem, with the introduction of a “budget parameter” as a measure of conservatism Based on their approach, any uncertain parameter can either be at its nominal or worst-case value but the total number of uncertain parameters that can take their worst value is controlled by the budget parameter. Another difference between stochastic programming and robust optimisation is that in the latter, the uncertainty is assumed to lie within a AIChE Journal

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