Sensitivity analysis of multiobjective linear programming from a geometric perspective

TL;DR

This paper introduces a geometric approach to sensitivity analysis in multiobjective linear programming, classifying problems into equivalence classes based on boundary subsets to understand solution behavior under parameter changes.

Contribution

It presents a novel, computationally feasible method for classifying MOLP problems in two dimensions using an equivalence relation on linear maps.

Findings

Classifies MOLP problems into finite equivalence classes.

Links each class to a specific boundary subset of the feasible region.

Provides a numerical example illustrating the approach.

Abstract

Sensitivity analysis plays a crucial role in multiobjective linear programming (MOLP), where understanding the impact of parameter changes on efficient solutions is essential. This work builds upon and extends previous investigations. In this paper, we introduce a novel approach to sensitivity analysis in MOLP, designed to be computationally feasible for decision-makers studying the behavior of efficient solutions under perturbations of objective function coefficients in a two-dimensional variable space. This approach classifies all MOLP problems in $S \subset \mathbb{R}^{2}$ by defining an equivalence relation that partitions the space of linear maps$-$comprising all sequences of linear forms on $\mathbb{R}^2$ of length $K \geq 2-$into a finite number of equivalence classes. Each equivalence class is associated with a unique subset of the boundary of $S$. For any MOLP with $K$ objective functions belonging to the same equivalence class, its set of efficient solutions corresponds to the associated subset of the boundary of $S$. This approach is detailed and illustrated with a numerical example.