A new geometric approach to multiobjective linear programming problems

TL;DR

This paper introduces a geometric method for solving multiobjective linear programming problems that avoids calculating each objective's optimal value, reducing computational effort and complexity.

Contribution

It presents a novel geometric approach based on equivalence classes and sensitivity analysis to efficiently identify ideal solutions without extensive calculations.

Findings

Reduces computational complexity in MOLPP solutions

Effectively identifies ideal solutions without optimal value calculations

Demonstrated success with numerical example

Abstract

In this paper, we present a novel method for solving multiobjective linear programming problems (MOLPP) that overcomes the need to calculate the optimal value of each objective function. This method is a follow-up to our previous work on sensitivity analysis, where we developed a new geometric approach. The first step of our approach is to divide the space of linear forms into a finite number of sets based on a fixed convex polygonal subset of $\mathbb{R}^{2}$. This is done using an equivalence relationship, which ensures that all the elements from a given equivalence class have the same optimal solution. We then characterize the equivalence classes of the quotient set using a geometric approach to sensitivity analysis. This step is crucial in identifying the ideal solution to the MOLPP. By using this approach, we can determine whether a given MOLPP has an ideal solution without the need to calculate the optimal value of each objective function. This is a significant improvement over existing methods, as it significantly reduces the computational complexity and time required to solve MOLPP. To illustrate our method, we provide a numerical example that demonstrates its effectiveness. Our method is simple, yet powerful, and can be easily applied to a wide range of MOLPP. This paper contributes to the field of optimization by presenting a new approach to solving MOLPP that is efficient, effective, and easy to implement.