Solving Minimum Cost Flow Problem under Neutrosophic Environment Using the Lexicographic Approach

Abstract

In decision-making, linear programming is one of the most useful models for obtaining the optimal solution. A crucial element of the linear programming (LP) model is the minimum cost flow (MCF). The objective of the MCF is to reduce the transportation cost of a single product across a network with capacity constraints. Recently, neutrosophic set theory has become a strong way to deal with the uncertainty that often comes with trying to optimize things. This manuscript explores how neutrosophic set theory can be applied to the MCF problem which has caught the interest of some researchers. The primary objective of this study is threefold: firstly, to tackle the MCF problem considering the uncertainty of the neutrosophic set, focusing especially on the cost. Secondly, to introduce an innovative lexicographical method tailored for the MCF problem, marking a first in the field of neutrosphic sets. Lastly, to combine this new method with a multi-objective optimization approach, improving the way we solve the MCF problem in various ways at once. This thorough method is meant to lead to more detailed and effective ways of solving optimization problems when there is uncertainty. To show how our method works, we will go through some numerical examples related to the MCF problem with cost defined by neutrosophic numbers.