A constrained tropical optimization problem: complete solution and application example

TL;DR

This paper presents a complete solution to a multidimensional tropical optimization problem with linear constraints, and demonstrates its application in project scheduling to optimize activity flow time.

Contribution

It introduces a novel approach using an additional variable to reduce the problem to linear inequalities, providing a complete direct solution in a compact form.

Findings

Derived necessary and sufficient conditions for the inequalities to hold.

Provided a general solution expressed in vector form.

Applied the method to an optimal project scheduling problem.

Abstract

The paper focuses on a multidimensional optimization problem, which is formulated in terms of tropical mathematics and consists in minimizing a nonlinear objective function subject to linear inequality constraints. To solve the problem, we follow an approach based on the introduction of an additional unknown variable to reduce the problem to solving linear inequalities, where the variable plays the role of a parameter. A necessary and sufficient condition for the inequalities to hold is used to evaluate the parameter, whereas the general solution of the inequalities is taken as a solution of the original problem. Under fairly general assumptions, a complete direct solution to the problem is obtained in a compact vector form. The result is applied to solve a problem in project scheduling when an optimal schedule is given by minimizing the flow time of activities in a project under various activity precedence constraints. As an illustration, a numerical example of optimal scheduling is also presented.