# Tropical optimization technique in bi-objective project scheduling under temporal constraints

## TL;DR

This paper introduces a tropical optimization approach to solve bi-objective project scheduling problems with temporal constraints, providing explicit Pareto-optimal solutions for minimizing flow-time and makespan.

## Contribution

The paper applies tropical mathematics to formulate and solve bi-objective scheduling problems, deriving explicit Pareto-optimal solutions and analyzing their computational complexity.

## Key findings

- Explicit Pareto-optimal solutions derived using tropical optimization.
- The approach handles complex temporal constraints efficiently.
- Illustrative examples demonstrate practical applicability.

## Abstract

We consider a project that consists of a set of activities performed in parallel under constraints on their start and finish times, including start-finish precedence relationships, release start times, release end times, and deadlines. The problems of interest are to decide on the optimal schedule of the activities to minimize both the maximum flow-time over all activities, and the project makespan. We formulate these problems as bi-objective optimization problems in the framework of tropical mathematics which investigates the theory and applications of algebraic systems with idempotent operations and has various applications in management science and operations research. Then, the use of methods and techniques of tropical optimization allows to derive complete Pareto-optimal solutions of the problems in a direct explicit form ready for further analysis and straightforward computation. We discuss the computational complexity of the solution and give illustrative examples.