Solving a tropical optimization problem with application to optimal scheduling

Abstract

A multidimensional optimization problem is considered, which is formulated and solved in terms of tropical mathematics focused on the theory and applications of semirings with idempotent addition. To solve the problem, which has an objective function given by a matrix, methods and results of idempotent algebra and tropical optimization are used. A strict lower bound for the objective function of the problem is first derived to allow the evaluation of the minimum value of the objective function. Then, an equation is formed and solved for the objective function and its minimum value, from which a complete solution is obtained in the form of all eigenvectors of the matrix in the problem. As an application of the result obtained, an explicit solution is given to the problem of optimal scheduling of a project that consists of a set of activities to be done under given constraints on the start and finish times of the activities. The optimality criterion for scheduling is defined as the minimum of maximal deviation, over all activities, of the working cycle time, which is given by the time interval between start and finish of the activity. The analytical result obtained extends and supplements the existing algorithmic numerical solutions to optimal scheduling problems. An example is presented to illustrate application of the result with a scheduling problem for a project consisting of three activities.