The use of tropical optimization methods in problems of project scheduling

The paper is devoted to the solution of problems in project scheduling by using methods of tropical optimization. Problems are examined that are to develop an optimal schedule for a project consisting in the execution of a set of interrelated tasks under given constraints on their initiation and completion time. The optimal schedule criteria are considered, which require the maximization of the deviation of the initiation or the deviation of the completion time of tasks. Such problems arise when, for some reason (such as a lack of resources, technical constraints, security requirements, and the like), there is a need to avoid a simultaneous start or finish for all tasks in the project. The paper begins with the formulation of scheduling problems in the form of usual optimization problems. Then, definitions and results of tropical mathematics are given, which are used in the subsequent analysis and solution of tropical optimization problems. New constrained problems of tropical optimization are considered, and their solutions are obtained. The scheduling problems are solved by reducing to tropical optimization problems. To illustrate the results obtained, numerical examples are presented. Refs 15.