Tropical optimization problems in time-constrained project scheduling

Abstract

We consider a project that consists of activities to be performed in parallel under various temporal constraints, which include start-start, start-finish and finish-start precedence relationships, release times, deadlines, and due dates. Scheduling problems are formulated to find optimal schedules for the project with respect to different objective functions to be minimized, such as the project makespan, the maximum deviation from the due dates, the maximum flow-time, and the maximum deviation of finish times. We represent these problems as optimization problems in terms of tropical mathematics, and then solve them by applying direct solution methods of tropical optimization. As a result, new direct solutions of the scheduling problems are obtained in a compact vector form, which is ready for further analysis and practical implementation. The solutions are illustrated by simple numerical examples.Comment: 28 page

Algebraic solution of tropical optimization problems via matrix sparsification with application to scheduling

“…The solutions obtained for the scheduling problems under both mini-mization and maximization of the maximum deviation of finish times present quite new results. For instance, we derive a complete solution of the scheduling problem under the minimization criterion, which significantly extends previously known partial solutions [25,27]. The maximization problem under study generalizes those considered in [26] by taking into consideration additional constraints.…”

“…Solutions were given within a combined framework that involves two reciprocally dual idempotent semifields. Similar problems in the general setting of tropical mathematics were examined by Krivulin in [25,26,27], where direct, explicit solutions were suggested. However, the results obtained present a partial solution, rather than a complete solution, or offer a solution in scalar terms, rather than in a compact vector form.…”

“…We consider the tropical optimization problems formulated in [25,26,27] as extensions of the problems of minimizing and maximizing the span seminorm, and represent them in a slightly different form to minimize (maximize) q − x(Ax) − p,…”

Minimizing maximum lateness in two-stage projects by tropical optimization

“…In the framework of tropical algebra, many temporal project scheduling problems can be represented as optimization problems and then analytically solved using methods and techniques of tropical optimization in explicit form. Examples of the solution includes results on both single-criterion problems [21,23,24,25] and multicriteria problems [26].…”

Solution of minimax location problem in three-dimensional space with rectilinear metric

“…Подход на основе методов тропической математики находит свое применение при решении задач в различных областях науки и техники, включая задачи принятия решений при анализе результатов оценки альтернатив на основе парных сравнений [20], и задачи планирования сроков выполнения проектов [21]. На основе этого подхода в работах [22][23][24][25][26] получены решения минимаксных задач размещения точечного объекта на плоскости с прямоугольной и чебышевской метрикой без ограничений и с ограничениями на допустимую область размещения.…”