Application of an optimization problem in Max-Plus algebra to scheduling problems

In this paper we discuss a modeling framework for model predictive scheduling of a class of semi-cyclic discrete event systems that can be described by switching max-plus linear models. We study the structure of the system matrices and derive how routing, ordering, and synchronization can be manipulated by a set of control variables. In addition, we show that this leads to a system matrix that is linear in the control variables. We define the model predictive scheduling design problem to optimize the schedule, and we show that the problem can be recast as a mixed integer linear programming (MILP) problem.

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“…We have derived four conditions (1), (5), (7), and (9) for x(k). We also have a set of scheduling decision variables from • Routing: ω(k).…”

Section: Overall Mpl Systemmentioning

confidence: 99%

“…However, [6], [7], [18], [19], [24], [30] already showed that for many scheduling problems, above all for semi-cyclic discrete event systems, max-plus algebra is better suited for solving sequencing problems than the classical algebra. An overview of the cyclic scheduling problem is available in [10], [12].…”

Section: Introductionmentioning

confidence: 96%

A modeling framework for model predictive scheduling using switching max-plus linear models

Boom

Lopes

Schutter

2013

52nd IEEE Conference on Decision and Control

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“…A direct solution, which uses algebraic properties of the multiplicative conjugate transposition to reduce (5) to an equivalent inequality, is obtained in [26] as follows.…”

Section: Solution To Linear Inequalitiesmentioning

confidence: 99%

“…Tropical optimization problems were investigated in a number of works, in which scheduling issues frequently helped to motivate and illustrate the study. Specifically, tropical mathematics (max-plus algebra) served as a solution framework for scheduling problems in [8,42,40,4,41,5,12,16,7,6,18,2].…”

Section: Introductionmentioning

confidence: 99%

Tropical optimization problems in time-constrained project scheduling

Krivulin

2016

Optimization

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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

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“…It is the case for some single-machine problems (1||L max , 1|| C j , ...) and for other two-machine flow shop scheduling problems [11].…”

Section: Conclusion and Further Research Directionsmentioning

confidence: 99%

Complete Characterization of Near-Optimal Sequences for the Two-Machine Flow Shop Scheduling Problem

Billaut

Hébrard

López

2012

Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems

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Application of an optimization problem in Max-Plus algebra to scheduling problems

Cited by 30 publications

References 10 publications

A modeling framework for model predictive scheduling using switching max-plus linear models

A modeling framework for model predictive scheduling using switching max-plus linear models

Tropical optimization problems in time-constrained project scheduling

Complete Characterization of Near-Optimal Sequences for the Two-Machine Flow Shop Scheduling Problem

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