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Complexity of comparing expressions in max-plus algebra

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Frontiers of Electrical and Electronic Engineering in China 2009年第4期4卷 392-396页

作者： Qianchuan ZHAO Department of Automation Tsinghua UniversityBeijing 100084China

max-plus algebra has been widely used in the study of discrete-event dynamic *** max-plus algebra makes it easy to specify safety constraints on events since they can be described as a set of inequalities of state variables,i.e.,firing times of relevant *** paper proves that the problem of solving max-plus inequalities in a cube(maxINEQ)is nondeterministic polynomial-time hard(NP-hard)in strong sense and the problem of verifying max-plus inequalities(VERmaxINEQ)is *** a corollary,the problem of solving a system of multivariate max-algebraic polynomial equalities and inequalities(MPEI)is shown to be NP-hard in strong *** results indicate the difficulties in comparing max-plus formulas in *** structures of specific systems have to be explored to enable the development of efficient algorithms.

关键词： max-plus algebra NP-hard discrete event dynamic systems

Timetable Design for a Helicopter Maintenance Process using Timed Event Petri Nets and max-plus algebra

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Timetable Design for a Helicopter Maintenance Process using ...

2010 Chinese Control and Decision Conference

作者： Zvi Retchkiman Knigsberg Instituto Politecnico Nacional CIC Mineria 17-2Col.EscandonMexico D.F 11800Mexico

＜正＞In this paper an algorithm for computing a generalized eigenmode of reducible regular matrices over the max-plus algebra is applied to a helicopter maintenance process.A timed event Petri net model is constructed from the state transition dynamics table that characterizes the transport system.A max-plus recurrence equation,with a reducible and regular matrix,is associated to the timed event Petri ***,given the reducible and regular matrix,the problem consists in giving an algorithm which will tell us how to compute its generalized eigenmode over the max plus algebra. The solution to the problem is achieved by studying some type of recurrence *** fact,by transforming the reducible regular matrix into its normal form,and considering a very specific recurrence equation, an explicit mathematical characterization is obtained, upon which the algorithm is *** generalized eigenmode obtained sets a timetable for the helicopter maintenance process.

关键词： max-plus algebra Reducible Matrices Eigenmode Recurrent Equations Algorithm Helicopter Maintenance Process

Cascading Delays for the High-Speed Rail Network Under Different Emergencies:A Double Layer Network Approach

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自动化学报:英文版 2023年第10期10卷 2014-2025页

作者： Xingtang Wu Mingkun Yang Wenbo Lian Min Zhou Hongwei Wang Hairong Dong School of Automation Science and Electrical Engineering Beihang UniversityBeijing 100191China School of Control and Computer Engineering North China Electric Power UniversityBeijing 100096China State Key Laboratory of Advanced Rail Autonomous Operation Beijing Jiaotong UniversityBeijing 100044China National Research Center of Railway Safety Assessment Beijing Jiaotong UniversityBeijing 100044China

High-speed rail(HSR) has formed a networked operational scale in China. Any internal or external disturbance may deviate trains’ operation from the planned schedules, resulting in primary delays or even cascading delays on a network scale. Studying the delay propagation mechanism could help to improve the timetable resilience in the planning stage and realize cooperative rescheduling for dispatchers. To quickly and effectively predict the spatial-temporal range of cascading delays, this paper proposes a max-plus algebra based delay propagation model considering trains’ operation strategy and the systems’ constraints. A double-layer network based breadth-first search algorithm based on the constraint network and the timetable network is further proposed to solve the delay propagation process for different kinds of emergencies. The proposed model could deal with the delay propagation problem when emergencies occur in sections or stations and is suitable for static emergencies and dynamic emergencies. Case studies show that the proposed algorithm can significantly improve the computational efficiency of the large-scale HSR network. Moreover, the real operational data of China HSR is adopted to verify the proposed model, and the results show that the cascading delays can be timely and accurately inferred, and the delay propagation characteristics under three kinds of emergencies are unfolded.

关键词： Delay propagation double layer network high speed rail network max-plus algebra

Modeling and Stability Analysis of a Communication Network System

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Journal of Computer and Communications 2015年第11期3卷 176-183页

作者： Zvi Retchkiman Konigsberg Centro de Investigacion en Computacion Instituto Politecnico NacionalMexico CityMexico

In this work, the modeling and stability problem for a communication network system is addressed. The communication network system consists of a transmitter which sends messages to a receiver. The proposed model considers two possibilities. The first one, that messages are successfully received, while in the second one, during the sending process the transmitter breaks down and as a result the message does not reach the receiver. Timed Petrinets is the mathematical and graphical modeling technique utilized. Lyapunov stability theory provides the required tools needed to aboard the stability problem. Employing Lyapunov methods, a sufficient condition for stabilization is obtained. It is shown that it is possible to restrict the communication network system state space in such a way that boundedness is guaranteed. However, this restriction results to be vague. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graphical model.

关键词： Communication Network System Transmitter Breakdown Discrete Event Dynamical Systems max-plus algebra Lyapunov Method Timed Petri Nets

Resolution of Resource Contentions in the CCPM-MPL Using Simulated Annealing and Genetic Algorithm

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American Journal of Operations Research 2016年第6期6卷 480-488页

作者： Hajime Yokoyama Hiroyuki Goto Department of Industrial & System Engineering Hosei University Tokyo Japan

This research aims to plan a “good-enough” schedule with leveling of resource contentions. We use the existing critical chain project management-max-plus linear framework. Critical chain project management is known as a technique used to both shorten the makespan and observe the due date under limited resources;the max-plus linear representation is an approach for modeling discrete event systems as production systems and project scheduling. If a contention arises within a single resource, we must resolve it by appending precedence relations. Thus, the resolution framework is reduced to a combinatorial optimization. If we aim to obtain the exact optimal solution, the maximum computation time is longer than 10 hours for 20 jobs. We thus experiment with Simulated Annealing (SA) and Genetic Algorithm (GA) to obtain an approximate solution within a practical time. Comparing the two methods, the former was beneficial in computation time, whereas the latter was better in terms of the performance of the solution. If the number of tasks is 50, the solution using SA is better than that using GA.

关键词： Critical Chain Project Management max-plus algebra CCPM-MPL Simulated Annealing Genetic Algorithm

A delay propagation algorithm for trains on a single railway line

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A delay propagation algorithm for trains on a single railway...

第32届中国控制与决策会议

作者： Wentao Xu Chaoyu Tang Junfeng Ma School of Automation Central South University

Due to the large passenger flow and the high traffic density of the Chinese high-speed railway network,the number of train delay incidents is increasing in recent *** a railway network,an initial train delay may bring a cascade effect of secondary delays over the following *** this paper,we analyse the delay propagation among trains on a single railway line and aim to establish a train delay propagation algorithm using the max-plus algebra *** we illustrate the proposed algorithm by an *** study presented here could be potentially useful in large-scale railway networks,and may serve as a useful component in decision support systems to assist train dispatchers.

关键词： Train delay Railway traffic max-plus algebra

Logic-based Solvability of max-plus Linear Equations

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Logic-based Solvability of Max-plus Linear Equations

第三十九届中国控制会议

作者： CHEN Qi LI Haitao LI Ping YANG Xinrong School of Mathematics and Statistics Shandong Normal University

max-plus algebra is an important tool for studying various types of discrete event *** this paper,a logic-based solvability method is proposed to study several kinds of max-plus linear ***,an equivalent logical equation is constructed for the max-plus linear equation A■x=b,based on which,some new necessary and sufficient conditions are presented for the existence and uniqueness of ***,the solvability of max-plus linear equation A■x=b is given by applying semi-tensor product of matrices to solve the equivalent logical ***,using the column stacking approach,A■X⊕X■B=C is converted into the form of A■x=*** examples are given to illustrate the obtained results.

关键词： max-plus algebra Logical equation Solvability Semi-tensor product of matrices

Modelling E/T scheduling problem as optimal tracking problem in assemble lines

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Modelling E/T scheduling problem as optimal tracking problem...

第二十一届中国控制会议

作者： ZHAO Qian-chuan, MAO Jian-feng (Center for Intelligent and Networked Systems Dept. Automation, Tsinghua University, Beijing 100084)

＜正＞In this paper, we propose to deal with E/T scheduling problem by introducing an equivalent assemble line model and solving a corresponding optimal tracking problem. The mean advantage of this method is that the well-developed max-plus algebra results can be used directly. The main contribution of this paper is to establish the equivalence between the original problem and the optimal tracking problem in the max-plus model of the assemble lines.

关键词： E/T scheduling max-plus algebra optimal tacking