• Title/Summary/Keyword: max-plus algebra.

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Timetabling and Analysis of Train Connection Schedule Using Max-Plus Algebra (Max-Plus 대수를 이용한 환승 스케줄 시간표 작성 및 분석)

  • Park, Bum-Hwan
    • Journal of the Korean Society for Railway
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    • v.12 no.2
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    • pp.267-275
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    • 2009
  • Max-plus algebra is a nonlinear system comprised of two operations, maximization (max) and addition (Plus), which are corresponding to the addition and the multiplication in conventional algebra, respectively. This methodology is applicable to many discrete event systems containing the state transition with the maximization and addition operation. Timetable with connection is one of such systems. We present the method based on max-plus algebra, which can make up timetable considering transfer and analyse its stability and robustness. In this study, it will be shown how to make up the timetable of the urban train and analyse its stability using Max-Plus algebra.

Derivation Algorithm of State-Space Equation for Production Systems Based on Max-Plus Algebra

  • Goto, Hiroyuki;Masuda, Shiro
    • Industrial Engineering and Management Systems
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    • v.3 no.1
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    • pp.1-11
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    • 2004
  • This paper proposes a new algorithm for determining an optimal control input for production systems. In many production systems, completion time should be planned within the due dates by taking into account precedence constraints and processing times. To solve this problem, the max-plus algebra is an effective approach. The max-plus algebra is an algebraic system in which the max operation is addition and the plus operation is multiplication, and similar operation rules to conventional algebra are followed. Utilizing the max-plus algebra, constraints of the system are expressed in an analogous way to the state-space description in modern control theory. Nevertheless, the formulation of a system is currently performed manually, which is very inefficient when applied to practical systems. Hence, in this paper, we propose a new algorithm for deriving a state-space description and determining an optimal control input with several constraint matrices and parameter vectors. Furthermore, the effectiveness of this proposed algorithm is verified through execution examples.

AN OVERVIEW OF MAX-PLUS LINEAR SYSTEMS

  • Kim, Yong-Gu;Shin, Hyun-Hee
    • Honam Mathematical Journal
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    • v.33 no.1
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    • pp.93-113
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    • 2011
  • Let $a{\oplus}b$ = max(a, b), $a{\otimes}b$=a+b, a, $b\in\mathbb{R}_{\varepsilon}\;:=\cup\{-\infty\}$. In max-plus algebra we work on the linear algebra structure for the pair of operations (${\oplus},{\otimes}$) extended to matrices and vectors over $\mathbb{R}_{\varepsilon}$. In this paper our main aim is to reproduce the work of R. A. Cuninghame-Green [3] on the linear systems over a max-plus semi-field $\mathbb{R}_{\varepsilon}$.

Eigenspaces of Max-Plus Matrices: An Overview

  • Kim, Yonggu;Shin, Hyun Hee
    • Journal for History of Mathematics
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    • v.31 no.1
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    • pp.1-17
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    • 2018
  • In this expository paper, we present an abridged report on the max-plus eigenspaces of max-plus matrices with its brief history. At the end of our work, a number of examples are presented with maple codes, and then we make a claim from the observation of these examples, which is on the euclidean dimension of the max-plus eigenspaces of strongly definite matrices.

Model Predictive Control for Productions Systems Based on Max-plus Algebra

  • Hiroyuki, Goto;Shiro, Masuda;Kazuhiro, Takeyasu;Takashi, Amemiya
    • Industrial Engineering and Management Systems
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    • v.1 no.1
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    • pp.1-9
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    • 2002
  • Among the state-space description of discrete vent systems, the max-plus algebra is known as one of the effective approach. This paper proposes a model predictive control (MPC) design method based on the max-plus algebra. Several studies related to these topics have been done so far under the constraints that system parameters are constant. However, in practical systems such as production systems, it is common and sometimes inevitable that system parameters vary by each event. Therefore, it is of worth to design a new MPC controller taking account of adjustable system parameters. In this paper, we formulate system parameters as adjustable ones, and they are solved by a linear programing method. Since MPC determines optimal control input considering future reference signals, the controller can be more robust and the operation cost can be reduced. Finally, the proposed method is applied to a production system with three machines, and the effectiveness of the proposed method is verified through a numerical simulation.

Explicit Expression for Moment of Waiting Time in a DBR Line Production System with Constant Processing Times Using Max-plus Algebra (Max-plus 대수를 이용한 상수 공정시간을 갖는 DBR 라인 생산시스템에서의 대기시간에 대한 간결한 표현식)

  • Park, Philip;Seo, Dong-Won
    • Journal of the Korea Society for Simulation
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    • v.24 no.2
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    • pp.11-17
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    • 2015
  • Although systems with finite capacities have been the topic of much study, there are as of yet no analytic expressions for (higher) moment and tail probability of stationary waiting times in systems with even constant processing times. The normal queueing theory cannot properly handle such systems due to the difficulties caused by finite capacity. In this study, for a DBR (Drum-Buffer-Rope) line production system with constant processing times, we introduce analytic expressions by using previous results obtained using a max-plus algebraic approach.

Monitoring and Scheduling Methods for MIMO-FIFO Systems Utilizing Max-Plus Linear Representation

  • Goto, Hiroyuki;Masuda, Shiro
    • Industrial Engineering and Management Systems
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    • v.7 no.1
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    • pp.23-33
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    • 2008
  • This paper proposes an approach to monitoring and scheduling methods for repetitive MIMO-FIFO DESs. We use max-plus algebra for modeling and formulation, known as an effective approach for controller design for this type of system. Because a certain type of linear equations in max-plus algebra can represent the system's behavior, the principal concerns in past researches were how to solve the equations. However, the researches focused mainly on analyses of the relation between inputs and outputs of the system, which implies that the changes or the slacks of internal states were not clarified well. We first examine several properties of the corresponding state variables, which contribute to finding and tracing the float times in each process. Moreover, we provide a rescheduling method that can take into account delays or changes of the internal states. These methods would be useful in schedule control or progress management.

Spreadsheet Model Approach for Buffer-Sharing Line Production Systems with General Processing Times (일반 공정시간을 갖는 버퍼 공유 라인 생산시스템의 스프레드시트 모형 분석)

  • Seo, Dong-Won
    • Journal of the Korea Society for Simulation
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    • v.28 no.2
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    • pp.119-129
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    • 2019
  • Although line production systems with finite buffers have been studied over several decades, except for some special cases there are no explicit expressions for system performances such as waiting times(or response time) and blocking probability. Recently, a max-plus algebraic approach for buffer-sharing systems with constant processing times was introduced and it can lead to analytic expressions for (higher) moment and tail probability of stationary waiting. Theoretically this approach can be applied to general processing times, but it cannot give a proper way for computing performance measures. To this end, in this study we developed simulation models using @RISK software and the expressions derived from max-plus algebra, and computed and compared blocking probability, waiting time (or response time) with respect to two blocking policies: communication(BBS: Blocking Before Service) and production(BAS: Blocking After Service). Moreover, an optimization problem which determines the minimum shared-buffer capacity satisfying a predetermined QoS(quality of service) is also considered.

Toward the Application of a Critical-Chain-Project-Management-based Framework on Max-plus Linear Systems

  • Takahashi, Hirotaka;Goto, Hiroyuki;Kasahara, Munenori
    • Industrial Engineering and Management Systems
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    • v.8 no.3
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    • pp.155-161
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    • 2009
  • We focus on discrete event systems with a structure of parallel processing, synchronization, and no-concurrency. We use max-plus algebra, which is an effective approach for controller design for this type of system, for modeling and formulation. Since a typical feature of this type of system is that the initial schedule is frequently changed due to unpredictable disturbances, we use a simple model and numerical examples to examine the possibility of applying the concepts of the feeding buffer and the project buffer of critical chain project management (CCPM) on max-plus linear discrete event systems in order to control the occurrence of an undesirable state change. The application of a CCPM-based framework on a max-plus linear discrete event system was proven to be effective.

Spreadsheet Model Approach for Buffer-Sharing Fork-Join Production Systems with General Processing Times and Structure (일반 공정시간과 구조를 갖는 버퍼 공유 분기-접합 생산시스템의 스프레드시트 모형 분석)

  • Seo, Dong-Won
    • Journal of the Korea Society for Simulation
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    • v.28 no.3
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    • pp.65-74
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    • 2019
  • For many years, it has been widely studied on fork-join production systems but there is not much literature focusing on the finite buffer(s) of either individuals or shared, and generally distributed processing times. Usually, it is difficult to handle finite buffer(s) through a standard queueing theoretical approach. In this study, by using the max-plus algebraic approach we studied buffer-shared fork-join production systems with general processing times. However, because it cannot provide proper computational ways for performance measures, we developed simulation models using @RISK software and the expressions derived from max-plus algebra. From the simulation experiments, we compared some properties on waiting time with respect to a buffer capacity under two blocking policies: BBS (Blocking Before Service) and BAS (Blocking After Service).