Max-plus Algebra Research Articles

Modelling and feedback control for a class of Petri Nets with shared resources subject to strict time constraints using Max-plus algebra

We address a modelling framework and a state feedback control problem of discrete event systems with shared resources using Max-plus algebra. In this paper, we consider the class of Networked Conflicting Timed Event Graphs (NCTEGs), which are timed Petri nets with conflicts, subjected to strict temporal constraints. Firstly, an algebraic formalisation in terms of switching Max-plus linear systems is proposed to describe the dynamic behaviour of a NCTEG. Secondly, closed-loop control laws are calculated to ensure the compliance of these time constraints imposed on a certain number of places. For this, sufficient conditions for the existence of such control laws have been provided. Finally, these theoretical approaches are illustrated by a realistic example of a time-critical railway network crossing system.

Max-plus approximation for reinforcement learning

Max-Plus Algebra has been applied in several contexts, especially in the control of discrete events systems. In this article, we discuss another application closely related to control: the use of Max-Plus algebra concepts in the context of reinforcement learning. Max-Plus Algebra and reinforcement learning are strongly linked due to the latter’s dependence on the Bellman Equation which, in some cases, is a linear Max-Plus equation. This fact motivates the application of Max-Plus algebra to approximate the value function, central to the Bellman Equation and thus also to reinforcement learning. This article proposes conditions so that this approach can be done in a simple way and following the philosophy of reinforcement learning: explore the environment, receive the rewards and use this information to improve the knowledge of the value function. The proposed conditions are related to two matrices and impose on them a relationship that is analogous to the concept of weak inverses in traditional algebra.

Tropical Geometry and Machine Learning

Tropical geometry is a relatively recent field in mathematics and computer science, combining elements of algebraic geometry and polyhedral geometry. The scalar arithmetic of its analytic part preexisted in the form of max-plus and min-plus semiring arithmetic used in finite automata, nonlinear image processing, convex analysis, nonlinear control, optimization, and idempotent mathematics. Tropical geometry recently emerged in the analysis and extension of several classes of problems and systems in both classical machine learning and deep learning. Three such areas include: 1) deep neural networks with piecewise linear (PWL) activation functions; 2) probabilistic graphical models; and 3) nonlinear regression with PWL functions. In this article, we first summarize introductory ideas and objects of tropical geometry, providing a theoretical framework for both the max-plus algebra that underlies tropical geometry and its extensions to general max algebras. This unifies scalar and vector/signal operations over a class of nonlinear spaces, called weighted lattices, and allows us to provide optimal solutions for algebraic equations used in tropical geometry and generalize tropical geometric objects. Then, we survey the state of the art and recent progress in the aforementioned areas. First, we illustrate a purely geometric approach for studying the representation power of neural networks with PWL activations. Then, we review the tropical geometric analysis of parametric statistical models, such as HMMs; later, we focus on the Viterbi algorithm and related methods for weighted finite-state transducers and provide compact and elegant representations via their formal tropical modeling. Finally, we provide optimal solutions and an efficient algorithm for the convex regression problem, using concepts and tools from tropical geometry and max-plus algebra. Throughout this article, we also outline problems and future directions in machine learning that can benefit from the tropical-geometric point of view.

A discrete-event model of the train traffic on a linear metro line

We present in this article a mathematical model for the traffic on a linear metro line (a metro line without junction), taking into account the train dynamics and the passenger travel demand. The train dynamics are modeled as a discrete event dynamic system, written linearly in the Max-plus algebra. The model permits an analytic derivation of phase diagrams for the train dynamics depending on the passenger travel demand. By this, the physics of traffic on linear metro lines is wholly understood and interpreted. We introduce into the model the passenger capacity of trains, and derive analytically an indicator for the passenger comfort inside the trains, as a function of the number of trains and of the level of the passenger demand. By this model, metro operators can adjust the train dwell times at platforms depending on the level of the passenger demand. They can also optimize the train frequency with respect to the number of trains, and depending on the level of the passenger demand. Furthermore, the level of the passenger comfort inside the trains can be taken as the result of a compromise between the passengers’ satisfaction, in one side, and the number of trains as well as the passenger capacity of trains, in terms of the economic criteria of the operator, in the other side.

On Application of Max-Plus Algebra to Synchronized Discrete Event System

Max-plus algebra is a discrete algebraic system developed on the operations max (<img src=image/13420822_01.gif>) and plus (<img src=image/13420822_03.gif>), where the max and plus operations are defined as addition and multiplication in conventional algebra. This algebraic structure is a semi-ring with its elements being real numbers along with ε=-∞ and e=0. On the other hand, the synchronized discrete event problem is a problem in which an event is scheduled to meet a deadline. There are two aspects of this problem. They include the events running simultaneously and the completion of the lengthiest event at the deadline. A recent survey on max-plus linear algebra shows that the operations max (<img src=image/13420822_01.gif>) and plus (<img src=image/13420822_03.gif>) play a significant role in modeling of human activities. However, numerous studies have shown that there are very limited literatures on the application of the max-plus algebra to real-life problems. This idea motivates the basic algebraic results and techniques of this research. This paper proposed the discrepancy method of max-plus for solving n×n system of linear equations with n≤n, and further show that an nxn linear system of equations will have either a unique solution, an infinitely many solutions or no solution whiles nxn linear system of equations has either an infinitely many solutions or no solution in (<img src=image/13420822_02.gif>). Also, the proposed concept was extended to the job-shop problem in a synchronized event. The results obtained have shown that the method is very efficient for solving n×n system of linear equations and is also applicable to job-shop problems.

Assessing manufacturing flow lines under uncertainties in processing time: An application based on max-plus equations, multicriteria decisions, and global sensitivity analysis

In this paper, a novel application on how uncertainties in a manufacturing flow line − MFL (e.g., times required to perform an action) could be analyzed and what the benefits are of such analysis. The approach proposed investigates three main goals: i) Uncertainty analysis, ii) Stochastic dominance, and iii) Sensitivity analysis. In particular, this paper extends the application of max-plus algebra to model MFL with different flow configurations and buffer capacities and provides the approximated probability density functions (PDFs) of selected performance indicators (e.g., the total idle time in the whole line, output rates, throughputs, among others). As a result, it is possible to quantify the variability of the selected output, compare different possible configurations among MFL, choose the best one, and identify critical variables and risk drivers (e.g., the processing times that affect the most a KPI − key performance indicator). The approach, illustrated by analyzing a case study of the literature, emphasizes the benefits for a decision-maker in charge of the design or managing of the manufacturing system.

Resilience dynamics modeling and control for a reconfigurable electronic assembly line under spatio-temporal disruptions

This paper proposes a resilience dynamics modeling and control approach for a reconfigurable electronic assembly line under disruptions. A Digital Twin (DT) platform is developed as the basis for resilience analysis, and open reconfigurable architectures (ORAs) are introduced to support reconfiguration of the assembly line. The time-delays of disruptions are identified and used to characterize their spatio-temporal attributes. A systematic method based on max-plus algebra is proposed to model resilience dynamics under disruptions. The resilience control policy used in the DT platform is developed to minimize production losses, and it is tested on a smart phone assembly line, with its effectiveness validated by comparative analysis.

On the tightness of bounds for transients of weak CSR expansions and periodicity transients of critical rows and columns of tropical matrix powers

We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by max{T 1, T 2}, where T 1 is the weak CSR threshold and T 2 is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for T 1 and T 2, naturally leading to the question which matrices, if any, attain these bounds. In the present paper, we characterize the matrices attaining two particular bounds on T 1, which are generalizations of the bounds of Wielandt and Dulmage–Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case.

Petri Net Model and Max-Plus Algebra in Outpatient Care at Solo Peduli Clinic, Surakarta

Patients come to SoloPeduli Clinic to get medical treatment. Medical treatment at SoloPeduli Clinic is divided into two parts namely outpatient and inpatient health services. Patients generally do not know how long it will take for service from arrival to discharge at the clinic. The queue flow is called a system, while the patients queuing at each service are called states, the number of states cannot be limited as health services must be available 24 hours a day. This issue includes a Discrete Event System which can be modeled with a Petri Net. Petri Net is a mathematical model that can be used to describe certain events in a Discrete Event System. In this research, service flow is modeled using Petri Net which is then represented in the form of a matrix. Coverability Tree was created to define life and deadlock. Scheduling is modeled using Max-Plus Algebra. Service times will be randomly selected to facilitate scheduling and length of service time.

A Note on Eigenvalue of Matrices over The Symmetrized Max-Plus Algebra

Max-plus algebra is the structure that doesn’t have an inverse of additive. Therefore, there exists an equation that doesn’t have a solution. For example, equation 3 ®x=2 has no solution because there is no x such that max(3,x) = 2. The max-plus will have an inverse element of addition if that structure is extended to the symmetrized max-plus algebra. The expansion into a larger system is the same as the expansion of the natural number into an integer number. This paper describes the necessary or sufficient condition of the eigenvalue of matrices over the symmetrized max-plus algebra using the linear balance systems A®xV b with V as a balance relation.

Max-Plus Linear Algebra and Its Applications

Max-Plus Linear Algebra and Its Applications

A Novel Model for Train Operation Adjustment in High-Speed Railway based on Max-plus Algebra

The running time of high-speed train is generally not late, and it can run normally in most cases. However, when severe weather conditions, train components and equipment, accidents or emergencies occur, it will lead to train operation delay and traffic congestion. Therefore, when an accident occurs, we need to adjust the train time or route timely and accurately. As an important algebraic system, max-plus algebra is widely used in the field of industrial production control. In industrial production, the most production mode is the discrete system , but the characteristics and the ability of discrete systems depends on the periodic of system and the number of workpiece produced by the system in unit time, and the characteristics of the system are closely related to the properties of the matrix, especially, the eigenvalues and eigenvectors of the matrix in the sense of max-plus algebra. Therefore, this paper studies the max-plus algebra theory and the solution of eigenvalues and eigenvectors of matrices in the sense of max-plus algebra, establish the operation time matrix to optimize the train operation adjustment model of high-speed railway, and analyze the failure propagation model.

MODELING PREDICTIVE TRACKING CONTROL FOR MAX-PLUS LINEAR SYSTEMS IN MANUFACTURING

Discrete event systems, also known as DES, are class of system that can be applied to systems having an event that occurred instantaneously and may change the state. It can also be said that a discrete event system occurs under certain conditions for a certain period because of the network that describes the process flow or sequence of events. Discrete event systems belong to class of nonlinear systems in classical algebra. Based on this situation, it is necessary to do some treatments, one of which is linearization process. In the other hand, a Max-Plus Linear system is known as a system that produces linear models. This system is a development of a discrete event system that contains synchronization when it is modeled in Max-Plus Algebra. This paper discusses the production system model in manufacturing industries where the model pays the attention into the process flow or sequence of events at each time step. In particular, Model Predictive Control (MPC) is a popular control design method used in many fields including manufacturing systems. MPC for Max-Plus-Linear Systems is used here as the approach that can be used to model the optimal input and output sequences of discrete event systems. The main advantage of MPC is its ability to provide certain constraints on the input and output control signals. While deciding the optimal control value, a cost criterion is minimized by determining the optimal time in the production system that modeled as a Max-Plus Linear (MPL) system. A numerical experiment is performed in the end of this paper for tracking control purposes of a production system. The results were good that is the controlled system showed a good performance.

Analysis of Global and Local Synchronization in Parallel Computing

In a parallel computing scenario, the synchronization overhead, needed to coordinate the execution on the parallel computing nodes, can significantly impair the overall execution performance. Typically, synchronization is achieved by adopting a global synchronization schema involving all the nodes. In many application domains, though, a looser synchronization schema, namely, local synchronization, can be exploited, in which each node needs to synchronize only with a subset of the other nodes. In this work, we compare the performance of global and local synchronization using the efficiency, i.e., the ratio between the useful computing time and the total computing time, including the synchronization overhead, as a key performance indicator. We present an analytical study of the asymptotic behavior of the efficiency when the number of nodes increases. As an original contribution, we prove, using the Max-Plus algebra, that there is a non-zero lower bound on the efficiency in the case of local synchronization and we present a statistical procedure to find a value of this bound. This outcome marks a significant advantage of local synchronization with respect to global synchronization, for which the efficiency tends to zero when increasing the number of nodes.

Scaling, Modeling and Traffic Control of a Real Railway Network using Max-plus Algebra and Model Predictive Control

Scaling, Modeling and Traffic Control of a Real Railway Network using Max-plus Algebra and Model Predictive Control

A uniform synchronization problem over max-plus algebra

The solution of $$A\otimes x{=}B{\otimes} y$$ has been considered in the literature and various methods have been established. However, a solution such that the resulting product is a vector having all its components equal has not been treated to the best of our knowledge. In this paper, we study a synchronization problem $$A\otimes x{=}B\otimes y{=}\alpha $$ and proposed an $$O(mn+mk)$$ algorithm for its solution. Where $$m$$ is the number of rows of the matrices and $$n \, and \, k$$ are the number of columns of $$A \, and \, B$$ respectively. That is, given any Two matrices that have the same number of rows, $$m$$ we introduce some algorithm that generates two column vectors $$x\, and \, y$$ such that $$A\otimes x=B\otimes y=\alpha $$ , where $$A \, and \, B$$ assumed to be P-doubly G-astic matrices having the same number of rows and $$\alpha $$ is a column vector having all its components equal.

Factor rank preservers of matrices over additively-idempotent multiplicatively-cancellative semirings

Here, we characterize the linear operators that preserve factor rank of matrices over additively-idempotent multiplicatively-cancellative semirings. The main results in this paper generalize the corresponding results on the two element Boolean algebra [L. B. Beasley and N. J. Pullman, Boolean-rank-preserving opeartors and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984) 55–77] and on the max algebra [R. B. Bapat, S. Pati and S.-Z. Song, Rank preservers of matrices over max algebra, Linear Multilinear Algebra 48(2) (2000) 149–164]; and hold on max-plus algebra and some other tropical semirings.

English

We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation A ⊗ x = λ ⊗ x for usual eigenvectors. Under a suitable assumption, the equation has a nontrivial solution if and only if λ is a root of the characteristic polynomial. (2) The set of algebraic eigenvectors forms a max-plus subspace called algebraic eigenspace. (3) The dimension of each algebraic eigenspace is at most the multiplicity of the corresponding root of the characteristic polynomial.

Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

Matrix representation of formal polynomials over max-plus algebra

This paper proposes the matrix representation of formal polynomials over max-plus algebra and obtains the maximum and minimum canonical forms of a polynomial function by standardizing this representation into a canonical form. A necessary and sufficient condition for two formal polynomials corresponding to the same polynomial function is derived. Such a matrix method is constructive and intuitive, and leads to a polynomial algorithm for factorization of polynomial functions. Some illustrative examples are presented to demonstrate the results.