Max-plus Algebra Research Articles

The non-positive circuit weight problem in parametric graphs: A solution based on dioid theory

Let us consider a parametric weighted directed graph in which every arc (j,i) has weight of the form w((j,i))=max(Pij+λ,Iij−λ,Cij), where λ is a real parameter and P, I and C are arbitrary square matrices with elements in R∪{−∞}. In this paper, we design an algorithm that solves the Non-positive Circuit weight Problem (NCP) on this class of parametric graphs, which consists in finding all values of λ such that the graph does not contain circuits with positive weight. This problem, which generalizes other instances of the NCP previously investigated in the literature, has applications in the consistency analysis of a class of discrete-event systems called P-time event graphs. The proposed algorithm is based on max-plus algebra and formal languages, and improves the worst-case complexity of other existing approaches, achieving strongly polynomial time complexity O(n4) (where n is the number of nodes in the graph).

MAX-PLUS ALGEBRA MODEL ON INAPORTNET SYSTEM SHIPS SERVICE SCHEME

Queues are often found in public service providers, including port services. Therefore, the InaPortNet scheme was developed to facilitate and minimize queues in and out of ships. Also, service performance was optimized through behavior analysis and queuing system stability. This study focused on designing the max-plus algebra model and time estimation on the incoming ship service scheme. The results showed that the InaPortNet system is useful for modeling. Furthermore, the max-plus algebra matrix is used to obtain an estimated service time from registration to the ship docking process at the port. However, further study needs to be carried out by scheduling the max-plus algebraic matrix in order to analyze the behavior and stability of the queuing system

Tree Topologies along a Tropical Line Segment

Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.

The Model Matching Problem for Switching Max-Plus Systems: a Geometric Approach

Linear systems over the max-plus algebra can model discrete event systems where synchronization, without competition, is involved. The lack of competition can be partly circumvented by considering multiple linear models, each representing a possible choice in resource allocation, and a switching mechanism, thus obtaining a switching linear max-plus system. We propose a formulation of the model matching problem for systems of such kind. The aim is to force a given plant to match exactly the output of a given model. A sufficient condition for the solvability of the problem is obtained by extending the geometric approach to switching systems over the max-plus algebra.

Switched Max-Plus Linear-Dual Inequalities: Application in Scheduling of Multi-Product Processing Networks

P-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by max-plus linear-dual inequalities (LDIs), i.e., systems of linear dynamical inequalities in the max-plus algebra and its dual, the min-plus algebra. We define a new class of models called switched LDIs (SLDIs), which allow to switch between different modes of operations, each corresponding to an LDI, according to an infinite sequence of modes called schedule. In this paper, we focus on the analysis of SLDIs when the schedule is fixed and periodic. We show that SLDIs can model single-robot multi-product processing networks, in which every product has different processing requirements and corresponds to a specific mode of operation. Based on the analysis of SLDIs, we propose an algorithm to compute minimum and maximum cycle times for these processes that improves the time complexity of other existing approaches.

Scheduling Inland Waterway Transport Vessels and Locks Using a Switching Max-Plus-Linear Systems Approach

This paper considers the inland waterborne transport (IWT) problem, and presents a scheduling approach for inland vessels and locks to generate optimal vessel and lock timetables. The scheduling strategy is designed in the switching max-plus-linear (SMPL) systems framework, as these are characterized by a number of features that make them well suited to represent the IWT problem. In particular, the resulting model is linear in the max-plus algebra, and SMPL systems can switch between modes, an interesting feature due to the presence of vessel routing and ordering constraints in the model. Moreover, SMPL systems can be transformed into mixed-integer linear programming (MILP) problems, for which efficient solvers are available. Finally, a realistic case study is used to test the approach and assess its effectiveness.

Systems synchronisation in Max-Plus algebra: a controlled invariance perspective In memoriam Édouard Wagneur

We study the concept of systems synchronisation in Max-Plus algebra. We show that the problem of synchronising Max-Plus linear systems is twofold. It can be stated in terms of controlled invariance, and of coreachability. The controlled invariance is the property of a set for which a suitable control exists, that maintains the trajectory in the set. We recall some properties of controlled invariance, in particular the existence of a maximal controlled invariant set included into any given set. Beyond the determination of the admissible controlled invariant set, we define and study a problem of coreachability. The controlled invariant set is the target and we study the existence of a control law that brings the trajectory to the target from the initial state within a specific number of steps. We illustrate the study with an example from the literature.

Periodic Trajectories in P-Time Event Graphs and the Non-Positive Circuit Weight Problem

P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers d, the existence of consistent d-periodic trajectories for a given P-TEG. In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP) consists in finding all the values of a parameter such that a particular parametric weighted directed graph does not contain circuits with positive weight. In a related paper, we have proposed a strongly polynomial algorithm that solves the PIC-NCP in lower worst-case complexity compared to other algorithms reported in literature. In this letter, we show that the first problem can be formulated as an instance of the second; consequently, we prove that the same algorithm can be used to find d-periodic trajectories in P-TEGs. Moreover, exploiting the connection between the PIC-NCP and max-plus algebra we prove that, given a P-TEG, the existence of a consistent 1-periodic trajectory of a certain period is a necessary and sufficient condition for the existence of a consistent d-periodic trajectory of the same period, for any value of d.

Simultaneously Updating Multiple Performance Measures of Manufacturing Systems after Scheduling Perturbations

We simultaneously update multiple performance measures of manufacturing systems after several scheduling perturbations through a single pass without calculating from scratch. Our approach reuses information from the schedule before the perturbations. It utilizes Max-Plus algebra and system matrix to calculate changes propagated from initially perturbed operations to their affected successors. Then, by using these changes, it evaluates new measures for affected operations. Previous algorithms either need to calculate multiple performance measures from scratch by disregarding information before perturbations; or calculate only a single performance measure at a time while reusing system information before perturbations. Our approach addresses both issues.

Max-Plus Algebraic Description of Evolutions of Weighted Timed Event Graphs

Max-Plus Algebraic Description of Evolutions of Weighted Timed Event Graphs

MASALAH EIGEN DAN EIGENMODE MATRIKS ATAS ALJABAR MIN-PLUS

Eigen problems and eigenmode are important components related to square matrices. In max-plus algebra, a square matrix can be represented in the form of a graph called a communication graph. The communication graph can be strongly connected graph and a not strongly connected graph. The representation matrix of a strongly connected graph is called an irreducible matrix, while the representation matrix of a graph that is not strongly connected is called a reduced matrix. The purpose of this research is set the steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and also eigenmode of the regular reduced matrix over min-plus algebra. Min-plus algebra has an ispmorphic structure with max-plus algebra. Therefore, eigen problems and eigenmode matrices over min-plus algebra can be determined based on the theory of eigenvalues, eigenvectors and eigenmode matrices over max-plus algebra. The results of this research obtained steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and eigenmode algorithm of the regular reduced matrix over min-plus algebra

Eigenvalue decomposition of a symmetric matrix over the symmetrized max-plus algebra

This paper discusses topics in the symmetrized max-plus algebra. In this study, it will be shown the existence of eigenvalue decomposition of a symmetric matrix over symmetrized max-plus algebra. Eigenvalue decomposition is shown by using a function that corresponds to the symmetrized max-plus algebra with conventional algebra. The result obtained is the existence of eigenvalue decomposition of a symmetric matrix over symmetrized max-plus algebra and its application to determine eigenvalues and eigenvectors.

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

Motion Modeling and Control of Lower Limb Exoskeleton Based on Max-Plus Algebra

Max-plus algebra is a special method to describe the discrete event system. In this paper, it is introduced to describe the motion of lower limb exoskeleton. Based on the max-plus algebra and the timed event graph, the walking process of exoskeleton is modelled. The max-plus algebra approach can describe the logical sequence and safety condition in the walking process, which cannot be achieved via other conventional modelling approaches. The autonomous control of lower limb exoskeleton system is studied via the model based on max-plus algebra. In the end, an FSM (finite state machine) controller embedded with the max-plus algebra model is proposed, and the experiments show ideal speed and gait/phase period control effect, as well as the good safety and stable performance.

Scheduling‐measure dependent modelling of delay propagation on a single railway line: A matrix transformation approach

This paper proposes an analytical delay propagation model for single railway lines based on the max-plus algebra theory. The scheduling measures taken by dispatchers, including re-timing and re-ordering, will be incorporated into our delay propagation model using a matrix transformation method. An analysis of delay propagation under some typical emergencies such as segment blockages and train speed limitation is performed. Numerical simulations show that the proposed train delay propagation model can predict emergency-induced train delays under different scheduling strategies, thus may give a guidance to improve the traffic management. In the high-speed railway train system, the scheduling measures taken by dispatchers, such as re-timing and re-ordering, can be formulated as a delay propagation model using a matrix transformation method.

Matrix roots in the max-plus algebra

We define positive integer roots of finite matrices in the 2×2 case. Where possible, we try to generalise results to finite n×n matrices.

Cayley-Hamilton theorem in the min-plus algebra

Max-plus algebra is the set of ℝ ϵ = ℝ ∪ {ϵ} where ℝ is a set of all real numbers and ϵ = -∞ which is equipped with max (⊕) and plus (⊗) operations. The structure of max-plus algebra is semi-field. Max-plus algebra has structural similarities to conventional algebra. Because of these similarities, concept in conventional algebra such as the Cayley-Hamilton theorem has max-algebraic equivalence. Another semi-field is min-plus algebra, so that the Cayley-Hamilton theorem also has a min-plus algebraic equivalence. In this paper, we will show how the Cayley-Hamilton theorem can be proved in the min-plus algebra.

Determinant of a matrix over min-plus algebra

Max-plus algebra is a set ℝ max = ∪ ℝ {ϵ}, with ℝ is the set of all real numbers and e = -∞ equipped with ⊕ (maximum) and ⊗ (plus) operation. Another algebra that can be learned is the min-plus algebra. Min-plus algebra is a set ℝ min = ℝ ∪ {ϵ′}, with ℝ is the set of all real number and ϵ = +∞ equipped with ⊕′ (minimum) and ⊗ (plus) operation. Any square matrices over max-plus algebra can be related to the determinant. The determinant term of max-plus algebra is represented by the permanent and dominant. This study will discuss the determinant of a matrix over min-plus algebra.

Scheduling tank trucks at a fuel distribution terminal using max-plus model-based predictive control

This study addresses the problem of scheduling tank trucks at a fuel distribution terminal. The plant was modeled in the max-plus algebra, applying machine learning to determine process times. With this model, and based on a just-in-time approach, we have developed a predictive controller with two operation modes. This control system aims to prevent the excess of tank trucks inside the loading yard, thus achieving a better flow, efficiency, and safety in the process. Next, we have investigated the case study of a realistic and representative fuel distribution terminal, developing a simulator to enable a performance comparison between the proposed algorithms and the current heuristic. There was a 42.7% reduction in the work-in-progress (WIP) and 41.4% in the lead time, while productivity suffered a 2.8% loss. Bearing in mind, however, that there is flexibility in parametrization to mitigate this loss of productivity. In doing so, the reductions in WIP and lead time are slightly lower, at 34.7% for both metrics. The results show that the proposed control system can contribute significantly to improving the company’s performance indicators.

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.