Max-plus Algebra Research Articles

Min–max consensus of multi-agent systems in random networks

This paper focuses on the study of the min–max consensus problem for multi-agent systems in random networks. A min–max consensus algorithm is proposed, which incorporates the weighted average of the agent’s state, the maximum value of its neighbors’ states, and the minimum value of its neighbors’ states. The interaction network is composed of a directed random network where edges exist with probability and are independent of each other. Then, the convergence of the min–max consensus algorithm is investigated utilizing max-plus algebra, min-plus algebra, and stochastic analysis theory. Sufficient and necessary conditions are given to ensure that the multi-agent systems achieve min–max consensus almost surely and in mean-square, respectively. Finally, some numerical simulations are conducted to confirm the effectiveness of the proposed consensus algorithm.

Tropical Logistic Regression Model on Space of Phylogenetic Trees

Classification of gene trees is an important task both in the analysis of multi-locus phylogenetic data, and assessment of the convergence of Markov Chain Monte Carlo (MCMC) analyses used in Bayesian phylogenetic tree reconstruction. The logistic regression model is one of the most popular classification models in statistical learning, thanks to its computational speed and interpretability. However, it is not appropriate to directly apply the standard logistic regression model to a set of phylogenetic trees, as the space of phylogenetic trees is non-Euclidean and thus contradicts the standard assumptions on covariates. It is well-known in tropical geometry and phylogenetics that the space of phylogenetic trees is a tropical linear space in terms of the max-plus algebra. Therefore, in this paper, we propose an analogue approach of the logistic regression model in the setting of tropical geometry. Our proposed method outperforms classical logistic regression in terms of Area under the ROC Curve in numerical examples, including with data generated by the multi-species coalescent model. Theoretical properties such as statistical consistency have been proved and generalization error rates have been derived. Finally, our classification algorithm is proposed as an MCMC convergence criterion for Mr Bayes. Unlike the convergence metric used by Mr Bayes which is only dependent on tree topologies, our method is sensitive to branch lengths and therefore provides a more robust metric for convergence. In a test case, it is illustrated that the tropical logistic regression can differentiate between two independently run MCMC chains, even when the standard metric cannot.

CRAMER’S RULE IN MIN-PLUS ALGEBRA

Cramer’s rule is one of a method for solving a system of linear equations in conventional algebra. The system of linear equation can be solved using Cramer’s rule if . Max-plus algebra is a set where is a set of real numbers, equipped with biner operations and where and . Min-plus Algebra is a set where is a set of real numbers, equipped with biner operations and where and . In max-plus algebra has been formulated Cramer’s rule to solve a system of linear equations. Because max-plus algebra is isomorphic to min-plus algebra, Cramer’s rule can be formulated into min-plus algebra. The purpose of this research is to determine the sufficient conditions for a system of linear equations can be solved using Cramer’s rule. The method used in this research is a literature study that reviews previous research related to min-plus algebra, max-plus algebra, and Cramer’s rule in max-plus algebra. By using the appropriate analogy in max-plus algebra, we can determine the sufficient conditions so that a system of linear equations in min-plus algebra can be solved using Cramer’s rule. Based on the research, the sufficient conditions for a system of linear equations can be solved using Cramer’s rule are for and with the Cramer’s rule is . For an invertible matrix A, Cramer’s rule can be written as .

On max-plus two-sided linear systems whose solution sets are min-plus linear

The max-plus algebra R∪{−∞} is defined in terms of a combination of the following two operations: addition a⊕b:=max⁡(a,b) and multiplication a⊗b:=a+b. In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system A⊗x=B⊗x. We demonstrate that the minimum “min-plus” linear subspace containing the “max-plus” solution space can be computed by applying the alternating method, which is a well-known algorithm to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the “min-plus” and “max-plus” subspaces to be identical. The computational complexity of the algorithm presented in this study is pseudo-polynomial.

Input-Output Analysis on Pia Saronde Production Process Scheduling with Invariant Max-Plus Linear System

Max-plus algebra is one of the analysis methods of discrete event systems which has many applications on systems theory and graph theory. Max-plus algebra is a set of real numbers R combined with =-∞ equipped with operations max (⊕) and plus (⊗), can be denoted [(R]_ε,⊕,⊗) with [(R]_ε=R⋃{ε}) . The production process of pia saronde is one of the problems that can be analyzed using max-plus algebra. The production process of this product is sequentially carried out by making skin dough, filling, baking, cooling, and packaging the pia. The max-plus algebra theory was used in this research to determine the optimal time in the production scheduling of pia saronde. Meanwhile, the Invarian Max-plus Linear System (IMLS), max-plus algebraic theory, and the Discrete Event System (DES) were used to solve the production-related problems. IMLS analysis produces eigenvalues that represent the optimum production time. The results obtained the max-plus algebra model of x(k+1)=A x(k), where A =A⊕B⊗C and y=K⊗x_0⊕H⊗u for input-output IMLS analysis. From the matrix A, eigenvalue λ= 226 and eigenvector v=[278 278 278 279 299 302 324 356 488] were obtained. Furthermore, the value of λ describes the pia production schedule at a time span of 226 minutes.
 Keywords: input-output analysis, pia saronde, scheduling, max-plus linear system

Max-consensus of multi-agent systems in random networks

This paper considers max-consensus of a discrete-time multi-agent system (MAS) in directed random networks. Interactions among agents in the MAS are probabilistic and independent with each other. By using max-plus algebra and random theory, a sufficient and necessary condition is given for achieving max-consensus of the MAS. Moreover, we demonstrate that the max-consensus in four probabilistic senses (almost surely, in probability, expectation and mean square) is equivalent when expected graph is strongly connected. This ensures that max-consensus can be achieved in multi-agent systems even if random failures occur in the communication network, which is of practical importance in the fields of wireless sensor networks and distributed computing. A simulation example is presented to illustrate the effectiveness of theoretical results.

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

The max-plus algebra R ∪ { − ∞ } is a semiring with the two operations: addition a ⊕ b := max ( a , b ) and multiplication a ⊗ b := a + b . Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that algebraic eigenvectors of symmetric matrices with respect to distinct algebraic eigenvalues are orthogonal to each other.

Switched max-plus linear-dual inequalities: cycle time analysis and applications

AbstractP-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 primal and dual operations of the max-plus algebra. We define a new class of models called switched LDIs (SLDIs), which allow to switch between different modes of operation, each corresponding to a set of LDIs, according to a sequence of modes called schedule. In this paper, we focus on the analysis of SLDIs when the considered schedule is fixed and either periodic or intermittently periodic. We show that SLDIs can model a wide range of applications including 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 algorithms to compute: i. minimum and maximum cycle times for these processes, improving the time complexity of other existing approaches; ii. a complete trajectory of the robot including start-up and shut-down transients.

Determining the Inverse of a Matrix over Min-Plus Algebra

Linear algebra over the semiring R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' ) is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i<ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.

Exact and heuristic max-plus strategies for makespan minimization in permutation flow shops with time-window constraints

We consider permutation flow shops with time-window constraints, which consist of several jobs to be processed in a sequence of machines in the same order, where no job overtaking is allowed, and with restrictions on the timing of the operations. The investigated class of permutation flow shops is larger than the one typically considered in scheduling literature. This paper introduces two strategies, based on previous results in the max-plus algebra, to obtain a schedule minimizing the makespan: an exact branch and bound method, and a heuristics inspired from receding-horizon control. Simulations demonstrate the advantages and limitations of the proposed approaches.

A max-plus algebra approach to model laboratory experiences in blended control engineering education

Internet-based control engineering education combines traditional methods with modern technology, creating a dynamic and accessible learning environment. Both online and traditional approaches have unique advantages and challenges, with the choice depending on factors like student preferences, geographic location, and available resources. A blended approach, integrating online and in-person activities, leverages the strengths of both methods to offer a comprehensive and flexible learning experience. In this context, managing laboratory resources and activities scheduling may be challenging. For this reason, this paper introduces a novel approach to modelling these activities and resources through the mathematical framework of max-plus algebra. This algebra can improve the use of finite resources, meeting educational goals without concurrency issues. Moreover, the resolution of a model matching problem shows how lessons can be structured to meet time constraints and maximize flexibility, accessibility, and resource efficiency, thus preparing students effectively for the field.

RETRACTED: Data-driven model predictive control for real-time planned lead time optimization in a reconfigurable flow line

Mass customization has enabled smart manufacturing systems to develop reconfigurable machines and fixtures to fulfill customized products with high process variety. Coupled with these reconfigurable components, diversified design and operational problems should be dealt with such that the efficiency of reconfigurations can be maximized. This paper investigates a planned lead time (PLT) optimization problem after addressing the pre-determined reconfigurations and real-time disruptions in a flow line. A data-driven model predictive control (d-MPC) is developed for this problem such that the total expected cost for work-in-process inventory, final-product inventory, and tardiness penalty can be minimized. Considering the predictive multi-stage transitions and reconfigurable behaviors, d-MPC utilizes max-plus algebra to generate a time-varying state-space equation for a flow line. In this regard, a systematic method for permanent production loss identification can be formulated for real-time disruption diagnosis. This identification is useful in providing feedback signals for d-MPC and thus a closed-loop feedback control for PLT optimization can be achieved. A case study of an industrial welding line is reported to illustrate the operational benefits of d-MPC. The results demonstrate that d-MPC can efficiently cooperate with PLT optimization with machine reconfiguration such that the total expected cost can be minimized. Compared with continuous re-optimization, d-MPC adopts an event-driven mechanism to respond to real-time disruptions and can decrease the nervousness of flow lines with lower operational costs.

MASALAH NILAI EIGEN ATAS ALJABAR MAX-PLUS PADA SISTEM PRODUKSI BULU MATA

A discrete event system is a system whose behavior is determined by events and their events change at discrete times. The eyelash production system at PT Bio Takara Purwokerto is a discrete event system problem and can be expresses in a max-plus linear problem. By using the power algorithm, it is determined eigen values ​​and eigen vectors over max-plus algebra which will be used as production time periodization. The results obtained that the periodic time for the eyelash production system at PT Bio Takara Purwokerto is 78.488 minutes. Based on this periodic time, we can then obtain the time each processing unit starts working to produce eyelashes.

Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra

Let R_ε=R∪{-∞}, with R being a set of all real numbers. The algebraic structure (R_ε,⊕,⊗) is called max-plus algebra. The task of finding the eigenvalue and eigenvector is called the eigenproblem. There are several methods developed to solve the eigenproblem of A∈R_ε^(n×n), one of them is by using the characteristic max-polynomial. There is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is a set of R_(ε^' )=R∪{+∞} that uses minimum (⊕^' ) and addition (⊗) operations. The eigenproblem in min-plus algebra is to determine λ∈R_(ε^' ) and v∈R_(ε^')^n such that A⊗v=λ⊗v. In this paper, we provide a method for determining the characteristic min-polynomial and solving the eigenproblem by using the characteristic min-polynomial. We show that the characteristic min-polynomial of A∈R_(ε^')^(n×n) is the permanent of I⊗x⊕^' A, the smallest corner of χ_A (x) is the principal eigenvalue (λ(A)), and the columns of A_λ^+ with zero diagonal elements are eigenvectors corresponding to the principal eigenvalue.

Quality analysis of distributed max-plus algebra based morphological wavelet transform

This proposed technique employs a disseminated data embedding watermarking strategy that is based on a morphological wavelet transform and uses Max-plus algebra to achieve high image quality, big data capacity, and reasonable operational cost. Previously, we had to use a simple MMT watermarking methodology and estimated that using image quality parameters, primarily considering the diagonal component (HH). For embedding watermarks after the decomposition process carried through MMT, the presented distributed data embedding strategy, however, emphasises on middle frequency signal-groups. We have used hypothesized watermarking technique to process five benchmark images during the experiments to examine the scheme’s capability. The PSNR and the SSIM values are elevated by the proposed approach. The combined (high and low) frequency’s PSNR value is preferable towards the high frequency’s. It is also remarked that the middle-frequency SSIM value is higher than the high-frequency value.

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

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.

THE EXISTENCE OF SOLUTION OF GENERALIZED EIGENPROBLEM IN INTERVAL MAX-PLUS ALGEBRA

An eigenproblem of a matrix is where and . Vector and are eigenvector and eigenvalue, respectively. General form of eigenvalue problem is with , . Interval maks-plus algebra is and equipped with a maximum ( and plus operations. The set of matrices which its component elements of is called matrices over interval max-plus algebra and denoted by . Let , eigenproblem in interval max-plus algebra is with and . Vector and are eigenvector and eigenvalue, respectively. In this research, we will discuss the generalization of the eigenproblem in interval max-plus algebra. Especially about the existence of solution of generalized eigenproblem in interval max-plus algebra. Keywords: interval max-plus algebra, generalized eigenproblem, the existence of the solution.

Regularity of interval max-plus matrices

Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and addition, respectively. We say that the columns of a real matrix A are strongly independent if the max-plus linear system A⊗x=b has a unique solution for at least one real vector b. A square matrix A with strongly independent columns is called strongly regular. The investigation of the properties of regularity is important for applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. The present paper studies three versions of the regularity of matrices and interval matrices, namely, strong regularity, von Neumann regularity and Gondran-Minoux regularity. For each concept of regularity we will present equivalent conditions which can be verified in polynomial time.

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

Pura Vida Neutrosophic Algebra

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors.