Max-Plus Algebra and Discrete Event Systems: Development of Algorithms and Their Applications

Abstract

In this thesis, we consider discrete event systems (DES). DES are the systems who’s time evolution can be described by the occurrence of events. An important class of systems consists of min-max-plus systems in which the operations: the maximization, the minimization and the addition, can describe DES. The bipartite min-max-plus systems are determined by the union of two sets of equations: one set of equations containing maximization and addition; and another set of equations containing minimization and addition. System that contains only maximization and addition is called max-plus system and the system that contains minimization and addition is called min-plus system. For further study about DES, we refer [5, 6, 19]. There are various applications of max-plus algebra in mathematics as well as in different areas such as mathematical physics, optimization, combinatorics, and algebraic geometry [18]. Max-plus algebra is also used in traffic control, parallel processing systems, telecommunication networks, manufacturing systems, machine scheduling, and control theory [9, 12, 14]. In [35], the author has described the model of the whole Dutch railway system with all train types by a max-plus system. Max-plus algebra has been used in steganography [31], where matrix multiplication operations are used to hide an image into another image. In [3] the authors have shown how max-plus algebra can be helpful for the dynamic programming of algorithms. The primary reason for the utilization of max-plus algebra in different fields is that in max-plus algebra many equations become linear which are non-linear in conventional algebra [12]. We will focus on two classes of min-max-plus systems. One is max-plus system and other is bipartite min-max-plus system. The eigenvalue problem is considered in max-plus algebra for different types of matrices. As per our knowledge and from literature review, an open problem in max-plus algebra is to propose an efficient and fast algorithm to solve the eigenvalue problems iv for these systems. So, this problem motivates us to develop new algorithms to find the eigenvalue and eigenvectors of DES. To evaluate the eigenvalue and the corresponding eigenvectors of discrete event systems, various algorithms have been designed [3, 13, 27, 36]. Karp gave the very first algorithm to solve the eigenvalue problem [23]. This algorithm computes the eigenvalue by finding the maximum cycle mean in the associated directed graph. An algorithm based on policy iteration is proposed in [7]. An algorithm was proposed to solve linear inequalities in max-plus algebra, see for more details [1]. In [16], there is a method for determining the eigenvectors of monotone and homogenous functions. A general method for determining the eigenvalue is presented in [16], which computes the eigenvalue by calculating the cycle time vector. In this thesis, we propose algorithms to solve the eigenvalue problems for max-plus and bipartite min-max-plus systems. Here we give a brief overview of the thesis: In Chapter 1, historical background of max-plus algebra is introduced. Then some preliminaries notions of max-plus algebra are recalled. This chapter also introduces some important notions of bipartite min-max-plus systems. In Chapter 2, the eigenvalue problem is solved for matrices in max-plus systems. A robust algorithm is proposed to solve the eigenvalue problem in max-plus system. This algorithm computes the eigenvalue by finding the maximal cycle mean in the digraph of a given matrix, then computes the eigenvectors in an iterative way. In particular, we extend the algorithm for Latin squares and compared it with some of the existing algorithm. In Chapter 3, we consider Latin squares in max-plus algebra. We define a cycle vector corresponding to a cycle in a permutation. We compute the eigenvalue and determine the eigenvectors (trivial and non-trivial) by considering the cycle vector corresponding to each cycle in permutation of in Latin square. v In Chapter 4, we propose an algorithm to solve the eigenvalue problem for bipartite min-maxplus systems. This algorithm computes the eigenvalue by calculating the cycle time vector, then computes the eigenvectors by using an iterative approach. This algorithm is extended for Latin squares in a bipartite min-max-plus system. Computational efficiency of this algorithm is shown by comparing it with some of the existing algorithms. In Chapter 5, we present bipartite min-max-plus systems for the permutative matrices and compute the eigenvalue and the corresponding trivial and non-trivial eigenvectors for such models. We demonstrate that the presence of eigenvectors (non-trivial) depends upon the position of the greatest and the least elements in the permutative matrices. In Chapter 6, we conclude the findings and give future directions. At the end of the thesis references are included.