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

Abstract

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.