ALJABAR MAX-PLUS BILANGAN KABUR DAN PENERAPANNYA PADA MASALAH PENJADWALAN DAN JARINGAN ANTRIAN KABUR; FUZZY NUMBER MAX-PLUS ALGEBRA AND ITS APPLICATIONS TO FUZZY SCHEDULING AND QUEUING NETWORK PROBLEMS

M. ANDY RUDHITO, Sri Wahyuni

2011 | Tesis | PROGRAM STUDI S3 MATEMATIKA

The activities time that are not known exactly in a network, can be modeled with fuzzy numbers. Modeling and analysis of existing fuzzy networks using PERT-CPM, mathematical programming and MVA. This dissertation will discusses the max-plus algebra approach for modeling and analysis of fuzzy networks, which are expected to provide a more compact model and the results are more analytical. This approach requires the concepts of max-plus algebra in the set of all fuzzy numbers. Will be discussed firstly generalize the concepts of max-plus algebra into the max-plus algebra fuzzy numbers which include systems of linear equations and eigenvalue. Operations on fuzzy numbers are done through ?-cut in the form of intervals on real numbers. The finding shows that the fuzzy numbers max-plus algebra structure is a commutative idempotent semiring. The set of all matrices over fuzzy number max-plus algebra is a semimodule over fuzzy number max-plus algebra. Solution of system of fuzzy number max-plus linear equations can be determined by first determining the solution of each ?-cut interval system of the fuzzy system. If necessary, it can be modified such that the ?-cut of each entry of the solution is the ?-cut of a fuzzy numbers. A fuzzy scalar, where the lower bound and upper bound of its ?-cut are maximum mean weight of elementary circuit in precedence graph of lower bound and upper bound of ?-cut matrix of a square fuzzy number matrix, respectively, is a fuzzy numbers max-plus eigenvalue of the fuzzy numbers matrix. Fuzzy number max-plus eigenvector associated with these eigenvalues can be determined through fundamental eigenvectors associated with the interval eigenvalues of each ?-cut matrix of a square fuzzy number matrix. If necessary, it can be modified such that the ?-cut of each entry of the eigenvector is the ?-cut of a fuzzy number. Fuzzy project network dynamics can be modeled as an iterative system of fuzzy number max-plus linear equations where the solution of the system is the fuzzy earliest starting time from every node. In a fuzzy serial closed queuing network, for a given level of risk, it can be determined the earliest of early departure time of a customer, so that the customer's departure interval time will be in the smallest interval where the lower bound and upper bound are periodic.

Kata Kunci : max-plus algebra, fuzzy number, system of linear equations, eigenvalue, fuzzy scheduling and fuzzy queuing network.