Business

JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009

331

Multiplication Operation on Fuzzy Numbers

Shang Gao and Zaiyue Zhang

School of Computer Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China Email: gao_shang@hotmail.com yzzjzzy@sina.com

Cungen Cao

Institute of Computing Technology,The Chinese Academy of Sciences,Beijing 100080, China Email: cgcao@ict.ac.cn

Abstract—A fuzzy number is simply an ordinary number whose precise value is somewhat uncertain. Fuzzy numbers are used in statistics, computer programming, engineering, and experimental science. The arithmetic operators on fuzzy numbers are basic content in fuzzy mathematics. Operation of fuzzy number can be generalized from that of crisp interval. The operations of interval are discussed. Multiplication operation on fuzzy numbers is defined by the extension principle. Based on extension principle, nonlinear programming method, analytical method, computer drawing method and computer simulation method are used for solving multiplication operation of two fuzzy numbers. The nonlinear programming method is a precise method also, but it only gets a membership value as given number and it is a difficult problem for solving nonlinear programming. The analytical method is most precise, but it is hard to α -cuts interval when the membership function is complicated. The computer drawing method is simple, but it need calculate the α -cuts interval. The computer simulation method is the most simple, and it has wide applicability, but the membership function is rough. Each method is illuminated by examples. Index Terms—fuzzy number, membership function, extension principle, α -cuts; nonlinear programming

I. INTRODUCTION In most of cases in our life, the data obtained for decision making are only approximately known. In1965, Zadeh [1] introduced the concept of fuzzy set theory to meet those problems. In 1978, Dubois and Prade defined any of the fuzzy numbers as a...