Global Journal of Pure and Applied Mathematics • ISSN 0973-1768 • Volume 12, Number 3 (2016) © Research India Publications • http://www.ripublication.com/gjpam.htm 875 A New α β γ δ       -cuts in Trapezoidal Fuzzy Number and its Approach in Decomposition Theorem M. Clement Joe Anand Assistant Professor (S.G.) Department of Mathematics Hindustan University, Chennai-603103 mclementj@hindustanuniv.ac.in Janani Bharatraj Ph.D. Research Scholar (Full-Time) Department of Mathematics Hindustan University, Chennai-603103 jananichari@gmail.com Abstract— In this paper we are introducing a new cut called the matrix cut in Trapezoidal Fuzzy Number based on α -cuts. We have proposed the properties of the matrix cut and applied the same in the Decomposition theorems. Keywords—Fuzzy Number, Trapezoidal Fuzzy Number, α - cuts, Decomposition theorem. I. INTRODUCTION The original formulation of the theory of fuzzy sets contains the concept of a fuzzy number as a convex fuzzy subset of the real line. It started gaining momentum with the publication of seminal papers by A. Kaufmann, H. Dubois and D. Parade, and others. Zadeh [13] introduced the concept of α -cut or α -level set to bridge the concepts of fuzzy set theory and traditional set theory. Let X be an universe and let F(X) denote the set of all fuzzy sets of X. Let   A F X  and [0,1] α  . Then the crisp set   / () A A x X x α μ α    is called the α -cut or α -level set of A [8, 13]. Any fuzzy set can be formed from a family of nested strong α -cut satisfying the following equation.   / () A A x X x α μ α    . The decision making problems in the context of fuzzy sets can be solved by transforming these sets into their families of nested α -cuts. The solutions to each can be found using traditional methods and then all the partial results are merged to reconstruct a solution to the problem. The similarity relations and fuzzy orderings have been discussed by Zadeh using α -cut sets [13]. But the following question has to be answered while introducing α -cuts. Is a single cut enough to resolve a particular real-world problem? We now present a real-world problem to stress on the importance of having two or more cuts. Let us consider Bharat Stage emission standards. In order to regulate the output of air-pollutants from internal combustion engine equipment these emission standards are instituted by the Government of India. The emission standards for diesel truck and bus engines under EuroIV are given below. EuroIV Test CO HC NOx PM ESC 1.6 0.46 3.5 0.02 ETC 4.0 0.55 3.5 0.03 Here CO, HC, NOx and PM are considered as our proposed α β γ δ       -cuts for a trapezoidal fuzzy number. II. PRELIMINARIES: In this section the basic properties and decomposition theorems for α -cuts have been presented. A. Properties Let   , AB F X  . Then the following properties holds for all , [0,1] αβ  . i. ; A A α α   ii. and A A A A α β α β α β       iii. ( ) A B A B α α α    and ( ) A B A B α α α    ; iv. ( ) A B A B α α α       and ( ) A B A B α α α       ; v. (1 ) A A α α    . Figure 2.1 Trapezoidal Fuzzy Number with α -cut