Generalized And Trapezoid Discrete Fuzzy Numbers Theories And Their Applications

Since1965, professor Zadeh put forward the concept of fuzzy set in “Fuzzy sets”[1], more andmore researchers devoted themselves into the theories of fuzzy set and their applications. Discretefuzzy numbers as one kind of special fuzzy sets, have some application backgrounds. For example,discrete fuzzy number can be used to estimate the number of trees and animals which are alive inthe jungle, and we can use discrete fuzzy number to express the pixel value in the center point of awindow for a black and white image so that it can be applied to digital image processing. Also wecan use discrete fuzzy numbers to discuss digital image filtering algorithm in[43]. And we can use2-dimensional discrete fuzzy numbers to evaluate the natural environment in[42]. So studyingdiscrete fuzzy number is significant.The concept of1-dimensional discrete fuzzy numbers was defined by Voxman in [44] in2001,then more and more researchers devoted themselves into the theories of discrete fuzzy numbersin[45]-[53]. In this paper, we study two aspects about discrete fuzzy numbers. On the one hand, thediscrete fuzzy numbers which are studied by the above author are all normal discrete fuzzy numbers,but in applications, the generalized discrete fuzzy numbers are often used(see example3.1.1,3.1.2,3.3.1). So we study generalized discrete fuzzy numbers and the operations whichpreserve the closeness. We give some weak orders on generalized discrete fuzzy number space andestablish an algorithmic version of risk evaluation via constructing and ranking generalized discretefuzzy numbers to represent uncertain or imprecise information, and give a practical example toshow the application of the algorithmic version. On the other hand, in applications, we often meetsuch concepts which have no a clear boundary about integers and some problems in relateduncertain or imprecise integer information, such as comprehensive evaluation of this information.So we study two types of fuzzy integers, trapezoid fuzzy integers and triangle fuzzy integers, showthat they are all discrete fuzzy numbers. Then we define operations which not only preserve thecloseness of the operation, but also make the calculation very simple. We give a practical exampleto show the application of trapezoid fuzzy integers. As follows, we will show the main works of thispaper:1. We mainly introduce the research background of fuzzy numbers and discrete fuzzy numbers,the research purpose and meaning in the chapter1.2. Some basic concepts and properties of fuzzy numbers and discrete fuzzy numbers are listedin the chapter2. 3. In the third chapter, we give the concept and the representation theorem of generalizeddiscrete fuzzy numbers. Then define a new addition operation and multiplication operation whichnot only preserve the closeness of the operation, but also make the calculation easy. We give someweak orders on generalized discrete fuzzy number space, and obtain some properties of weak orders.Then we establish an algorithmic version of risk evaluation via constructing and rankinggeneralized discrete fuzzy numbers to represent uncertain or imprecise information, and give apractical example to show the application of the algorithmic version.4. In the forth chapter, we study two types of fuzzy integers which can be used to express theuncertain or imprecise integer information. Then we give the concept of trapezoid fuzzy integersand triangle fuzzy integers, investigate their properties, show that they are all discrete fuzzynumbers, and define operations which preserve the closeness of the operation. At lat, we give apractical example to show the application of trapezoid fuzzy integers.5．In the fifth chapter, we make a summary and propose some further research contents.