Ranking Function to Solve a Fuzzy Multiple Objective Function

In this paper two ranking functions are employed to treat the fuzzy multiple objective (FMO) programming model, then using two kinds of membership function, the first one is trapezoidal fuzzy (TF) ordinary membership function, the second one is trapezoidal fuzzy weighted membership function. When the objective function is fuzzy, then should transform and shrinkage the fuzzy model to traditional model, finally solving these models to know which one is better.

The most important techniques in operational research are fuzzy programming problems, in which many real data in the programming model deal with vague numbers and variables by using fuzzy programming models. Bellman and Zadeh (1) suggested the fuzzy decision making. The fuzzy programming problem is focused by Zimmermann (2) making application in fuzzy multiple objective programming techniques Maleki (3) suggested many methods to find the solution for the programming model. Iden and Anfal (4) used an algorithm of ranking function (RF) to solve fuzzy transportation problem. Nehi and Hajmohamadi (5) employed ranking function (RF) method to solve the fuzzy multi objective (FMO) programming problem. Porcheivi and Vasanthi (6) used a new method to treat the fuzzy variable programming problems (PP) based on ranking function (RF) for multiple objective function. Kiruthiga and Loganathan (7) employed the fuzzy multi objective programming problems (FMOPP) by fuzzy programming model and reduced it to crisp model. Mitlif (8) used an efficient method to solve the fully fuzzy multi objective programming problems (FFMOPP). Iden and Abood (9) used fully fuzzy games problems while there is uncertainty in data by applying the fuzzy payoffs problem by using three different ranking functions. Ingle and Ghadle (10) used a new ranking function method to solve the fully fuzzy programming problems (FFPP) with hexagonal (HFN). The aim of this paper is employing two ranking functions (RF) for fuzzy variable in multiple objective functions by using trapezoidal fuzzy (TF) membership function when the objective variables are fuzzy. This paper presents in section two multiple objective programming (MOP). In section three the concepts of fuzzy theory and definitions are introduced. Section four deals with the triangular fuzzy number .In section five the fuzzy multiple objective problem is tackled. Section six presents ranking function of triangular fuzzy number and algorithms of ranking function (RF), and section seven involves the test example. Finally, section eight presents conclusions.

Multiple Objective Programming (MOP)
This section describes many decision problems, which have multiple objectives to be optimized simultaneously. If the constraints of the problems and its multiple objectives can be expressed as functions, then it is called as a MOP problem. The basic approach to solve multi objective problems is to determine a solution that represents an acceptable tradeoff, compromise between the objectives, or determine a set of such solutions that allow the decision marker to choose among them. Defining the multi objective function problem as follows: there are P cost vectors denoted by = ( 1 , ... , ) , = 1 , … , , the usual set × constraints = and ≥ 0. Each where = 1 , . . . . , defines the objective function, subject to the constraints; the problem is stated as below:

Definition 1 (11):
is said to be an efficient solution to (1) if is feasible and no other feasible solution exists such that ≥ ≠ . Theorem 2 (11): let be one of the objective function of the vector maximum (1). Suppose the programming problem of maximizing together with = ≥ 0 and let be its unique solution. Then is an efficient of (1).

Concept of Fuzzy Theory
Some definitions of fuzzy theory are listed throughout this section.

Definition 3 (12):
Assume is a non-empty set, a fuzzy set ̃ in is characterized by the membership function ̃ = → [0,1] ( ) is explained as the degree of membership of element in ̃ , ϵ . Definition 4 (13): The crisp set of elements which belong to ̃ at least to the degree is named aslevel set where a, b, c are core ( ), left width and right width respectively.

Fuzzy Multiple Objective Problem (FMOP)
The fuzzy multiple objective problems are the multiple objective problems with triangular fuzzy quantities. Now the formula of the fuzzy objective function problem is described by: Applying the ranking function.
 Algorithm 2 In this algorithm, we used the weighted triangular membership function, will be used below:

Conclusions:
In this paper, a fuzzy variable in (MOP) problems is considered with fuzzy variables of multiple objectives. Then two examples are solved, the first one is maximized and second one is minimized. The first example applies the traditional programming model and ordinary ranking function, the second example applies the traditional programming model and weighted ranking function. Finally, it is noted that the ordinary ranking function is equal to weighted ranking function when