A New Hendecagonal Fuzzy Number for Optimization Problems

A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number.


INTRODUCTION
A fuzzy number is a quantity whose values are imprecise, rather than exact as in the case with singlevalued function. The generalization of real number is the main concept of fuzzy number. In real world applications all the parameters may not be known precisely due to uncontrollable factors. L.A.Zadeh introduced fuzzy set theory in 1965. Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number, hexagonal fuzzy number, heptagonal fuzzy number, octagonal fuzzy number, nanagonal fuzzy number, decagonal fuzzy number have been introduced with its membership function.
In much decision analysis, the uncertainty existing in input information is usually represented as fuzzy numbers [1],. S.H.Chen introduced maximization and minimization of fuzzy set, uncertainty and information [4]. The arithmetic operations , alpha cut and ranking function are already introduced for existing fuzzy numbers by C.B.Chen and C.M.Klein,T.S.Liou and M.J.Wang [6], [11]. When the vagueness arises in eleven different points it is difficult to represent the fuzzy number. In this paper a new type of fuzzy number named as hendecagonal fuzzy number is defined with its membership function. The arithmetic operations, alpha cut and ranking procedure for hendecagonal fuzzy numbers are introduced to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman problem (FTSP).In literature, many methods were proposed for fuzzy optimization with uncertain parameters [3], [5], [10], [13].Here uncertainty in assignment cost and travelling distance are represented by new fuzzy number named as Hendecagonal fuzzy number, which are ranked using the ranking function introduced by R.R.Yager [16], [7]. Numerical examples show the effectiveness of the proposed method and the new fuzzy number, It is simple and very easy to understand and can be applied in many real life problems.

3.2:
A fuzzy number ) , , , is said to be a Hendecagonal fuzzy number if its membership function is given by

HENDECAGONAL FUZZY NUMBERS
In this section, arithmetic operations between two Hendecagonal fuzzy numbers, defined on universal set of real numbers R, are presented.
be two hendecagonal fuzzy number then

RANKING HENDECAGONAL FUZZY NUMBERS
The ranking method proposed in [4] is used to rank the hendecagonal fuzzy numbers.
The ranking function where F(R) is a set of fuzzy number defined on set of real numbers, which maps each fuzzy number into the real line, where the natural order exists, i.e.

VI. FUZZY ASSIGNMENT PROBLEM AND FUZZY TRAVELLING SALESMAN PROBLEM
In this section, mathematical formulation of fuzzy assignment problem is given and a direct approach is proposed to solve FAP and FTSP. The method is applicable for all optimization problems.

A. Formulation of Fuzzy Assignment Problem
Let there be m Tasks and m Workers , ij C be the cost of assigning i th Worker to the j th Task and the uncertainty in cost is here represented as Hendecagonal fuzzy numbers. Let ij x be the decision variable define

B. Formulation of Fuzzy Travelling Salesman Problem
The travelling salesman problem deals with finding shortest path in a n-city where each city is visited exactly once. The travelling salesman problem is similar to assignment problem that excludes sub paths. Specifically in an n-city situation define

VII. NUMERICAL EXAMPLE
In this section numerical examples are given to illustrate the proposed method and it is shown that the proposed method offers an effective way for handling FAP as well as FTSP.
Example 7.1: A manufacturing company manufactures a certain type of spare parts with three different machines. The company official has to execute three jobs with three machines. The information about the cost of assignment is imprecise and here Hendecagonal Fuzzy numbers are used to represent the cost. The fuzzy assignment problem is given in Table 1.

Solution:
Step 1: Calculate the ranking value of each fuzzy cost is given in Table 2.
Step 2: Encircle the fuzzy cost with least ranking value in each row and examine all the encircled fuzzy costs and identify the encircled fuzzy cost that is uniquely encircled in both row wise and column wise. Assign it and delete the corresponding row and column. The resultant table is given in Table 3.
Step 3:If the cost is not uniquely selected both row wise and column wise then choose next minimum and proceed as in step 2.This process is continued until the fuzzy cost is uniquely selected row and column wise.

Solution:
Step 1: Calculate the ranking value of each fuzzy distance is given in Table 5.
Step 2: Encircle the fuzzy distance with least ranking value in each row and examine all the encircled fuzzy distance to find the uniquely encircled fuzzy distance in both row wise and column wise. Assign it and delete the corresponding row and column. The resultant table is given in Table 6.
Step 3:If the distance is not uniquely selected both row wise and column wise then choose next minimum and repeat the step 2.This process is continued until the fuzzy distance is uniquely selected row and column wise.
Thus the optimal assignment is with the optimal distance (5,9,16,20

VIII. CONCLUSION AND FUTURE ENHANCEMENT
In this paper, a new fuzzy number is developed for solving optimization problem with Hendecagonal fuzzy cost and fuzzy distance. The optimal solution to FAP and FTSP obtained by the proposed method is same as that of the optimal solution obtained by the existing methods. However the proposed method is simpler, easy to understand and it takes few steps for obtaining the fuzzy optimal solution. Numerical example shows that the proposed method offers an effective tool for handling the fuzzy assignment problem. In future, the generalization of Hendecagonal fuzzy number is developed to solve optimization problems.