A Generalized Polygon Fuzzy Number for Fuzzy Multi

A Generalized Polygon Fuzzy Number for Fuzzy Multi Criteria Decision Making Samah Bekheet, Ammar Mohammed, and Hesham A. Hefny Department of Computer & Information Sciences, Institute of Statistical Studies and Research, Cairo University, Giza, Egypt [email protected], [email protected], [email protected]

Abstract. Allowing various forms of fuzzy numbers to be adopted in fuzzy multi criteria decision making (FMCDM) problems adds more flexibility to decision makers to represent their own opinions to handle uncertainty. For most cases uncertain numbers of the forms of: interval, triangle, or trapezoidal are used. In this paper, polygon fuzzy numbers (PFNs) are introduced so as to allow decision makers to adopt other forms of numbers such as: pentagon, hexagon, heptagon, octagon, etc, to provide more flexibility to represent uncertainty. A case study is given to illustrate the way of manipulation of the proposed PFN. Keywords: Polygon Fuzzy Number, Multi Criteria Decision Making, Generalized Fuzzy Number.

1

Introduction

Decisions in real world applications are often made under the presence of conflicting, uncertain, incomplete and imprecise information. Decision making is the procedure to find the best alternatives among a set of feasible alternatives and also ranking them as their priorities. Fuzzy multi Criteria Decision making (FMCDM) provides a powerful approach for drawing rational decisions under uncertainty given in the form of linguistic values (e.g. excellent, very good, good, and bad). To enable the decision makers to express their own opinions. Such linguistic values need fuzzy tools to evaluate their calculations [1]. Examples of FMCDM tools are T-Norm Based, Gaussian fuzzy numbers, Interval fuzzy numbers, Interval type two fuzzy number, Triangle fuzzy numbers and Trapezoidal fuzzy numbers [2]. Interval, Triangle and Trapezoidal fuzzy numbers are more popular due to their conveniences of the arithmetic operations such as: addition, subtraction, multiplication, division, reciprocal, geometric mean, etc. Such operations enable the decision makers to determine the rank of criteria (alternatives) powerfully [3]. Several researchers consider Trapezoidal fuzzy numbers as Generalized fuzzy numbers (GFNs) [4,5,6]. This mainly due to the fact that other popular forms of specific fuzzy numbers including: triangles, intervals, or even singleton can be obtained as special cases of Trapezoidal fuzzy numbers. In this paper, we introduce Polygon Fuzzy Number (PFN) as the actual form of GFN. The proposed form of PFN provides A.E. Hassanien et al. (Eds.): AMLTA 2014, CCIS 488, pp. 415–423, 2014. © Springer International Publishing Switzerland 2014

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and H.A. Hefny

higher flexibility to decision makers to express their own linguistic values rather than other form of fuzzy numbers. Using PFNs, decision makers can freely express their own linguistic values using fuzzy numbers of various shapes e.g. triangle, trapezoidal, pentagon, hexagon, heptagon, octagon and etc. This provides a powerful tool for solving various decision making problems which are based on different views of decision makers [7, 8, 9, 13]. The rest of the paper is organized as follows: Section 2 defines the problem. Section 3 introduces the proposed model and the required definitions of the proposed model. Section 4 introduces a numerical example. Finally section 5 presents the conclusion.

2

Problem Definition

A Trapezoidal fuzzy number with its four vertices is considered a generalized number for other forms including: triangles, intervals and also singletons (i.e. crisp numbers) [4,5,6]. However, it is intuitively clear that allowing more vertices to the fuzzy number adds more flexibility to the decision maker to represent his own opinion to deal with the considered FMCDM problem. Therefore, the ability to introduce generalized piece-wise membership function with n-vertices as a fuzzy number with its own arithmetic operations represents the typical unification of all other forms of fuzzy numbers. Adopting such new forms of generalized fuzzy numbers should considerably enhance modeling and solving FMCDM problems.

3

Polygon Fuzzy Number

3.1

Basic Definitions

A polygon fuzzy number (PFN) is defined as a convex and normal polygon fuzzy set, where Polygon fuzzy sets are firstly addressed in the context of fuzzy interpolative reasoning [10]. A polygon fuzzy set A has n characteristic points (a0, a1 ,...., an-1) as shown in fig. 1. The core of the fuzzy set, at which the membership equals one, is represented by the interval [a (n-1/2)! , a "(n-1/2)# ]. There are (n-1)/2!+1 membership levels including bottom and top levels. Thus the cardinality of the level set of a polygon fuzzy set is denoted by V as given in (1), It is clear that V represents the number of -cuts of the polygon fuzzy sets, namely: 0= 0, ...., (n-1)/2!+1=1. V = (n-1)/2! +1

(1)

A Generalized Polygon Fuzzy Number for Fuzzy Multi Criteria Decision Making

417

Fig. 1. Polygon fuzzy set

3.2

Ranking of Polygon Fuzzy Number

The centre of area of a fuzzy number is considered the most popular ranking method [11, 12]. Computing the centroid of a general polygon fuzzy number can be obtained by the following theorem. 3.2.1 The Centroid Theorem The centroid x (center of area ) of the polygon which characterized by the n-points (x0, x1,.., xn-1) with area A and membership level yi = µA(xi), can be computed using the following formula: n

x =

$ X A $ A i

i =1

X

i

1 [( + 3 xi

A

i

=

(2)

n

i =1

=

i

x i −1) + (

( xi −

i

x y +x y y+y i

i −1

i

)( y +

i −1

i

y

)]

(3)

i −1

i

x

i −1

)

i −1

(4)

2

3.2.2 The Theorem Proof The polygon fuzzy number A with its n vertices sees Figure 2, can be divided into (n-1) sub-polygons.

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Each sub-polygon can generally be represented as a trapezoidal with 4-vertices. Thus the ith sub-polygon as shown in Fig.3, has an area equals Ai and its centroid x i is computed as follows:

Fig. 2. Polygon characterized

Fig. 3. Sub- polygon area

xi

%x f

x

i

( x ) dx

= x i −1 xi

i

%f

( x ) dx

i

x i −1

=

f

i

I1 = I2

( x) =

f

( x) −

y

x −x

i −1

i

y−y x −x i i

i −1 i −1

i −1

(x

=

−

y −y x −x x

i

i −1

i

i −1

i −1

)+ y

i −1

A Generalized Polygon Fuzzy Number for Fuzzy Multi Criteria Decision Making

=

419

yi − yi−1 y −y − xi−1 + yi−1 x x −x − xi xi−1 i

i−1

i

i−1

∆y x ∆x i

=

+

y

i −1

∆yi

−

x i −1

∆xi

i

Therefore,

f

i

(x) =

α x

+

i

β

i

Where

α

i

=

∆y ∆x

i

,

β

i

=

y

∆yi

−

i -1

∆ xi

i

x i -1

Then, performing the integrations for both I1 and I2, we have:

x

i

=

1 [( 3

x

i

+

x

) + ( i−1

x y y i

i

+ i

x y + y i−1

i−1

)]

i−1

Also, it is clear that the area of the ith sub-polygon shown in fig. 3 is:

A 3.3

i

=

(xi −

x

i −1

)(

y

i

+

y

i −1

)

2

Arithmetic Operations

The arithmetic operations of two polygon fuzzy numbers should satisfy the following two rules: i - Both fuzzy numbers should have the same number of vertices. ii- Both fuzzy numbers should have the same level set. Thus, if we have to add two different polygon fuzzy numbers, e.g triangle T(a0, a1, a2) and hexagon H(b0, b1, b2, b3, b4, b5). If the level set of the triangle is: {T(a0)=T(a2)=0, (a1)=1}, and the level set of the hexagon is: {H(b0)=H(b5)=0, H(b1)=H(b4)=0.6, H(b2)=H(b3)=1 }. Then additional level set value at 0.6 should be added to the triangle fuzzy number and consequently more vertices appears for that triangle to be in the form: T(a0, a1, a2, a3, a4, a5) so that its level set becomes: {T(a0)=T(a5)=0, T(a1)=T(a4)=0.6, T(a2)=T(a3)=1}.

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Thus, keeping the above two rules in mind, assume that there two PFNs A(a0, a1, a2, ...., an-1) and B(b0, b1, b2, ...., bn-1), then the arithmetic operations can be defined as follows: PFNs Addition A ⊕ B = (a0+b0, a1+b1, a2+b2, ....., an-1+bn-1)

PFNs Subtraction A

B = (a0-bn-1, a1-bn-2, a2-bn-3, ....., an-1-b0)

PFNs Multiplication A ⊗ B = (a0×b0, a1×b1, a2×b2, ....., an-1×bn-1)

PFNs Divisions A Φ B = (a0/bn-1, a1/bn-2, a2/bn-3, ....., an-1/b0 ) Where b0 !0, b1!0, ....... and bn-1 !0.

4

Illustrative Example

Assume three alternatives and A1, A2 A3 are evaluated with respect to five criteria C1, C2, C3, C4 and C5 as shown in table1. Table 1. Criteria-Alternatives Evaluation Matrix

Alternatives A1 A2 A3

C1 A11 A21 A31

C2 A12 A22 A32

Criteria C3 A13 A23 A33

C4 A14 A24 A34

C5 A15 A25 A35

Let the decision maker put his evaluation values in the form of polygon fuzzy numbers as shown in table2. According to table 2, the whole level set is {0, 0.4, 0.5, 0.6, 1}. Therefore, all the above PFNs should be rewritten according to the whole level set. This of course will add more vertices as shown in Table 3. It is clear, that the obtained unified PFNs are all having ten vertices at the above five values of the level sets (five Left, five Right ).

A Generalized Polygon Fuzzy Number for Fuzzy Multi Criteria Decision Making

421

Table 2. The Evaluation PFNs

Type Quadrilateral Triangle Hexagon Triangle Hexagon Quadrilateral Pentagon Triangle Quadrilateral Pentagon Triangle Pentagon Quadrilateral Pentagon Pentagon

Level Sets A(1)=A(4)=0, A(2)=A(3)=1 A(1)=A(3)=0, A(2)=1 A(1)=A(6)=0, A(2)=0.4, A(3)=A(4)=1, A(5)=0.6 A(1)=A(3)=0, A(2)=1 A(1)=A(6)=0, A(2)=0.4, A(3)=A(4)=1, A(5)=0.6 A(1)=A(4)=0, A(2)=A(3)=1 A(1)=A(5)=0, A(2)=0.5, A(3)=1, A(4)=0.6 A(1)=A(3)=0, A(2)=1 A(1)=A(4)=0, A(2)=0.5, A(3)=1 A(1)=A(5)=0, A(2)=0.4, A(4)=4.6 ,A(3)=1 A(1)=A(3)=0, A(2)=1 A(1)=A(5)=0, A(2)=0.6, A(3)=1, A(4)=0.4 A(1)=A(4)=0, A(2)=0.4, A(3)=1 A(1)=A(5)=0, A(2)= A(4)=0.5, A(3)=1, A(1)=A(5)=0, A(2)=0.6, A(3)=1, A(4)=0.6

Fuzzy No. A11 A12 A13 A14 A15 A21 A22 A23 A24 A25 A31 A32 A33 A34 A35

Table 3. The new forms of Evaluation PFNs

level set

The fuzzy numbers corresponding to the level sets A11

A12

A13

A14

A15

A21

A22

A23

A24

A25

A31

A32

A33

A34

A35

0L

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0.4L

1.4

1.4

2

1.4

2

1.4

1.8

1.4

1.8

2

1.4

1.7

2

1.8

1.7

0.5L

1.5

1.5

2.2

1.5

2.2

1.5

2

1.5

2

2.2

1.5

1.8

2.2

2

1.8

0.6L

1.6

1.6

2.3

1.6

2.3

1.6

2.2

1.6

2.2

2.3

1.6

2

2.3

2.2

2

1L

2

2

3

2

3

2

3

2

3

3

2

3

3

3

3

1R

3

2

4

2

4

3

3

2

3

3

2

3

3

3

3

0.6R

3.4

2.4

4.8

2.4

5

3.4

4

2.4

3.4

4

2.4

3.7

3.4

3.8

3.8

0.5R

3.5

2.5

5

2.5

5.2

3.5

4.2

2.5

3.5

4.2

2.5

3.8

3.5

4

4

0.4R

3.6

2.6

5.2

2.6

5.3

3.6

4.3

2.6

3.6

4.3

2.6

4

3.6

4.2

4.2

0R

4

3

6

3

6

4

5

3

4

5

3

5

4

5

5

Then, get the normalized ranked PFN for each alternative as in (5) as show in Table 4: R(Ai) = " Aik / ""Aik Where k=1, 2,..,5 and i=1,2,3

(5)

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and H.A. Hefny

Table 4. The Normalized Ranking PFNs

level set 0L 0.4L 0.5L 0.6L 1L 1R 0.6R 0.5R 0.4R 0R

R(A1) 0.077 0.145 0.163 0.181 0.279 0.385 0.610 0.683 0.769 1.467

R(A2) 0.077 0.149 0.169 0.190 0.302 0.359 0.583 0.653 0.734 1.400

R(A3) 0.077 0.151 0.172 0.194 0.326 0.359 0.578 0.652 0.740 1.467

Finally, applying the above centroid theorem for PFN to get the final crisp ranking value for each alternative as follows: COA(R(A1)) = 0.476 COA(R(A2)) = 0.462 COA(R(A3)) = 0.472 Thus it is clear that alternative A1 should be selected as the best choice, and we can arrange them as the highest priority as A1>A3>A2.

5

Conclusions

This paper introduced a way for adopting PFNs in FMCDM problems. The proposed form of PFN ensures its generality over other popular forms of fuzzy numbers. A general formula for ranking PFNs is given using the centroid method. The arithmetic operations for PFNs are presented and an explanation example shows how to adopt such generalized fuzzy numbers for solving FMCDM problems.

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