Ranking nonnormal p-norm trapezoidal fuzzy numbers with integral value

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Computers and Mathematics with Applications 56 (2008) 2340–2346

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Ranking nonnormal p-norm trapezoidal fuzzy numbers with

integral value

Chi-Chi Chen

a

, Hui-Chin Tang

b,∗

a_{Department of Industrial Engineering and Management, Cheng Shiu University, Kaohsiung 833, Taiwan}

b_{Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung 80778, Taiwan}

a r t i c l e i n f o Article history: Received 24 April 2008 Accepted 27 May 2008 Keywords: Ranking Integral value

Nonnormal fuzzy numbers

p-norm trapezoidal fuzzy numbers

a b s t r a c t

This paper considers the ranking fuzzy numbers with integral value, proposed by Liou and Wang, for the nonnormal p-norm trapezoidal fuzzy numbers. Two interesting special cases of p-norm trapezoidal fuzzy numbers are the well-known triangular and trapezoidal fuzzy numbers. For the nonnormal fuzzy numbers, the differences exist among the membership functions for the triangular, trapezoidal and p-norm trapezoidal fuzzy numbers. These differences can affect their left, right and total integral values, so we establish the relationship between these three types of fuzzy numbers.

Since the publication of Zadeh’s 1965 paper [1] on fuzzy set and Jain’s 1978 and Dubois and Prade’s 1978 papers [2,3] on fuzzy number (FN), the fuzzy theory and application literature has grown explosively. In a fuzzy environment, ranking fuzzy numbers is a prerequisite procedure for the decision-making problem. The method of ranking the FNs has been proposed first by Jain (1976) [4]. Since then, a large variety of methods have been developed in an attempt to rank the FNs. According to Chen and Hwang [5], important methods may be categorized into four classes (1) preference relation [6], (2) fuzzy mean and spread [7], (3) fuzzy scoring [8–10] and (4) linguistic express [11]. Wang and Kerre [11,12] classified the important 35 ordering indices into three categories: (1) ranking functions [8–10,13], (2) reference sets [14] and (3) linguistic approach [12]. This paper deals with the fuzzy scoring and ranking function methods from Chen and Hwang [5] and Wang and Kerre [11,12] point of view, respectively, especially for the integral value proposed by Liou and Wang [10]. For the triangular and trapezoidal FNs, Liou and Wang [10] showed that the integral values of normal and nonnormal FNs are equal. Cheng [13] indicated that Liou and Wang’s method cannot rank normal and nonnormal triangular/trapezoidal FNs because of the equivalence of the normal and nonnormal triangular/trapezoidal FNs. Wang and Kerre [12] proposed seven axioms which serve as the reasonable properties to evaluate the ordering procedures. They showed that the integral value of Liou and Wang [10] satisfies all the axioms except that the fuzzy product is compatible with the order. The unrestricted axiom of this exception is the linearity property. In this paper, we analyze the linearity of the integral value of a normal and nonnormal FN. For a nonnormal FN, we propose a more generalized trapezoidal FN, and compare and establish the relationship between the integral values of triangular, trapezoidal and generalized trapezoidal FNs.

The remainder of the paper is organized as follows. Firstly, we provide a concise review of the integral value of Liou and Wang [10]. Next, we analyze the behaviors of the integral values for the nonnormal FNs. Analyses are given to compare and evaluate the integral values of triangular, trapezoidal and generalized trapezoidal FNs. Finally, some concluding remarks are given.

∗_{Corresponding author.}

E-mail address:tang@cc.kuas.edu.tw(H.-C. Tang).

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C.-C. Chen, H.-C. Tang / Computers and Mathematics with Applications 56 (2008) 2340–2346 2341

2. Total integral value

In 1978, Dubois and Prade [3] gave the definition of a real FN, defined as follows.

A real FN A

= [

a

,

b

,

c

,

d

;

w]

is a fuzzy subset of the real R with membership function fA

(

x

)

, defined as

fA

(

x

) =







f_AL

(

x

)

a

≤

x

≤

b 1 b

≤

x

≤

c f_AR

(

x

)

c

≤

x

≤

d

,

(1)

which is convex and bounded, where

−∞

<

a

≤

b

≤

c

≤

d

< ∞

and 0

< w ≤

1. Assume that the left membership function fL

A

(

x

) : [

a

,

b

] → [

0

, w]

is continuous and strictly increasing function and the right membership function f_AR

(

x

) : [

c

,

d

] → [

0

, w]

is continuous and strictly decreasing function. Let Supp

(

A

) = {

x

∈

R

|

fA

(

x

) >

0

}

. When

w =

1, a FN A is called the normal FN. The inverse functions of fL

A

(

x

)

and fAR

(

x

)

, denoted by gAL

(

y

)

and gAR

(

y

)

, are continuous and strictly increasing g_AL

(

y

) : [

0

, w] → [

a

,

b

]

and continuous and strictly decreasing g_AR

(

y

) : [

0

, w] → [

c

,

d

]

, respectively.

Combining the left and right integral values, Liou and Wang [10] suggested a method of ranking FNs with an index of optimism

α ∈ [

0

,

1

]

. More precisely, the left and right integral values of a FN A are defined as IL

(

A

) = R

₀wgAL

(

y

)

dy and

IR

(

A

) = R

0wg

R

A

(

y

)

dy, which reflect the pessimistic and optimistic viewpoint of the decision maker, respectively. The total integral value with index of optimism

α ∈ [

0

,

1

]

is defined as

I_Tα

(

A

) = α

IR

(

A

) + (

1

−

α)

IL

(

A

) = α

Z

w 0 g_AR

(

y

)

dy

+

(

1

−

α)

Z

w 0 g_AL

(

y

)

dy

.

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In the literature, two interesting special cases are trapezoidal FNs and triangular FNs. A trapezoidal FN A

= [

a

,

b

,

c

,

d

;

w]

is defined as follows fA

(

x

) =











w

x

−

a b

−

a a

≤

x

≤

b 1 b

≤

x

≤

c

w

d

−

x d

−

c c

≤

x

≤

d

.

(3)

Then the inverse of fA

(

x

)

is

gA

(

y

) =











a

+

b

−

a

w

y 0

≤

y

≤

w

d

−

d

−

c

w

y 0

≤

y

≤

w,

(4)

so the left, right and total integral values with index of optimism

α

are

I_Lα

(

A

) =

w

2

(

a

+

b

)

I_Rα

(

A

) =

w

2

(

c

+

d

)

and I_Tα

(

A

) =

w

2

{

α(

c

+

d

) + (

1

−

α)(

a

+

b

)} .

(5)

After some calculation, it is trivially shown that the total integral value of a nonnormal trapezoidal FN is not a linear function. However, for a normal trapezoidal FN, the total integral value is a linear function. This result is stated formally below.

Proposition 1. Let

γ

and q be two real numbers. For a nonnormal trapezoidal FN A, we have I_Tα

(γ

A

+ [

q

,

q

,

q

,

q

;

w]) =

γ

I_Tα

(

A

) + w

q. Moreover, if

w =

1, then we get I_Tα

(γ

A

+ [

q

,

q

,

q

,

q

;

1

]

) = γ

I_Tα

(

A

) +

q

.

When b

=

c, a trapezoidal FN A

= [

a

,

b

,

c

,

d

;

w]

is called a triangular FN. A triangular FN, denoted symbolically by

A

= [

a

,

b

,

c

;

w]

, is defined as fA

(

x

) =











w

x

−

a b

−

a a

≤

x

≤

b

w

c

−

x c

−

b b

≤

x

≤

c

.

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Its inverse function is

gA

(

y

) =











a

+

b

−

a

w

y 0

≤

y

≤

w

c

−

c

−

b

w

y 0

≤

y

≤

w,

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2346 C.-C. Chen, H.-C. Tang / Computers and Mathematics with Applications 56 (2008) 2340–2346

main conclusions can be drawn from this paper. Firstly, the total integral value is not a linear function for a nonnormal trapezoidal/triangular FN, and is a linear function for a normal trapezoidal/triangular FN. Secondly, we propose a p-norm trapezoidal FN which is a nonlinear FN. Eqs.(11)–(13)are its left, right and total integral values, respectively. When p

=

1, it becomes the well-known trapezoidal FN. Thirdly, for a nonnormal FN B, under the assumptions IL

(

B

) =

IL

(¯

B

)

and

IR

(

B

) =

IR

(¯

B

)

, proposed by Liou and Wang [10],Proposition 3shows that IL

(

B

) =

IL

(

A

)

, IR

(

B

) =

IR

(

A

)

and ITα

(

B

) =

ITα

(

A

)

for a normal FN A with membership function fA

(

x

) =

fB_w(x), where

w =

maxx∈Supp(B)fB

(

x

)

and 0

< w <

1. If these two assumptions are not satisfied, the left, right and total integral values are increasing functions on

w

, so we have IL

(

B

) ≤

IL

(

A

)

,

IR

(

B

) ≤

IR

(

A

)

and ITα

(

B

) ≤

ITα

(

A

)

for all

α ∈ [

0

,

1

]

. Fourthly, for the nonnormal FNs, the left integral value of the triangular FN is larger than that of trapezoidal FN which in turn is larger than that of p-norm trapezoidal FN. The larger value of p results in the smaller the left integral value. Conversely, the right integral value of the p-norm trapezoidal FN outperforms that of trapezoidal FN, which in turn outperforms that of triangular FN. Fifthly, for the nonnormal triangular and trapezoidal FNs, the order of the total integral values is dependent on the relationship between the values attaining the maximum membership grade of the triangular and trapezoidal FNs. When the left and right spreads of the p-norm trapezoidal FNs are equal, as the value of p increases, the total integral value decreases for

α <

1₂, is constant for

α =

1₂and increases for

α >

1₂.

References

[1] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.

[2] R. Jain, A procedure for multi-aspect decision making using fuzzy sets, Internat. J. Systems Sci. 8 (1978) 1–7. [3] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (1978) 613–626.

[4] R. Jain, Decision-making in the presence of fuzzy variables, IEEE Trans. Syst. Man Cybern. 6 (1976) 698–703. [5] S.J. Chen, C.L. Hwang, Fuzzy Multiple Attribute Decision Making, Springer, New York, 1992.

[6] M. Modarres, S. Sadi-Nezhad, Ranking fuzzy numbers by preference ratio, Fuzzy Sets and Systems 118 (2001) 429–436.

[7] E.S. Lee, R.J. Li, Comparison of fuzzy numbers based on the probability measure of fuzzy events, Comput. Math. Appl. 15 (1988) 887–896.

[8] L.H. Chen, H.W. Lu, An approximate approach for ranking fuzzy numbers based on left and right dominance, Comput. Math. Appl. 41 (2001) 1589–1602. [9] T.C. Chu, C.T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, Comput. Math. Appl. 43 (2002) 111–117. [10] T.S. Liou, M.J.J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems 50 (1992) 247–255.

[11] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II), Fuzzy Sets and Systems 118 (2001) 387–405. [12] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems 118 (2001) 375–385. [13] C.H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems 95 (1998) 307–317.

[14] P.A. Raj, D.N. Kumar, Ranking alternatives with fuzzy weights using maximizing set and minimizing set, Fuzzy Sets and Systems 105 (1999) 365–375. [15] Wolfram Research, Inc., Mathematica 4.1, Champaign, IL, 1999.