analyzing fuzzy risk based on a new similarity measure between interval-valued fuzzy numbers

ANALYZING FUZZY RISK BASED ON A NEW SIMILARITY MEASURE BETWEEN INTERVAL-VALUED FUZZY NUMBERS

KATA SANGUANSAT1, SHYI-MING CHEN1,2

1 Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology,

Taipei, Taiwan.2 Department of Computer Science and Information Engineering,

Jinwen University of Science and Technology, Taipei County, Taiwan.

2

Outline

Introduction Interval-Valued Fuzzy Numbers The Proposed Similarity Measure Between Interval-Valued

Fuzzy Numbers A Comparison with the Existing Similarity Measures Fuzzy Risk Analysis Based on the Proposed Similarity

Measure Conclusions

3

Introduction

There have been several researches regarding fuzzy risk analysis [1984] Schmucker presented a method for fuzzy risk analysis based

on fuzzy number arithmetic operations. [1989] Kangari and Riggs presented a method for constructing risk

assessment by using linguistic terms. [2005] Tang and Chi presented a method for predicting the

multilateral trade credit risk by the ROC curve analysis. [2007] Chen and Chen presented a method for fuzzy risk analysis

based on the ranking of generalized trapezoidal fuzzy numbers. Etc.

4

Introduction (cont.) Recent researches found that interval-valued fuzzy

numbers are effective for representing evaluating terms in fuzzy risk analysis problems.

Some researchers presented fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers. [2009] Chen and Chen [2009] Wei and Chen Etc.

In this paper, we present a new similarity measure between interval-valued fuzzy numbers.

5

Interval-Valued Fuzzy Numbers In 1987, Gorzalczany presented the concept of interval-

valued fuzzy sets. Based on the representation presented by Yao and Lin

[2002], we can see that an interval-valued trapezoidal fuzzy number can be represented by

where and denote the lower and the upper interval-valued trapezoidal fuzzy numbers, respectively,

)],ˆ;,,,(),ˆ;,,,[(]~~

,~~

[~~

~~4321~~4321U

A

UUUUL

A

LLLLUL waaaawaaaaAAA

LA~~ UA

~~

.~~~~ UL AA

A~~

6

Interval-Valued Fuzzy Numbers (cont.)

)]ˆ;,,,(),ˆ;,,,[(]~~

,~~

[~~

~~4321~~4321U

A

UUUUL

A

LLLLUL waaaawaaaaAAA

Fig. 1. Interval-valued trapezoidal fuzzy number

7

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers

The proposed method combines the concepts of geometric distance, the perimeters and the spreads of the differences between interval-valued fuzzy numbers on both the X-axis and the Y-axis

Assume there are two interval-valued trapezoidal fuzzy numbers and , where

The proposed method for calculating the degree of similarity between and is presented as follows.

A~~

B~~

A~~

B~~

8

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Step 1: Calculate the degree of closeness between the upper interval-valued fuzzy numbers of and , respectively, where

and . The larger the value of , the closer the interval-valued fuzzy numbers and .

A~~

B~~

)~~

,~~

( UUUX BAS

]1 ,0[)~~

,~~

( UUUX BAS 41 i

(1)

)~~

,~~

( UUUX BAS

A~~

B~~

9

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Step 2: Let be an array of differences between the corresponding values of the interval-valued fuzzy numbers and

on the X-axis,

Let be the mean of the elements in the array , where

A~~

B~~

P

X P

(2)

(3)

10

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Step 3: Let be an array of differences between the membership degrees of the corresponding points of the interval-valued fuzzy numbers and ,

where and denote the membership functions of the interval-valued fuzzy numbers and , respectively, and

.

A~~

B~~

Q

A~~

B~~

Af ~~

Bf ~~

]1 ,0[:~~ XfA

]1 ,0[:~~ XfB

(4)

11

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Let be the mean of the elements in the array , where

Step 4: Calculate the spread of the differences between the interval-valued fuzzy numbers and on the X-axis,

where denotes the element of the array defined in Eq. (2), , and denotes the mean of the elements in the array , as defined in Eq. (3). The lower the value of , the more similarity between the shapes of and on the X-axis.

QY

.8

8

1

iiy

Y

)~~

,~~

( BASTDXA~~

B~~

,18

)()

~~,

~~(

8

1

2

i

i

X

XxBASTD

(5)

(6)

ix thi P

81 i X

P )~~

,~~

( BASTDXA~~

B~~

12

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Step 5: Calculate the spread of the differences between the interval-valued fuzzy numbers and on the Y-axis,

where denotes the element of the array defined in Eq. (4), , and denotes the mean of the elements in the array , as defined in Eq. (5). The lower the value of , the more similarity between the shapes of and on the Y-axis.

A~~

B~~

(7)

thi

81 i

)~~

,~~

( BASTDY

,18

)()

~~,

~~(

8

1

2

i

i

Y

YyBASTD

iy Q

Y

Q )~~

,~~

( BASTDY

A~~

B~~

13

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Step 6: Calculate the perimeters and of the upper interval-valued fuzzy numbers and , respectively, where

(8)

UA~~ UB

~~)

~~( UAL )

~~( UBL

2~~

243

2~~

221

ˆ)(ˆ)()~~

(UU A

UU

A

UUU waawaaAL ),()( 1423UUUU aaaa

2~~

243

2~~

221

ˆ)(ˆ)()~~

(UU B

UU

B

UUU wbbwbbBL ).()( 1423UUUU bbbb (9)

14

The Proposed Similarity Measure Between Interval-Valued Fuzzy Numbers (cont.)

Step 6: Calculate the degree of similarity between the interval-valued fuzzy numbers and ,

The larger the value of , the more the similarity

between the interval-valued trapezoidal fuzzy numbers and .

(10)

)~~

,~~

( BAS

A~~

B~~

,))

~~(),

~~(min())

~~(),

~~(max(

))~~

,~~

(1())~~

,~~

(1()~~

,~~

()

~~,

~~(

1

UUUU

YXUUU

X

BLALBLAL

BASTDBASTDBASBAS

].1 ,0[)~~

,~~

( BAS )~~

,~~

( BAS

A~~

B~~

15

A Comparison with the Existing Similarity Measures

A~~

B~~C~~

A~~B~~

C~~

A~~

B~~

C~~

A~~B~~C~~

Fig. 2. Four sets of interval-valued fuzzy numbers

Table 1. Comparison of the calculation results of the proposed similarity measure and the existing methods.

Note: “N/A” denotes cannot be calculated; “ ” denotes unreasonable results.

16

Fuzzy Risk Analysis Based on the Proposed Similarity Measure

Fig. 3. The structure of for fuzzy risk analysis

Assume that there are n manufactories and and assume that each component produced by manufactory consists of sub-components and , where .

..., ,, 21 CC nC

iA iC p..., ,, 21 ii AA ipA ni1

iC

17

Fuzzy Risk Analysis Based on the Proposed Similarity Measure (cont.)

Table 2. Linguistic terms and their corresponding fuzzy numbers

A nine-members linguistic term set shown in Table 2 is used to represent the linguistic terms and their corresponding fuzzy numbers.

18

Fuzzy Risk Analysis Based on the Proposed Similarity Measure (cont.)

The arithmetic operations between interval-valued trapezoidal fuzzy numbers and are defined by Chen [1997] and Wei and Chen [2009] as follows:

A~~

B~~

where

19

Fuzzy Risk Analysis Based on the Proposed Similarity Measure (cont.)

Based on the proposed similarity measure, the new algorithm for fuzzy risk analysis is presented as follows:

Step 1: Based on fuzzy weighted mean method presented by Schmucker [1984], aggregate the evaluating items and of sub-component of each component made by manufactory , where and , to get the probability of failure of each component made by manufactory , where

where is an interval-valued fuzzy number and .

ikR~~

ikW~~

ikA iA iC

pk 1 ni 1 iR~~

iA iC

iR~~

ni 1

20

Fuzzy Risk Analysis Based on the Proposed Similarity Measure (cont.)

Step 2: Based on the proposed similarity measure, calculate the degree of similarity between the interval-valued fuzzy numbers and , respectively, where and . If is the largest value among the values

then is transformed into the linguistic term corresponding to .

)~~

,~~

( ji HRS

`

~~iR jH

~~ni1 91 j )

~~,

~~( ji HRS

),~~

,~~

( ..., ),~~

,~~

( ),~~

,~~

( 21 jiii HRSHRSHRS `

~~iR

jH~~

21

Fuzzy Risk Analysis Based on the Proposed Similarity Measure: Example

Table 3. Linguistic values of the evaluating items of the sub-components made by manufactories

The linguistic values of evaluating items and of the sub-component made by manufactory are shown in Table 3.

ikR~~

ikW~~

ikA iC

22

Fuzzy Risk Analysis Based on the Proposed Similarity Measure: Example (cont.)

[Step 1] The probability of failure of each component made

by manufactory is shown as follows:iR

~~iA

iC

23

Fuzzy Risk Analysis Based on the Proposed Similarity Measure: Example (cont.)

[Step 2] The calculated degree of similarity between each pair of the interval-valued fuzzy numbers and is shown as follows:

iR~~

)~~

,~~

( ji HRS

jH~~

24

Fuzzy Risk Analysis Based on the Proposed Similarity Measure: Example (cont.) Because = 0.6264 is the largest value among

, the probability of failure of the component made by the manufactory is transformed into the linguistic term “Medium”.

Because = 0.7783 is the largest value among , the probability of failure of the component made by the manufactory is transformed into the linguistic term “Fairly-High”.

Because = 0.6424 is the largest value among , the probability of failure of the component made by the manufactory is transformed into the linguistic term “Fairly-High”.

The results of the proposed method coincide with the ones presented in Chen and Chen [2009].

)~~

,~~

( 51 HRS ),~~

,~~

( 1 jHRS

91 j 1

~~R

1C1A

)~~

,~~

( 62 HRS ),~~

,~~

( 2 jHRS

91 j 2

~~R 2A

2C

)~~

,~~

( 63 HRS ),~~

,~~

( 3 jHRS

91 j 3

~~R 3A

3C

25

Conclusions

In this paper, we presented a new similarity measure between interval-valued fuzzy numbers to overcome the drawbacks of the existing methods.

The proposed similarity measure is applied to develop a new algorithm for dealing with fuzzy risk analysis problems.

Based on the new similarity measure, the proposed algorithm for fuzzy risk analysis can provide us with a simple, useful and more flexible way to deal with fuzzy risk analysis problems.