analyzing fuzzy risk based on a new similarity measure between interval-valued fuzzy numbers
DESCRIPTION
ANALYZING FUZZY RISK BASED ON A NEW SIMILARITY MEASURE BETWEEN INTERVAL-VALUED FUZZY NUMBERS. KATA SANGUANSAT 1 , SHYI-MING CHEN 1,2 1 Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. - PowerPoint PPT PresentationTRANSCRIPT
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.