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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 SANGUANSAT 1 , SHYI-MING CHEN 1,2 1 Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan.

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

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  1. 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,

  6. Interval-Valued Fuzzy Numbers (cont.) 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.

  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 . (1)

  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 (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 . (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. (5) (6)

  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. (7)

  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) (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)

  15. A Comparison with the Existing Similarity Measures 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. Fig. 2. Four sets of interval-valued fuzzy numbers

  16. Fuzzy Risk Analysis Based on the Proposed Similarity Measure • Assume that there are n manufactories and and assume that each component produced by manufactory consists of sub-components and , where . Fig. 3. The structure of for fuzzy risk analysis

  17. Fuzzy Risk Analysis Based on the Proposed Similarity Measure(cont.) • A nine-members linguistic term set shown in Table 2 is used to represent the linguistic terms and their corresponding fuzzy numbers. Table 2. 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: 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 .

  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 .

  21. Fuzzy Risk Analysis Based on the Proposed Similarity Measure: Example • The linguistic values of evaluating items and of the sub-component made by manufactory are shown in Table 3. Table 3. Linguistic values of the evaluating items of the sub-components made by manufactories

  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:

  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:

  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].

  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.

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