Assessment Model of Set Pair Analysis for Flood Loss Based on Triangular Fuzzy Intervals under α-cut

Transcription

1 Assessment Model of Set Pair Analysis for Flood Loss Based on Triangular Fuzzy Intervals under -Cut PAN Zheng-wei WU Kai-ya 2 JIN Ju-liang 3 LIU Xiao-wei. College of Natural Resources and Environmental Engineering Hefei University of Technology Hefei China Pzhwei023@63.com 2. National Innovative Institute for Public Management and Public Policy Fudan University Shanghai China Wuky2000@vip.sina.com 3. School of Civil Engineering Hefei University of Technology Hefei China JINJL66@26.com Abstract: The flood disaster is one of the most serious natural disasters. It is the important basis of making plan for flood control to assess flood losses accurately. The results of the common assessment methods are discrete values and low resolution which are applied widely at present. The research to more precise more intuitive method of loss evaluation for flood disaster is the focus. The set pair analysis assessment model was established by the -cut set of triangle fuzzy number à in this paper. The case study shows this method not only presents the concrete rank of flood lose but also finds that the evaluating rank is a confidence interval; in addition we can obtain priorities of different areas by the expectation and square difference method for priorities of interval numbers. Key word: flood lose; set pair analysis; triangle fuzzy number; -cut set; interval numbers; expectation and square difference method for priorities The flood disaster is one of the most serious natural disasters. The annual losses caused by flood account for about 3.5 % of the Gross National Product in China []. It is the important basis of making plan for flood control to assess flood losses accurately [2]. There isn't scientific rational and practical loss evaluation model of flood disaster is because the flood disaster has very complex characteristics such as unanticipated diversity stochastic etc [3 4]. The results are discrete values and low resolution for the assessment methods at present [5]. For example fuzzy comprehensive evaluation [6] object unit analysis method [7] BP neural network [8] etc. So these assessment methods have limitation while deal with the uncertain problem. The set pair analysis proposed by Chinese scholar Ke-qin Zhao in 989 has already been applied in numerous fields since 980s [9-]. In the connection degree formula µ = a + bi + cj the a and cj is certainty which the bi is uncertainty. In this paper the discrepancy degree i is acquired by the -cut set of the triangle fuzzy number Ã. Further the set pair analysis assessment model was established by using the -cut set of triangle fuzzy number Ã. INTRODUCTION. Set pair analysis Theoretical basis of SPA is the set pair. The set pair is defined as a pair that consists of two interrelated sets. Putting together set A and B to compose set pair H with This work is supported by the National Key Technologies R&D Program of China during the th Five-year Plan Period (No.2006BAB4B02) the National Natural Science Foundation of China (No ) and the Opening Foundation of Chengdu Institute of Plateau Meteorology China Meteorological Administration (LPM200808). respect to the problem W The connection degree formula is as following: µ = a + bi + cj () Eq. () is called identity-discrepancy-contrary connection degree [9] or three-element connection numbers [9 0]. In its practical application there are some problems that it is rough only to divide state-space of research object into three [ 2] which can not definitely describe these problems. Therefore it is necessary to expand basic formula of the connection degree on difference levels [3 4] and then on the same level to expand it to form a kind of multivariate connection number whose formula is as following [9 ] : µ = a + bi + b2i2 + + bnin + cj (2) At present the five-element connection numbers is adopted widely and the formula is as following [ 2] : µ = a + b i + b i + b i + cj (3) Where a b b b c [0 ] and a + b + b + b + c = a represents the identity degree of the set pair; b b 2 b 3 represent the discrepancy degree of the set pair; c represents the contrary degree of the set pair; i i 2 i 3 are the coefficient of the discrepancy degree and are some uncertain values between - and i.e. i [ ]. j is the coefficient of the contrary degree and is specified as -; the i i 2 i 3 and j can be regarded as the markers of the discrepancy degree and the contrary degree respectively..2 Triangular Fuzzy Intervals under -Cut Definition [5] : Given a fuzzy number à in the real number field and define a membership function: µ ( x ) : R [0 ] A x R if it has the following triangular-type membership function:

2 0 x < a x a a x b µ ( x) b a A = c x b x c c b 0 x > c then à is called the Triangle Fuzzy Number (TFN) denoted by A = ( a b c) where a b c ; if a = b = c à is a crisp set. The triangle fuzzy number is indicated in Fig.. Definition 2 [6] : Given à is the fuzzy set in the universe U to [0 ] common set A = { x µ ( x) } is called level set cut of the fuzzy set à for short -cut set is indicated in Fig.; when A = { x µ ( x) > } A A is called -strong cut set and is called confidence level. µ a L 0 a A b Given TFN is A = ( a b c) and confidence interval is [0 ] confidence level interval can be attained as follows [7] : A = [ A A ] = [( b a) + a ( c b) + c] (4) For the fuzzy number in the real number field -cut set A is the real number interval number [ A A ] which satisfies four arithmetic operation rules as follows [8] : [ a b] + [ c d] = [ a + c b + d] [ a b] [ c d] = [ a d b c] k [ c d] = [ k c k d] ( k > 0 ) [ a b] [ c d] = [min{ ac ad bc bd} max{ ac ad bc bd}] [ a b] [ c d] = [ a b] [ ] d c a a b b a a b b { d c d c} { d c d c} = [min max ] ([ c d] 0). A R A c x Fig. Triangular Fuzzy Intervals under -Cut.3 Expectation and square difference method for priorities Given two real number interval numbers a = [ a a + ] and b = [ b b + ] and random variables satisfies uniform + distribution on interval [ a a ] then the mathematical + ( a + a ) expectation is E = and the variance is ( a a ) D = by which a kind of priority method of 2 interval number is defined as follows [9] : () a superior to b means under two conditions: Ea > Eb 2 Ea = Eb and D a < D b denoted by a b ; (2) a equal to b means under condition: Ea = Eb and D a = D b denoted by a ~ b ; (3) a inferior to b means under two conditions: Ea < Eb 2 Ea = Eb and D a > D b denoted by a b. 2 SET PAIR ANALYSIS ASSESSMENT MODEL BASED ON TRIANGULAR FUZZY INTERVALS UNDER -CUT According to basic steps of SPA system analysis [2] the approach of the set pair analysis assessment model based on triangular fuzzy intervals under -cut can be established as follows: Step: Construct the study object-set pair and analyze the relations of identity discrepancy and contrary of the set pair. The set pair H = ( A B) of evaluation index set A and evaluation standard set B is generally established in the system analysis. By analyzing the characteristics involved in the set pair H the relations of relations of identity discrepancy and contrary can be established for the study object. To reach the objective of thinning the relations construct of identity discrepancy and contrary according to the hierarchy theory of set pair analysis theory we can establish the hierarchical structure which is fit for five grades evaluation standard as indicated in Fig.2. The relations of identity discrepancy and contrary system First Level Second level Identity Discrepancy Contrary Identity Partial identity in discrepancy Entirety discrepancy Partial contrary in discrepancy Contrary ( I ) ( II ) ( III ) ( IV ) ( V ) Third level Identity Partial identity in discrepancy Entirety discrepancy Partial contrary in discrepancy Contrary Fig.2 Relation frame of identity discrepancy and contrary Fig.2 shows: The relations of the first level are the ones of identity discrepancy and contrary in traditional. The relations of the second level thin the discrepancy into the partial identity in discrepancy the entirety discrepancy and the partial contrary in discrepancy which is relative to I-V grades evaluation standard. Accordingly i i 2 i 3 are the coefficient of the partial identity in discrepancy the coefficient of the entirety discrepancy and the coefficient of the partial contrary in discrepancy respectively. The relations of the third level make the partial identity in

3 discrepancy the entirety discrepancy and the partial contrary in discrepancy of the second level further thinning. For example the partial identity in discrepancy is divided into identity the partial identity in discrepancy and the entirety discrepancy according to a certain proportion. The entirety discrepancy is divided into the partial identity in discrepancy the entirety discrepancy and the partial contrary in discrepancy according to a certain proportion. The partial contrary in discrepancy is divided into the entirety discrepancy the partial contrary in discrepancy and contrary. Using the above method can deal with the difficult problem of measure scale too wide [20 2]. Step2: Determine the connection degree formula based on triangular fuzzy intervals under -cut () Graded connection degree. The key work for set pair analysis is to set up a connection degree formula. In system analysis evaluation some adopted methods are as following: According to the definition of connection degree adopting the probabilistic method calculate connection components and then to set up a connection degree formula [22]. 2 Adopting the triangle membership function calculate connection components and then to set up a connection [23 degree formula 24]. The essence of the triangle membership function calculating membership degree is dichotomy which is incompletely identical with the idea of trichotomy of SPA. Therefore the new model is proposed by distributing connection components according to a certain proportion which embody the idea of trichotomy of SPA. On the above analysis the graded connection degree is as follows: + 0i c I c sm (2-3) sm ( 2) c + i + i2 c II 2 sm (-2) sm (2-3) 2 2 sm (-2) sm (2-3) c sm (3-4) sm (2-3) c µ = 0 + i 2 + i2 + i sm (2-3) sm (3-4) c III (5) sm (2-3) sm (3-4) c sm (4-5) sm (3-4) c i + i2 + i3 + j c IV 2 sm (3-4) sm (4-5) 2 2 sm (3-4) sm (4-5) 0+ 0i + j c V Where m is the m-th evaluation index m = 2 M ; k is the k-th evaluation sample k = 2 K ; c is the k-th evaluation sample value; s m( x- y) is the critical value between x y grade-standard of the m-th index. (2) The coefficient of discrepancy degree basing on the triangular fuzzy intervals under -cut. According to the relation structure of identity discrepancy and contrary of the third level the identity degree a whose coefficient can be regarded as ; the contrary degree c whose coefficient often selects -; adopting analysis technique determines the coefficient of discrepancy degree i.e. i =0.5 i 2 =0 i 3 =-0.5. Thereby the coefficients of connection form a group of sequences: which are continuous in [- ]. So the coefficients of connection have the minimum value i min the optimum value i opt and the maximum value i max which can construct TFN A ( i) = ( imin iopt imax ). For example the minimum value optimum value and maximum value of the coefficient of the partial identity in discrepancy are and respectively which can construct TFN of the coefficient of the partial identity in discrepancy as A ( i ) = (0 0.5 ). Applying the same method with the above can construct TFN of the coefficient of the entirety discrepancy as A ( i 2) = ( ) and TFN of the coefficient of the partial contrary in discrepancy as A ( i 3) = ( 0.5 0). If the confidence level is given we can obtain the confidence interval under the confidence level through Eq. (4) as follows: A ( i ) = [ ] (6) A ( i2) = [ ( )] (7) A ( i3) = [ ] (8) If = then A ( i ) = 0.5 A ( i ) = 0 A ( i ) = which is 3 identical with the results of analysis technique. Step3: Given the confidence level and establish the connection degree matrix R Given the sample value of index is c then can obtain connection degree formula of the sample value c of evaluation index by Eq. (5). On the condition that the confidence level is given we can further obtain the connection degree value µ which is a confidence interval number of the confidence level of the sample value c and µ [ ]. The degree matrix R is established by the connection degree value µ as follows: [ µ µ ] [ µ 2 µ 2] [ µ K µ ] K [ 2 2] [ 22 22] [ 2K 2K ] R µ µ µ µ µ µ = [ µ M µ M ] [ µ M 2 µ M 2] [ µ MK µ MK ] Step4 : Confirm the weight vector W of evaluation index Given the weight vector of evaluation index is denoted by W = [ w w2 wm ]. Step5: Establishment of connection mathematical model and comprehensive assessment analysis () Results of comprehensive estimation. Adopting the comprehensive method of the addition weighting carry out the system comprehensive estimation as follows [25] : G = W R = [ g g2 gk ] (9) Where M g = [ g g ] = w [ µ µ ]. k k k m m= The results of comprehensive estimation of the confidence level can be obtained through Eq. (9) and it is similar to the connection degree value µ. The result of comprehensive estimation g k is also a confidence interval number and gk [ ]. (2) Comprehensive priority ordering. We can obtain priorities of comprehensive estimation results by the expectation and square difference method priorities of interval number.

4 (3) Grade of comprehensive estimation. According to the relation frame of identity discrepancy and contrary the projection function of the grade y k and the results g k of comprehensive estimation are established as follows: Evaluation index I (Small disaster) Tab. Assessment standards of flood loss [7] II (Middle disaster) yk = f ( gk ) = 2 gk + 3 ( gk [ ] ) (0) The grade of comprehensive estimation can be determined by projection function Eq. (0). Let the confidence level of interval number be and yk [ 5]. III (Great disaster) IV (Catastrophe) V (Supercatastrophe) B : Death toll 2.92 [ ] [ ] [ ] B 2 : Affected population [ ] [ ] [ ] B 3 : Collapsed building.80 [ ] [ ] [ ] B 4 : Damaged crop 27. [ ] [ ] [ ] B 5 : Direct loss of economy 2.86 [ ] [ ] [ ] Tab.2 Indexes data of flood loss for each province in 998 [7] Province B B 2 B 3 B 4 B 5 Province B B 2 B 3 B 4 B 5 Hebei Guangxi Inner Mongolia Hainan Heilongjiang Sichuan Zhejiang Chongqing Anhui Guizhou Fujian Yunnan Shandong Xizang Henan Gansu Hubei Qinghai Hunan Ningxia Guangdong Xinjiang CASE STUDY Now we can analyze to the flood losses of every provinces of China in 998 as follows by using the set pair analysis assessment model based on the triangular fuzzy intervals under -cut. Flood disaster is divided into 0 I-V grade according to the division method of Investigation Group of National Natural Disaster. i.e. slight disaster small disaster middle disaster great disaster catastrophe and super-catastrophe [26] and the assessment standards are listed in Tab. [7]. The flood losses in Beijing Tianjin Shanxi Liaoning Jilin Shanghai Jiangsu Jiangxi Shanxi etc are less by the disaster situation statistical data [7]. This can be directly evaluated as slight disaster. The index data of other provinces are listed in Tab.2. Step: Construct the set pair The sample values of the m-th index constitute the evaluation index set Am = [ cm cm 2 cm22] and the evaluation standards of m-th index constitute the evaluation standard set B = [ s (-2) s (2-3) s (3-4) s (4-5) ]. Then the set m m m m m pair H = ( Am Bm ) is constituted by A m and B m. Step2: Confirm the connection degree formula Taking Hebei province for example the connection degree formula of the evaluation index can be obtained through Eq. (5) (shown in Tab.3). Tab.3 Relational degree formula for indexes of Hebei Index Relational degree formula B µ = + 0i B 2 µ 2 = i i 2 B 3 µ 3 = + 0i B 4 µ 4 = 0 + 0i i i j B 5 µ 5 = i i 2 Step3: Given the confidence level and construct the connection degree matrix R If the confidence level is 0.75 in this article we can obtain the confidence interval of the coefficient of the partial identity in discrepancy A ( i ) = [ ] the confidence interval of the coefficient of the entirety discrepancy A ( i2 ) = [ ] and the confidence interval of the coefficient of the partial contrary in discrepancy A ( i3) = [ ] through Eqs. (6) (7) and (8). Therefore the connection degree value of the evaluation index sample values µ can be obtained (shown in Tab.4) then the connection degree matrix R is constructed by µ.

5 Tab.4 Evaluation results and ranking of flood loss for each province using different methods Province Result Expectation Variance Priority Grade Ref. [7] Hebei [ ] [ ] II Inner Mongolia [ ] [ ] III Heilongjiang [ ] [ ] IV Zhejiang [ ] [ ] II Anhui [ ] [ ] II Fujian [ ] [.97.3 ] I Shandong [ ] [ ] II Henan [ ] [ ] II Hubei [ ] [ ] IV Hunan [ ] [ ] IV Guangdong [ ] [ ] II Guangxi [ ] [ ] III Hainan [ ] [ ] I Sichuan [ ] [ ] II Chongqing [ ] [ ] II Guizhou [ ] [ ] I Yunnan [ ] [ ] IV Xizang [ ] [ ] I Gansu [ ] [ ] I Qinghai [ ] [ ] I Ningxia [ ] [ ] I Xinjiang [ ] [ ] I Step4: Confirm the weight vector W of evaluation index The weight vector of evaluation index was build up by the method of index number heavy as follows [7] : W = [ ]. Step5: Comprehensive assessment and analysis () Results of comprehensive assessment. The comprehensive assessment results of the confidence level =0.75 were obtained through Eq. (9) (shown in Tab.4). (2) Comprehensive priority ordering. The comprehensive assessment results gk = [ gk gk ] are confidence interval numbers and can be calculated using its mathematics expectation and variance (shown in Tab.4). We can obtain priorities of the flood losses of every provinces of China in 998 according to the expectation and square difference method for priorities of interval number (shown in Tab.4). (3) Grades of comprehensive assessment. The comprehensive assessment grades of the confidence level =0.75 were obtained through Eq. (0) according to the comprehensive assessment results gk = [ gk gk ] (shown in Tab.4). Meanwhile the results of other assessment methods are listed in the Tab.4. (4) Comparative analysis of results. Tab.4 shows the flood losses of every provinces of China in 998 are as following: Hainan Xizang Ningxia Gansu Qinghai Fujian Xinjiang Guizhou etc. belong to I-grade i.e. small disaster; Chongqing Guangdong Hebei Anhui Zhejiang Henan Shandong Sichuan etc. belong to II-grade i.e. middle disaster; Inner Mongolia Guangxi etc. belong to III-grade i.e. great disaster; Heilongjiang Hunan Yunnan Hubei etc. belong to IV-grade i.e. catastrophe. The appraisal results are unanimous basically with by the matter element method and are consistent with the actual situation in CONCLUSIONS The case analysis of flood losses of every provinces of China in 998 shows: () The grades of flood loss are imprecise by the traditional methods of loss assessment for flood disaster. Such as Inner Mongolia and Guangxi belong to great disaster and can not more finely differentiate the disaster loss extent of two provinces. But we can further obtain the result of the loss of Inner Mongolia less than Guangxi according to the set pair analysis assessment model based on triangular fuzzy intervals under -cut. (2) The results are point values by the traditional method of loss assessment for flood disaster which the results are confidence interval numbers according to the set pair analysis assessment model based on triangular fuzzy intervals under -cut and the confidence interval of results are also different while the confidence level taking different values between 0 and. The results are not a confidence interval but a precise value while =. (3) It is not only can be obtain the confidence interval of results according to the set pair analysis assessment model based on triangular fuzzy intervals under -cut but also can obtain priorities of the flood loss according to the

6 expectation and square difference method for priorities of interval numbers. Which the model has theoretically important applied value and it can be widely used in loss assessment for different natural disasters. References [] Kang Xiang-wu Wu Shao-hong Dai Er-fu et al. Large-scale flood lose and pre-estimation of influencing [J]. Chinese Science Bulletin (B07): (in Chinese) [2] Fu Xiang Ji Chang-ming. Study on Estimating Index of Flood Damage [J]. Advances In Water Science 2000 (4): (in Chinese) [3] Wan Qing et al. Systems analysis and evaluation for flood disaster [M]. Science Press Beijing (in Chinese) [4] Wang Bao-hua Fu Qiang Xie Yong-gang et al. A review on evaluation method of economic loss of flood in the world [J]. Journal of Catastrophology (3): (in Chinese) [5] Yang Xiao-hua Yang Zhi-feng Shen Zhen-yao et al. Genetic projection pursuit method for evaluating water resources reproducible ability [J]. Advances In Water Science (): (in Chinese) [6] Pan Hua-sheng Zhang Gui-hua Dong Shu-hua. An estimation model of flood grade in Heilongjiang province-the method of fuzzy overall evaluation [J]. Heilongjing Meteorology : (in Chinese) [7] Li Chao. Flood damage comprehensive evaluation based on matter-element analysis of China [J]. Statistical Education (3): (in Chinese) [8] Huang Tao-zhen Wang Xiao-dong. Fast evaluation of flood and water-logging losses by BP network [J]. Journal of Hehai University: Natural Sciences (4): (in Chinese) [9] Zhao Ke-qin. Set pair analysis and its preliminary applications [M]. Zhejiang Science and Technology Press Hangzhou (in Chinese) [0] Wang Wan-jun. The set pair analysis model for evaluation of multi-factor number and its application [J]. Journal of Gansu Lianhe University: Natural Sciences (4): (in Chinese) [] Jiang Yun-liang XU Cong-fu. Advances in set pair analysis theory and its applications [J]. Computer Science (): (in Chinese) [2] Zhao Ke-qin. The application of SPA-based identical-discrepancy-contrary system theory in artificial intelligence research [J]. CAAL Transactions on Intelligent Systems (5): (in Chinese) [3] Yu Guo-xiang. The model of pluralistic connection number for comprehensive evaluation and its application [J]. Journal of Shaoxing College of Arts and Sciences (9): (in Chinese) [4] Qin Jie Zhao Ke-qin. The application of multi-element connection number in synthetic evaluation and sorting of medical quality of hospital [J]. Chinese Journal of Hospital Statistics 2004 (2): (in Chinese) [5] Michael Angelo B. Promentilla T. Furuichi K. Ishii N. Tanikawa. A Fuzzy Analytic Network Process for Multi-criteria Evaluation of Contaminated Site Remedial Countermeasures [J]. Journal of Environmental Management (3): [6] Nang-Fei Pan. Fuzzy AHP Approach for Selecting the Suitable Bridge Construction Method [J]. Automation in Construction (8): [7] Ronald E G Robert E Y. Analysis of the Error in the Standard Approximation Used for Multiplication of Triangular an Trapezoidal Fuzzy Numbers and the Development of a New Approximation [J]. Fuzzy Sets and Systems 997 9(): -3. [8] Zeng Wen-yi Luo Cheng-zhong Rozi Haji. Comprehensive decision model of interval number [J]. Systems Engineering Theory & Practice 997 (): (in Chinese) [9] Zhou Guang-ming Cheng Yang-jin. The preference of interval number in uncertain multi-attribute decision-making: square difference method for priorities [J]. Xiangtan University Journal of Philosohp & Social Sciences Edition 2002 (S): (in Chinese) [20] Tang Yun-ting. Several basic conceptions haven't been make clear in connection mathematics [C]. Collection of thesis of academic annual meeting of computer science of the national theory of 2003 Qingdao (in Chinese) [2] Liu Ying Yu Dong-mei Cheng Xian-yi. Several questions in connection mathematics [J]. Mathematics In Practice and Theory (0): (in Chinese) [22] Wang Hong-guang Wang Guo-ping. Applying pairing analysis to air quality [J]. Arid Environmental Monitoring (3): (in Chinese) [23] Deng Hong-xia Li Cun-jun Zhu Bing et al. Integrative assessment of eco-carrying capacity based on set pair analysis [J]. Journal of Yangtze River Scientific Research Institute (6): (in Chinese) [24] Shao Jin-hua Liu Xian-zhao. Comprehensive evaluation of regional water resources development and utilization based on set pair analysis [J]. China Rural Water and Hydropower : (in Chinese) [25] Jin Ju-liang Wei Yi-ming. Generalized intelligent assessment methods for complex systems and applications [M]. Science Press Beijing (in Chinese) [26] Investigation Group of National the Magnitude Natural Disaster. Natural disaster and decreasing I [M]. Earthquake Press Beijing 990. (in Chinese)