GROUP JUDGMENTS IN FLOOD RISK MANAGEMENT: SENSITIVITY ANALYSIS AND THE AHP

Transcription

1 IHP 3, Bali, Indonesia, ugust 7-9, 3. GROUP JUDGMENT IN FLOOD RIK MNGEMENT: ENITIVITY NLYI ND THE HP Jason K. Lev, Jens Hartmann, Hirokazu Tatano, Norio Okada, hogo Tanaka Disaster Prevention Research Institute, Koto Universit, Japan Keith W. Hipel Universit of Waterloo, Waterloo, Ontario Canada D. Marc Kilgour Wilfrid Laurier Universit, Waterloo, Ontario, Canada Chennat Gopalakrishnan Universit of Hawaii, Manoa, Hawaii Kewords: uncertaint, sensitivit analsis, flood management, consistenc ratio ummar: Flood management is an important issue in Japan as Japanese rivers are steep in gradient and short in length, and some million people populate the river basin densel. The HP is shown to be a valuable technolog for aggregating group judgments. ensitivit measures are developed to determine the robustness of the consistenc ratio and the principal right eigenvector to perturbations in the group judgments of the pairwise comparison matrix. This paper investigates how uncertaint in each of the input affects the derived output variables.. Introduction MCD for flood management is one of the fastest growing areas in operations research, borrowing heavil from fields such as hdrolog, geolog, pscholog and computer science. Most flood management problems in Japan are participator processes which inherentl involve input from a variet of decision makers, whose judgments must be aggregated. In the context of HP, the aggregation of group judgments in discussed ection. Next, sensitivit metrics are proposed to quantif the notion of robustness to uncertaint. It is shown that the derived consistenc ratio ection 3 and principal eigenvector ection are insensitive to small perturbation of the consistent pairwise comparison matrix.. ggregation of Individual Judgments and ensitivit nalsis in HP n important problem in decision analsis involves the combination or aggregation of problem-solving knowledge and judgments. In the HP it is well known that that an m x m pairwise comparison matrix requires m m such input judgments. Bolloju 997 emphasizes the importance of combining or aggregating individual judgments into a group score: Man tpical business environments require such combination or aggregation of decision making for man reasons such as validation, consistenc verification, and training. n technique for elicitation ggregation of problem-solving knowledge should deal with inconsistencies, conflicts, and decision makers' subjectivit. In the context of HP, this paper uses the geometric mean of individual judgments to obtain a combined group judgment. This ma help to overcome difficulties arising from a lack of group consensus Davies, 99. Other techniques for aggregation include conceptual aggregation based on conceptual clustering and case-based learning for real-time dnamic decision making Chaturvedi et al., 993; the flexible modeling approach based on Baesian analsis for aggregation of point estimates Clemen and Winkler, 993; ggregation of preference patterns using social choice framework Dubois and Koning, 99. Comparative studies on Proceedings 7 th IHP 3 Bali, Indonesia 9

2 group preference aggregation are reported b Ramanathan and Ganesh 99 and Perez and Barba- Romero Mean of Individual Judgments In the Tokai flooding problem, three important criteria are considered: flood frequenc criteria, water velocit criteria, and inundation depth criteria 3. Consider judgment a, the relative importance of the flood frequenc compared with water velocit. The sample geometric mean of a set 3 N, a, a, K a of judgments from N experts is defined b: { } a, < L N 3 N a a a a a In such a wa, a consensus matrix can be established b aggregating the individual judgments of the N decision makers using the geometric mean approach. While the geometric mean method is the most widel used and theoreticall accepted approach in HP for combining individual judgments to form a group opinion, Ramanathan and Ganesh 99 showed, using counter-examples, that the geometric mean method does not alwas satisf the Pareto optimalit axiom, one of the prominent social choice axioms.. Variance of Individual Judgments It is also important to consider a measure of data variabilit. For illustrative purposes, consider again judgment a, the relative importance of the flood frequenc versus water velocit. n unbiased estimator, the sample variance of the judgments, can be established as the mean square deviation of a values from the geometric mean, : var[ N i a a ] σ [ a ] N i ince we are eliciting opinions from a sample of N decision makers from a possibl large population, the geometric mean is onl an approximation to the "true mean." Because this mean must be estimated from available judgments data, the number of degrees of freedom is reduced b, hence the factor of /N- in the variance. If we have judgments available from the total population of decision makers, or if the mean was known a priori, then the factor would be /N. Of course the standard deviation of judgment a is well-known to be σ [ a ], the square root of the variance. The standard deviation gives a measure of the "spread" of the judgments and can also be used to assign probabilities for being within a certain range of values. Consider now the snthesis of individual judgments for entr a, assuming that the judgments of three decision makers are as follows: 3 a, a, a,, 3 { } { } The combined group judgment, using the geometric mean should be: 3 with corresponding variance: Proceedings 7 th IHP 3 Bali, Indonesia 9

3 For convenience, assume that we have the following geometric mean values for each entr in the pairwise comparison matrix for the Tokai flooding problem, giving rise to a perfectl consistent matrix: imilarl, let us assume the following pairwise comparison matrix of variances: 3 3 /5 7 5/ 9 7 / 9 / 7.3 ensitivit Metrics In this section we develop sensitivit metrics to quantif how output variables are affected b uncertaint in the group judgments of the pairwise comparison matrix. To obtain the eigenvalue and eigenvectors of a 3 b 3 matrix, we must solve the problem a a w w a a w λ max w a 3 a3 w3 w3 In the cubic and quartic cases, derived output variables, such the consistenc ratio, can be explicitl written as a function of the judgments in the pairwise comparison matrix. For example, in the cubic case, aat 3 has explicitl derived CR f a, a, as follows: a CR CI MRCI n 3 a / 3 t a t a t + a t a t / a t. 9 where CR, CI, and MRCIn are the Consistenc Ratio, Consistenc Index and Mean Random Consistenc Index. imilarl, the weights, w, and the largest eigenvalue, λ, are also a function of the input i a judgments, hence w i f a, a, and λ f a, a, a. scenario can be defined as a max max vector of values for the pairwise comparison matrix input judgments, a, where: a a, a, a For each input, the geometric mean of the group judgments represent the nominal or base case value for each input. Denote these nominal input values,,, and. Together, these three input values specif the nominal scenario:,, Proceedings 7 th IHP 3 Bali, Indonesia 93

4 It is of interest to note how an output variable, such as the consistenc ratio, changes with the value of the inputs. The corresponding nominal output variable is defined as: < f sensitivit metric,, is used to quantif the degree to which each of the group input judgments aij contributes to the uncertaint in the output variable : ij aij < ij Note that all of the partial derivatives are evaluated at the nominal i.e. geometric mean scenario,. These measures can be combined to determine the total sensitivit, : T + a T a + a 5 Morgan and Henrion 99 refer to the metric in Eq. as a measure of elasticit, and discuss the use of additional related sensitivit metrics. 3. ensitivit of the Consistenc Ratio to Uncertaint in Group Judgments This section illustrates that the consistenc ratio is robust to uncertaint small perturbations in the group judgments of a 3 b 3 pairwise comparison matrix. Using Eq., the sensitivit in the CR arising from uncertainties in judgment a is denoted: < a CR The factor in Eq ensures that this measure of uncertaint is unaffected b the unit of < measurement of both the input and output variables. imilar measures can be defined for the two remaining T group judgments a. s shown in Figures through 3 CR should be relativel small because the individual sensitivities CR a, CR a, and CR a are shown to be small, when the consistenc ratio is the output variable. However, a drawback of this approach is that it does not consider the degree of variation in each input. group input judgment that has a small sensitivit, but a large variation about its nominal value due to uncertaint and perhaps disagreements among decision makers in the group ma lead to significant variation in the derived consistenc ratio and eigenvector. Proceedings 7 th IHP 3 Bali, Indonesia 9

5 a_ a_ a_ a b Figure. CR a as a function of a a and b a a_. -. a_ a_.. a_ a_ a b Figure. CR a as a function of a a and b a a_ a_ a_ a_ a b Figure 3. CR a as a function of a a and b a Proceedings 7 th IHP 3 Bali, Indonesia 95

6 3. Gaussian pproximation to measure the variance of the consistenc ratio The Talor series expansion provides a wa to express deviations of an output from nominal values, <, in terms of deviations of inputs from their nominal values: a. uccessive terms of the Talor series contain higher order powers of deviations and higher order derivatives of the function with respect to each input. If the deviations aij ij are relativel small, then higher powers will become ver small. nd if the output function is relativel smooth in the region of, the higher derivatives will also be small and hence higher order terms can be safel ignored. Under these conditions, the Talor series produces a good approximation. For example, consider an approximation for the variance of assuming independence of the a, a, input judgments, using onl Talor series terms up to the second order: var a From Eq. 5 one can see that if the input judgments are independent, then the variance of the output is approximatel the sum of squares of the products of the standard deviation, σ a, and the sensitivit a ij of each input: var σ a + σ a + σ a This implies that the variance of the output is given b a Gaussian approximation, where total uncertaint in the output, expressed as variance, is explicitl decomposed as the sum of uncertaint contributions from the input. We can prove that ever pair of input judgments, for example a are independent b showing that their covariance is zero. The covariance of two elements a is defined b: cov a, a < a µ a µ 7 a where µ and µ are the means of a, respectivel. Now, it is true that for an two distinct judgments in a pairwise comparison matrix, the geometric mean of the product of two judgments is equal to the product of their geometric means. For example: a a < < a, a cov, a It follows that for a 3x3 matrix cov a, a, and cov a, a. Finall, the covariance of two identical judgments can be expressed as the variance of a single judgment. For example, the covariance between judgment a and itself becomes: cov a, a ij ij ij 9 Proceedings 7 th IHP 3 Bali, Indonesia 9

7 .5..5 varha_l varha_l Fig. Variance of the Consistenc Ratio, varcr, as a function of and The relationship between variations in the inputs, and, and variations in the output, varcr, is given in Figure where it is assumed, for illustrative purposes, that and CR a.. ensitivit of Criteria Weights to Perturbations in Input Judgments ssume that the pairwise comparisons in Figure 5 are obtained b taking the geometric mean and variance of the group judgments. Note that in Figure 5 there are three criteria: flood frequenc criteria, flood velocit criteria B, and flood depth criteria C. Fig 5. Consistent Matrix for the Flooding Criteria: Frequenc, Velocit, and Depth Proceedings 7 th IHP 3 Bali, Indonesia 97

8 The weights of criteria frequenc, B velocit, and C depth are denoted as w, w B, and w c. Using the sensitivit notation developed earlier, for group input a the sensitivit in w would be: w < w w a The flooding criteria w should also be examined for sensitivit to changes in group inputs a : w w a < w and w w a < w We next investigate how uncertaint in the aggregated judgment a affects the sensitivit of the flooding, velocit, and depth criteria. This is illustrated in Figures through respectivel. Note that in Figure 7 the sensitivit metric: w < w B w B a B is used because we are considering the sensitivit of the velocit criteria criteria B. Finall, Figure considers the sensitivit of depth criteria criteria C to uncertainties in judgment a and hence uses the metric: w < w C w C a C a_ a_ a_ a_ Figure. w a as a function of a a and b a Proceedings 7 th IHP 3 Bali, Indonesia 9

9 a_ a_.3.5. a_ a_ Figure 7. w a as a function of a a and b a B a_ a_ a_ a_ Figure. w a as a function of a a and b a C Note that: w a a C wb w f a f Hence, we can conclude that the weight of the depth criteria w c is most affected b uncertainties in the group judgment a. lso, from Eq. we can conclude that the weight of the velocit criteria w B is more sensitive to uncertainties in a than is the depth criteria w c. References Bolloju, N On Problems and Issues in Discover of Multi-ggregations of Classification Problem-olving Knowledge from Multiple Decision Makers. ssociation for Information stems. 997 mericas Conference. Indianapolis, Indiana. ugust 5-7,997. Chan, C. and Wong.K.C., 99. tatistical Technique for Extracting Classificator Knowledge from Databases, Knowledge Discover in Databases, G. Piatetsk-hapiro, and W.J. Frawle, eds., I Press, Proceedings 7 th IHP 3 Bali, Indonesia 99

10 Chaturvedi, C. Hutchinson, G.K., and Nazareth, D.L. 993, upporting complex real-time decision making through machine learning, Decision upport stems,, -3. Clemen, R. and Winkler R.L. 993, ggregating Point Estimates: Flexible Modeling pproach, Management cience, 39:, Davies, 993. Multicriteria Decision Model pplication for Managing Group Decisions, Journal of the Operational Research ociet, 5: Markham, C. and Ragsdale, C.T. 995, Combining Neural Networks and tatistical Predictions to olve the Classification Problem in Discriminant nalsis, Decision ciences, : Perez, C. and Barba-Romero,. 995, Three practical criteria of comparison among ordinal preference aggregating rules, European Journal of Operational Research, 5, Ramanathan, R. and Ganesh, L.. 99, Group preference aggregation methods emploed in HP: n evaluation n intrinsic process for deriving members' weightages, European Journal of Operational Research, 79, 9-5. aat, T. 3, n xiomatic Framework for ggregating Weights and Weight-Ratio Matrices, Proceedings of the eventh International mposium on the naltic Hierarch Process, ugust 7-9, 3, Bali, Indonesia. Proceedings 7 th IHP 3 Bali, Indonesia 3