A LOSS FUNCTION APPROACH TO GROUP PREFERENCE AGGREGATION IN THE AHP

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1 ISAHP 003, Bali, Indonesia, August 7-9, 003 A OSS FUNCTION APPROACH TO GROUP PREFERENCE AGGREGATION IN THE AHP Keun-Tae Cho and Yong-Gon Cho School of Systes Engineering Manageent, Sungkyunkwan University 300 Chonchon-dong, Jangan-gu, Suwon, Kyounggi-do, , South Korea ktcho@yuri.skku.ac.kr, ygjo94@iesys.skku.ac.kr Keywords: AHP, consistency ratio, evaluation quality, group decision, group weight, loss function Suary: The Analytic Hierarchy Process (AHP) is a useful ethod in aggregating group preference. We suggest a new ethod that uses consistency ratio as group evaluation quality. For this ethod, we introduce Taguchi s loss function. We also develop an evaluation reliability function to derive group weight. astly, we perfor four experients in order to confir validity of this ethod.. Introduction The essentiality of ulti-criteria decision aking is to choose the best alternative fro a set of copeting alternatives that are evaluated under conflicting criteria. The Analytic Hierarchy Process as a ulti-criteria decision aking ethod provides us with a coprehensive fraework for solving such decision probles by quantifying the subjective judgents. The AHP as a growing field in both its theoretical and applied raifications has been applied widely in decision aking (Vargas, 990; Zahedi, 986; Shi, 989). One of the topics on which research concentrates is the proble of group judgents aggregation and consistency ratio. The AHP has been criticized and enhanced on aggregation and consistency of individual judgents necessary for group decision aking by any researchers (Aczel and Saaty, 983, 986, 987; Foran, 990; Foran and Peniwati, 998; Ko and ee, 00; ane and Verdini, 989; Saaty and Mariano, 979; Vargas, 98). However, they have focused ainly on the rules of aggregation such as geoetric ean or arithetic ean as well as consistency ratio itself. Thus, there was little research to use consistency ratio as evaluation quality of a group in aggregating group judgents. The previous group aggregation ethods keeps Saaty s rule that consistency ratio of individual pairwise coparison should be less than 0.. That is, in aggregating individual judgents to a group opinion, one takes only individual judgents whose consistency ratio is less than 0.. However, judgents are frequently inconsistent, and practically, pairwise coparison atrices rarely satisfy the consistency criterion (Murphy, 993). Thus these ethods tend to ignore any inforation of evaluation by using Saaty s consistency ratio. While previous ethods dealt with weight of each evaluator, there have been no studies that tried to aggregate group priority by using weight of a group unit in aggregating individual judgent. In order to overcoe such probles, we introduce the concept of Taguchi s loss function (Antony and Kaye, 000) and develop a loss function approach, which is a new ethod for iproving group judgents aggregation. Taguchi defines quality as the loss. The saller the loss, the higher the desirability. We define consistency ratio as the loss of evaluation quality. The loss of evaluation quality will be used as the weight of group in aggregating group judgents. We call this ethod the Weighted Proceedings 7 th ISAHP 003 Bali, Indonesia 79

2 After Geoetric Mean Method (WAGMM) and the Weighted After Arithetic Mean Method (WAAMM) for convenience.. Group Decision Making in the AHP Group decision aking involves weighted aggregation of different individual preferences to obtain a single collective preference. This subject has received a great deal of attention fro researchers in any disciplines. Unfortunately, it is extreely difficult to accurately assess and quantify changing preferences, and to aggregate conflicting opinions held by diverse group.. Geoetric Mean Method Aczel et al. (983, 987) proposed a functional equation approach to aggregate the ratio judgents. et us suppose that the nuerical judgents x, x,, x given by persons lie in a continuu (interval) P of positive nubers so that P ay contain x, x,, x as well as their powers, reciprocals and geoetric eans, etc. The aggregating function will ap P into a proper interval J, and f ( x, x,, x ) will be called the result of the aggregation for the judgents x, x,, x. The function, which should satisfy the separability condition, unaniity condition and reciprocal condition, is the geoetric ean as the following equation () / f ( x, x,, x ) = ( x, x,, () x ) { A i Approach A: The approach A is to derive A fro }. By applying equation () to every eleent of the pairwise coparison atrix { A i }, we will have the following expression. a a A = M an a a a M n a n a n = M ann { a} { a} M { an} / / / { a} { a} M { an} Once A is obtained, the priority V can be derived fro V = f (A). / / / { an } { an} M { ann} Approach B: As an alternative to approach A, the aggregated group priority vector V can be obtained fro the priority vector of each person in the group. The priority vector fro each individual V i of the group is derived fro A i. V = f ( A ) = ({ v},,{ v } ), i =,, (3). Arithetic Mean Method i i i n i, / / / V = ( v =, v,, vn) { v} i, { v} i,, { vn} i (4) In addition to the geoetric ean ethod discussed above, the arithetic ean ay also be used to aggregate group judgents. The only difference is that the arithetic ean can only be applied on the final priority weights, i.e. Approach B. This is because of the reciprocal property of pairwise coparison and / a jki / a jki. The arithetic ean ethod cannot be used to aggregate the pairwise coparison atrix A. So we have the atheatical for of the arithetic ean operated on the priority weight as follows: V = f ( A ) = ({ v},,{ v } ), i =,, (5) i i i n i, / / / () Proceedings 7 th ISAHP 003 Bali, Indonesia 80

3 V = ( v, v,, vn) = { v} i, { v} i,, { vn} (6) 3. oss Function Approach 3. oss Function oss function has three types of characteristics such as noinal-is- best characteristics, Saller-is-better characteristics and arge-is-better characteristics. 3.. Noinal-is-best Characteristics Taguchi suggests a quadratic loss function. Taguchi's loss function shows that as a characteristic deviates or oves further away fro its target value, an increasing loss will be incurred. The saller the characteristic variation about the target value, the better. et y be a characteristic and let denote the target value. Taylor series expansion loss function (y) for is given by equation (7). ( ) ( y) = ( ) + ( )( y ) + ( y ) + (7) As shown in Fig (a), ( ) = 0, ( ) = 0 (8) Thus, if one ignore ters of third order, then loss function for noinal-is-best characteristics is given by equation (9) ( y) = k( y ) (9) where k is () /. et A be loss of custoer when tolerance liit is +. k = A (0) where is the distance fro the target to a tolerance liit and A is the cost when characteristic exceeds the tolerance liits (i.e. + ). The loss function for this characteristic is shown in Fig (a). et E ( y), V ( y) be µ,σ, respectively. The average loss can be obtained by equation (). 3.. Saller-is-better Characteristics = E[ ( y)] = E[ k( y ) ] = ke( y ) = ke{[ y E( y)] + [ E( y) ]} = k[ σ + ( µ ) In this case, the characteristic y is continuous and positive with the ost desired value of zero. Here the loss function (y) increases as y increases fro zero. The loss function for saller-is-better characteristics is given by equation (). A ( y) = ky, k = () Because y is continuous and positive, the loss function (y) is a one-sided function and therefore cannot accepts negative values. The loss junction for this characteristic is shown in Fig (b). The average loss can be obtained by equation (3). = ke( y ) = k( σ + µ ) (3) 3..3 arge-is-better Characteristics In this case, the characteristic is continuous where one would like the characteristic to be as large as possible. The loss function (y) becoes progressively saller as the value of the characteristic y ] () Proceedings 7 th ISAHP 003 Bali, Indonesia 8

4 increases along the x-axis. The ideal value of this type of characteristic is infinity and the loss at that point is zero. The loss function for large-is-better characteristics is given by equation (4). ( y) = k (4) y where the loss coefficient k is deterined by equation (5). k = A 0 0 (5) The loss function for large-is-better characteristics is shown in Fig (c) (a) Noinal-is-best (b) Saller-is-better (c) arge-is-better Fig.. oss Function 3. The Concept of a oss Function Approach Consistency ratio in the AHP is a unique tool, which can easure consistency of each individual evaluator. We here use the consistency ratio as an evaluation quality. It is acceptable that if consistency ratio is low, then an evaluation quality is high, and that if consistency ratio is high, then an evaluation quality is low. As a way to easure an evaluation quality of a group by using consistency ratio, a loss function is introduced. Taguchi s loss function is faous as a useful ethod in the area of quality control. Taguchi defines the quality as the loss iparted by any product to society after being shipped to a custoer, other than any loss caused by its intrinsic function (Antony and Kaye, 000; Ross, 996). Taguchi believes that the desirability of a product is deterined by the social loss it generates fro the tie it is shipped to the custoer. Thus, the saller the social loss, the higher the desirability. By loss, Taguchi refers to the following two categories. loss caused by variability of product functional perforance; loss caused by harful side-effects. According to Taguchi s concept, we define consistency ratio as the loss, which ipacts on evaluation quality. The loss is occurred by inconsistency of individual evaluator. Now, we first introduce an expected loss derived fro Taguchi s loss function in order to obtain a collective and aggregated weight of an evaluation group. Taguchi's loss function is classified into three types of functions such as noinal-is-best characteristics, saller-is-better characteristics and large-isbetter characteristics. Aong the three of the, we take a loss function for saller-is-better characteristics, according to the fact that it is good judgents, as consistency ratio is near to zero. An expected loss for this characteristic is obtained by the su of ean and variance as the following equation (6). = ke( y ) = k( σ + µ ) (6) As shown in the above equation (6), the greater the ean or the variance for consistency ratio of each individual, the greater the expected loss of the group. We next propose a new function called evaluation reliability function, which transfors an expected loss to a collective weight of the group. The evaluation reliability function is defined as the following equation (7). Proceedings 7 th ISAHP 003 Bali, Indonesia 8

5 X ) = X ) = Exp( α X ) 0 < X < Toleranceliit (7) X ) = 0 X = 0 X Toleranceliit where, α is coefficient by each diension. The characteristic of this function is that the greater the expected loss of the group, the saller the group weight. Many different types of evaluation reliability function can be defined according to the diension of pairwise coparison atrix. The tolerance liits are required to define the function. Thus, we use the square of Ko s consistency ratio (Ko and ee, 00) as tolerance liits for evaluation reliability function. Tolerance liits of evaluation reliability function are shown in Table. Table. Tolerance liits of evaluation reliability function Diension Tolerance iit 3 0 < X < < X < < X < < X < < X < When an expected loss for consistency ratio of a group becoes 0, the weight of the group has a value of. When an expected loss is beyond tolerance liit, the group has a value of The Procedure Step. Copute C R and V CR as ean and variance of consistency ratio for pairwise coparison judgents of each individual evaluator as shown the following equation (8). n CR n i ( CRi CR ) CR =, VCR = (8) n ( n ) where, CR is consistency ratio for evaluator i ( i =,,, n ), CR i is ean of consistency ratio, V CR is variance of consistency ratio and n is the nuber of evaluator. Step. Calculate an expected loss estiate X of consistency ratio fro ean and variance obtained in Step. An expected loss estiate is given by the following equation (9). where, E () is an expected loss estiate of consistency ratio. X = E( ) = VCR + ( CR) (9) Step3. Calculate a group weight fro evaluation reliability function. A graph and equation of diension of evaluation reliability function are as follows: Table. Evaluation reliability function n Evaluation reliability function n Evaluation reliability function 3 X ) = X = 0 4 X ) = X = X ) = Exp( 750X ) 0 < X < X ) = 0 X ) = X ) = 0 X X = 0 X ) = Exp( 7X ) 0 < X < X (0) () 6 X ) = Exp( 70X ) 0 < X < X ) = 0 X ) = X ) = Exp( 0X ) X ) = 0 X ) = X ) = 0 X X = 0 0 < X < X X = 0 () (3) X ) = Exp( 8X ) 0 < X < X Proceedings 7 th ISAHP 003 Bali, Indonesia 83

6 (4) n=4 n=3 n=5 n=7 n= Fig.. Evaluation Reliability Function Step4. Aggregate group priorities to consider an evaluation quality. i n A = α X ) A C (5) j= where, A i is total priority ( i =,,, ), X j ) is jth group weight ( j =,,, n ), Aij is priority for alternatives, C j is priority for criteria, i is the nuber of alternative, j is the nuber of criterion and α is noralizing constant. 3.4 Experiental Design We perfored four experients in order to confir validity of this approach. Experient (GMM and AMM ): we use geoetric ean ethod (GMM ) and arithetic ean ethod (AMM ) for aggregating pairwise coparison judgents. In experient, we take only judgents of evaluators whose consistency ratio pairwise coparison atrix is less than 0.. Experient (GMM and AMM ): we also use geoetric ean ethod (GMM ) and arithetic ean ethod (AMM ) for aggregating pairwise coparison judgents. However, in experient, any consistency ratio would be accepted. Experient 3(WGMM and WAMM): In experient 3, we use weighted geoetric ean ethod (WGMM) and weighted arithetic ean ethod (WAMM) for aggregating pairwise coparison judgents. We use consistency ratio as a weight of individual evaluator, which is calculated by Ki s ethod (Ki and Eo, 994). Experient3 also accept any consistency ratio. Experient 4(A loss function approach): We define WAGMM as weighted after geoetric ean ethod and WAAMM as weighted after arithetic ean ethod. We use consistency ratio as a weight of evaluators. Weights are calculated by a new ethod. Experient4 also accept any consistency ratio. j ij j Proceedings 7 th ISAHP 003 Bali, Indonesia 84

7 4. Nuerical Exaple Fig. 3. Experiental Design 4. Situation et a decision hierarchy be as shown in Fig 4. It has six alternatives (A, A, A 3, A 4, A 5, A 6 ), which is copared with respect to both of two criteria C, C. It is assued that the criteria have equal weights as 0.5 respectively. et an evaluation group consist of ten evaluators E,, E0. Fig. 4. A decision hierarchy for a nuerical exaple Consistency ratios for each individual evaluator are shown in Table 3. Five evaluators fro evaluator to evaluator 6 have consistency ratio less than 0. with regard to both two criteria. Table 3. Suary of consistency ratio for each individual E E E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 0 For C For C Results We perfored four experients designed in section 3. Alt. Table 4. Priority and rank for the alternatives using the GMM and AMM Priority(GMM ) Total Priority(AMM ) Total Rank Alt C C priority C C priority Rank A A A A A A A A A A A A Proceedings 7 th ISAHP 003 Bali, Indonesia 85

8 Alt. Table 5. Priority and rank for the alternatives using the GMM and AMM Priority(GMM ) Total Priority(AMM ) Total Rank Alt C C priority C C priority Rank A A A A A A A A A A A A Table 6. Weight of each individual evaluator E E E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 0 For C For C Alt. Table 7. Priority and rank for the alternatives using the WGMM and WAMM Priority(WGMM) Total Priority(WAMM) Total Rank Alt C C priority C C priority Rank A A A A A A A A A A A A Table 8. Weight of an evaluation quality Mean Variance Expected loss, X i F ( X i ) Table 9. Priority and rank for the alternatives using the WAGMM and WAAMM Alt. Pri.(WAGMM) Total Pri.(WAAMM) Total Nor. rank Alt C C priority C C priority Nor. rank A A A A A A A A A A A A Findings The findings are classified into five categories through coparison of results. Coparison of the result is suarized in Table 0 and. GMM versus AMM: Our experiental results show that the rank of geoetric ean ethod and arithetic ean ethod coonly is not different. It is certain that the ranks of GMM and AMM are sae as entioned Aczel and Saaty (983, 986, 987). The other coparisons confired us that GMM and AMM are not different in aggregating individual judgents. Proceedings 7 th ISAHP 003 Bali, Indonesia 86

9 Application of CR versus Not Application of CR: The coparison results for GMM (AMM ) and GMM (AMM ) are little different. These results indicate that ethods by Saaty and the other ethod are not different. That is, in aggregating individual judgents to a group opinion, consistency ratio does not ipact on group priority. GMM (AMM) versus WGMM (WAMM): The ranks GMM (AMM) and WGMM (WAMM) are the sae. Individual weight by consistency ratio does not ipact on group priority. Particularly, the ore nuber of evaluator, the lower ipact on weight. GMM (AMM) versus WAGMM (WAAMM): The rank of GMM (AMM) differs fro the rank of WAGMM (WAAMM). This result indicates that a loss function approach is appropriate for aggregating individual judgents. WGMM (WAMM) versus WAGMM (WAAMM): The coparison results for WGMM (WAMM) and WAGMM (WAAMM) are different. While the weight for each evaluator is calculated in WGMM (WAMM), the weight for a group is calculated in WAGMM (WAAMM). This result indicates that the difference is occurred by a way to derive weight. It is proved that loss function approach is better ethod than others. Table 0. Coparison of geoetric ean ethods Alt. GMM GMM WGMM WAGMM priority rank Priority rank priority rank priority Rank A A A A A A Table. Coparison of arithetic ean ethods Alt. AMM AMM WAMM WAAMM priority rank Priority rank priority rank priority Rank A A A A A A Conclusion This study has developed a new ethod to derive group priority. For group decisions, we used an expected loss of a group and evaluation reliability function. As shown above, this process of proposed ethod is very siple and appropriated to derive group priority. This ethod has soe liitation. First of all, the loss function approach proposed in this study is not applied to the highest level of a hierarchy. Thus, it is necessary to develop a way to be applied to the highest level. We only proposed Evaluation Reliability Function for pairwise coparison atrix fro three diensions to seven diensions. However, it is necessary to extend it to eight diensions and nine diensions. We ade an evaluation reliability function into exponential function. It is necessary to develop various for of evaluation reliability function according to characteristic of a group. Proceedings 7 th ISAHP 003 Bali, Indonesia 87

10 References Antony, J. and Kaye, M. (000) Experiatal Quality, Kluwer Acadeic Publishers. Aczel, J. and Saaty, T.. (983) Procedures for synthesizing ratio judgents, Journal of Matheatical Psychology, Vol.7, Aczel, J. and Saaty, T.. (987) Synthesizing judgent: a functional equations approach, Journal of Matheatical Psychology, Vol.7, Aczel, J. and Saaty, T.. (986) On synthesis of judgents, Socio-Econoic Planning Sciences, Vol.0, Cho, S.H., et al (998) A study on the aggregation of ulti-experts priorities using copatibility in the AHP, Jorunal of the Korean OR/MS Society, Vol.3, No.4, Dadkah, K.M. and Zahedi, F. (993) A atheatical treatent of inconsistency in the anlaytic hierarchy process, Matheatical and Coputer Modelling, Vol.7, No.4/5, -. Finan, J.S. and Hurley, W.J. (997) The analytic hierarchy process: does adjusting a pairwise coparison atrix to iprove the consistency ratio help?, Coputers and Operations Research, Vol.4, No.8, Festervand, T.A., Bryan, K., Waller, R. and Bennie D. (000) The arketing of industrial real estate: application of Taguchi loss functions, Journal of Multi-Criteria Decision Analysis, Vol.0, 9-8. Foran, E. (990) Rando indices for incoplete pairwise coparison atrices, European Journal of Operational Research, Vol.48, Foran, E. and Peniwati, K. (998) Aggregating individual judgents and priorities with the analytic hierarchy process, European Journal of Operational Research, Vol.08, Karapetrovic, S. and Rosenbloo, E.S. (999) A quality control approach to consistency paradoxes in AHP, European Journal of Operational Research, Vol.9, Ki, S.C. and Eo, H.J. (994) Priority aggregation for AHP based on experts opinions, Jorunal of the Korean OR/MS Society, Vol.9, No.3, 4-5. Ko, K.G. and ee, K.J. (00) Statistical characteristics of response consistency paraeters in anlaytic hierarchy process, Jorunal of the Korean OR/MS Society, Vol. 6, No. 4, 7-8. Koczkodaj, W.W. (993) A new definition of consistency of pairwise coparsions, Matheatical and Coputer Modelling, Vol.8, No.7, ane, E.F. and Verdini, W.A. (989) A consistency test for AHP decision akers, Decision Sciences, Vol.0, ee, K. I. (998) The optial preferred alternatives for MNDM probles using the Taguchi's loss function, Journal of the Korean Institute of Industrial Engineers, Vol.4, No.4, Murphy, C.K. (993) iits on the analytic hierarchy process fro its consistency index, European Journal of Operational Research, Vol.65, Proceedings 7 th ISAHP 003 Bali, Indonesia 88

11 Monsuur, H. (996) An intrinsic consistency threshold for reciprocal atrices, European Journal of Operational Research, Vol.96, Raanathan, R. and Ganesh,.S. (994) Group preference aggregation ethods eployed in AHP: an evaluation and an intrinsic process for deriving ebers' weightages, European Journal of Operational Research, Vol.79, Ross, P. J. (996) Taguchi Techniques for Quality Engineering, McGraw-Hill. Saaty, T.. (983) Priority setting in coplex probles, IEEE Transactions on Engineering Manageent, Vol.30, No.3, Saaty, T.. (990) How to ake a decision: the analytic hierarchy process, European Journal of Operational Research, Vol.48, 9-6. Saaty, T.. (980) The Analytic Hierarchy Process, McGraw-Hill, New York. Saaty, T.. (995) Decision Making for eaders, RWS Publications. Saaty, T.. and Mariano, R.S. (979) Rationing energy to industries: priorities and input-output dependence, Energy Systes and Policy, Vol.3, No., 85-. Saaty, T.. and Vargas,.G. (00) Models, Methods, Concepts & Applications of the Analytic Hierarchy Process, Kluwer Acadeic Publishers. Shi, J. P. (989) Bibliographical research on the analytic hierarchy process(ahp), Socio-Econoic Planning Sciences, Vol.3, No.3, Souder, W.E. and Mandakovic, T. (986) R&D project selection odels, Research Manageent, Vol.4, No.4, Van Den Honert, R.C. and ootsa, F.A. (996) Group preference aggregation in the ultiplicative AHP; the odel of the group decision process and Pareto optiality, European Journal of Operational Research, Vol.96, Vargas,.G. (990) An overview of the analytic hierarchy process and its applications, European journal of Operational Research, Vol.48, -8. Vargas,.G. (98) Reciprocal atrices with rando coefficients, Matheatical Modelling, Vol.3, Xu, Z. (000) On consistency of the weighted geoetric ean coplex judgeent atrix in AHP, European Journal of Operational Research, Vol.6, Zahedi, F. (986) The analytic hierarchy process - a survey of the ethod and its application, INTERFACE, Vol.6, No.4, Proceedings 7 th ISAHP 003 Bali, Indonesia 89

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