Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation

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1 Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation ISSN Report of the Research Institute of Industrial Technology, Nihon University Number 77, 2005 Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation Kazutomo NISHIZAWA* ( Received November 6, 2004 ) Abstract This paper proposes a method to estimate unknown paired comparisons and compensation in incomplete AHP. The typical methods in incomplete AHP are Harker method and Two-stage method. In Harker method, however, weights are calculated without estimate unknown comparisons. In Two-Stage method, estimation for unknown comparisons is carried out, but the priority of known comparisons and estimated comparisons are treated with equal importance. As a result, some undesirable problems occurred. In this study, unknown comparisons are estimated based on repeating geometric mean and compensated by difficulty of estimation based on consistent binary AHP. Keyword: Incomplete AHP, Unknown comparisons, Estimation, Compensation 1. Introduction AHP (Analytic Hierarchy Process) 1) is a useful tool for decision makers in various fields. However, in some real problems, it is impossible or difficult to have comparisons of some pairs of alternatives. Let us call such cases incomplete AHP. It is very important to estimate incomplete comparisons data to have alternative s weights. In this paper, an estimation method in incomplete information AHP, and a compensation method are proposed and apply proposed methods to the ranking estimation in sports field. In tournament games, for example of baseball, the rankings of two teams fighting in the final competition are clear; the winning team is first and the defeated team is second, however ranking of other teams are unknown. Especially it is not clear the ranking of the team defeated in the first game. We can estimate the ranking of all teams by proposed methods. The typical methods in incomplete AHP are Two- Stage method 2), 3) and Harker method 4). Unknown comparisons data, in incomplete comparisons, are estimated by Two-Stage method. However the results, if Two-Stage method is applied to the ranking estimation of tournament games, are unstable with increasing the number of participant teams. In Harker method, obtained results are stable even if the participant teams are increasing. It directly gives weights of alternatives without estimation of comparisons data. Therefore, an estimation * Associate Professor, Department of Mathematical Information Engineering, College of Industrial Technology, Nihon University

2 Kazutomo NISHIZAWA method, improving Two-Stage method, is proposed. Furthermore, in both typical methods, the priority of known comparisons and estimated comparisons are equally important. This causes some-what undesiable problems. We need to make clear the distinction of the priority between known comparisons and estimated comparisons. A compensation method for estimated comparisons is also proposed based on perfect consistent binary AHP. In this paper, our estimation method, based on geometric mean, is described in section 2. In section 3, the compensation method is described, and in section 4, illustrates two examples by proposed methods. One is four teams tournament and other is 49 teams. Finally conclude this investigation in section Estimation In this section, an estimation method in incomplete information case, based on Two-Stage method, is proposed. In complete AHP of n alternatives, we denote the element of comparison matrix A by a ij for i, j = 1 ~ n. And we also denote obtained weight by w i for i = 1 ~ n. In perfect consistent case, we have the following relation between w i and w j, for i, j = 1 ~ n. a ij = w i / w j (1) So for any k, we also have the relation a ij = a kj / a ki (2) then assume a kj / a ki = 1. The estimated a ij by (3) is treated as known comparison, and the a ij with m = 0 in (3) is treated as unknown in the next level. Repeat above procedure until unknown elements completely estimated. 3. Compensation In this section, the compensation method for estimated complete comparison matrix is proposed. Assume we have estimated complete comparison matrix by repeating l time estimation we need (l + 1)-level compensation. To make clear the distinction between known comparisons data and estimated comparisons, we introduce compensation weight p k for k = 0 ~ l. If estimated comparison a ij > 1 (< 1) then we compensate by a ij p k (a ij / p k ). There are various methods to decide the value of p k. In this study, we have p k based on consistent binary AHP. Then for (l + 1)-level compensation we introduce (l + 1) (l + 1) consistent binary matrix B and denote the element of B by b ij, where b ij = α (i < j), b ji = 1 / b ij, b ii = 1 (i, j = 0 ~ l) and α (> 1) is a parameter. For example l = 2 (three-level compensation), consistent comparison matrix B is as follows: 1 α α B = 1/α 1 α. (4) 1/α 1/α 1 From B, by using geometric mean, we have compensation weights p k for k = 0 ~ l, as follows: based on above. On the other hand, assume a ij is unknown comparison in incomplete AHP. There are various methods to estimate value of a ij based on known comparisons, for example arithmetic mean, harmonic mean and so on. In this study we propose the following estimation method by using geometric mean, for unknown a ij : p k = α (l - 2k) / (l + 1). (5) Note that if compensation weight p k < 1 then result of compensation a ij p k < 1 even if estimated element a ij > 1. The reversal of judge often occurs by compensation. Therefore normalizing p k with p l = 1, shown as below: a ij = ( n Π k = 1 a kj / a ki ) 1/m, (3) where m is the number of known a kj / a ki, k = 1 ~ n. If unknown comparisons in factors of a kj / a ki are included, p k = α 2 (l - k) / (l + 1). (6) Furthermore we introduce x (= p 0 /p l ), the ratio of maximum compensation weight and minimum compensation weight.

3 Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation For (l + 1)-levels compensation of x, the value of parameter α calculates by (7). α = x (l + 1) / 2l (7) From (6) and (7), as a result, for k = 0 ~ l, we have p k from x and k without α. Table 1 The ranking in Example-1 team ranking Team 1 1 Team 4 2 Team 2? Team 3? 4. Example p k = x (l - k) / l (8) In this section, to explain proposed methods, two examples are illustrated with ranking estimation of tournament games. Example-1 is estimation of ranking among four teams in tournament, and Example-2 is that for 49 teams. The value of a ij, the element of comparison matrix A, is competition result of between team i and team j, shown as below: a ij = θ fij, (9) where f ij = (score of team i - score of team j) / (score of team i + score of team j) and θ (> 1) is a parameter. Usually we use θ = 2 in calculation Example-1 At first, illustrate ranking estimation with four-team tournament. The result of Example-1 is shown in Fig. 1 and Table 1. Assume each competition is determined by 1 to 0. As a result the champion team is Team 1 and Team 4 is the second, however ranking of Team 2 and Team 3 are not clear. Based on (9), incomplete comparison matrix A 0 of this example, is shown in (10). A 0 = 1 θ () θ 1/θ 1 () () () () 1 1/θ 1/θ () θ 1 (10) For example, Team 1 defeats Team 2 by 1 to 0, then a 012 = θ 1, and a 021 = θ -1. That is if team i defeats j then a 0ij = θ and a 0ji =1/θ, and unknown by ( ). Based on (3), we can estimate unknown elements in (10), and have estimated comparison matrix A, shown as (11). A 0 = 1 θ θ 2 θ 1/θ 1 θ 1 1/θ 2 1/θ 1 1/θ 1/θ 1 θ 1 (11) And next, from (11), we calculate the principle eigen vector W by power method with θ = 2. The convergence limit in this study is The result is shown as (12). W = (12) On the other hand, estimated matrix from (10) by Harker method is easily obtained. The result is shown as below: Team 1 Team 2 Team 3 Team Team 1 0 Team 4 Team 1 A Harker = 2 θ 0 θ 1/θ /θ 1/θ 0 θ 2. (13) Fig. 1 The result of tournament in Example-1

4 Kazutomo NISHIZAWA And the resulting weight vector W Harker is obtained by power method as follows: W Harker = (12) The values of W coincide with W Harker except the computational error. In (12), obtained by proposed method, the value of w 2 is equal to w 4 then Team 2 and Team 4 are regarded equally strong. It does not coincide with Fig. 1 and Table 1. But this result is natural, because second row, in (11), coincide with fourth row. However second row include two estimated elements and fourth row include one estimated element. Therefore, we can consider Team 4 stronger than Team 2. Now, compensate (11) by proposed method. Corresponding estimation number of upper triangular element is shown in (15), where we denote known comparison by 0, first estimation by 1 and second estimation by 2. In (15), lower triangular elements are symmetric. * * 2 1 * 0 * (15) This example needs three-level compensation. Based on (15), each element of estimated comparison matrix is compensated by below p k, where p 2 = 1. 1 θ p 0 θ 2 p 1 θ p 0 A = 1/(θ p 0 ) 1 θ p 2 1 1/(θ 2 p 1 )1/(θ p 2 ) 1 1/(θ p 0 ) 1/(θ p 0 ) 1 θ p 0 1. (17) As a result, in this example, typical values of α and corresponding p k (k = 0 ~ 2) and recalculated w i (i = 1 ~ 4) from (17) are obtained in Table 4. For various values of α, in Table 4, the value of w 2 is not equal to w 4 and there are no changes in ranking. Table 2 The value of x and α on three-level x α Table 3 The value of α and x on three-level α x * p 0 p 1 p 0 * p 2 p 1 (16) * p 0 * To calculate p k, based on (8), typical ratio of x and corresponding value of α in compensation binary AHP is shown in Table 2. In general we usually used α = 2 in binary AHP. In this case, corresponding value of x is approximately 2.5. Compensated comparison matrix, based on (11) and (16), as follows: Table 4 Compensation weights in Example-1 α p p p w w w w

5 Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation The result of estimated ranking by proposed compensation method is shown in Table 5. This result not differs from Fig. 1. However, we do not know suitable value of x. Furthermore, the typical value of α, x and corresponding CI (Consistency Index) in this compensation example are shown in Table 6. Table 5 The result ranking by proposed methods in Example-1 Increasing value of x, in Table 6, the value of CI becomes larger. The case of x > 3, in this example, CI > 0.1 then compensated comparison matrix becomes inconsistent Example-2 Estimated ranking team 1 Team 1 2 Team 4 3 Team 2 4 Team 3 Table 6 The value of CI in Example-1 α x CI Next example is ranking estimation of 49 teams (T0 ~ T48) in tournament games. All competitions of 49 teams are 1176 games in round robin. On the other hand, the real competitions are 48 games in this tournament. Then the ratio of non-competition is 95.91%. The main result of this tournament is shows in Table 7. In Table 7, we do not know the ranking of all teams, except T40 and T31. We can easily construct incomplete comparison matrix A 0. For example, Team i won Team j by 5 to 2, then a0ij = θ 3/7, and a0ji = θ -3/7. Based on proposed estima- Table 7 The result of tournament in Example-2 team result T40 The champion team T31 Defeated in the final T13 Defeated in the semifinal T39 Defeated in the semifinal T20 Defeated in the quarterfinal T35 Defeated in the quarterfinal T37 Defeated in the quarterfinal T38 Defeated in the quarterfinal T8 T15 T17 T21 T22 T23 T30 T34 T0 T2 T3 T5 T9 T10 T11 T14 T16 T24 T26 T27 T41 T43 T45 T46 T1 T4 T6 T7 T12 T18 T19 T25 T28 T29 T32 T33 T36 T42 T44 T47 T48

6 Kazutomo NISHIZAWA Table 8 The result of estimation before compensation in Example-2 estimated k ranking team T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T tion method, in (3), the result of estimation before compensation is shown in Table 8. For example in Table 8, team T32 played one real competition (k = 0) and estimated ranking is 47. For non-competition 48 games, in this team, estimated one competition at first estimation (k = 1), six at second (k = 2), 38 at third (k = 3) and two at fourth (k = 4) estimation. In this example, non-competition games are completely estimated by four times estimation (l = 4) then we need five-level compensation. Typical ratio of x and corresponding value of α in compensation binary AHP is shown in Table 9. Based on (6) and (7), the value of α and compensation weights and value of p 0 to p 4, are obtained. The result, corresponding x on 2 to 5 are shown in Table 10. The results of compensation for x = 2 to 5 are shown in Table 11. For higher ranked teams and lower ranked teams, in Table 11, there are almost no ranking changes in this example. However, increasing x, the ranking of team T36 was down. This team defeat in the first game. Table 9 The value of x and α on five-level x α Table 10 Compensation weights in Example-2 α p p p p p

7 Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation Table 11 The result of ranking by proposed methods in Example-2 estimated x ranking T40 T40 T40 T40 2 T8 T8 T8 T8 3 T36 T31 T31 T31 4 T31 T36 T36 T2 5 T2 T2 T2 T36 6 T35 T35 T35 T35 7 T1 T1 T1 T1 8 T39 T39 T39 T39 9 T16 T16 T16 T16 10 T33 T30 T30 T30 11 T11 T11 T11 T11 12 T30 T33 T33 T33 13 T15 T15 T15 T15 14 T6 T6 T6 T6 15 T20 T20 T20 T20 16 T41 T24 T24 T13 17 T24 T41 T13 T24 18 T22 T13 T41 T41 19 T13 T22 T22 T37 20 T37 T37 T37 T22 21 T29 T38 T38 T38 22 T38 T29 T23 T23 23 T23 T23 T29 T29 24 T45 T45 T45 T48 25 T28 T28 T28 T28 26 T48 T48 T48 T45 27 T46 T46 T46 T46 28 T10 T3 T3 T3 29 T3 T10 T10 T10 30 T4 T4 T4 T34 31 T21 T34 T34 T21 32 T34 T21 T21 T4 33 T19 T19 T19 T9 34 T9 T9 T9 T19 35 T17 T17 T17 T17 36 T7 T7 T7 T7 37 T43 T43 T43 T43 38 T14 T14 T14 T14 39 T12 T27 T27 T27 40 T42 T12 T12 T12 41 T27 T42 T42 T42 42 T25 T25 T25 T25 43 T44 T44 T44 T44 44 T5 T5 T5 T5 45 T26 T26 T26 T26 46 T47 T47 T47 T47 47 T32 T32 T32 T32 48 T0 T0 T0 T0 49 T18 T18 T18 T18 Table12 The value of CI corresponding x in Example-2 x CI The relation between x and CI, in Example-2, shows in Table 12. As similar Example-1, increasing the value of x, the value of CI becomes larger. In this example the case of x > 4 then CI > Conclusion In this paper, the methods to estimate unknown comparisons and to compensate in incomplete AHP were proposed and applied to ranking estimation in tournament games. The compensation weights were calculated by consistent binary AHP based on the ratio of maximum compensation weight and minimum compensation weight. In this example, however, there is no necessity too large value of the ratio x (= p 0 /p l ). In Example-1, the same rank teams, obtained by Harker method, could be given reasonable rank by our method. In Example-2, for large amount of unknown comparison case, unreasonable results, for instance the champion team ranked low, often occurred by Two- Stage method, however, reasonable results obtained by our method. Considering these results, we can insist that our proposed methods are superior to Harker method and Two- Stage method. In future, we need more discussion for inconsistent and incomplete AHP. Acnowledgement This work was supported by Nihon University Individual Research Grant (2003).

8 Kazutomo NISHIZAWA References 1) Saaty, T. L. : The Analytic Hierarchy Process, (McGraw-Hill, New York, 1980). 2) Takahashi, I. : AHP Applied to Binary and Ternary Comparisons, Journal of Operations Research Society of Japan, Vol. 33, No. 3, (1990) ) Takahashi, I and M. Fukuda : Comparisons of AHP with other methods in binary paired comparisons, Proceedings of the Second Conference of the Association of Asian-Pacific Operational Research Societies within IFORS, (1991) ) Harker P. T : Incomplete Pairwise Comparisons in the Analytic Hierarchy Process. Math. modeling, Vol.9, (1987)

9 Estimation of Unknown Comparisons in Incomplete AHP and It s Compensation

10 Kazutomo NISHIZAWA Biographical Sketches of the Author Kazutomo Nishizawa is associate Professor of college of Industrial Technology, Nihon University. He was born in Nagano, Japan on January 1, He received his degree of B. eng. In 1977, M. Eng. in 1979, and Dr. Eng. in 1997 from Nihon University. He is interested in AHP, ANP, graph theory and quantification theory. He is a member of The Operations Research Society of Japan (ORSJ), Information Processing Society of Japan (IPSJ), and The Japan Society of Mechanical Engineers (JSME) and so on.