Note on Deriving Weights from Pairwise Comparison Matrices in AHP

M19N38 2008/8/15 12:18 page 507 #1 Information and Management Sciences Volume 19, Number 3, pp. 507-517, 2008 Note on Deriving Weights from Pairwise Comparison Matrices in AHP S. K. Jason Chang Hsiu-li Lei National Taiwan University Oriental Institute of Technology Sung-Te Jung Oriental Institute of Technology Robert H.-J. Lin Oriental Institute of Technology Jennifer Shu-Jen Lin National Taipei Univ. of Tech. Chun-Hsiung Lan Nanhua University Yen-Chieh Yu Nanhua University Jones P. C. Chuang National Taiwan University Abstract The paper of Barzilai published in the Journal of Operational Research Society is studied. In his paper, a fundamental theorem was established to claim that the geometric mean is the only method satisfying the two proposed axioms. However, the proof contained some questionable results and is therefore amended in our report. Keywords: Analytic Hierarchy Process AHP), Normalization, Geometric Mean, Relative Weight, Comparison Matrices. 1. Introduction Since the funding researcher, Thomas Saaty [19], developed the Analytic Hierarchy Process AHP), it becomes a useful tool for estimating judgment elements by quantifying Received May 2006; Revised November 2006; Accepted March 2007. Supported in part by OIT grant and by the National Science Council, Taiwan, ROC, under NSC 95-2416-H-015-002.

M19N38 2008/8/15 12:18 page 508 #2 508 Information and Management Sciences, Vol. 19, No. 3, September, 2008 subjective decisions. It is used to derive relative weights of decision elements and then synthesize them to obtain the corresponding weights for alternatives and criteria. It is a building block for decision making. There are many papers applied AHP to solve decision problem. For example, Zanakis et al. [28] studied over 100 applications of AHP in the service and government sectors. On the other hand, other researchers provided different approach to enhance the theoretical background of AHP to refine its analytical derivation. One open question in AHP is how to derive the relative weights from a comparison matrix. The majority practitioner agreed to use the eigenvector method proposed by Saaty [19]. However, some researchers suggested other choice, for example, Belton and Gear [6 7] developed a refined method to adjust the maximum entry being one for the weight of alternatives; Barzilai [4] recommended the geometric mean method to instead of the eigenvector method. There are 23 papers [1 3, 5, 8 18, 20 27] that have quoted Barzilai s paper [4]. However, there are some questionable results did not point out by those 23 papers such that none of them tried to revise Barzilai s paper. In this note, we will discuss four questionable results in the paper of Barzilai [4] as follows 1) The componentwise operation of matrices may lead unreasonable results. 2) The first improvement of the fundamental theorem of Barzilai [4]. 3) The second revision for the fundamental theorem of Barzilai [4]. 4) The numerical examples provided by Barzilai have calculation errors such that his conclusion is inadequate to criticize the eigenvector method. 2. The Componentwise Operation of Matrices In Barzilai [4], he used the matrix operations that apply componentwise. It means that two matrices, A = a ij m n and B = b ij m n then we execute A B = C where C = c ij m n, with c ij = a ij b ij. The componentwise operation is an elegant algebraic operation such that it preserves the beautiful group homomorphism structure. However, in the next, we will prepare an example to demonstrate that this kind of componentwise operation is questionable in the practical sense. For example, we faced a decision problem, under criterion C, to attain a goal, by two possible alternatives, A 1 and A 2. The goal is to improve the efficiency of traffic between two cities, under the criterion, C, the budget and the two alternatives: A 1 is to build a railroad and A 2 is to construct a highway. If the relative weights for two alternatives are A 1 = 2 3 and A 2 = 1 3, then

M19N38 2008/8/15 12:18 page 509 #3 Note on Deriving Weights from Pairwise Comparison Matrices in AHP 509 A) For the best alternative problem, we select A 1 to build a railroad, and B) For the allocation problem, we spend two-thirds the budget to build a railroad, and one-third the budget to construct a highway. If there are three decision makers, namely Robert, Ronald; Vern, Paulsen and Dana, Ter and they have the unanimous [ opinion ] for alternatives A 1 and A 2 for criteria C with 1 2 the following comparison matrix. 1/2 1 For simplifying the future computation, we denoted the comparison matrices for Robert, Ronald as M R, for Vern, Paulsen as M P and for Dana, Ter as M T, respectively. Hence, it yields [ ] 1 2 M R = = M P = M T. 1) 1/2 1 We may say that the relative ratio between A 1 and A 2 as A 1 = 2 such that the A 2 2 normalized relative weights for A 1 and A 2 is 3, 1 by the unanimous decision of Ronald, 3) Paulsen and Ter. [ ] 1 8 If we perform the componentwise operation, then M R M P M T =. 1/8 1 Hence, after the componentwise operation, the normalized relative weights for A 1 and A 2 8 is 9, 1. It may indicate that the componentwise operation is a questionable operation 9) without practical use. The proper way to synthesis the group decision for Ronald, Paulsen and Ter, is followed the rule of the conventional AHP with the relative weights for these three decision makers are w 1,w 2,w 3 ) with w 1 +w 2 +w 3 = 1 such that the relative weight for alternative A 1 is computer as 2 3 w 1 + 2 3 w 2 + 2 3 w 3 = 2 3 w 1 + w 2 + w 3 ) = 2 and then the synthesized 3 2 weights are 3, 1 as we expected by the unanimous decision of Ronald, Paulsen and 3) Ter. Our discussion may provide an open question to debate. If those researchers can not provide some appropriate explanation to clarify what situation the componentwise operation is suitable, then we may claim that a) Theorem 2 of Barzilai [4] for the independent of scale-inversion, b) Theorem 3 of Barzilai [4] for the homomorphism of group structure, c) Theorem 4 of Barzilai [4] for the measure of estimation of the additive pairwise comparison matrix, and

M19N38 2008/8/15 12:18 page 510 #4 510 Information and Management Sciences, Vol. 19, No. 3, September, 2008 d) Theorem 5 of Barzilai [4] for the measure of estimation of the multiplicative pairwise comparison matrix, all contained questionable operations. 3. Discuss of the Fundamental Theorem of Barzilai We begin to consider the fundamental theorem Theorem 1) of Barzilai [4]. He tried to solve the problem to find a special mapping f from A to w where A is the set of all comparison matrix and w is the set of all multiplicative weight vectors as w = w k ) with w k > 0, w k = 1. k=1 We first recall the two proposed axioms in his paper. Axiom 1 If A is a consistent comparison matrix with an underlying multiplicative weight vector w where w = w i ) 1 n and A = a ij ) n n with a ij = w i /w j then the solution is the vector w, that is fa) = w. Axiom 2 The weight w i attributed to alternative i is independent of relative measurements among alternatives other than i. With the restriction in Axiom 1, Barzilai [4] derived that n ) 1/n n ) w 1/n i a ij = = w i ) n/ n w j w j ) 1/n = w i. 2) He claimed that the geometric mean satisfies Axiom 1. Next, he admitted that the geometric mean is not the only solution satisfying Axiom 1. He predicted that the Fundamental Theorem Theorem 1) will establish that the geometric mean is the only procedure that satisfies both axioms. We quote the fundamental theorem of Barzilai [4] as the next theorem. Theorem 1 of Barzilai [4]). There is exactly one mapping w = fa) satisfying ) 1/n. Axioms 1 and 2, namely, the geometric mean, given by w i = a ij We begin to recall his proof. For a given pairwise comparison matrix A = a ij, and a fixed k with 1 k n. Barzilai [4] defined a matrix B = b ij by b ij = a ik a kj for 1 i, j n. Note that b kj = a kj and b ik = a ik for 1 i, j n and for all 1 i, j n, b ij = u i u j, where u i = 1/a ki. He knew that B is consistent. By Axiom 1, it implies

M19N38 2008/8/15 12:18 page 511 #5 Note on Deriving Weights from Pairwise Comparison Matrices in AHP 511 fb) = u where the vector u is given by n ) 1/n u i = b ij 3) for 1 i n. In particular, he obtained that n ) 1/n n ) 1/n u k = b kj = a kj. 4) Since b kj = a kj and b ik = a ik for 1 i, j n, by Axiom 2, w k = u k, he combined previous results to derive that n ) 1/n w k = a kj 5) which completes the proof since k can assume any value between 1 and n. 4. Improvement for Proof of Theorem 1 of Barzilai In this section, for the time being, we assume that Equation 3) is the desired results that will be pursued for the proof of Theorem 1. In the next section, we will point out that Equation 3) contains questionable results even after our improvement. Here, we point out the questionable results in his proof. We find that n ) 1/n n ) u 1/n i u i b ij = = u j ) 1/n. 6) u j ) 1/n, Hence, his result in Equation 3), say u i = b ij is valid if and only if It means that n ) 1/n u j = 1. 7) n ) 1 = u j = 1 a kj = a jk. 8) Equation 8) holds if and only if the kth column of A is consistent of a multiplicative weight vector. However, A is an arbitrary comparison matrix and k is a fixed number with 1 k n such that Barzilai [4] wanted that a jk = 1 9)

M19N38 2008/8/15 12:18 page 512 #6 512 Information and Management Sciences, Vol. 19, No. 3, September, 2008 is false. We will modify his proof to change u i from u i = 1/a ki to u i = ) 1/n a kj From our modification of Equation 10), we can derive that b ij = u i u j = a ki. 10) ) 1/n a ks /a ki s=1 s=1 a ks ) 1/n / a kj = a ik a kj. 11) According to Equation 11), we obtain the same new comparison matrix as Barzilai [4]. Moreover, we find that n ) 1/n u j = ) 1/n a ks s=1 a kj 1/n ) 1/n a ks s=1 = ) 1/n = 1. 12) a kj By Equation 12), our new definition of u i then the criterion of Equation 7) is ) 1/n satisfied. Therefore, the desire results of u i = b ij is preserved by us. 5. Revision for the Theorem 1 of Barzilai Let us examine Equation 3) again, even after our revision to define u i according to Equation 10). The derivation of Equation 3) still looks like with strong relations to the geometric mean. By the way, Barzilai [4] also mentioned that the geometric mean is not the only method that satisfies Axiom 1. Consequently, we will try to revise the proof of the fundamental theorem by another approach, namely, the right eigenvector method of Saaty [19]. If w is a multiplicative weight vector with w = w k ) and w k > 0, w k = 1 and A is the perfectly consistent comparison matrix with the underlying multiplicative weight vector w with a ij = w i /w j then by Axiom 1, fa) = w. We know that w 1 /w 1 w 1 /w 2 w 1 /w n w 1 w 1 w 2 /w 1 w 2 /w 2 w 2 /w n w 2 w 2 Aw = = n. 13)...... w n /w 1 w n /w 2 w n /w n w n w n k=1

M19N38 2008/8/15 12:18 page 513 #7 Note on Deriving Weights from Pairwise Comparison Matrices in AHP 513 Our approach is to accept that when A is a perfectly consistent comparison matrix then the following statement fa) = the right eigenvector corresponding to the maximum eigenvalue such that the multiplication of all entries of the right eigenvector is one is valid. For an arbitrary comparison matrix, say C = c ij ), for example, we will use the kth row, say R k = c k1,c k2,...,c kn ), to construct a new comparison matrix, say D = d ij ), with d ij = c kj c ki such that D is perfectly consistent. From c k1 /c k1 c k2 /c k1 c kn /c k1 1/c k1 1/c k1 c k1 /c k2 c k2 /c k2 c kn /c k2 1/c k2 1/c k2 = n. 14)...... c k1 /c kn c k2 /c kn c kn /c kn 1/c kn 1/c kn it yields that fd) = θ[1/c k1 1/c k2 1/c kn ] T with θ n = 1. It yields c kj n ) 1/n θ = c kj. 15) Since the kth row and the kth column of C and D are all the same, by Axiom 2, the kth component of fc) and fd) are the same. Using Equation 15), It implies that the kth component of fc) is θ n ) 1/n = θ = c kj. 16) c kk We finish the proof for the fundamental theorem of Barzilai [4] from our approach by the right eigenvector method of Saaty [19]. Our approach provides a substitutive approach that is no relation to the geometric mean. It avoids the shortcomings in Equation 3) as in advance predicted the desired results and then verified them. 6. The Revision of the Numerical Examples of Barzilai [4] Next, we begin to analyze the numerical examples of Barzilai [4]. We quote his

M19N38 2008/8/15 12:18 page 514 #8 514 Information and Management Sciences, Vol. 19, No. 3, September, 2008 results in detail for further explanation. He assumed 1 1/7 1/2 1/8 2 1 7 2 8 1/2 7 1 3 1 8 1/7 1 1/3 1 1/8 X = 2 1/3 1 1/4 5 and Y = 1/2 3 1 4 1/5 8 1 4 1 5 1/8 1 1/4 1 1/5 1/2 1/8 1/5 1/5 1 2 8 5 5 1 17) where Y is the scale inversion of X. He found that the right eigenvector of X, say ex), normalized multiplicatively, is given by ex) = 0.441, 2.696, 0.966, 2.786, 0.313) 18) which ranks the alternatives 4, 2, 3, 1, 5) and while ey ) = 2.197, 0.348, 1.032, 0.373, 3.397) 19) corresponding to the ranking 2,4,3,1,5). Barzilai [4] noted that the ranking of alternatives 2 and 4 is reversed and that 1 ey ) = 0.455,2.873,0.969, 2.680,0.294) 20) while the geometric mean solutions are given by 1 gy ) = gx) = 0.447,2.873,0.964,2.759, 0.302) 21) which consistently correspond to the ranking 2,4,3,1,5) for both X and Y. We quote his assertion and, in terms of row dominance, according to the right eigenvector method 8,1,4,1,5) dominates 7,1,3,1,8), and at the same time, 1 8,1,4,1,5) dominates 1 7, 1, 3, 1, 8) a clear contradiction. Interestingly, both examples provided by Saaty 1990) display the same rank reversals. By Equation 19), Barzilai 1997) derived the ranking 2,4,3,1,5) that is false. According to Equation 20), the corrected ranking should be 4,1,3,2,5) such that his assertion that the rank reversal of alternatives 2 and 4 happens. Moreover, from Equation 21), the ranking corresponding to geometric mean method for X should be 4, 1, 3, 2, 5) such that his results of 1 ey ), gx) and 1 gy ) have the same ranking still valid. For the row dominance, Barzilai 1997) claimed that in ex) and 1 ey ), alternative 4 is the top rank. However, for 1 ey ), the top rank is alternative 2 such

M19N38 2008/8/15 12:18 page 515 #9 Note on Deriving Weights from Pairwise Comparison Matrices in AHP 515 that his assertion about Saaty s eigenvector method is based on false result. Hence, we advise practitioners do not adopt his opinion of the row dominance to criticize Saaty s eigenvector method. 7. Conclusion We provide a patchwork to improve the verification for fundamental theorem of Barzilai to show that the geometric mean is the only method satisfying the two Axioms proposed by him. Our work may help researchers to accept the geometric mean method is an alternative approach to apply in the Analytic Hierarchy Process. Acknowledgements The authors are grateful to the referees for their constructive comments that lead to important improvements in this manuscript. In addition, the authors wish to express appreciations to Ms. Dana Ter for her assistance in English revisions. References [1] Aguarón, J., Escobar, M. T. and Moreno-Jiménez, J. M., Consistency stability intervals for a judgement in AHP decision support systems, European Journal of Operational Research, Vol. 145, No. 2, pp.382-393, 2003. [2] Aguarón, J. and Moreno-Jiménez, J. M., The geometric consistency index: Approximated thresholds, European Journal of Operational Research, Vol. 147, No. 1, pp.137-145, 2003. [3] Aguarón, J. and Moreno-Jiménez, J. M., Local stability intervals in the analytic hierarchy process, European Journal of Operational Research, Vol. 125, No. 1, pp.113-132, 2000. [4] Barzilai J, Deriving weights from pairwise comparison matrices, Journal of the Operational Research Society, Vol. 48, pp.1226-1232, 1997. [5] Barzilai, J., On the decomposition of value functions, Operations Research Letters, Vol. 22, No. 4-5, pp.159-170, 1998. [6] Belton, V. and Gear, T., On a shortcoming of Saaty s method of analytic hierarchies, Omega, Vol. 11, No. 3, pp.228-230, 1983. [7] Belton, V. and Gear, T., Feedback: the legitimacy of rank reversal- a comment, Omega, Vol. 13, No. 3, pp.143-144, 1985. [8] Brugha, C. M., Relative measurement and the power function, European Journal of Operational Research, Vol. 121, No. 3, pp.627-640, 2000. [9] Buckley, J. J., Feuring, T. and Hayashi, Y., Fuzzy hierarchical analysis revisited, European Journal of Operational Research, Vol. 129, No. 1, pp.48-64, 2001. [10] Choo, E. U., and Wedley, W. C., A common framework for deriving preference values from pairwise comparison matrices, Computers and Operations Research, Vol. 31, No. 6, pp.893-908.2004. [11] Dembczyński, K., Greco, S. and S lowiński, R., Methodology of rough-set-based classification and sorting with hierarchical structure of attributes and criteria, Control and Cybernetics, Vol. 31, No. 4, pp.891-920, 2002. [12] Escobar, M. T. and Moreno-Jiménez, J. M., Reciprocal distributions in the analytic hierarchy process, European Journal of Operational Research, Vol. 123, No. 1, pp.154-174, 2000.

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M19N38 2008/8/15 12:18 page 517 #11 Note on Deriving Weights from Pairwise Comparison Matrices in AHP 517 Sung-Te Jung is currently a lecturer in the Department of Communication Engineering, Oriental Institute of Technology, Taiwan, His research interests are operations research, geometry and applied mathematics. Department of Communication Engineering, Oriental Institute of Technology, Taiwan, E-mail: fb012@mail.oit.edu.tw Robert Huang-Jing Lin is currently a lecturer in the Department of Information Management, Oriental Institute of Technology, Taiwan, His research interests are operations research, statistics and applied mathematics. Department of Information Management, Oriental Institute of Technology, Taiwan, E-mail: hjlin@mail.oit.edu.tw Jennifer Shu-Jen Lin is currently an assistant professor of the Institute of Technological & Vocational Education, National Taipei University of Technology, Her research interests are inventory systems, analytic hierarchy process and Lanchester s equations. Institute of Technological & Vocational Education, National Taipei University of Technology, E-mail: jennifer1592001@seed.net.tw Chun-Hsiung Lan is currently a professor and chairman of the Institute of Management Sciences, Nanhua University, Taiwan, His research interests are operations research, production design and control, intelligent computing and material strategy. Institute of Management Sciences, Nanhua University, Taiwan, E-mail: chlan@mail.nhu.edu.tw Yen-Chieh Yu is currently a PhD student in the Institute of Management Sciences, Nanhua University, Taiwan, His research interests are management sciences and business administration. Institute of Management Sciences, Nanhua University, Taiwan, E-mail: ycyu6886@yahoo.com.tw Jones P.C. Chuang is currently a PhD student in the Department of Civil Engineering, National Taiwan University, Taiwan, His research interests are transportation, traffic models and management sciences. Department of Civil Engineering, National Taiwan University, Taiwan, E-mail: una050@mail.cpu.edu.tw