This paper describes a logarithmic least squares method or a geometric mean method for estimating the relative weight of alternatives when some entries of the pairwise comparisons matrix are missing

TONE Kaoru

This paper describes a logarithmic least squares method or a geometric mean method for estimating the relative weight of alternatives when some entries of the pairwise comparisons matrix are missing.departmental bulletin pape

TONE, Kaoru

Insitutional Repository at the National Graduate Institute for Policy Studies

A Logarithmic Least Squares Method for Incomplete Pairwise Comparisons in the Analytic Hierarchy Process Kaorn TONE' December 20, 1993 Abslrnct This paper describes a. logadtl1mic lea.st squares metl1od or a. geo-mettic lneari method for cslimallng the relative ~eight or alletnalives when some'entrics of the pairwise compaliso11s ntatrix a.re missing, 94-B-2 94-B-2 A Logarithmic Least Squares Method for Incomplete Pairwise Comparisons in the Analytic Hierarchy Process Kaoru TONE* December 20, 1993 Abstract This paper describes a logarithmic least squares method or a geo-metric mean method for estimating the relative weight of alternatives when some entries of the pairwise comparisons matrix are missing. 1 Introduction Saaty's Analytic Hierarchy Process (AHP) [3] is now being widely used for decision making purposes. At each level in the hierarchy, the AHP uses a pairwise comparisons matrix A = (a;;) E Rnxn representing the relative importance of alternatives i against j, with the properties: (1) a;;= 1(i=1, ... ,n), a;;= l/a;; (V(i,j)), a;;> 0 (V(i,j)). There are two frequently used methods for estimating the relative weight of n alternatives from the matrix A. One is the eigenvalue method (EM) which *Graduate School of Policy Science, Saitama University, Urawa, Sa.itama 338, Japan. e-ma.il:tone@poli-sci.sa.itama-u.ac.jp 1 solves the principal eigenvector v = ( v;) E Rn in Av = AmaxV with Amax the principal eigenvalue of A. The other is the geometric mean method (GM) or the logarithmic least squares method (LLSM) which minimizes, with respect tog= (g;) E Rn, n (2) I: (log a;; - log g;f g; )2 • i,j=1 It turns out that the GM (LLSM) solution g is given by the normalized geometric mean of elements in each row: (3) qIT'J=i a;; . g; = . (i = 1, ... ,n) Ei:=1 \JIT'l=1 ak; The two methods are not the same and the solutions v and g are different generally. However, it was showed that the two approaches give almost the same weight v and g, if the matrix A is nearly consistent. (See Golden and Wang [1] and Tone [5]. See Saaty [4], too, for further discussions.) One major drawback of the AHP is that at each level in the hierarchy, n(n - 1)/2 questions must be answered. The number of questions grows very large with n. In addition, for certain pairs (i,j), it is very difficult to answer the question "compare i against j". This results in some entries of A being missing. Therefore, methods for estimating the weight of alternatives from the incomplete matrix are requested. Harker [2] solved this problem effectively in the framework of the eigenvalue method. The main purpose of this paper addresses the solution to the incomplete matrix problem by the logarithmic least squares principle. 2 2 Logarithmic Least Squares for Incomplete Pairwise Comparisons Matrix We can define an undirected graph corresponding to the paired comparisons with the vertices 1, 2, ... , n and with arcs ( i, j) if i and j are compared directly. Definition 1 We call a pairwise comparisons matrix incomplete, if 1. the corresponding graph is connected and 2. is not a perfect graph. Let an incomplete matrix be A= (a;;). For each vertex i, we define P; as the set of vertices adjacent to i and N; as the degree of i , i.e., the number of arcs connected to i. Since the graph is connected, for each i, P; is nonempty and N; <:: 1. For the missing matrix entries a;;, let us approximate their value by the ratio of the (yet unknown) weights g;f g;. For the purpose of obtaining the weight g, we solve the following logarithmic least squares problem: (4) minimize (5) The problem results in a set of linear equations in (log g;) as (6) N;logg;-°L:logg;= "L:loga;;. (i=l,2, ... ,n) jEPi jEPi 3 Example 1 The matrix below has entries (1, 3), (2, 4) and (3, 4) missing. The corresponding linear equations are ( 2 -1 0 -1 ) ( log 91 ) -1 2 -1 0 logg2 0 -1 1 0 logg3 -1 0 0 1 logg4 ( log a12a14 ) = log a21 a23 . log a32 loga41 The rule for constructing the coefficient matrix of the linear equations is: (1) Put -1 on the compared entries and 0 on the missing ones, and (2) on the diagonal entries, put the number of -ls on the row. Let the coefficient matrix be D. Then, we have Theorem 1 The rank of the matrix D is n - 1, if and only if the graph of the pairwise comparisons is connected. Proof. First, we show the 'if-part'. Since the sum of n row vectors of D is zero, the rank of D is less than n - 1. Let Dn-l be the left upper (n - 1) x (n - 1) matrix of D. For a vector :v =(xi) E Rn-1, the quadratic form associated with Dn-l is: n-1 (7) Q = :VT Dn-1:V = L N;x; + 2 L d;jXiXj i:::l 1::;;i<j::;;n-l (8) - 1:5i<j:5n~, d;;=-1 (x; - Xj)2 + ~ ( N; + j£i d;j) x; We observe the case Q = 0. (i) If the first term (x; - Xj)2 on the right-hand side of (8) is not vacant, 4 then, for each i, we have, under the condition Q = 0, (9) x; = Xj. (\/j E P;) Furthermore, at least one of x; and (xj) (j E P;) has the term x~ or x] in the second term on the right-hand side of (8), since otherwise the vertices x; and (xi) (j E P;) are disconnected to the remaining ones and this contradicts the connected graph hypothesis. Thus, we have, for each i in the first term, (10) x; = Xj (\/j E P;) = 0. (ii) For xk not included in the first term, we have x~ in the second term. Hence Xk = 0. Thus, if Q = 0, then x = 0. Therefore, all the eigenvalue of Dn-i is positive and the rank of Dn-i is n - 1. The 'only-if' part can be demonstrated as follows. Suppose the graph is disconnected. Then, the matrix D can be decomposed, after rearrangement, into (11) where Di E Rn1 xni, D~n-ni)x(n-ni) and ni > 0. The ranks of Di and D2 are less than or equal to ni - 1 and n - ni - 1, respectively. Hence, the rank of D must be less than or equal to n - 2, since the rank is the maximum number of linearly independent columns (or rows) of the matrix. D 3 A Geometric Mean Method for Incom-plete Pairwise Comparisons Based on the preceding theorem, a geometric mean method for incomplete pairwise comparisons goes as follows: 5 1. Let any one of (log g;) (j = 1, ... , n) be zero and solve the equations (6) in remaining (n - 1) unknowns. 2. Obtain the weight g; from logg; for (j = 1, ... , n). 3. Normalize (g;) so that (12) gl = g; (. 1 ) J "'n J = ' ... ' n .Gk=l gk Example 2 Let an incomplete pairwise comparisons matrix A be as below, where the symbol - stands for uncompared entries: 1 1/9 1/6 The corresponding linear equations are ( 2 0 -1 -1 l ( log g1 l 0 2 -1 -1 logg2 -1 -1 2 0 logg3 -1 -1 0 2 logg4 ( log3 x 2 l = log 9 x 6 log(l/3) x (1/9) · log(l/2) x (1/6) We assume logg4 = 0 and solve the system for logg; (j = 1, 2, 3) which gives logg1 = log2, logg2 = log6, logg3 = log(2/3), logg4 = 0. Thus, we obtain the normalized weight (13) g' = (0.207, 0.621, 0.069, 0.103). 6 4 Concluding Remarks This paper dealt with the incomplete pairwise comparisons in the AHP within the framework of the logarithmic least squares method. It is easy to see that the weight thus obtained has perfect consistency, if the estimates in the compared entries are consistent. A measure of consistency can be defined by 2:::'=1 ( LjEP; a;;g; / g; - N;) G = __ _,___ _____ ,__ 2:::'=1 N; (14) which is an average deviation of the compared estimate a;; from gif g;. Ob-viously, G is nonnegative and equal to zero if and only if the estimate a;; satisfies (15) a;;= g; (V(i,j)). 9j However, it should be noted that, if, in the most incomplete case, the graph is a spanning tree, the calculated weight (g;) is always consistent and hence G = 0. This observation suggests the need for other indices of accu-racy of measurement for incomplete comparisons. This is a future research subject. References (1] Golden, B.L. and Q. Wang, "An Alternate Measure of Consistency", in B.L.Golden, E.A.Wasil and P.T.Harker (Eds), The Analytic Hierarchy Process:Application and Studies, pp.68-81, Springer, (1989). [2] Harker, P.T., "Incomplete Pairwise Comparisons in the Analytic Hier-archy Process", Mathematical Modelling, 9, 837-848 (1987). (3] Saaty, T.L., The Analytic Hierarchy Process, McGraw-Hill, (1980). 7 [4] Saaty, T.L., "Eigenvalue and Logarithmic Least Squares", European Journal of Operations research, 48, 156-160 (1990). [5] Tone,K., "A Practical Procedure for Finding Copy-like Alternatives in AHP", (in Japanese), Communications of the Operations Research Soci-ety of Japan, 36, 4, 190-194 (1991). 8 

A Logarithmic Least Squares Method for Incomplete Pairwise Comparisons in the Analytic Hierarchy Process

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A Logarithmic Least Squares Method for Incomplete Pairwise Comparisons in the Analytic Hierarchy Process

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