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Incomplete pairwise comparison and consistency optimization
European Journal of Operational Research 183 (2007) 303–313 www.elsevier.com/locate/ejor
Decision Support
Incomplete pairwise comparison and consistency optimization Michele Fedrizzi a
a,*
, Silvio Giove
b
Dipartimento di Informatica e Studi Aziendali, Universita` di Trento Via Inama 5, TN 38100 Trento, Italy b Dipartimento di Matematica Applicata, Universita` di Venezia Dorsoduro, 3825/E Venezia, Italy Received 20 April 2006; accepted 20 September 2006 Available online 8 December 2006
Abstract This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented. 2006 Elsevier B.V. All rights reserved. Keywords: Consistency; Pairwise comparison matrices; Incomplete preference relation
1. Introduction Pairwise comparison is a well established technique in decision making. In Saaty’s AHP [17], as an example, pairwise comparison matrices (PCM in the following) are used to derive the priorities for n alternatives by means of the so-called eigenvector method. Nevertheless, in some cases we have to face a problem with missing judgements, thus obtaining an incomplete comparison matrix. This may happen, for instance, when the number of the alternatives, n, is large. In such cases it may be practically impossible, or at least unacceptable from the point of view of the decision maker, to perform all the n(n 1)/2 required comparisons to complete the PCM. A trade-off between the completeness of the available information and the need to keep reasonably small the number of questions to be submitted to the decision maker is then unavoidable. Moreover, as it has been pointed out in [9], it can be convenient/necessary to skip some direct critical comparison between alternatives, even if the total number of alternatives is small. Some methods have been proposed in the literature to derive the priorities of n alternatives from an incomplete n · n PCM, [1,3,9,10,14,19,23,24]. In this paper we define a measure of the inconsistency of a PCM using an index introduced in [6] and then we propose to calculate the missing elements of an incomplete PCM by maximizing the global consistency (i.e. *
Corresponding author. Tel.: +39 0461 882148. E-mail address: [email protected] (M. Fedrizzi).
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.09.065
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minimizing the inconsistency) of the ‘completed’ matrix. The paper is organized as follows: in Section 2 we define the problem, introduce the necessary notations and briefly summarize some methods proposed in the literature to solve the problem of incomplete comparisons. In Section 3 we describe our method and we solve the one and two-dimensional cases (i.e. when one or two comparisons between the alternatives are missing); then we extend our results for the general case of p missing comparisons. Finally, we present some results obtained by applying our method on a case proposed in [18] by Saaty. 2. Incomplete comparisons 2.1. The problem formulation Let K = {A1, A2, . . . , An} be a set of n alternatives and let the judgements of a decision maker be expressed by pairwise comparisons. If all pairs of alternatives (Ai, Aj) with i < j are considered, then it is necessary to perform n(n 1)/2 comparisons. With this data, one can obtain the upper diagonal triangle of an n · n matrix. The remaining elements of the matrix are easily derived, as it is usually assumed that by comparing Ai with Aj, the comparison of Aj with Ai is automatically assigned. Clearly, the elements of the diagonal need not to be computed. There are two main approaches to the problem of quantifying the comparative judgements. The first is the Saaty’s approach, which is also called multiplicative. In this framework, aij estimates the relative preference of the alternative Ai when it is compared with the alternative Aj. The obtained multiplicative PCM A = [aij] is positive, reciprocal (aji = 1/aij), with aii = 1, i = 1, . . . , n [16,17]. To estimate the comparison aij, Saaty proposes to use the rational numbers 1/9, 1/8, . . . , 1, 2, . . . , 9 with the following meaning: if Ai is absolutely preferred to Aj, then aij = 9, in the opposite case aij = 1/9, while indifference is indicated by aij = 1. Intermediate values correspond to intermediate judgements. A multiplicative PCM is called consistent if and only if aih ahj ¼ aij ;
i; j; h ¼ 1; . . . ; n:
ð1Þ
If a multiplicative PCM A = [aij] is consistent, then a positive vector w = [wi] exists such that aij = wi/wj, i, j = 1, . . . , n. In the second approach, also called additive approach [2], the expert’s preferences are described by a fuzzy preference relation r : K · K ! [0, 1] [4,20–22]. In this framework, r(Ai, Aj) (for conciseness denoted by rij in the following) indicates the preference degree of alternative Ai over alternative Aj. It follows that rij = 0.5 indicates indifference between Ai and Aj, rij = 1 indicates that Ai is definitely preferred over Aj, and rij = 0 indicates the opposite case. The n · n matrix R = [rij] is called additive PCM. Matrix R is nonnegative and reciprocal in the additive sense: rji = 1 rij, i, j = 1, . . . , n. An additive PCM R = [rij] is called consistent (in additive sense) if and only if ðrih 0:5Þ þ ðrhj 0:5Þ ¼ ðrij 0:5Þ i; j; h ¼ 1; . . . ; n:
ð2Þ
Property (2) can also be written as rih þ rhj rij 0:5 ¼ 0
i; j; h ¼ 1; . . . ; n:
ð3Þ
By writing (2), the role of the differences from indifference value 0.5 is emphasized. If an additive PCM R = [rij] is consistent, then a positive vector u = [ui] of utility values exists such that rij ¼ 0:5 þ 0:5ðui uj Þ i; j ¼ 1; . . . ; n
ð4Þ
with jui ujj 6 1 "ij [20,21]. In the new method proposed in this paper, we assume that the preferences are expressed following this additive approach. Nevertheless, it can be shown that the two approaches are equivalent: the simple function introduced in [5] 1 rij ¼ f ðaij Þ ¼ ð1 þ log9 aij Þ 2
ð5Þ
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transforms the aij values into the rij values in such a way that all the relevant properties of A = [aij] are transformed into the corresponding properties for R = [rij] in the additive sense. In particular, multiplicative reciprocity is transformed into additive reciprocity and multiplicative consistency is transformed into additive consistency. This topic has been then developed in several other papers [2,4,11–13,25]. A comprehensive and elegant survey on the relationships between the additive and the multiplicative approach (with slightly different definitions) can be found in [2]. 2.2. Some known methods for incomplete comparisons Let us now assume that one or more comparisons are missing. As a consequence, the PCM is incomplete and it is no longer possible to derive the priorities for the alternatives using the well known methods of the eigenvector or the geometric mean, to cite the most popular ones. Some methods have been proposed in the literature to solve the incomplete comparison problem. Most of these methods are formulated in the multiplicative framework [3,9,10,14,19], some other in the additive framework [1,23,24]. An incomplete additive PCM is also referred to as incomplete fuzzy preference relation. Let us very briefly describe the ideas that, in our opinion, are most relevant among those presented in the above-mentioned literature. Two methods have been proposed by Harker. The first one [10], called the geometric mean method, is based on the concept of ‘‘connecting path’’. If alternatives Ai and Aj are not compared with each other, let us denote by {Ai, Aj} the missing comparison (MC in the following) and let xij be the corresponding numerical value to be estimated; a connecting path of size r has the following form xij ¼ aik1 ak1 k2 akr j ;
ð6Þ
where the comparison values at the right hand side are known. The connecting path of size two, also called elementary connecting path, corresponds to the more familiar expression (see (1)) xij ¼ aik akj :
ð7Þ
Note that each connecting path corresponds to an indirect comparison between Ai and Aj. Harker proposes that the value xij of the MC should be equal to the geometric mean of all connecting paths related to this MC. The drawback of this method is that the number of connecting paths grows with the number n of the alternatives in such a way that the method becomes computationally intractable for many real world problems. The second Harker’s method [9] is based on the following idea. The missing (i, j)-component is set to be equal to wi/wj, where the components of the vector w are not known and are to be calculated. In other words, the missing entries are completed by setting them equal to the value they should approximate. The matrix obtained with the described substitution is called C. An auxiliary nonnegative numerical matrix A is then associated to C satisfying Aw = Cw. The matrix A is nonnegative and quasi reciprocal, in the sense that all its positive entries are reciprocal, but it contains entries equal to zero. In this way, Harker transforms the original problem in that of computing the principal eigenvector of a nonnegative quasi reciprocal matrix. In order to justify his method, Harker develops a theory for such type of matrices, following the Saaty’s one for positive reciprocal matrices. Shiraishi et al. in [19] propose a heuristic method which is based on a property of a coefficient of the characteristic polynomial of a PCM A. More precisely, the coefficient c3 of kn3 is viewed as an index of consistency for A. Therefore, in order to maximize the consistency of the PCM, the authors consider the missing entries in the PCM as variables x1, . . . , xm and propose to maximize c3(x1, . . . , xm) as a function of these variables. In [3] a Least Squares type method is proposed. Instead of first calculating the missing entries of a PCM, the priority vector w = [wi] is directly calculated as the solution of a constrained optimization problem. Here the variables are the n components wi of w and only the known entries aij are approximated by wi/wj. The corresponding error is minimized as a function of w1, . . . , wn. In [23], Xu proposes to calculate the priority vector w = [wi] of incomplete fuzzy preference relation by a goal programming approach. His method refers to (4) and minimizes the errors eij = jrij 0.5 + 0.5(ui uj)j for all missing entries (i, j). He also proposes his goal programming approach with another type of consistency, different from (4).
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In his second proposal, Xu [24] develops a method, for incomplete fuzzy preference relations, similar to that introduced by Harker [9] for incomplete multiplicative PCM. In [24] the priority vector w = [wi] is calculated by solving a system of equations which corresponds to the Harker’s auxiliary eigenproblem. In [1] an iterative method is proposed to evaluate the MCs in an incomplete fuzzy preference relations. The main idea is the following. If rij is unknown, the corresponding consistent value is calculated for each known indirect comparison between Ai and Aj (i.e. for each elementary connecting path); the arithmetic mean of all these values is the estimated value for rij (the approach is therefore in the spirit of [10]). The estimated values for the MCs are then iteratively utilized as known entries. 3. A new method for incomplete comparisons Our method is based on an (in)consistency index introduced in [6], which directly refers to the definition (3) of consistency for a PCM. As mentioned before, we assume that the preferences are expressed by an additive PCM R = [rij], rij 2 [0, 1]. Following [6] and taking into account (3), let Lijh ¼ ðrih þ rhj rij 0:5Þ
2
ð8Þ
be the inconsistency contribution associated with the triplet of alternatives {Ai, Aj, Ah}. This definition is meaningful, as the following lemma holds. Lemma 1. Lijh is invariant under permutations of the indices. Proof. The proof is based on the additive reciprocity of R : rji = 1 rij. Let us rewrite Lijh in the form Lijh = (rih + rhj + rji 1.5)2. It follows that 2
Lhij ¼ ðrhj þ rji þ rih 1:5Þ ¼ Lijh ; Ljhi ¼ ðrji þ rih þ rhj 1:5Þ2 ¼ Lijh ; 2
2
Lihj ¼ ðrij þ rjh þ rhi 1:5Þ ¼ ð1:5 rji rhj rih Þ ¼ Lijh ; Lhji ¼ ðrhi þ rij þ rjh 1:5Þ2 ¼ ð1:5 rih rji rhj Þ2 ¼ Lijh ; 2
2
Ljih ¼ ðrjh þ rhi þ rij 1:5Þ ¼ ð1:5 rhj rih rji Þ ¼ Lijh :
Two other useful properties of Lijh are very easy to check: • If i 5 j 5 h, then Lijh = 0 if the alternatives Ai, Aj, Ah are compared in a perfectly consistent way. • If at least two indices are equal, then Lijh ¼ 0:
ð9Þ
As Lijh corresponds to a local inconsistency contribution, the global inconsistency index q of matrix R is defined as (see [6]) n X Lijh : ð10Þ q¼ i;j;h¼1
P Taking (9) into account, q can be expressed as q ¼ i6¼j6¼h Lijh . Moreover, in (10), it would be possible to con n sider the sum of only the terms corresponding to all the (non ordered) triplets of alternatives, thus avoid3 P ing repetitions; from lemma 1 we have: q ¼ 6 i
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307
3.1. The single missing comparison case Let us assume that only one comparison is missing: {As, At}, i.e. the comparison between As and At, s 5 t. Then the two elements rst and rts = 1 rst are unknown in the n · n matrix R. The optimal value rst is the solution of the following problem: min qðrst Þ:
ð11Þ
rst 2½0;1
In order to minimize q = q(rst), let us calculate the stationary point(s) of q. Taking into account Lemma 1, we have (in the following all the sums are made with respect to the index h) dq d X ¼6 Lsth ; drst drst h6¼s h6¼t
! X X X 1 dq ¼ ð2rst 2rsh 2rht þ 1Þ ¼ 2 ðn 2Þrst rsh rht þ n=2 : 6 drst h6¼s h6¼t h6¼s
ð12Þ
h6¼t
By setting the derivative equal to zero we obtain " # X X 1 ^rst ¼ rsh þ rht n=2 : n 2 h6¼t h6¼s
ð13Þ
Function q(rst) is a sum of strictly convex functions Lsth, so it is itself a strictly convex function. As a consequence, (13) gives the global minimum of q(rst). The optimal value ^rst given by (13) can be outside the interval [0, 1], as it is well known that consistency may be incompatible with the use of a bounded scale; more precisely, given rih, rhj 2 [0, 1], it is possible that no rij 2 [0, 1] exists such that (3) holds. As an example, consider the extreme values rih = rhj = 0 or rih = rhj = 1. Nevertheless, in this one-dimensional convex case, if ^rst 62 ½0; 1, then the (bounded) optimal solution rst of (11) is simply obtained by taking rst ¼ 0 if ^rst < 0 and rst ¼ 1 if ^rst > 1. 3.2. The two missing comparisons case As the general case of p missing comparisons needs a rather complex notation, let us first assume that only two comparisons are missing, say {As, At} and {Au, Av}. If the two MCs do not share any alternative, i.e. indices s,t,u,v are all different from each other, the optimal values of rst and ruv can be calculated independently as described in the previous section, i.e. using (13). Let us then assume that an alternative, say At, is involved in both MCs, which become {As, At} and {Au, At}. The four unknown elements in the n · n matrix R are rst, rts = 1 rst, rut and rtu = 1 rut. The optimal values rst and rut are computed by solving the following problem: min
ðrst ;rut Þ2½0;12
qðrst ; rut Þ:
ð14Þ
In order to minimize q = q(rst, rut), let us first calculate the stationary point(s) of q(rst, rut) oq o X ¼6 Lsth ; orst orst h6¼s;h6¼t X oLsth 1 oq ¼ 6 orst h6¼s;h6¼t orst X ¼ ð2rst 2rsh 2rht þ 1Þ h6¼s;h6¼t
¼ 2ðn 2Þrst 2
X h6¼s;h6¼t
rsh 2
X h6¼s;h6¼u;h6¼t
rht 2rut þ ðn 2Þ:
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By setting oq/orst = 0 we obtain X 2ðn 2Þrst 2rut ¼ 2 rsh þ 2 h6¼s;h6¼t
X
ðn 2Þrst rut ¼ ðn 2Þrst rut ¼
X
rsh þ
h6¼t
rht n þ 2;
h6¼s;h6¼u;h6¼t
X
rsh þ
h6¼s;h6¼t
X
rht n=2 þ 1;
h6¼s;h6¼u;h6¼t
X
rht n=2:
h6¼s;h6¼u
Analogously, we obtain X X 1 oq ¼ 2ðn 2Þrut 2 ruh 2 rht 2rst þ ðn 2Þ 6 orut h6¼u;h6¼t h6¼s;h6¼u;h6¼t and, by setting oq/orut = 0, X X ruh þ rht n=2: ðn 2Þrut rst ¼ h6¼t
h6¼s;h6¼u
By imposing both derivatives equal to zero, we obtain the system 8 P P > < ðn 2Þrst rut ¼ h6¼t rsh þ h6¼s;h6¼u rht n=2; P P > ruh þ rht n=2; : rst þ ðn 2Þrut ¼ h6¼t
ð15Þ
h6¼s;h6¼u
where the coefficient matrix is n 2 1 Q¼ 1 n 2 and the r.h.s. vector is "P # P h6¼t r sh þ h6¼s;h6¼u r ht n=2 P b¼ P : h6¼t ruh þ h6¼s;h6¼u rht n=2
ð16Þ
ð17Þ
The coefficient matrix is clearly nonsingular and therefore system (15) has a unique solution ð^rst ; ^rut Þ ¼ Q1 b which is the single stationary point of the function q(rst, rut) . The function q(rst, rut) is convex, as it is a sum of convex functions, so that the stationary point is a global minimum point. The convexity of q(rst, rut) can be also directly checked by its (constant) hessian matrix 2ðn 2Þ 2 H ¼6 ; ð18Þ 2 2ðn 2Þ which is positive definite for n > 3. If ð^rst ; ^rut Þ belongs to [0, 1]2, it is the solution of problem (14). As in the onemissing comparison case, when the available judgements are seriously inconsistent, it may happen that 2 ð^rst ; ^rut Þ 62 ½0; 1 . In this case the constrained problem should be solved by means of a quadratic programming algorithm [8]. 3.3. The p missing comparisons case As in the previous cases, let us associate a variable rst to each MC {As, At}. If p comparisons are missing, then the matrix R contains 2p unknown elements, due to its reciprocity. In this general case, analogously to the two-dimensional one, it is necessary to distinguish between two different type of MCs. The first one, which we call independent, occurs when in a MC {As, At} neither of the alternatives As and At is involved in any other MC. That is: {As, At} \ {Ai, Aj} = ; for any other MC {Ai, Aj}. In this case the optimal value ^rst of the variable rst is simply given by (13).
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We now focus on sets of dependent MCs; we call dependent the MCs of a set such that for every partition into two subsets, there exists at least one alternative which is in both subsets. As an example, {{A1, A2}, {A2, A4}, {A4, A7}} is a dependent set of MCs, while {{A1, A2},{A2, A4},{A5, A7}, {A5, A8}} is not, as the two subsets {{A1, A2}, {A2, A4}} and {{A5, A7}, {A5, A8}} do not contain a common alternative. It is possible to clarify this definition of dependency with the help of graph theory. If nodes are associated to alternatives and arcs to MCs, then a set of dependent MCs corresponds to a connected (sub)graph. As a consequence of these definitions, the p MCs can be divided in a certain number of independent MCs and some disjoint sets of dependent MCs. As mentioned before, it is easy to find the optimal value of a variable rst associated with an independent MC, as it can be calculated independently from the other variables. Analogously, we can calculate the optimal value of the variables associated with a set of dependent MCs independently from all the other variables not referring to that set. In the following, we describe how to calculate the optimal values of a set of m 6 n 2 variables associated with a set of dependent MCs. In order to simplify the indices notation, we can ignore all the other p m MCs not in the dependent set, assuming that only those m comparisons are missing in the problem. We denote the m pair of alternatives corresponding to the m MCs by fAsk ; Atk g; k ¼ 1; . . . ; m. Then q is a function of the m variables xk ¼ rsk tk ; k ¼ 1; . . . ; m. Without loss of generality, we assume, that sk < tk. The matrix R has 2m unknown elements: the m variables rsk tk ; k ¼ 1; . . . ; m above the diagonal and the m corresponding reciprocal elements below the diagonal. For each variable rsk tk , let us denote with IRþ sk and IRsk the sets of column indices of the other unknown elements which are on the same row sk respectively above and below the diagonal. Analogously, let us denote with IC þ tk and IC tk the sets of row indices of the other unknown elements which are on the same column tk respectively above and below the diagonal. The example in the next subsection will clarify this notation. 3.3.1. Consistency optimization Analogously to the one and two-dimensional cases, in order to solve the problem min qðrs1 t1 ; . . . ; rsm tm Þ; m ðrs1 t1 ;...;rsm tm Þ2½0;1
ð19Þ
let us calculate the m partial derivatives of q and set them equal to zero. From (12) and (13) we have oq ¼ 0; orsk tk X X r sk h rhtk þ n=2 ¼ 0 ðn 2Þrsk tk h6¼tk
h6¼sk
and, after some calculation, X X X X r sk h þ rhsk rhtk þ rtk h ðn 2Þrsk tk h2IR s
h2IRþ s
¼
CardðIR sk Þ
þ
CardðIC tk Þ
h2IC t
h2IC þ t
k
k
k
k
X
þ
h62ðIRþ s [IRs [ft k gÞ k
X
r sk h þ
k
rhtk n=2:
ð20Þ
h62ðIC þ t [IC t [fsk gÞ k
k
Taking (20) for k = 1, . . . ,m, we impose the m derivatives oq=orsk tk equal to zero, thus obtaining a linear system with m equations in the m variables rsk tk ; k ¼ 1; . . . ; m : Qx ¼ b;
ð21Þ
where 2
n2 6 q 6 21 6 6 ... Q¼6 6 ... 6 6 4 qm11 qm1
q12 n2
q13 q23
qm12
n2
qm2
qmm1
3 q1m q2m 7 7 7 ... 7 7; ... 7 7 7 qm1m 5 n2
ð22Þ
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2
r s1 t 1
6 6 6 6 6 6 x ¼ 6 r sk t k 6 6 6 6 4
3 7 7 7 7 7 7 7; 7 7 7 7 5
2
6 6 6 6 6 6 b ¼ 6 bk 6 6 6 6 4
r sm t m bk ¼
CardðIR sk Þ
8 n2 > > > < 1 qkh ¼ > þ1 > > : 0
b1
3 7 7 7 7 7 7 7; 7 7 7 7 5
bm þ CardðIC tk Þ þ
X h62ðIRþ sk [IRs [ft k gÞ k
X
r sk h þ
rhtk n=2;
h62ðIC þ t [IC t [fsk gÞ k
k
if h ¼ k; þ if th 2 IRþ sk or sh 2 IC tk ; if th 2 IR sk or sh 2 IC tk ; otherwise:
It can be observed that in the above definition the conditions of four cases are disjoint and that the definition itself can be equivalently be written in the simpler form: 8 n 2 if h ¼ k; > > > < 1 if sh ¼ sk or th ¼ tk ; qkh ¼ > if sh ¼ tk or th ¼ sk ; > > þ1 : 0 otherwise: Note that all information on the known comparisons are summarized in vector b, while matrix Q = [qi j] encodes the interdependence structure of the m MCs, with qij 2 { 1, 0, 1} for i 5 j and qii = n 2 "i. The following theorem on the optimal values of the missing comparisons is the main result of the paper. Theorem 1. If m 6 n 2, then (a) Matrix Q is nonsingular and therefore system (21) has a unique solution given by ^x ¼ Q1 b. (b) Solution ^x is the global minimum point for function q(x).
Proof Pm (a) From the assumption m 6 n 2, it follows that Q is strictly diagonally dominant,1 j¼1j6¼i jqij j < m 6 n 2 ¼ jqii j for all i. A strictly diagonally dominant matrix is nonsingular (Levy–Desplanques theorem) and therefore invertible [15]. Then system (21) has a unique solution given by ^x ¼ Q1 b, i.e. the unique stationary point of q(x). The first part of the theorem is then proved. (b) From the definition of qhk, it follows that qhk = qkh. Then Q is a symmetric matrix, strictly diagonally dominant, with all the diagonal elements positive; this is sufficient to conclude that Q is positive definite. The hessian matrix of q(x) is itself positive definite, being a positive multiple of Q. Then q(x) is a strictly convex function and the stationary point ^x is its global minimum point. h As in the previous cases, when the available judgements are seriously inconsistent, it may happen that ^x 62 ½0; 1m . Also in this case the minimum point in [0, 1]m can be obtained by solving a quadratic programming problem.
1
A square m · m matrix Q = [qij] is called diagonally dominant if jqii j P dominant if strict inequality holds.
P
j6¼i jqij j; i
¼ 1; . . . ; m. The matrix is called strictly diagonally
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311
It is well known that it is possible to obtain a numerical result not belonging to the a-priori chosen feasible set of values. This is unavoidable when dealing with inconsistency and with a bounded scale. Clearly the problem does not exist when an open scale is used [2,7]. On the other hand, despite the elegant mathematical results, every unbounded scale yields serious drawbacks in practical applications, but we do not want to dwell deeper on this issue. Example 1. As the notation required by the general p-dimensional case described above is rather complex, we clarify the preceding issues by the following example. Let us consider a case with nine alternatives {A1, A2, . . . , A9}; we do not take into account the known entries in the 9 · 9 matrix, but we focus on the following five MCs with their corresponding variables Missing comparison
Variable
k
{A1, A5} {A1, A9} {A3, A5} {A5, A7} {A5, A9}
r15 = x1 r19 = x2 r35 = x3 r57 = x4 r59 = x5
1 2 3 4 5
In the resulting PCM R we indicate only the unknown entries 3 2 r15 r19 7 6 7 6 7 6 7 6 r35 7 6 7 6 7 6 7 6 7 6 R ¼ 6 1 r15 1 r35 r57 r59 7: 7 6 7 6 7 6 7 6 7 6 1 r 57 7 6 7 6 5 4 1 r19 1 r59 The left hand 8 7r15 > > > > > > < r15 r15 > > > > r15 > > : r15
ð23Þ
side part of system (21) becomes r19
r35
þr57
þr59
¼
7r19
0
0
r59
¼
0
7r35
þr57
þr59
¼
0
þr35
þ7r57
r59
¼
r19
þr35
r57
þ7r59
¼
ð24Þ
Clearly, we cannot write the vector b on the r.h.s. of the system unless we assign all the known entries. The coefficient matrix of the system (24) is 3 2 7 1 1 1 1 6 1 7 0 0 1 7 7 6 7 6 7; ð25Þ 1 0 7 1 1 Q¼6 7 6 7 6 4 1 0 1 7 1 5 1
1
1
1
7
which is symmetric, positive definite and therefore nonsingular. With the help of the known entries of PCM R (whatever they are) is then possible to calculate vector b, so that the solution of the system (24) is given by ^x ¼ Q1 b.
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4. Numerical results We propose some numerical experiences on a problem, concerning the choice of a job, which has been proposed and studied by Saaty in [18], page 85. The pairwise comparison matrix obtained by Saaty is transformed, by means of (5), into the equivalent additive matrix 3 2 0:5 0:5 0:5 0:8155 0:5 0:3423 6 0:5 0:5 0:6577 0:8155 0:5 0:3423 7 7 6 6 0:5 0:3423 0:5 0:8662 0:75 0:3423 7 7 6 R¼6 ð26Þ 7: 6 0:1845 0:1845 0:1338 0:5 0:25 0:25 7 6 7 4 0:5 0:5 0:25 0:75 0:5 0:25 5 0:6577 0:6577 0:6577 0:75 0:75 0:5 The inconsistency index (10) of R is q = 3.78. Now, let us assume that the decision maker is not able to perform the comparison between alternatives A2 and A3. Then R is incomplete, as the two elements r23 and r32 = 1 r23 are unknown. We complete R using our method, i.e. by minimizing q(r23), as explained in Section 3.1. We obtain r23 = 0.4248 and the inconsistency index of R becomes q = 2.478. If, in addition to {A2, A3}, the comparison {A3, A5} is missing as well, we analogously calculate the optimal values of r23 and r35 by solving (15). The obtained values are r23 = 0.4726 and r35 = 0.559 with the inconsistency decreased to q = 1.6577. Finally, if we consider as missing also a third comparison {A4, A6}, we have to solve the system (21) to calculate the optimal values of the three variables, obtaining: r23 = 0.4726, r35 = 0.559 and r46 = 0.0074. The matrix R, completed with these values, is very close to consistency, being q = 0.245. Note that in all the cases mentioned above the obtained values belong to [0, 1]. 5. Conclusions and final remarks We think that our proposal is a natural way to solve the problem of missing data in pairwise comparison. We propose to complete the PCM coherently with the available judgements by directly referring to the definition (3) of consistency. From the computational point of view our method is rather simple: to calculate the optimal values, we only have to solve a nonsingular linear system. Our future research effort will be directed at comparing our method with other approaches by suitable numerical experiments, in order to highlight different properties/characteristics. By means of (10) we define the global inconsistency index q of a PCM R simply by summing all the local inconsistency contributions Lijh. If we are interested in defining an index independent of the order n of the matrix, it is clearly possible to apply the necessary normalization. By taking into account only the non trivial P n elements in (10) and dividing by their number, a mean value of Lijh is obtained, L ¼ i
M. Fedrizzi, S. Giove / European Journal of Operational Research 183 (2007) 303–313 [7] [8] [9] [10] [11] [12]
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[23] [24] [25]
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P.C. Fishburn, SSB utility theory: An economic perspective, Mathematical Social Sciences 8 (1984) 63–94. R. FletcherPractical Methods of Optimization, vols. 1–2, John Wiley & Sons, New York, 1981. P.T. Harker, Alternative modes of questioning in the analytic hierarchy process, Mathematical Modelling 9 (3–5) (1987) 353–360. P.T. Harker, Incomplete pairwise comparisons in the analytic hierarchy process, Mathematical Modelling 9 (11) (1987) 837–848. F. Herrera, E. Herrera-Viedma, F. Chiclana, Multiperson decision-making based on multiplicative preference relations, European Journal of Operational Research 129 (2001) 372–385. F. Herrera, E. Herrera-Viedma, F. Chiclana, Additive consistency of fuzzy preference relations: Characterization and construction, in: Proc. 4th International Workshop on Preferences and Decisions, Trento 2003, Universita` di Trento, Italy, September 2003, pp. 39– 46. E. Herrera-Viedma, F. Herrera, F. Chiclana, M. Luque, Some issues on consistency of fuzzy preference relations, European Journal of Operational Research 154 (2004) 98–109. K. Nishizawa, Estimation of unknown comparisons in incomplete AHP and it’s compensation, Report of the Research Institute of Industrial Technology, Nihon University, no. 77, 2004. J.M. Ortega, Matrix Theory, Plenum Press, New York, 1988. T.L. Saaty, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977) 234–281. T.L. Saaty, The Analytical Hierarchy Process, McGraw-Hill, New York, 1980. T.L. Saaty, Decision Making for Leaders: The Analytical Hierarchy Process for Decisions in a Complex World, University of Pittsburgh Press, Pittsburgh, 1988. S. Shiraishi, T. Obata, M. Daigo, Properties of a positive reciprocal matrix and their application to AHP, Journal of Operational Research Society of Japan 41 (1998) 404–414. T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems 12 (1984) 117–131. T. Tanino, Fuzzy preference relations in group decision making, in: J. Kacprzyk, M. Roubens (Eds.), Non-Conventional Preference Relations in Decision Making, Springer-Verlag, 1988. T. Tanino, On group decision making under fuzzy preferences, in: J. Kacprzyk, M. Fedrizzi (Eds.), Multiperson Decision Making Models using Fuzzy Sets and Possibility Theory, Theory and Decision Library, series B: Mathematical and Statistical Methods, vol. 18, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. Z.S. Xu, Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation, International Journal of Approximate Reasoning 36 (2004) 261–270. Z.S. Xu, A procedure for decision making based on incomplete fuzzy preference relation, Fuzzy Optimization and Decision Making 4 (2005) 175–189. Z.S. Xu, A least deviation method to obtain a priority vector of a fuzzy preference relation, European Journal of Operational Research 164 (2005) 206–216.
Decision Support
Incomplete pairwise comparison and consistency optimization Michele Fedrizzi a
a,*
, Silvio Giove
b
Dipartimento di Informatica e Studi Aziendali, Universita` di Trento Via Inama 5, TN 38100 Trento, Italy b Dipartimento di Matematica Applicata, Universita` di Venezia Dorsoduro, 3825/E Venezia, Italy Received 20 April 2006; accepted 20 September 2006 Available online 8 December 2006
Abstract This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented. 2006 Elsevier B.V. All rights reserved. Keywords: Consistency; Pairwise comparison matrices; Incomplete preference relation
1. Introduction Pairwise comparison is a well established technique in decision making. In Saaty’s AHP [17], as an example, pairwise comparison matrices (PCM in the following) are used to derive the priorities for n alternatives by means of the so-called eigenvector method. Nevertheless, in some cases we have to face a problem with missing judgements, thus obtaining an incomplete comparison matrix. This may happen, for instance, when the number of the alternatives, n, is large. In such cases it may be practically impossible, or at least unacceptable from the point of view of the decision maker, to perform all the n(n 1)/2 required comparisons to complete the PCM. A trade-off between the completeness of the available information and the need to keep reasonably small the number of questions to be submitted to the decision maker is then unavoidable. Moreover, as it has been pointed out in [9], it can be convenient/necessary to skip some direct critical comparison between alternatives, even if the total number of alternatives is small. Some methods have been proposed in the literature to derive the priorities of n alternatives from an incomplete n · n PCM, [1,3,9,10,14,19,23,24]. In this paper we define a measure of the inconsistency of a PCM using an index introduced in [6] and then we propose to calculate the missing elements of an incomplete PCM by maximizing the global consistency (i.e. *
Corresponding author. Tel.: +39 0461 882148. E-mail address: [email protected] (M. Fedrizzi).
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.09.065
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minimizing the inconsistency) of the ‘completed’ matrix. The paper is organized as follows: in Section 2 we define the problem, introduce the necessary notations and briefly summarize some methods proposed in the literature to solve the problem of incomplete comparisons. In Section 3 we describe our method and we solve the one and two-dimensional cases (i.e. when one or two comparisons between the alternatives are missing); then we extend our results for the general case of p missing comparisons. Finally, we present some results obtained by applying our method on a case proposed in [18] by Saaty. 2. Incomplete comparisons 2.1. The problem formulation Let K = {A1, A2, . . . , An} be a set of n alternatives and let the judgements of a decision maker be expressed by pairwise comparisons. If all pairs of alternatives (Ai, Aj) with i < j are considered, then it is necessary to perform n(n 1)/2 comparisons. With this data, one can obtain the upper diagonal triangle of an n · n matrix. The remaining elements of the matrix are easily derived, as it is usually assumed that by comparing Ai with Aj, the comparison of Aj with Ai is automatically assigned. Clearly, the elements of the diagonal need not to be computed. There are two main approaches to the problem of quantifying the comparative judgements. The first is the Saaty’s approach, which is also called multiplicative. In this framework, aij estimates the relative preference of the alternative Ai when it is compared with the alternative Aj. The obtained multiplicative PCM A = [aij] is positive, reciprocal (aji = 1/aij), with aii = 1, i = 1, . . . , n [16,17]. To estimate the comparison aij, Saaty proposes to use the rational numbers 1/9, 1/8, . . . , 1, 2, . . . , 9 with the following meaning: if Ai is absolutely preferred to Aj, then aij = 9, in the opposite case aij = 1/9, while indifference is indicated by aij = 1. Intermediate values correspond to intermediate judgements. A multiplicative PCM is called consistent if and only if aih ahj ¼ aij ;
i; j; h ¼ 1; . . . ; n:
ð1Þ
If a multiplicative PCM A = [aij] is consistent, then a positive vector w = [wi] exists such that aij = wi/wj, i, j = 1, . . . , n. In the second approach, also called additive approach [2], the expert’s preferences are described by a fuzzy preference relation r : K · K ! [0, 1] [4,20–22]. In this framework, r(Ai, Aj) (for conciseness denoted by rij in the following) indicates the preference degree of alternative Ai over alternative Aj. It follows that rij = 0.5 indicates indifference between Ai and Aj, rij = 1 indicates that Ai is definitely preferred over Aj, and rij = 0 indicates the opposite case. The n · n matrix R = [rij] is called additive PCM. Matrix R is nonnegative and reciprocal in the additive sense: rji = 1 rij, i, j = 1, . . . , n. An additive PCM R = [rij] is called consistent (in additive sense) if and only if ðrih 0:5Þ þ ðrhj 0:5Þ ¼ ðrij 0:5Þ i; j; h ¼ 1; . . . ; n:
ð2Þ
Property (2) can also be written as rih þ rhj rij 0:5 ¼ 0
i; j; h ¼ 1; . . . ; n:
ð3Þ
By writing (2), the role of the differences from indifference value 0.5 is emphasized. If an additive PCM R = [rij] is consistent, then a positive vector u = [ui] of utility values exists such that rij ¼ 0:5 þ 0:5ðui uj Þ i; j ¼ 1; . . . ; n
ð4Þ
with jui ujj 6 1 "ij [20,21]. In the new method proposed in this paper, we assume that the preferences are expressed following this additive approach. Nevertheless, it can be shown that the two approaches are equivalent: the simple function introduced in [5] 1 rij ¼ f ðaij Þ ¼ ð1 þ log9 aij Þ 2
ð5Þ
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305
transforms the aij values into the rij values in such a way that all the relevant properties of A = [aij] are transformed into the corresponding properties for R = [rij] in the additive sense. In particular, multiplicative reciprocity is transformed into additive reciprocity and multiplicative consistency is transformed into additive consistency. This topic has been then developed in several other papers [2,4,11–13,25]. A comprehensive and elegant survey on the relationships between the additive and the multiplicative approach (with slightly different definitions) can be found in [2]. 2.2. Some known methods for incomplete comparisons Let us now assume that one or more comparisons are missing. As a consequence, the PCM is incomplete and it is no longer possible to derive the priorities for the alternatives using the well known methods of the eigenvector or the geometric mean, to cite the most popular ones. Some methods have been proposed in the literature to solve the incomplete comparison problem. Most of these methods are formulated in the multiplicative framework [3,9,10,14,19], some other in the additive framework [1,23,24]. An incomplete additive PCM is also referred to as incomplete fuzzy preference relation. Let us very briefly describe the ideas that, in our opinion, are most relevant among those presented in the above-mentioned literature. Two methods have been proposed by Harker. The first one [10], called the geometric mean method, is based on the concept of ‘‘connecting path’’. If alternatives Ai and Aj are not compared with each other, let us denote by {Ai, Aj} the missing comparison (MC in the following) and let xij be the corresponding numerical value to be estimated; a connecting path of size r has the following form xij ¼ aik1 ak1 k2 akr j ;
ð6Þ
where the comparison values at the right hand side are known. The connecting path of size two, also called elementary connecting path, corresponds to the more familiar expression (see (1)) xij ¼ aik akj :
ð7Þ
Note that each connecting path corresponds to an indirect comparison between Ai and Aj. Harker proposes that the value xij of the MC should be equal to the geometric mean of all connecting paths related to this MC. The drawback of this method is that the number of connecting paths grows with the number n of the alternatives in such a way that the method becomes computationally intractable for many real world problems. The second Harker’s method [9] is based on the following idea. The missing (i, j)-component is set to be equal to wi/wj, where the components of the vector w are not known and are to be calculated. In other words, the missing entries are completed by setting them equal to the value they should approximate. The matrix obtained with the described substitution is called C. An auxiliary nonnegative numerical matrix A is then associated to C satisfying Aw = Cw. The matrix A is nonnegative and quasi reciprocal, in the sense that all its positive entries are reciprocal, but it contains entries equal to zero. In this way, Harker transforms the original problem in that of computing the principal eigenvector of a nonnegative quasi reciprocal matrix. In order to justify his method, Harker develops a theory for such type of matrices, following the Saaty’s one for positive reciprocal matrices. Shiraishi et al. in [19] propose a heuristic method which is based on a property of a coefficient of the characteristic polynomial of a PCM A. More precisely, the coefficient c3 of kn3 is viewed as an index of consistency for A. Therefore, in order to maximize the consistency of the PCM, the authors consider the missing entries in the PCM as variables x1, . . . , xm and propose to maximize c3(x1, . . . , xm) as a function of these variables. In [3] a Least Squares type method is proposed. Instead of first calculating the missing entries of a PCM, the priority vector w = [wi] is directly calculated as the solution of a constrained optimization problem. Here the variables are the n components wi of w and only the known entries aij are approximated by wi/wj. The corresponding error is minimized as a function of w1, . . . , wn. In [23], Xu proposes to calculate the priority vector w = [wi] of incomplete fuzzy preference relation by a goal programming approach. His method refers to (4) and minimizes the errors eij = jrij 0.5 + 0.5(ui uj)j for all missing entries (i, j). He also proposes his goal programming approach with another type of consistency, different from (4).
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In his second proposal, Xu [24] develops a method, for incomplete fuzzy preference relations, similar to that introduced by Harker [9] for incomplete multiplicative PCM. In [24] the priority vector w = [wi] is calculated by solving a system of equations which corresponds to the Harker’s auxiliary eigenproblem. In [1] an iterative method is proposed to evaluate the MCs in an incomplete fuzzy preference relations. The main idea is the following. If rij is unknown, the corresponding consistent value is calculated for each known indirect comparison between Ai and Aj (i.e. for each elementary connecting path); the arithmetic mean of all these values is the estimated value for rij (the approach is therefore in the spirit of [10]). The estimated values for the MCs are then iteratively utilized as known entries. 3. A new method for incomplete comparisons Our method is based on an (in)consistency index introduced in [6], which directly refers to the definition (3) of consistency for a PCM. As mentioned before, we assume that the preferences are expressed by an additive PCM R = [rij], rij 2 [0, 1]. Following [6] and taking into account (3), let Lijh ¼ ðrih þ rhj rij 0:5Þ
2
ð8Þ
be the inconsistency contribution associated with the triplet of alternatives {Ai, Aj, Ah}. This definition is meaningful, as the following lemma holds. Lemma 1. Lijh is invariant under permutations of the indices. Proof. The proof is based on the additive reciprocity of R : rji = 1 rij. Let us rewrite Lijh in the form Lijh = (rih + rhj + rji 1.5)2. It follows that 2
Lhij ¼ ðrhj þ rji þ rih 1:5Þ ¼ Lijh ; Ljhi ¼ ðrji þ rih þ rhj 1:5Þ2 ¼ Lijh ; 2
2
Lihj ¼ ðrij þ rjh þ rhi 1:5Þ ¼ ð1:5 rji rhj rih Þ ¼ Lijh ; Lhji ¼ ðrhi þ rij þ rjh 1:5Þ2 ¼ ð1:5 rih rji rhj Þ2 ¼ Lijh ; 2
2
Ljih ¼ ðrjh þ rhi þ rij 1:5Þ ¼ ð1:5 rhj rih rji Þ ¼ Lijh :
Two other useful properties of Lijh are very easy to check: • If i 5 j 5 h, then Lijh = 0 if the alternatives Ai, Aj, Ah are compared in a perfectly consistent way. • If at least two indices are equal, then Lijh ¼ 0:
ð9Þ
As Lijh corresponds to a local inconsistency contribution, the global inconsistency index q of matrix R is defined as (see [6]) n X Lijh : ð10Þ q¼ i;j;h¼1
P Taking (9) into account, q can be expressed as q ¼ i6¼j6¼h Lijh . Moreover, in (10), it would be possible to con n sider the sum of only the terms corresponding to all the (non ordered) triplets of alternatives, thus avoid3 P ing repetitions; from lemma 1 we have: q ¼ 6 i
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307
3.1. The single missing comparison case Let us assume that only one comparison is missing: {As, At}, i.e. the comparison between As and At, s 5 t. Then the two elements rst and rts = 1 rst are unknown in the n · n matrix R. The optimal value rst is the solution of the following problem: min qðrst Þ:
ð11Þ
rst 2½0;1
In order to minimize q = q(rst), let us calculate the stationary point(s) of q. Taking into account Lemma 1, we have (in the following all the sums are made with respect to the index h) dq d X ¼6 Lsth ; drst drst h6¼s h6¼t
! X X X 1 dq ¼ ð2rst 2rsh 2rht þ 1Þ ¼ 2 ðn 2Þrst rsh rht þ n=2 : 6 drst h6¼s h6¼t h6¼s
ð12Þ
h6¼t
By setting the derivative equal to zero we obtain " # X X 1 ^rst ¼ rsh þ rht n=2 : n 2 h6¼t h6¼s
ð13Þ
Function q(rst) is a sum of strictly convex functions Lsth, so it is itself a strictly convex function. As a consequence, (13) gives the global minimum of q(rst). The optimal value ^rst given by (13) can be outside the interval [0, 1], as it is well known that consistency may be incompatible with the use of a bounded scale; more precisely, given rih, rhj 2 [0, 1], it is possible that no rij 2 [0, 1] exists such that (3) holds. As an example, consider the extreme values rih = rhj = 0 or rih = rhj = 1. Nevertheless, in this one-dimensional convex case, if ^rst 62 ½0; 1, then the (bounded) optimal solution rst of (11) is simply obtained by taking rst ¼ 0 if ^rst < 0 and rst ¼ 1 if ^rst > 1. 3.2. The two missing comparisons case As the general case of p missing comparisons needs a rather complex notation, let us first assume that only two comparisons are missing, say {As, At} and {Au, Av}. If the two MCs do not share any alternative, i.e. indices s,t,u,v are all different from each other, the optimal values of rst and ruv can be calculated independently as described in the previous section, i.e. using (13). Let us then assume that an alternative, say At, is involved in both MCs, which become {As, At} and {Au, At}. The four unknown elements in the n · n matrix R are rst, rts = 1 rst, rut and rtu = 1 rut. The optimal values rst and rut are computed by solving the following problem: min
ðrst ;rut Þ2½0;12
qðrst ; rut Þ:
ð14Þ
In order to minimize q = q(rst, rut), let us first calculate the stationary point(s) of q(rst, rut) oq o X ¼6 Lsth ; orst orst h6¼s;h6¼t X oLsth 1 oq ¼ 6 orst h6¼s;h6¼t orst X ¼ ð2rst 2rsh 2rht þ 1Þ h6¼s;h6¼t
¼ 2ðn 2Þrst 2
X h6¼s;h6¼t
rsh 2
X h6¼s;h6¼u;h6¼t
rht 2rut þ ðn 2Þ:
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By setting oq/orst = 0 we obtain X 2ðn 2Þrst 2rut ¼ 2 rsh þ 2 h6¼s;h6¼t
X
ðn 2Þrst rut ¼ ðn 2Þrst rut ¼
X
rsh þ
h6¼t
rht n þ 2;
h6¼s;h6¼u;h6¼t
X
rsh þ
h6¼s;h6¼t
X
rht n=2 þ 1;
h6¼s;h6¼u;h6¼t
X
rht n=2:
h6¼s;h6¼u
Analogously, we obtain X X 1 oq ¼ 2ðn 2Þrut 2 ruh 2 rht 2rst þ ðn 2Þ 6 orut h6¼u;h6¼t h6¼s;h6¼u;h6¼t and, by setting oq/orut = 0, X X ruh þ rht n=2: ðn 2Þrut rst ¼ h6¼t
h6¼s;h6¼u
By imposing both derivatives equal to zero, we obtain the system 8 P P > < ðn 2Þrst rut ¼ h6¼t rsh þ h6¼s;h6¼u rht n=2; P P > ruh þ rht n=2; : rst þ ðn 2Þrut ¼ h6¼t
ð15Þ
h6¼s;h6¼u
where the coefficient matrix is n 2 1 Q¼ 1 n 2 and the r.h.s. vector is "P # P h6¼t r sh þ h6¼s;h6¼u r ht n=2 P b¼ P : h6¼t ruh þ h6¼s;h6¼u rht n=2
ð16Þ
ð17Þ
The coefficient matrix is clearly nonsingular and therefore system (15) has a unique solution ð^rst ; ^rut Þ ¼ Q1 b which is the single stationary point of the function q(rst, rut) . The function q(rst, rut) is convex, as it is a sum of convex functions, so that the stationary point is a global minimum point. The convexity of q(rst, rut) can be also directly checked by its (constant) hessian matrix 2ðn 2Þ 2 H ¼6 ; ð18Þ 2 2ðn 2Þ which is positive definite for n > 3. If ð^rst ; ^rut Þ belongs to [0, 1]2, it is the solution of problem (14). As in the onemissing comparison case, when the available judgements are seriously inconsistent, it may happen that 2 ð^rst ; ^rut Þ 62 ½0; 1 . In this case the constrained problem should be solved by means of a quadratic programming algorithm [8]. 3.3. The p missing comparisons case As in the previous cases, let us associate a variable rst to each MC {As, At}. If p comparisons are missing, then the matrix R contains 2p unknown elements, due to its reciprocity. In this general case, analogously to the two-dimensional one, it is necessary to distinguish between two different type of MCs. The first one, which we call independent, occurs when in a MC {As, At} neither of the alternatives As and At is involved in any other MC. That is: {As, At} \ {Ai, Aj} = ; for any other MC {Ai, Aj}. In this case the optimal value ^rst of the variable rst is simply given by (13).
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We now focus on sets of dependent MCs; we call dependent the MCs of a set such that for every partition into two subsets, there exists at least one alternative which is in both subsets. As an example, {{A1, A2}, {A2, A4}, {A4, A7}} is a dependent set of MCs, while {{A1, A2},{A2, A4},{A5, A7}, {A5, A8}} is not, as the two subsets {{A1, A2}, {A2, A4}} and {{A5, A7}, {A5, A8}} do not contain a common alternative. It is possible to clarify this definition of dependency with the help of graph theory. If nodes are associated to alternatives and arcs to MCs, then a set of dependent MCs corresponds to a connected (sub)graph. As a consequence of these definitions, the p MCs can be divided in a certain number of independent MCs and some disjoint sets of dependent MCs. As mentioned before, it is easy to find the optimal value of a variable rst associated with an independent MC, as it can be calculated independently from the other variables. Analogously, we can calculate the optimal value of the variables associated with a set of dependent MCs independently from all the other variables not referring to that set. In the following, we describe how to calculate the optimal values of a set of m 6 n 2 variables associated with a set of dependent MCs. In order to simplify the indices notation, we can ignore all the other p m MCs not in the dependent set, assuming that only those m comparisons are missing in the problem. We denote the m pair of alternatives corresponding to the m MCs by fAsk ; Atk g; k ¼ 1; . . . ; m. Then q is a function of the m variables xk ¼ rsk tk ; k ¼ 1; . . . ; m. Without loss of generality, we assume, that sk < tk. The matrix R has 2m unknown elements: the m variables rsk tk ; k ¼ 1; . . . ; m above the diagonal and the m corresponding reciprocal elements below the diagonal. For each variable rsk tk , let us denote with IRþ sk and IRsk the sets of column indices of the other unknown elements which are on the same row sk respectively above and below the diagonal. Analogously, let us denote with IC þ tk and IC tk the sets of row indices of the other unknown elements which are on the same column tk respectively above and below the diagonal. The example in the next subsection will clarify this notation. 3.3.1. Consistency optimization Analogously to the one and two-dimensional cases, in order to solve the problem min qðrs1 t1 ; . . . ; rsm tm Þ; m ðrs1 t1 ;...;rsm tm Þ2½0;1
ð19Þ
let us calculate the m partial derivatives of q and set them equal to zero. From (12) and (13) we have oq ¼ 0; orsk tk X X r sk h rhtk þ n=2 ¼ 0 ðn 2Þrsk tk h6¼tk
h6¼sk
and, after some calculation, X X X X r sk h þ rhsk rhtk þ rtk h ðn 2Þrsk tk h2IR s
h2IRþ s
¼
CardðIR sk Þ
þ
CardðIC tk Þ
h2IC t
h2IC þ t
k
k
k
k
X
þ
h62ðIRþ s [IRs [ft k gÞ k
X
r sk h þ
k
rhtk n=2:
ð20Þ
h62ðIC þ t [IC t [fsk gÞ k
k
Taking (20) for k = 1, . . . ,m, we impose the m derivatives oq=orsk tk equal to zero, thus obtaining a linear system with m equations in the m variables rsk tk ; k ¼ 1; . . . ; m : Qx ¼ b;
ð21Þ
where 2
n2 6 q 6 21 6 6 ... Q¼6 6 ... 6 6 4 qm11 qm1
q12 n2
q13 q23
qm12
n2
qm2
qmm1
3 q1m q2m 7 7 7 ... 7 7; ... 7 7 7 qm1m 5 n2
ð22Þ
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2
r s1 t 1
6 6 6 6 6 6 x ¼ 6 r sk t k 6 6 6 6 4
3 7 7 7 7 7 7 7; 7 7 7 7 5
2
6 6 6 6 6 6 b ¼ 6 bk 6 6 6 6 4
r sm t m bk ¼
CardðIR sk Þ
8 n2 > > > < 1 qkh ¼ > þ1 > > : 0
b1
3 7 7 7 7 7 7 7; 7 7 7 7 5
bm þ CardðIC tk Þ þ
X h62ðIRþ sk [IRs [ft k gÞ k
X
r sk h þ
rhtk n=2;
h62ðIC þ t [IC t [fsk gÞ k
k
if h ¼ k; þ if th 2 IRþ sk or sh 2 IC tk ; if th 2 IR sk or sh 2 IC tk ; otherwise:
It can be observed that in the above definition the conditions of four cases are disjoint and that the definition itself can be equivalently be written in the simpler form: 8 n 2 if h ¼ k; > > > < 1 if sh ¼ sk or th ¼ tk ; qkh ¼ > if sh ¼ tk or th ¼ sk ; > > þ1 : 0 otherwise: Note that all information on the known comparisons are summarized in vector b, while matrix Q = [qi j] encodes the interdependence structure of the m MCs, with qij 2 { 1, 0, 1} for i 5 j and qii = n 2 "i. The following theorem on the optimal values of the missing comparisons is the main result of the paper. Theorem 1. If m 6 n 2, then (a) Matrix Q is nonsingular and therefore system (21) has a unique solution given by ^x ¼ Q1 b. (b) Solution ^x is the global minimum point for function q(x).
Proof Pm (a) From the assumption m 6 n 2, it follows that Q is strictly diagonally dominant,1 j¼1j6¼i jqij j < m 6 n 2 ¼ jqii j for all i. A strictly diagonally dominant matrix is nonsingular (Levy–Desplanques theorem) and therefore invertible [15]. Then system (21) has a unique solution given by ^x ¼ Q1 b, i.e. the unique stationary point of q(x). The first part of the theorem is then proved. (b) From the definition of qhk, it follows that qhk = qkh. Then Q is a symmetric matrix, strictly diagonally dominant, with all the diagonal elements positive; this is sufficient to conclude that Q is positive definite. The hessian matrix of q(x) is itself positive definite, being a positive multiple of Q. Then q(x) is a strictly convex function and the stationary point ^x is its global minimum point. h As in the previous cases, when the available judgements are seriously inconsistent, it may happen that ^x 62 ½0; 1m . Also in this case the minimum point in [0, 1]m can be obtained by solving a quadratic programming problem.
1
A square m · m matrix Q = [qij] is called diagonally dominant if jqii j P dominant if strict inequality holds.
P
j6¼i jqij j; i
¼ 1; . . . ; m. The matrix is called strictly diagonally
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It is well known that it is possible to obtain a numerical result not belonging to the a-priori chosen feasible set of values. This is unavoidable when dealing with inconsistency and with a bounded scale. Clearly the problem does not exist when an open scale is used [2,7]. On the other hand, despite the elegant mathematical results, every unbounded scale yields serious drawbacks in practical applications, but we do not want to dwell deeper on this issue. Example 1. As the notation required by the general p-dimensional case described above is rather complex, we clarify the preceding issues by the following example. Let us consider a case with nine alternatives {A1, A2, . . . , A9}; we do not take into account the known entries in the 9 · 9 matrix, but we focus on the following five MCs with their corresponding variables Missing comparison
Variable
k
{A1, A5} {A1, A9} {A3, A5} {A5, A7} {A5, A9}
r15 = x1 r19 = x2 r35 = x3 r57 = x4 r59 = x5
1 2 3 4 5
In the resulting PCM R we indicate only the unknown entries 3 2 r15 r19 7 6 7 6 7 6 7 6 r35 7 6 7 6 7 6 7 6 7 6 R ¼ 6 1 r15 1 r35 r57 r59 7: 7 6 7 6 7 6 7 6 7 6 1 r 57 7 6 7 6 5 4 1 r19 1 r59 The left hand 8 7r15 > > > > > > < r15 r15 > > > > r15 > > : r15
ð23Þ
side part of system (21) becomes r19
r35
þr57
þr59
¼
7r19
0
0
r59
¼
0
7r35
þr57
þr59
¼
0
þr35
þ7r57
r59
¼
r19
þr35
r57
þ7r59
¼
ð24Þ
Clearly, we cannot write the vector b on the r.h.s. of the system unless we assign all the known entries. The coefficient matrix of the system (24) is 3 2 7 1 1 1 1 6 1 7 0 0 1 7 7 6 7 6 7; ð25Þ 1 0 7 1 1 Q¼6 7 6 7 6 4 1 0 1 7 1 5 1
1
1
1
7
which is symmetric, positive definite and therefore nonsingular. With the help of the known entries of PCM R (whatever they are) is then possible to calculate vector b, so that the solution of the system (24) is given by ^x ¼ Q1 b.
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4. Numerical results We propose some numerical experiences on a problem, concerning the choice of a job, which has been proposed and studied by Saaty in [18], page 85. The pairwise comparison matrix obtained by Saaty is transformed, by means of (5), into the equivalent additive matrix 3 2 0:5 0:5 0:5 0:8155 0:5 0:3423 6 0:5 0:5 0:6577 0:8155 0:5 0:3423 7 7 6 6 0:5 0:3423 0:5 0:8662 0:75 0:3423 7 7 6 R¼6 ð26Þ 7: 6 0:1845 0:1845 0:1338 0:5 0:25 0:25 7 6 7 4 0:5 0:5 0:25 0:75 0:5 0:25 5 0:6577 0:6577 0:6577 0:75 0:75 0:5 The inconsistency index (10) of R is q = 3.78. Now, let us assume that the decision maker is not able to perform the comparison between alternatives A2 and A3. Then R is incomplete, as the two elements r23 and r32 = 1 r23 are unknown. We complete R using our method, i.e. by minimizing q(r23), as explained in Section 3.1. We obtain r23 = 0.4248 and the inconsistency index of R becomes q = 2.478. If, in addition to {A2, A3}, the comparison {A3, A5} is missing as well, we analogously calculate the optimal values of r23 and r35 by solving (15). The obtained values are r23 = 0.4726 and r35 = 0.559 with the inconsistency decreased to q = 1.6577. Finally, if we consider as missing also a third comparison {A4, A6}, we have to solve the system (21) to calculate the optimal values of the three variables, obtaining: r23 = 0.4726, r35 = 0.559 and r46 = 0.0074. The matrix R, completed with these values, is very close to consistency, being q = 0.245. Note that in all the cases mentioned above the obtained values belong to [0, 1]. 5. Conclusions and final remarks We think that our proposal is a natural way to solve the problem of missing data in pairwise comparison. We propose to complete the PCM coherently with the available judgements by directly referring to the definition (3) of consistency. From the computational point of view our method is rather simple: to calculate the optimal values, we only have to solve a nonsingular linear system. Our future research effort will be directed at comparing our method with other approaches by suitable numerical experiments, in order to highlight different properties/characteristics. By means of (10) we define the global inconsistency index q of a PCM R simply by summing all the local inconsistency contributions Lijh. If we are interested in defining an index independent of the order n of the matrix, it is clearly possible to apply the necessary normalization. By taking into account only the non trivial P n elements in (10) and dividing by their number, a mean value of Lijh is obtained, L ¼ i
M. Fedrizzi, S. Giove / European Journal of Operational Research 183 (2007) 303–313 [7] [8] [9] [10] [11] [12]
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