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The synthetic hierarchy method: An optimizing approach to obtaining priorities in the AHP
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
ELSEVIER
European Journal of Operational Research 93 (1996) 550-564
Theory and Methodology
The synthetic hierarchy method: An optimizing approach to obtaining priorities in the AHP Stan Lipovetsky * Faculty of Management, Tel Aviv Unioersity, Ramat Aviv, Tel Aviv 69978, Israel
Received 9 September 1994; revised 22 February 1995
Abstract
A new method of synthesizing local and criteria priorities into global priorities is suggested. This approach is a development of the Analytic Hierarchy Process enabling the united consideration of all horizontal and vertical connections of a hierarchical system in a single optimizing objective function based on statistical models of the synthesis process. The solution can be reduced to a linear system or to an eigenproblem of a special matrix constructed as a combination of Kronecker's sums and products of pairwise judgement matrices. A numerical example shows that the optimizing approach produces a ranking of global priorities that may be different from the ranking produced by the classical AHP. Keywords: Multiple criteria; Decision analysis; AHP; Synthesizing priorities; Statistical optimization
1. Introduction
The Analytic Hierarchy Process (AHP) has found widespread application in decision making problems involving multiple criteria in systems of many levels, and for priority-modeling in complex systems. These uses are enabled by the methodological strength of AHP: rendering a complex system into the form of a structured hierarchy. The levels of the hierarchy describe a system from the lowest level (sets of alternatives), through the intermediate levels (subcriteria and criteria), to the highest level (general objective). Using the AHP methodology, priorities of alternatives are estimated independently for every criterion at each level. The weight (or priority) of each criterion is defined by the same AHP procedure. Afterwards, summing the priorities of every alternative with the weights of every criterion creates a composite priority for the highest level - this procedure is called the hierarchical composition, or synthesis. It yields the overall priorities of alternatives on each successively higher level as a linear combination of the subpriorities derived for the previous level. Such summing through the whole hierarchical structure produces a synthesized judgement for all alternatives under the stated goal (see Saaty (1980), Saaty (1994), Harker (1989), Forman (1993)). The strongest features of the AHP are that it generates numerical priorities from the subjective knowledge inherent in the estimates of
* Present address: 71-11 Yellowstone Blvd., #6-E, Forest Hills, New York, NY 11375, USA. 0377-2217/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSD1 0377-2217(95)00085-2
s. Lipovetsky/ European Journal of Operational Research 93 (1996) 550-564
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pairwise comparisons matrices, and that it organizes a complex sYstem into a number of ordered subsystems arranged in a hierarchy. The aim of the current work is to modify the AHP procedure to obtain all partial and global priorities in a simultaneous and connected estimation. We achieve this by optimizing an objective function. Earlier attempts of this kind can be found in Chu et al. (1979), Jensen (1983), Jensen (1984), Triantaphyllou et al. (1990), Triantaphyllou and Mann (1994), Schoner et al. (1992), Schoner et al. (1993), Basak and Saaty (1993) (See also Zahedi (1986), Saaty (1994)). We term our approach the Synthetic Hierarchy Method, SHM, because it obtains a single united priority vector describing all hierarchical levels of a complex system. (This method should be distinguished from SHP the Systemic Hierarchy Process - a global approach to MCDM, suggested by Marzen (1994)). A development of the AHP priority estimation, the SHM yields global priorities which may differ from those given by the classical AHP. This paper is structured in the following way. Section I is an introduction. Section 2 describes the AHP methodology for the synthesizing local priorities into global priorities, and introduces the technique of a weighted matrix of united priorities. Section 3 presents the SHM approach. Introduced in this section is a statistical model encompassing all priorities. This model allows the formulation of an objective function that can be reduced to a linear system of equations or to an eigenproblem with a combined matrix of Kronecker's kind. There is a discussion of the different properties of these solutions, and of the place of the classical AHP solution. Additionally, for the purpose of comparing the described methods, in Sections 2 and 3 each method is applied to the same numerical example. Section 4 suggests several different directions in which the SHM may be developed further. Section 5 is a summary.
2. A generalized description of priority-setting for alternatives and criteria In order to make the forthcoming theoretical derivation easier to follow, let us begin with a numerical example discussed by Saaty and Keams (1985), Chapter 3 and by Saaty and Vargas (1994), Chapter 1. They posed the problem of the "Best House to Buy". On the upper level was the goal "satisfaction with house". There were three houses (alternatives) to choose from, and the choice was made according to eight criteria. Local priorities obtained by the AHP for each of the eight criteria, stacked in the priority matrix a , are set out in the following columns: 0.754 u = | 0.181 ~0.065
0.233 0.055 0.713
0.754 0.065 0.181
0.333 0.333 0.333
0.674 0.101 0.226
0.747 0.060 0.193
0.200 0.400 0.400
0.072 0.650) • 0.278
(1)
0.167
0.333).
(2)
The vector of preferences among these eight criteria is: /3'=(0.173
0.054
0,188
0.018
0.031
0.036
Priorities by each criterion are normalized by one, i.e., the sum of the elements in each column in (1), or the row in (2), equals one. The elements of (2) are the weights of the columns in (1). In order to compare the elements of these eight columns among themselves, we need to multiply them by their weights. This yields a matrix of weighted columns:
G=
0.130 0.031 0.011
0.013 0.003 0.039
0.142 0.012 0.034
0.006 0.006 0.006
0.021 0.003 0.007
0.027 0.002 0.007
0.033 0.067 0.067
0.024~ 0.216 . / 0.093
(3)
552
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
In order to establish the composite priorities for the three alternatives, we make a summation of the weighted scores of these alternatives by all criteria, i.e., we calculate the sum in each row of matrix (3): 7r' = ( 0 . 3 9 6
0.341
0.263).
(4)
Thus, by these synthesized priorities 7r, the most preferable alternative is no. 1. Next comes no. 2, and no. 3 is the least preferable of the three alternatives. Now let us present a more generalized description of this process of synthesizing priorities. Suppose, we have i = 1,2 . . . . . n alternatives and j = 1,2 . . . . . m criteria (n = 3 and m = 8 in the example given above). For each jth criterion, we use the corresponding matrix A (j), of order n × n, for pairwise comparisons among n alternatives, in order to estimate a vector of priorities a {j) by the A H P eigenproblem:
A(J)oL(j)
= A(J)a (j),
j = 1 . . . . . m.
(5)
The priority vector a w corresponds to the maximal eigenvalue A°) for the jth criterion. Similarly, we use the matrix B, of order m × m, for pairwise comparisons among all rn criteria, in order to obtain a vector of criteria priorities. To this end, we solve the A H P eigenproblem for the maximal eigenvalue /x: B/3 =/z/3.
(6)
Then we gather all priority vectors a {j) (5) in a priority matrix: = (
.....
.....
(7)
that is the n × m matrix corresponding to (1) in the example. We present the vector of criteria priorities in the following form: /3 = d i a g ( / 3 ) e,,,
(8)
where the diagonal matrix contains all the elements of this vector/3, i.e., diag(/3 ) - diag{/31 ,/32 . . . . . /3j . . . . . /3m},
(9)
and e,, is a uniform vector of the ruth order: e'-
(1,1 . . . . . 1).
(10)
(8) corresponds to (2) in the example. (7) is a transposed stochastic matrix, because in accordance with the AHP, all priority vectors are normalized by one (the same with the vector/3), i.e., Or} j) = 1,
j = 1 . . . . . m,
(lla)
i=1
/3j= 1.
(llb)
j=l
The matrix of weighted priorities G (corresponding to (3) in the example) can be presented as the product of (7) and (9), i.e., ...
/
(12)
G = adiag(13) =
!
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
553
We also introduce the following notations for (12):
-=
.
(13)
where each ijth element of the matrix G is T} j) =
(14)
ol}J)[3j.
We can also present (13) as consisting of the vectors-columns
G - (T(O,T (2). . . . . T (j) . . . . . 3,(m)),
(15)
where these vectors are the priority vectors in (7) weighted with the criteria priorities flj, i.e.,
T(J)-a(J)/3j,
j=l
. . . . . m.
(16)
(13) can be used as a unified presentation of priority vectors ct O~ and criteria priorities t , instead of using all these partial vectors themselves. Indeed, to solve the AHP, we need to obtain a matrix a as in (7) with n X m parameters, plus m parameters in the vector fl as in (8), minus (m + 1) normalizing conditions, (1 la) and (l lb), i.e., for global priorities we have to estimate n m 1 free parameters. In (13), we have n x m parameters T} j), and with one normalizing condition for them, the number of free parameters becomes n m - I . Each column T (j) of the matrix G is proportional, with the term /31., to the vector ct (j) (16), i.e., all eigenproblems (5) can be presented as the following eigenproblems: A(J)T(J) = ~(J) y(J),
j = 1 . . . . . m.
(17)
After normalizing by one, that is, setting
Y'. T} j)= 1,
j= 1..... m
(18)
i=l
the vectors T ti~ for maximal eigenvalues A(j) will coincide with the vectors a u; of (5). Concerning the vector t , we can see that summing elements in each jth column of the matrix G (12) and (13) yields:
iffil
iffil
i=1
i.e., these sums are proportional to the elements of (8). With the normalizing conditions of (1 la) in (19), the vector fl coincides with the vector whose elements are the sums of the elements in the jth column of (13), flj= L T[ j),
J = 1 . . . . . m,
(20)
i=l
or in matrix notation,
f l = G'e.,
(21)
where e. is a uniform vector with n elements. Thus, (6) can be written in the form:
(
8 E v ? ) , E v [ " ..... E r r " ) i
i
i
)'
)'
=~
T}I).E3'} 2). . . . . E T [ '') , "
i
i
(22)
554
S. Lipovetsky / European J o u r n a l o f Operational Research 93 (1996) 5 5 0 - 5 6 4
where the vector of priorities corresponds to the maximal eigenvalue ~. Now we can consider the global, or synthesized, priorities. In the numerical example above, (4) can be constructed either as the product of (1) and (2), or as the sums of the elements in each row of (3). In the same way, according to (7), (8), (9) and (10), the global priority vector 1r equals the product of (7) and (8), that is zr= a/3 = adiag( f l ) e , , = G e m ,
(23)
where G is given by (12). Eq. (23) says that the vector of global priorities can be presented simply as the vector whose elements are the sums of the elements in the corresponding rows of (13): '77"i = ~ y~J), j=l
i = 1 . . . . . n.
(24)
Thus, we can summarize: For a matrix G of n × m coefficients (13), the columns of this matrix can be considered as priority vectors for each criterion (17). Sums of elements in each column of G can be applied for obtaining the vector of priorities among criteria (22). Sums in each row of G yield a vector of global preferences among all alternatives (23) and (24).
3. The synthetic hierarchy method: Minimization of the SHM objective function Now we shall describe the method for estimating all elements of (13) in the optimizing approach. This method corresponds to the solution that, in the introduction, we termed the Synthetic Hierarchy Method - the SHM. Let us construct an objective function for the united priorities of the SHM. First, we must recall that all elements of pairwise comparison matrices are defined as ratios of the corresponding priorities. By this premise, the following relations pertain: a(~) i,
-a~i) --
~j
-7(j), -
i,k=
i=1
(25a)
1 . . . . . n,
j,f=
(25b)
1 ..... m.
i=l
In (25a), ~-i,'~(J)means the ikth element of the jth matrix, A (j) (consisting of pairwise comparisons among all n alternatives according to the jth criterion). (14) is used for proportions between local priorities a~ J) and weighted local priorities y~J). In (25b), b j / means the j f t h element of the matrix B of, pairwise comparisons among all m criteria, and (20) is used for proportions between criteria priorities. The elements of empirical matrices of pairwise comparisons are inconsistent with (25a) and (25b). We propose the following model of connections between elements of the matrices and the corresponding priorities: a(J),,(J) = y[i) + ,:(J) ik I k ~ik '
bj/Y'- YY) = E y~i) + 6jr, i
(26a)
i,k = 1 " " " " ,n,
j,f=
1 . . . . . m.
(26b)
i
In (26a) and (26b) all coefficients y denote estimators for theoretical values of weighted priorities, and ¢~,J) and 61/ denote deviations from (25a) and (25b), expressed in linear form. Without additional assumptions concerning the distribution of the deviations in (26a) and (26b), we can apply a simple least-squares (LS) function to minimize these deviations: LS =
IIEII: + 116112 =
t,~(J).,(y) ( " i k r , - y/i)): + Z j=l
i,k=l
j,f~
bi~'Y'- y(~') _ i=1
TO) i=l
"~ min.
(27)
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
555
Because of the homogeneity of coefficients y in (27) we need to add a normalizing condition for them. It can be the simple condition
i=l j = l
Thus, the conditional problem for minimization is as follows:
F=-LS-2A(~
~y}J)-l)-*min, j=l
(29)
i=1
where LS is given by (27) and k is a Lagrange multiplier. The first order conditions 0F
Oy}p )
O, t
1 ..... n, p
1 ..... m
(30)
yield a linear system of n × m equations: tl
n
Y'.(alf)~'[P)- r,( p ) )-.~ ( p ) + y'(r}p)-..(p).,(p)~,,., ) i
k
+
bjp j
"y/(P)-
7 (i) bjp +
i
"
r/(p) - bp/ f
y/(/) = k,
(31)
"
that can be simplified to the following system for all indices t and p: n + ~ff~(a~)): i
+ m+
"
( e ) - ~[(a~,.p' + a ~ ) ) y / P ) l Tt
.
p 2 EY/P)i
j +bjp
y(i) = k. •
j
(32)
"
Let us represent this linear system in matrix notation. At first, we observe that because of the reciprocal symmetry of Saaty's pairwise comparison matrices ( a i t a t i = 1, bjpbpj - 1 ) , it is possible to write the relations: n
(p)
. + ~-~.(a~tP))2= Y'.(ai, i
(p)
(p)
+ a . )air,
(33a)
i
m
m + Y~ (bjp) 2= ~(bjp +bpj)bjp. J
(33b)
J
Now we introduce the symmetric matrices
A-(p) = A (p) + (A(P)) ',
p =
1 ..... m,
(34a)
= B + B'.
(34b)
Using (33a), (33b), (34a) and (34)b we unite for all t = 1. . . . . n the equations of (32), and present this system as follows:
diag(A-~e)A(P))y(P)-A-Ie)y(P)+
m + Y~ j
Jny(P)/
[(bpj+bjp)Jny(J)]=Ae.,
p = l . . . . . m.
j
(351
556
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
where J, is the matrix of the nth order whose elements all equal 1, i.e., J , = e,e',, where e, is a uniform vector of the nth order; 3, tp~ are the same vectors 3,°) . . . . . y(r,) as in (15), and diag denotes a diagonal matrix cut from the matrix in the brackets. All m equations (35) can be united into block matrices as follows:
I [diag( A-(I)AO)) -
0 }
0
...
diag( A-~2)A(2)) - A-(2)
...
0
0
...
diag(A-(,,)A(,,)) _ ~'(,n)
A- ( °
-(b,2 + b2,)J n
+
-(b~2 + b21)J.
m+
- (/,.. + b..)J.
[ ¢') / y(,,) ]
-(022+b22)
-(b2m
+ b,,,2)J n
J,
...
- ( b,m + b,n,)Jn
...
-(b2,,+bm2)J,
...
m+
-(bmm+bmm
J
[e,,
t
(36)
e,
Using (33a), (33b), (34a) and (34b), we introduce the set of matrices A"(j) = diag( A-IJ)A (i)) - A-~j),
j = 1 ..... m,
/~ = d i a g ( B B ) - B.
(37a) (37b)
Eq. (36) can now be written: S y - - Ae.,,,
(38)
where y is the vector, of n m t h order, of united priorities y = (yo),,y(2~ . . . . . y o . ~ ) ' ,
(39)
and e.,, is a uniform vector of the same order. S is a combined hyper-matrix constructed as follows: S=-ATI)~A'12)~
.-.
O A ' ( ' n ) + / ~ ® J n.
(40)
Its first part is a block-diagonal matrix, or the Kronecker sum of (37a),
Its second part is the Kronecker product of (37b) and the matrix J,. (On the properties of Kronecker sums and products - see Barnett (1990). Theil (1971).) Expression (40) can be written in the form: S=.4+B®J.,
(42)
corresponding with the two parts of the left-hand side of (36). It is interesting to note that (42) is symmetrical because its blocks ((37a) and (37b)) are symmetric matrices (due to (34a) and (34b)). This is in contrast to Saaty's pairwise comparison matrices A t j) and B, whose elements are reciprocally symmetrical. Solution of (38) yields a vector of weighted priorities for the SHM: y= AS-le,m.
(43)
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
557
With the normalizing condition of (28), expressed in vector notation, e'.,. 3' = 1, we determine the value of the term h=(e'..,S-le.,.) -l. Let us consider some properties of (43). Summing the elements in each subvector y(J) (39), we obtain the weights /3j (see Eq. (20)) of preferences among m criteria, and using them to normalize the subvectors y(J). we obtain the local priorities c~°) (see (14) and (16)) of the alternatives by all criteria. The synthesized priorities 7ri are determined by the same summation (24). If instead of a single normalization condition 28), we use m conditions (20) with constants /3j (these conditions ensure the fulfillment of m conditions (1 la) for local priorities), then instead of (29), we obtain the function
F= L S - 2 ~ Aj( ~ A~J)- ~j) ~ Iin j=l
(44)
i=1
with the same least square function (27). The derivation (30) till (38) with the objective function of (44) yields the following solution:
3' =
" y(")
= S-t
•
,
(45)
~ A~'en
with the uniform vectors e, of the nth order, and the same matrix S (see (42)). Thus, for each subvector 3'(J), solutions by (45) coincide with those of (43) up to the normalizing terms. If (41) is a nonsingular matrix, we can present S - l of (42) in the following form (see Barnett (1990), page 125, for the inversion formula). Suppose we solve an eigenproblem Bfi = A~, and present the matrix /~ in the form /~ = 17/117', where 17 is a matrix of the eigenvectors fi in its columns, and A is a diagonal matrix of the eigenvalues A. Using the matrix V = 17~1/2, we can write the following presentation of (42):
B®J. = (VV') ® ( e~e') = (V ® e.)(V' ® e') = (V ® e.)(V ® e~)'--RR', where e. is a uniform vector of the nth order; R denotes the matrix V ® e. of (nm) X (m)
(46)
order, constructed from matrix V using the properties of the Kronecker product and of its transposition. From (42) we can write the expression:
(47)
S - ' = ( .4 + R R ' ) - ' = .4-' - . 4 - ' R ( E , , + R ' . 4 - L R ) - ' R ' A - ' , where E,. is a uniform matrix of the mth order. Then (43) has the form: 3, =
AS-'e.,. = h/~-lento
(48)
-- A . ~ - I R ( E m + R ' t ~ - i R ) - I R ' A - l e . m .
The first item ?~- ~ e.,. with block-diagonal matrix ?~- 1 coincides with the set of separated solutions for local priorities. The second item corresponds to the connection of local priorities to criteria priorities. Together they produce the united vector 3' of weighted priorities. For the same numerical example that is used in Section 2 (see (1)-(4)), we calculate the solution for (43). It yields the matrix G of (13): G=
0.111 0.020 0.012
0.009 0.004 0.041
0.094 0.010 0.017
0.010 0.010 0.010
0.032 0.006 0.009
0.036 0.004 0.006
0.018 0.036 0.036
0.038 0.316 ,J 0.115
(49)
with sums in the columns of (49): /3'=(0.143
0.054
0.121
0.030
0.047
0.046
0.090
0.469),
(5o)
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
558
and with sums in the rows of (49): zr'=(0.348
0.406
0.246).
(51)
Dividing the elements in the columns of (49) by the corresponding sums of (50) we obtain the local priorities: a=
0.776 0.140 0.084
0.167 0.074 0.759
0.777 0.083 0.140
0.333 0.333 0.333
0.681 0.128 0.191
0.783 0.087 0.130
0.200 0.400 0.400
0.081 0.674 .J 0.245
(52)
The priorities of alternatives in column a coincide in their ranking with priorities in (1). But (50) and (51) differ in their ranking from the estimations produced by the classical AHP ((2) and (4)). We shall use the term inverted matrix SHM solution to denote (43), obtained by solving (38). Instead of the normalization in (28), we can take the condition
i=l j = l
that corresponds to normalizing by the variance of the vector 3' elements. With (53), (29) transforms into
F=LS-A[~(%(J))2-1]~min=1.i=lj
(54)
By the same derivation (30) till (38), we obtain from (54) the eigenproblem: s3' = A3',
(55)
with the same hyper-matrix (42). Eigenvalues A~,A2. . . . . A,,, (all of them are non-negative) correspond to the extreme values of the quadratic form LS (see (27)) conditioned by (53). The eigenvector 3' which relates to the minimal eigenvalue, A,,i,, gives the solution to (54), with F,,i, = A,,i,. This solution 3' for A,,~,, we shall call
the eigenvector SHM solution. For (1)-(4), (55) yields the following SHM G-matrix (additionally normalized by (28) for the sake of comparison): [0.114 G = [ 0.010 0.007
0.004 0.002 0.044
0.097 0.006 0.008
0.009 0.009 0.009
0.036 0.003 0.003
0.037 0.002 0.003
0.007 0.033 0.033
0.028/ 0.462 ). 0.033
(56)
0.042
0.073
0.523),
(57)
Summing elements in the columns of (56) yields: /3'=(0.131
0.050
0.111
0.027
0.042
and summing elements in the rows of (56) yields: 7r'=(0.332
0.527
0.140).
(58)
Dividing the elements of columns (56) by their sums (57), we produce the local priorities matrix: =
0.870 0.076 0.053
0.080 0.040 0.880
0.874 0.054 0.072
0.333 0.333 0.333
0.857 0.071 0.071
0.881 0.048 0.071
0.096 0.452 0.452
0.0541 0.8831. 0.063]
(59)
Both SHM solutions, the inverted matrix results, (49)-(52), and the eigenvector results, (56)-(59), are very similar. They demonstrate the same ranking order for local priorities a , weighted local priorities G, priorities among criteria /3, and global priorities ~. But the eigenvector SHM solution, obtained with (53) for variance of elements y~J), produces results with more variability among all coefficients. In other words, as can be seen by comparison of these two solutions, large (or small) elements in (49)-(52) become larger (or smaller) in
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
559
(56)-(59). Thus, the first SHM solution with the inverted matrix of (43) is simpler to calculate, but the second SHM solution with the eigenvector of (55) more clearly expresses the preferences among all objects under consideration. Let us consider how to explain the similarity of the results of these two variants, the inverted matrix SHM solution of (43) and the eigenvector SHM solution of (55). Suppose D is a non-singular positively defined matrix of the qth order which presents us with two problems - a linear system with a uniform vector on the right-hand side, O a = e,
(60a)
and an eigenproblem, O b = Ab.
(60b)
Eq. (60b) yields a set of eigenvalues A1 > Az > • - • > A0 > 0, and a set of eigenvectors b I . . . . . spectral decomposition of D is t
D = Aiblb' I + A2b2b' z + • "
bq. T h e
(61)
+Aqbqbq,
where each vector is normalized by the condition b ' b = 1. The inverted matrix can be presented in the following spectral form: 1
1
1
D - l = _ _ b i b , I + _ _A:2 bzb'2 + • • . + - - b q b q , ~
aI
Aq
because for )tq << Eq. (60a) gives
)tt,t-~
=
~q
b q b q, ,
(62)
1 . . . . . q - 1, the main input in D - l is given by the last item in (62) with a,.i. = h.q.
(b;e)
q a=O-le
1
t =12"- - ' ~ -
(b'qe)
A----~b q ,
b, =
(63)
where (62) is used for the matrix D-1. Thus, the solution a of the linear system is proportional (up to the c o n s t a n t (ffqe)//Aq) t o the eigenvector bq for the minimal eigenvalue Aq. But in (55), the focus of our interest is just the eigenvector y for the minimal eigenvalue. This explains the closeness of both solutions, (49)-(52) and (56)-(59) for the united vector Y. It is interesting to note that synthesizing AHP priority in our example by the so-called ideal mode (when the priorities of the alternatives for each criterion are first divided by the largest value among them, then multiplied by the priority of the corresponding criterion and summed) produces the global priorities as follows (see Saaty and Vargas (1994), p. 17, Table 5): ~" = (0.584, 0.782, 0,461). Normalizing the sum of these values to one, we obtain the vector of global priorities 7r' = (0.320, 0.428, 0.252), that is very similar to the solutions of (51) and (58), and corresponds to the ranking of preferences as ~2 >" ~J >" 7r3. This differs from the ranking of preferences 7r I >- ~e >" qr3 given by (4). The SHM was obtained from (26a) and (26b). Let us show to which model (5) and (6) can be related. Because of the theoretical assumption that all elements aij of a pairwise comparison Saaty matrix A are the ratios of the corresponding priorities o t i / ~ j, instead of the models with additive errors ((26a) and (26b)), let us consider the following model for an empirical matrix A:
aijot j =
Cti(1 -I-
•ij)"
i,j=
1 . . . . . n,
(64)
where errors are included in the theoretical model in multiplicative form. Then, summing by the index j produces the system of equations ~ a i j a ~= j~l
n+
Eij a i, j=l
i = l . . . . . n.
(65)
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If we assume that the sums of errors in (65) are equal for all i, E-
~Eij=const,
i = l . . . . . n,
(66)
jffil
then (65) reduces to the problem
Act = Act,
(67)
where A denotes an unknown parameter with the property A - n + E that expresses the deviation from the exact theoretical value A = n. (67) coincides with the classical AHP eigenproblem for the right eigenvector with maximal A of the pairwise comparison matrix A. Thus, the AHP corresponds to the model with multiplicative errors (64) on the assumption that summarized relative deviations are equal (66) for all rows of the matrix A. In other words, the AHP does not minimize all partial deviations Eij but equalizes sums of deviations in the rows of matrix A. In the same way, it can be seen that the left eigenvector of the nonsymmetric matrix A, i.e., the vector ~ of the AHP problem A'~ = An corresponds to the equalizing of the sums of deviations in the columns of the matrix A of pairwise comparisons. The methods of evaluation of unknown parameters by the minimization of an objective function generally produce better estimates than methods based on the equalizing of relations among parameters. Thus, we can infer that (27), with conditions of (29) or (54), yields solutions more reliable than those generated by an AHP eigenproblem of the sort of (67). Further, (27) includes all unknown parameters T~j) and produces simultaneously a united vector giving all priorities. Moreover, because of the symmetry of (42), the SHM yields a unique solution, whereas the AHP method for each partial matrix of pairwise comparison gives two different solutions, i.e., the left and the right eigenvectors, a cumbersome feature of AHP solutions for practical applications.
4. Further extensions o f the S H M
The SHM is a suitable approach for many different applications. This section outlines some directions for its further development.
4.1. A nonlinear model Instead of using (26a) and (26b), it is possible to consider a nonlinear model that resembles more closely (25a) and (25b). Thus:
,,o) ~ik = ~kj) + e~ ),
i,k = 1 . . . . n,
i=1____~_.__ bj:= ~z.., T~:) + 8j:'
(68a)
J ' f = 1. . . . . m.
(68b)
i=1
This model is more appropriate to the definition stating that elements in matrices of pairwise comparisons are ratios of priorities among alternatives or among criteria. Instead of (27), the nonlinear least squares objective is:
NLS=I['II2q-[[~[[2= ~ j=l
k ~,~ik[a(J)--~J)//~/(J)) i,k=l
£ j,/=l
bj:-
T/0'
i=l
~T~:'
li=l
~min.
(69)
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The coefficients T~j) could be estimated directly by this function with (28) or (53) with the help of existing computer programs for nonlinear estimations. Such NLS problems have been considered for a pairwise comparison matrix and the results have been compared systematically with other variants of priority vectors, in Lipovetsky (1994a). As shown in that work, such nonlinear minimization can be described as a special kind of regression on dummy variables. Together with the priorities themselves, the standard deviations and t-statistics can be obtained without difficulty. In this approach, different hypothesized values of priorities can be checked and corrections can be made in the ranking order of alternatives. Within standard assumptions concerning the distribution of errors, we can apply the maximum likelihood approach to estimate the vector elements. The maximum likelihood approach can be used both for the linear ((26a) and (26b)) and the nonlinear ((68a) and (68b)) models. The variances and covariances of errors, estimated by the residuals in these models, could be included as weights for different parts of objectives ((27) or (69)). Powerful software is able to construct these seemingly unrelated kinds of equations.
4.2. Extension to additional data sets The model can readily be extended to include additional data sets. Often encountered in practice is the case of many judges who estimate the same, or different, pairwise comparison matrices. Such a case can easily be described by linear or nonlinear objective functions. For example, if T l judges 01 = 1. . . . . T I) estimate alternatives by the first criterion, T2 judges (t 2 = 1. . . . . T2) by the second criterion, etc., till T,, judges (t,, = 1. . . . . T,,) estimate the alternatives by the ruth criterion and Q judges (q = 1. . . . . Q) estimate the criteria priorities themselves. These could be different judges, but assuming the existence of a unique vector 3' of all needed priorities, we can very easily write such a model with an NLS (or LS) objective that is an evident extension of (69):
NLS= ~
Y'~ ~
,(a(J't~)ik- y~j)/y~j))2+
j = l tj=l i,k=l
b~q/) q=l j,gffil
y/(J) iffil
Y'. y/(/)
---) min.
(70)
li=l
(70) differs from the previous, simpler (69) only in the presence of additional sums corresponding to information given by additional judges. The structure of priority ratios, constructed from the elements of the united priority vector y, is the same both in (69) and in (70). If the same judges each independently estimate all pairwise comparison matrices, there also exists another possibility, that is, to consider three-way matrices of pairwise comparisons where the third direction corresponds to the different judges. Four-way matrices can also be generated. For example, the fourth direction may correspond to judgements made at different times, or under different conditions. For evaluations using many-way matrices, see Lipovetsky and Tishler (1994), Lipovetsky (1994c).
4.3. A free parameter The range of applications of the SHM can be enlarged by including a free parameter ~o, indicating the significance of the various levels of the system under consideration. For example, in (69), or in (27), we can manipulate the relative weights of the alternatives and criteria:
NLS = (1 - ~)IIEII2 +
~I16112.
(71)
This multi-objective function can be tried with different values of the parameter ~p from the range [0, 1], corresponding to the importance of the different levels of decision making.
4.4. Linear and nonlinear estimations with missing data Linear and nonlinear estimations of the SHM can be carried out using the data in the upper triangle of pairwise judgement matrices, because only these data are given directly by judges. We can use the approach
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
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described in Lipovetsky (1994a) to evaluate the priorities by this method. The structure of SHM models tolerates the absence of all kinds of values in the matrices of pairwise comparisons, and even the complete absence of some of these matrices. The summation (27), or (69) and (70), for such cases is carried out simply on the items corresponding to the elements that are all present. 4.5• Fuzzified matrices
The SHM can be easily extended for fuzzified matrices of pairwise comparisons. Such a Fuzzy-SHM is described in Lipovetsky (1994b). In that work, several types of distribution of a random variable and its inverse were assessed, and Cauchy distribution was found to be the most appropriate description for fuzzified pairwise comparison matrices. Solutions by AHP and by SHM estimating approaches were modified for the evaluation of priorities using fuzzified data. 4.6. Intermediate levels
The SHM method can be applied rather well to the case with many intermediate levels of subcriteria. Let us describe the basic structure of the SHM for a system of n alternatives (i = 1. . . . . n), m subcriteria ( j = 1. . . . . m), and T criteria (t = 1. . . . . T), with the corresponding matrices of pairwise comparisons among alternatives A~j't) (of the nth order) for each j and t, among subcriteria B ~° (of the m the order) for each t, and among criteria C (of Tth order)• The global synthesized ~riorities 7r~ . . . . . 7r, for all alternatives can be obtained in the process: a(1,1)
•..
Or(re'l)
O~(1,1 )
• ..
O/(re'l)
~i,T, ... ~-.~, tU))
7"1"1 ,B"2
t~(l)
/'2
yl+...+ O~(1,1 )
• • •
O ,(rn'l)
•
(72)
"YT '
t3~')
where columns Ol ( j ' t ) correspond to the vectors of alternative priorities a} j'° related to the matrices A (j't), vectors /3mcontain priorities of subcriteria fl)t) related to the matrices B~t); and elements Tt . . . . . Yr are priorities of criteria related to the matrix C. Instead of (12) we have now the following matrix X of weighted priorities:
#(~,l)
...
#(,,,i)
I
I
#(~.r)
...
#(,.,T)
#1.,)
...
#~.1)
I
i
#,.T)
...
#-.n
X=
I ~.~")
...
~.~")
..•
I
(73)
I I
~. ~''~>
...
~.~'~)
where each element of this matrix corresponds to the related weighted priority coefficient from (72), i.e., ~i~j'') =- a~J'013~')%,
i = 1 . . . . . n, j = 1 . . . . . m t = 1 . . . . . T.
(74)
By estimating the values of (73) we can easily find all other needed priorities. For each subcriterion j and criterion t, the local priorities ct~j't) are proportional to (74). With normalization to one ((1 la) and (1 lb)) for each .
(75a)
i=1
~ ~,'J")= ~ f l ) t ) , t = Y,i=l j=l j
(75b)
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Thus, the subcriteria priorities [3Jt; of (75a) are proportional (up to the common constant 3,,) to the sums in the columns of (73) with each given t. Criteria priorities y, are proportional to the sums of all subcriteria (75b). The global priorities 7ri for all alternatives i = 1. . . . . n are defined by (72). Thus they are equal to the sums of the elements in the rows of (73): "n"i =
•
sri(j''), i = 1 . . . . . n.
(76)
j=l t=l
This is an evident generalization of (24) to the summing of the weighted local priorities by all m subcriteria and all T criteria. From (26a) and (26b), extended for all hierarchical levels under consideration, we obtain the objective which is a clear generalization of (27): [ ~ik
bk
t=l j = l i , k = l
+
E t,r=l
t=l j , f = t
c,T E E~:i (~'T) i
j
"
i
~i(~'') -*rain, i
(77)
j
where a~i't), b)5) and Cr,. are elements of the matrices A (J't), B (t) and C, respectively. With normalizing conditions of the sort of (28) or (53) for all coefficients ~i(j't) (73), we obtain the generalization of (29) or (54) with (77). These problems can in turn be reduced to the linear system of (38) or to the eigenproblem of (55) with a combined hyper-matrix S of order ( n m T ) X (nmT) to estimate the united vector ~, expressed by stacking the columns of (73). This matrix S could be constructed as Kronecker's sums and products of appropriate initial judgement matrices of three hierarchical levels. From (68a) and (68b), extended for this case of subcriteria and criteria, we can obtain the generalization of (69):
~'ik
- ~i(
) + E
t = l j = l i,k=l
+ ~ t,';= l
b~ -
t=l j,f=l
c,~-
~i(J'') i
~i(j'~) j
)
~i(i'')
~i(t'')
i
-)min.
(78)
In the same way we can consider complex systems with four or more levels of hierarchy.
5.
Summary
This paper describes a methodology for combining and estimating all local and criteria preferences in a matrix of weighted priorities. This method has proven to be convenient for considering synthesized priorities from a statistical point of view. Different extensions of this approach may be applied to hierarchical systems of many levels, judgements of many judges, multi-criteria objectives, fuzzified data and other applications. Statistical models produce conditional objective functions, minimization of which (using Kronecker's technique of sums and products of matrices) yields linear systems or eigenproblems for the estimation of aggregated priorities. Nonlinear models and nonlinear estimation can generate not only priorities but also their standard errors (which can be used for checking hypotheses on the global preferences). The numerical example indicates that the synthesized global priorities obtained using an optimizing approach may have an order of preferences that differs from the results of the classical AHP. So at least, if a decision-maker is not satisfied with the results
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obtained by AHP estimation, the problem can be reconsidered by applying the Synthetic Hierarchy Method optimizing approach. In general, the SHM promises to be very useful for the burgeoning field of theoretical investigation and for practical applications in numerous MCDM problems.
Acknowledgements The author wishes to thank David Mendels for editing the manuscript.
References Barnett, S. (1990), Matrices: Methods and Applications, Clarendon Press, Oxford. Basak, 1. and Saaty, T. (1993), "Group decision making using the analytic hierarchy process", Mathematical and Computer Modelling, 17(4-5) 101-109. Chu, A.T.W., Kalaba, R.E. and Spingarn, K. (1979), " A comparison of two methods for determining the weights of belonging to fuzzy sets", Journal of Optimization Theory and Applications, 27(4), 531-538. Forman, E. (1993), "Facts and fictions about the analytic hierarchy process", Mathematical and Computer Modelling, 17(4-5), 19-26. Harker, P.T. (1989), "The art and science of decision making: The analytic hierarchy process", in: B.L. Golden, E.A. Wasil and P.T. Harker, eds., The Analytic Hierarchy Process: Applications and Studies, Springer, Berlin, 3-36. Jensen, R.E. (1983), "Aggregation (composition) schema for eigenvector scaling of priorities in hierarchical structures", Multivariate Behavioral Research, 18(1), 63-84. Jensen, R.E. (1984). "An alternative scaling method for priorities in hierarchical structures", Journal of Mathematical Psychology, 28(3), 317-332. Lipovetsky, S. (1994a), "AHP priority vectors in linear and non-linear estimation", Faculty of Management, Tel Aviv University, Working Paper No. 8/94. Lipovetsky, S. (1994b), "The analytic and the synthetic hierarchy processes: Fuzzy applications", Faculty of Management, Tel Aviv University, Working Paper No. 7/94. Lipovetsky, S. (1994c), "Multimode data analysis for decision making", in: G.H. Tzeng et al., eds., Multiple Criteria Decision Making, Proceedings" of the lOth International Conference: Expand and Enrich the Domains of Thinking and Application, Springer, Berlin, 275-282. Lipovetsky, S. and Tishler, A. (1994), "Linear methods in multimode data analysis for decision making". Computers and Operation Research, 21(2), 169-183. Marzen, V. (1994), "'The enlargement of the analytic hierarchy process in respect to global planning and sustainable development", in: Proceedings of The 3rd International Symposium on The Analytic Hierarchy Process, Washington, I)C, USA, 75-88. Saaty, T.L. (1980), The Analytic Hierarchy Process, McGraw-Hill, New York. Saaty, T.L. (1994), Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process, RWS Publications, Pittsburgh. Saaty, T.L., and Kearns, K.P. (1985), Analytical Planning, Pergamon Press, New York. Saaty, T.L., and Vargas, L.G. (1994), Decision Making in Economic, Political, Social and Technological Environments with the Analytic Hierarchy Process, RWS Publications, Pittsburgh. Schoner, B., Wedley, W.C. and Choo, E.U. (1992), " A rejoinder to Forman on AHP, with emphasis on the requirements of composite ratio scales", Decision Sciences, 23(2), 509-517. Schoner, B., Wedley, W.C. and Choo, E.U. (1993). " A unified approach to AHP with linking pins", European Journal of Operational Research, 64, 384-392. Theil, M. (1971), Principles of Econometrics, John Wiley & Sons, New York. Triantaphyllou, E., and Mann, S.H. (1994), "Some critical issues in making decisions with pairwise comparisons", Proceedings of the 3rd International Symposium on The Analytic Hierarchy Process, Washington, DC, USA, 225-236. Triantaphyllou, E., Pardaios, P.M., and Mann, S.H. (1990), "The problem of determining membership values in fuzzy sets in real world situations' ', in: D.E. Brown and C.C. White (eds.), Operations Research and Artificial Intelligence: The Integration of Problem-solving Strategies, Kluwer Academic Publishers, Boston, 197-214. Zabedi, F. (1986), "The analytic hierarchy process - A survey of the method and applications", Interfaces, 16(4), 96-108.
ELSEVIER
European Journal of Operational Research 93 (1996) 550-564
Theory and Methodology
The synthetic hierarchy method: An optimizing approach to obtaining priorities in the AHP Stan Lipovetsky * Faculty of Management, Tel Aviv Unioersity, Ramat Aviv, Tel Aviv 69978, Israel
Received 9 September 1994; revised 22 February 1995
Abstract
A new method of synthesizing local and criteria priorities into global priorities is suggested. This approach is a development of the Analytic Hierarchy Process enabling the united consideration of all horizontal and vertical connections of a hierarchical system in a single optimizing objective function based on statistical models of the synthesis process. The solution can be reduced to a linear system or to an eigenproblem of a special matrix constructed as a combination of Kronecker's sums and products of pairwise judgement matrices. A numerical example shows that the optimizing approach produces a ranking of global priorities that may be different from the ranking produced by the classical AHP. Keywords: Multiple criteria; Decision analysis; AHP; Synthesizing priorities; Statistical optimization
1. Introduction
The Analytic Hierarchy Process (AHP) has found widespread application in decision making problems involving multiple criteria in systems of many levels, and for priority-modeling in complex systems. These uses are enabled by the methodological strength of AHP: rendering a complex system into the form of a structured hierarchy. The levels of the hierarchy describe a system from the lowest level (sets of alternatives), through the intermediate levels (subcriteria and criteria), to the highest level (general objective). Using the AHP methodology, priorities of alternatives are estimated independently for every criterion at each level. The weight (or priority) of each criterion is defined by the same AHP procedure. Afterwards, summing the priorities of every alternative with the weights of every criterion creates a composite priority for the highest level - this procedure is called the hierarchical composition, or synthesis. It yields the overall priorities of alternatives on each successively higher level as a linear combination of the subpriorities derived for the previous level. Such summing through the whole hierarchical structure produces a synthesized judgement for all alternatives under the stated goal (see Saaty (1980), Saaty (1994), Harker (1989), Forman (1993)). The strongest features of the AHP are that it generates numerical priorities from the subjective knowledge inherent in the estimates of
* Present address: 71-11 Yellowstone Blvd., #6-E, Forest Hills, New York, NY 11375, USA. 0377-2217/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSD1 0377-2217(95)00085-2
s. Lipovetsky/ European Journal of Operational Research 93 (1996) 550-564
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pairwise comparisons matrices, and that it organizes a complex sYstem into a number of ordered subsystems arranged in a hierarchy. The aim of the current work is to modify the AHP procedure to obtain all partial and global priorities in a simultaneous and connected estimation. We achieve this by optimizing an objective function. Earlier attempts of this kind can be found in Chu et al. (1979), Jensen (1983), Jensen (1984), Triantaphyllou et al. (1990), Triantaphyllou and Mann (1994), Schoner et al. (1992), Schoner et al. (1993), Basak and Saaty (1993) (See also Zahedi (1986), Saaty (1994)). We term our approach the Synthetic Hierarchy Method, SHM, because it obtains a single united priority vector describing all hierarchical levels of a complex system. (This method should be distinguished from SHP the Systemic Hierarchy Process - a global approach to MCDM, suggested by Marzen (1994)). A development of the AHP priority estimation, the SHM yields global priorities which may differ from those given by the classical AHP. This paper is structured in the following way. Section I is an introduction. Section 2 describes the AHP methodology for the synthesizing local priorities into global priorities, and introduces the technique of a weighted matrix of united priorities. Section 3 presents the SHM approach. Introduced in this section is a statistical model encompassing all priorities. This model allows the formulation of an objective function that can be reduced to a linear system of equations or to an eigenproblem with a combined matrix of Kronecker's kind. There is a discussion of the different properties of these solutions, and of the place of the classical AHP solution. Additionally, for the purpose of comparing the described methods, in Sections 2 and 3 each method is applied to the same numerical example. Section 4 suggests several different directions in which the SHM may be developed further. Section 5 is a summary.
2. A generalized description of priority-setting for alternatives and criteria In order to make the forthcoming theoretical derivation easier to follow, let us begin with a numerical example discussed by Saaty and Keams (1985), Chapter 3 and by Saaty and Vargas (1994), Chapter 1. They posed the problem of the "Best House to Buy". On the upper level was the goal "satisfaction with house". There were three houses (alternatives) to choose from, and the choice was made according to eight criteria. Local priorities obtained by the AHP for each of the eight criteria, stacked in the priority matrix a , are set out in the following columns: 0.754 u = | 0.181 ~0.065
0.233 0.055 0.713
0.754 0.065 0.181
0.333 0.333 0.333
0.674 0.101 0.226
0.747 0.060 0.193
0.200 0.400 0.400
0.072 0.650) • 0.278
(1)
0.167
0.333).
(2)
The vector of preferences among these eight criteria is: /3'=(0.173
0.054
0,188
0.018
0.031
0.036
Priorities by each criterion are normalized by one, i.e., the sum of the elements in each column in (1), or the row in (2), equals one. The elements of (2) are the weights of the columns in (1). In order to compare the elements of these eight columns among themselves, we need to multiply them by their weights. This yields a matrix of weighted columns:
G=
0.130 0.031 0.011
0.013 0.003 0.039
0.142 0.012 0.034
0.006 0.006 0.006
0.021 0.003 0.007
0.027 0.002 0.007
0.033 0.067 0.067
0.024~ 0.216 . / 0.093
(3)
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S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
In order to establish the composite priorities for the three alternatives, we make a summation of the weighted scores of these alternatives by all criteria, i.e., we calculate the sum in each row of matrix (3): 7r' = ( 0 . 3 9 6
0.341
0.263).
(4)
Thus, by these synthesized priorities 7r, the most preferable alternative is no. 1. Next comes no. 2, and no. 3 is the least preferable of the three alternatives. Now let us present a more generalized description of this process of synthesizing priorities. Suppose, we have i = 1,2 . . . . . n alternatives and j = 1,2 . . . . . m criteria (n = 3 and m = 8 in the example given above). For each jth criterion, we use the corresponding matrix A (j), of order n × n, for pairwise comparisons among n alternatives, in order to estimate a vector of priorities a {j) by the A H P eigenproblem:
A(J)oL(j)
= A(J)a (j),
j = 1 . . . . . m.
(5)
The priority vector a w corresponds to the maximal eigenvalue A°) for the jth criterion. Similarly, we use the matrix B, of order m × m, for pairwise comparisons among all rn criteria, in order to obtain a vector of criteria priorities. To this end, we solve the A H P eigenproblem for the maximal eigenvalue /x: B/3 =/z/3.
(6)
Then we gather all priority vectors a {j) (5) in a priority matrix: = (
.....
.....
(7)
that is the n × m matrix corresponding to (1) in the example. We present the vector of criteria priorities in the following form: /3 = d i a g ( / 3 ) e,,,
(8)
where the diagonal matrix contains all the elements of this vector/3, i.e., diag(/3 ) - diag{/31 ,/32 . . . . . /3j . . . . . /3m},
(9)
and e,, is a uniform vector of the ruth order: e'-
(1,1 . . . . . 1).
(10)
(8) corresponds to (2) in the example. (7) is a transposed stochastic matrix, because in accordance with the AHP, all priority vectors are normalized by one (the same with the vector/3), i.e., Or} j) = 1,
j = 1 . . . . . m,
(lla)
i=1
/3j= 1.
(llb)
j=l
The matrix of weighted priorities G (corresponding to (3) in the example) can be presented as the product of (7) and (9), i.e., ...
/
(12)
G = adiag(13) =
!
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
553
We also introduce the following notations for (12):
-=
.
(13)
where each ijth element of the matrix G is T} j) =
(14)
ol}J)[3j.
We can also present (13) as consisting of the vectors-columns
G - (T(O,T (2). . . . . T (j) . . . . . 3,(m)),
(15)
where these vectors are the priority vectors in (7) weighted with the criteria priorities flj, i.e.,
T(J)-a(J)/3j,
j=l
. . . . . m.
(16)
(13) can be used as a unified presentation of priority vectors ct O~ and criteria priorities t , instead of using all these partial vectors themselves. Indeed, to solve the AHP, we need to obtain a matrix a as in (7) with n X m parameters, plus m parameters in the vector fl as in (8), minus (m + 1) normalizing conditions, (1 la) and (l lb), i.e., for global priorities we have to estimate n m 1 free parameters. In (13), we have n x m parameters T} j), and with one normalizing condition for them, the number of free parameters becomes n m - I . Each column T (j) of the matrix G is proportional, with the term /31., to the vector ct (j) (16), i.e., all eigenproblems (5) can be presented as the following eigenproblems: A(J)T(J) = ~(J) y(J),
j = 1 . . . . . m.
(17)
After normalizing by one, that is, setting
Y'. T} j)= 1,
j= 1..... m
(18)
i=l
the vectors T ti~ for maximal eigenvalues A(j) will coincide with the vectors a u; of (5). Concerning the vector t , we can see that summing elements in each jth column of the matrix G (12) and (13) yields:
iffil
iffil
i=1
i.e., these sums are proportional to the elements of (8). With the normalizing conditions of (1 la) in (19), the vector fl coincides with the vector whose elements are the sums of the elements in the jth column of (13), flj= L T[ j),
J = 1 . . . . . m,
(20)
i=l
or in matrix notation,
f l = G'e.,
(21)
where e. is a uniform vector with n elements. Thus, (6) can be written in the form:
(
8 E v ? ) , E v [ " ..... E r r " ) i
i
i
)'
)'
=~
T}I).E3'} 2). . . . . E T [ '') , "
i
i
(22)
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S. Lipovetsky / European J o u r n a l o f Operational Research 93 (1996) 5 5 0 - 5 6 4
where the vector of priorities corresponds to the maximal eigenvalue ~. Now we can consider the global, or synthesized, priorities. In the numerical example above, (4) can be constructed either as the product of (1) and (2), or as the sums of the elements in each row of (3). In the same way, according to (7), (8), (9) and (10), the global priority vector 1r equals the product of (7) and (8), that is zr= a/3 = adiag( f l ) e , , = G e m ,
(23)
where G is given by (12). Eq. (23) says that the vector of global priorities can be presented simply as the vector whose elements are the sums of the elements in the corresponding rows of (13): '77"i = ~ y~J), j=l
i = 1 . . . . . n.
(24)
Thus, we can summarize: For a matrix G of n × m coefficients (13), the columns of this matrix can be considered as priority vectors for each criterion (17). Sums of elements in each column of G can be applied for obtaining the vector of priorities among criteria (22). Sums in each row of G yield a vector of global preferences among all alternatives (23) and (24).
3. The synthetic hierarchy method: Minimization of the SHM objective function Now we shall describe the method for estimating all elements of (13) in the optimizing approach. This method corresponds to the solution that, in the introduction, we termed the Synthetic Hierarchy Method - the SHM. Let us construct an objective function for the united priorities of the SHM. First, we must recall that all elements of pairwise comparison matrices are defined as ratios of the corresponding priorities. By this premise, the following relations pertain: a(~) i,
-a~i) --
~j
-7(j), -
i,k=
i=1
(25a)
1 . . . . . n,
j,f=
(25b)
1 ..... m.
i=l
In (25a), ~-i,'~(J)means the ikth element of the jth matrix, A (j) (consisting of pairwise comparisons among all n alternatives according to the jth criterion). (14) is used for proportions between local priorities a~ J) and weighted local priorities y~J). In (25b), b j / means the j f t h element of the matrix B of, pairwise comparisons among all m criteria, and (20) is used for proportions between criteria priorities. The elements of empirical matrices of pairwise comparisons are inconsistent with (25a) and (25b). We propose the following model of connections between elements of the matrices and the corresponding priorities: a(J),,(J) = y[i) + ,:(J) ik I k ~ik '
bj/Y'- YY) = E y~i) + 6jr, i
(26a)
i,k = 1 " " " " ,n,
j,f=
1 . . . . . m.
(26b)
i
In (26a) and (26b) all coefficients y denote estimators for theoretical values of weighted priorities, and ¢~,J) and 61/ denote deviations from (25a) and (25b), expressed in linear form. Without additional assumptions concerning the distribution of the deviations in (26a) and (26b), we can apply a simple least-squares (LS) function to minimize these deviations: LS =
IIEII: + 116112 =
t,~(J).,(y) ( " i k r , - y/i)): + Z j=l
i,k=l
j,f~
bi~'Y'- y(~') _ i=1
TO) i=l
"~ min.
(27)
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Because of the homogeneity of coefficients y in (27) we need to add a normalizing condition for them. It can be the simple condition
i=l j = l
Thus, the conditional problem for minimization is as follows:
F=-LS-2A(~
~y}J)-l)-*min, j=l
(29)
i=1
where LS is given by (27) and k is a Lagrange multiplier. The first order conditions 0F
Oy}p )
O, t
1 ..... n, p
1 ..... m
(30)
yield a linear system of n × m equations: tl
n
Y'.(alf)~'[P)- r,( p ) )-.~ ( p ) + y'(r}p)-..(p).,(p)~,,., ) i
k
+
bjp j
"y/(P)-
7 (i) bjp +
i
"
r/(p) - bp/ f
y/(/) = k,
(31)
"
that can be simplified to the following system for all indices t and p: n + ~ff~(a~)): i
+ m+
"
( e ) - ~[(a~,.p' + a ~ ) ) y / P ) l Tt
.
p 2 EY/P)i
j +bjp
y(i) = k. •
j
(32)
"
Let us represent this linear system in matrix notation. At first, we observe that because of the reciprocal symmetry of Saaty's pairwise comparison matrices ( a i t a t i = 1, bjpbpj - 1 ) , it is possible to write the relations: n
(p)
. + ~-~.(a~tP))2= Y'.(ai, i
(p)
(p)
+ a . )air,
(33a)
i
m
m + Y~ (bjp) 2= ~(bjp +bpj)bjp. J
(33b)
J
Now we introduce the symmetric matrices
A-(p) = A (p) + (A(P)) ',
p =
1 ..... m,
(34a)
= B + B'.
(34b)
Using (33a), (33b), (34a) and (34)b we unite for all t = 1. . . . . n the equations of (32), and present this system as follows:
diag(A-~e)A(P))y(P)-A-Ie)y(P)+
m + Y~ j
Jny(P)/
[(bpj+bjp)Jny(J)]=Ae.,
p = l . . . . . m.
j
(351
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S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
where J, is the matrix of the nth order whose elements all equal 1, i.e., J , = e,e',, where e, is a uniform vector of the nth order; 3, tp~ are the same vectors 3,°) . . . . . y(r,) as in (15), and diag denotes a diagonal matrix cut from the matrix in the brackets. All m equations (35) can be united into block matrices as follows:
I [diag( A-(I)AO)) -
0 }
0
...
diag( A-~2)A(2)) - A-(2)
...
0
0
...
diag(A-(,,)A(,,)) _ ~'(,n)
A- ( °
-(b,2 + b2,)J n
+
-(b~2 + b21)J.
m+
- (/,.. + b..)J.
[ ¢') / y(,,) ]
-(022+b22)
-(b2m
+ b,,,2)J n
J,
...
- ( b,m + b,n,)Jn
...
-(b2,,+bm2)J,
...
m+
-(bmm+bmm
J
[e,,
t
(36)
e,
Using (33a), (33b), (34a) and (34b), we introduce the set of matrices A"(j) = diag( A-IJ)A (i)) - A-~j),
j = 1 ..... m,
/~ = d i a g ( B B ) - B.
(37a) (37b)
Eq. (36) can now be written: S y - - Ae.,,,
(38)
where y is the vector, of n m t h order, of united priorities y = (yo),,y(2~ . . . . . y o . ~ ) ' ,
(39)
and e.,, is a uniform vector of the same order. S is a combined hyper-matrix constructed as follows: S=-ATI)~A'12)~
.-.
O A ' ( ' n ) + / ~ ® J n.
(40)
Its first part is a block-diagonal matrix, or the Kronecker sum of (37a),
Its second part is the Kronecker product of (37b) and the matrix J,. (On the properties of Kronecker sums and products - see Barnett (1990). Theil (1971).) Expression (40) can be written in the form: S=.4+B®J.,
(42)
corresponding with the two parts of the left-hand side of (36). It is interesting to note that (42) is symmetrical because its blocks ((37a) and (37b)) are symmetric matrices (due to (34a) and (34b)). This is in contrast to Saaty's pairwise comparison matrices A t j) and B, whose elements are reciprocally symmetrical. Solution of (38) yields a vector of weighted priorities for the SHM: y= AS-le,m.
(43)
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With the normalizing condition of (28), expressed in vector notation, e'.,. 3' = 1, we determine the value of the term h=(e'..,S-le.,.) -l. Let us consider some properties of (43). Summing the elements in each subvector y(J) (39), we obtain the weights /3j (see Eq. (20)) of preferences among m criteria, and using them to normalize the subvectors y(J). we obtain the local priorities c~°) (see (14) and (16)) of the alternatives by all criteria. The synthesized priorities 7ri are determined by the same summation (24). If instead of a single normalization condition 28), we use m conditions (20) with constants /3j (these conditions ensure the fulfillment of m conditions (1 la) for local priorities), then instead of (29), we obtain the function
F= L S - 2 ~ Aj( ~ A~J)- ~j) ~ Iin j=l
(44)
i=1
with the same least square function (27). The derivation (30) till (38) with the objective function of (44) yields the following solution:
3' =
" y(")
= S-t
•
,
(45)
~ A~'en
with the uniform vectors e, of the nth order, and the same matrix S (see (42)). Thus, for each subvector 3'(J), solutions by (45) coincide with those of (43) up to the normalizing terms. If (41) is a nonsingular matrix, we can present S - l of (42) in the following form (see Barnett (1990), page 125, for the inversion formula). Suppose we solve an eigenproblem Bfi = A~, and present the matrix /~ in the form /~ = 17/117', where 17 is a matrix of the eigenvectors fi in its columns, and A is a diagonal matrix of the eigenvalues A. Using the matrix V = 17~1/2, we can write the following presentation of (42):
B®J. = (VV') ® ( e~e') = (V ® e.)(V' ® e') = (V ® e.)(V ® e~)'--RR', where e. is a uniform vector of the nth order; R denotes the matrix V ® e. of (nm) X (m)
(46)
order, constructed from matrix V using the properties of the Kronecker product and of its transposition. From (42) we can write the expression:
(47)
S - ' = ( .4 + R R ' ) - ' = .4-' - . 4 - ' R ( E , , + R ' . 4 - L R ) - ' R ' A - ' , where E,. is a uniform matrix of the mth order. Then (43) has the form: 3, =
AS-'e.,. = h/~-lento
(48)
-- A . ~ - I R ( E m + R ' t ~ - i R ) - I R ' A - l e . m .
The first item ?~- ~ e.,. with block-diagonal matrix ?~- 1 coincides with the set of separated solutions for local priorities. The second item corresponds to the connection of local priorities to criteria priorities. Together they produce the united vector 3' of weighted priorities. For the same numerical example that is used in Section 2 (see (1)-(4)), we calculate the solution for (43). It yields the matrix G of (13): G=
0.111 0.020 0.012
0.009 0.004 0.041
0.094 0.010 0.017
0.010 0.010 0.010
0.032 0.006 0.009
0.036 0.004 0.006
0.018 0.036 0.036
0.038 0.316 ,J 0.115
(49)
with sums in the columns of (49): /3'=(0.143
0.054
0.121
0.030
0.047
0.046
0.090
0.469),
(5o)
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
558
and with sums in the rows of (49): zr'=(0.348
0.406
0.246).
(51)
Dividing the elements in the columns of (49) by the corresponding sums of (50) we obtain the local priorities: a=
0.776 0.140 0.084
0.167 0.074 0.759
0.777 0.083 0.140
0.333 0.333 0.333
0.681 0.128 0.191
0.783 0.087 0.130
0.200 0.400 0.400
0.081 0.674 .J 0.245
(52)
The priorities of alternatives in column a coincide in their ranking with priorities in (1). But (50) and (51) differ in their ranking from the estimations produced by the classical AHP ((2) and (4)). We shall use the term inverted matrix SHM solution to denote (43), obtained by solving (38). Instead of the normalization in (28), we can take the condition
i=l j = l
that corresponds to normalizing by the variance of the vector 3' elements. With (53), (29) transforms into
F=LS-A[~(%(J))2-1]~min=1.i=lj
(54)
By the same derivation (30) till (38), we obtain from (54) the eigenproblem: s3' = A3',
(55)
with the same hyper-matrix (42). Eigenvalues A~,A2. . . . . A,,, (all of them are non-negative) correspond to the extreme values of the quadratic form LS (see (27)) conditioned by (53). The eigenvector 3' which relates to the minimal eigenvalue, A,,i,, gives the solution to (54), with F,,i, = A,,i,. This solution 3' for A,,~,, we shall call
the eigenvector SHM solution. For (1)-(4), (55) yields the following SHM G-matrix (additionally normalized by (28) for the sake of comparison): [0.114 G = [ 0.010 0.007
0.004 0.002 0.044
0.097 0.006 0.008
0.009 0.009 0.009
0.036 0.003 0.003
0.037 0.002 0.003
0.007 0.033 0.033
0.028/ 0.462 ). 0.033
(56)
0.042
0.073
0.523),
(57)
Summing elements in the columns of (56) yields: /3'=(0.131
0.050
0.111
0.027
0.042
and summing elements in the rows of (56) yields: 7r'=(0.332
0.527
0.140).
(58)
Dividing the elements of columns (56) by their sums (57), we produce the local priorities matrix: =
0.870 0.076 0.053
0.080 0.040 0.880
0.874 0.054 0.072
0.333 0.333 0.333
0.857 0.071 0.071
0.881 0.048 0.071
0.096 0.452 0.452
0.0541 0.8831. 0.063]
(59)
Both SHM solutions, the inverted matrix results, (49)-(52), and the eigenvector results, (56)-(59), are very similar. They demonstrate the same ranking order for local priorities a , weighted local priorities G, priorities among criteria /3, and global priorities ~. But the eigenvector SHM solution, obtained with (53) for variance of elements y~J), produces results with more variability among all coefficients. In other words, as can be seen by comparison of these two solutions, large (or small) elements in (49)-(52) become larger (or smaller) in
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
559
(56)-(59). Thus, the first SHM solution with the inverted matrix of (43) is simpler to calculate, but the second SHM solution with the eigenvector of (55) more clearly expresses the preferences among all objects under consideration. Let us consider how to explain the similarity of the results of these two variants, the inverted matrix SHM solution of (43) and the eigenvector SHM solution of (55). Suppose D is a non-singular positively defined matrix of the qth order which presents us with two problems - a linear system with a uniform vector on the right-hand side, O a = e,
(60a)
and an eigenproblem, O b = Ab.
(60b)
Eq. (60b) yields a set of eigenvalues A1 > Az > • - • > A0 > 0, and a set of eigenvectors b I . . . . . spectral decomposition of D is t
D = Aiblb' I + A2b2b' z + • "
bq. T h e
(61)
+Aqbqbq,
where each vector is normalized by the condition b ' b = 1. The inverted matrix can be presented in the following spectral form: 1
1
1
D - l = _ _ b i b , I + _ _A:2 bzb'2 + • • . + - - b q b q , ~
aI
Aq
because for )tq << Eq. (60a) gives
)tt,t-~
=
~q
b q b q, ,
(62)
1 . . . . . q - 1, the main input in D - l is given by the last item in (62) with a,.i. = h.q.
(b;e)
q a=O-le
1
t =12"- - ' ~ -
(b'qe)
A----~b q ,
b, =
(63)
where (62) is used for the matrix D-1. Thus, the solution a of the linear system is proportional (up to the c o n s t a n t (ffqe)//Aq) t o the eigenvector bq for the minimal eigenvalue Aq. But in (55), the focus of our interest is just the eigenvector y for the minimal eigenvalue. This explains the closeness of both solutions, (49)-(52) and (56)-(59) for the united vector Y. It is interesting to note that synthesizing AHP priority in our example by the so-called ideal mode (when the priorities of the alternatives for each criterion are first divided by the largest value among them, then multiplied by the priority of the corresponding criterion and summed) produces the global priorities as follows (see Saaty and Vargas (1994), p. 17, Table 5): ~" = (0.584, 0.782, 0,461). Normalizing the sum of these values to one, we obtain the vector of global priorities 7r' = (0.320, 0.428, 0.252), that is very similar to the solutions of (51) and (58), and corresponds to the ranking of preferences as ~2 >" ~J >" 7r3. This differs from the ranking of preferences 7r I >- ~e >" qr3 given by (4). The SHM was obtained from (26a) and (26b). Let us show to which model (5) and (6) can be related. Because of the theoretical assumption that all elements aij of a pairwise comparison Saaty matrix A are the ratios of the corresponding priorities o t i / ~ j, instead of the models with additive errors ((26a) and (26b)), let us consider the following model for an empirical matrix A:
aijot j =
Cti(1 -I-
•ij)"
i,j=
1 . . . . . n,
(64)
where errors are included in the theoretical model in multiplicative form. Then, summing by the index j produces the system of equations ~ a i j a ~= j~l
n+
Eij a i, j=l
i = l . . . . . n.
(65)
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S. Lipovetsky/ EuropeanJournalof OperationalResearch93 (1996)550-564
If we assume that the sums of errors in (65) are equal for all i, E-
~Eij=const,
i = l . . . . . n,
(66)
jffil
then (65) reduces to the problem
Act = Act,
(67)
where A denotes an unknown parameter with the property A - n + E that expresses the deviation from the exact theoretical value A = n. (67) coincides with the classical AHP eigenproblem for the right eigenvector with maximal A of the pairwise comparison matrix A. Thus, the AHP corresponds to the model with multiplicative errors (64) on the assumption that summarized relative deviations are equal (66) for all rows of the matrix A. In other words, the AHP does not minimize all partial deviations Eij but equalizes sums of deviations in the rows of matrix A. In the same way, it can be seen that the left eigenvector of the nonsymmetric matrix A, i.e., the vector ~ of the AHP problem A'~ = An corresponds to the equalizing of the sums of deviations in the columns of the matrix A of pairwise comparisons. The methods of evaluation of unknown parameters by the minimization of an objective function generally produce better estimates than methods based on the equalizing of relations among parameters. Thus, we can infer that (27), with conditions of (29) or (54), yields solutions more reliable than those generated by an AHP eigenproblem of the sort of (67). Further, (27) includes all unknown parameters T~j) and produces simultaneously a united vector giving all priorities. Moreover, because of the symmetry of (42), the SHM yields a unique solution, whereas the AHP method for each partial matrix of pairwise comparison gives two different solutions, i.e., the left and the right eigenvectors, a cumbersome feature of AHP solutions for practical applications.
4. Further extensions o f the S H M
The SHM is a suitable approach for many different applications. This section outlines some directions for its further development.
4.1. A nonlinear model Instead of using (26a) and (26b), it is possible to consider a nonlinear model that resembles more closely (25a) and (25b). Thus:
,,o) ~ik = ~kj) + e~ ),
i,k = 1 . . . . n,
i=1____~_.__ bj:= ~z.., T~:) + 8j:'
(68a)
J ' f = 1. . . . . m.
(68b)
i=1
This model is more appropriate to the definition stating that elements in matrices of pairwise comparisons are ratios of priorities among alternatives or among criteria. Instead of (27), the nonlinear least squares objective is:
NLS=I['II2q-[[~[[2= ~ j=l
k ~,~ik[a(J)--~J)//~/(J)) i,k=l
£ j,/=l
bj:-
T/0'
i=l
~T~:'
li=l
~min.
(69)
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S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
The coefficients T~j) could be estimated directly by this function with (28) or (53) with the help of existing computer programs for nonlinear estimations. Such NLS problems have been considered for a pairwise comparison matrix and the results have been compared systematically with other variants of priority vectors, in Lipovetsky (1994a). As shown in that work, such nonlinear minimization can be described as a special kind of regression on dummy variables. Together with the priorities themselves, the standard deviations and t-statistics can be obtained without difficulty. In this approach, different hypothesized values of priorities can be checked and corrections can be made in the ranking order of alternatives. Within standard assumptions concerning the distribution of errors, we can apply the maximum likelihood approach to estimate the vector elements. The maximum likelihood approach can be used both for the linear ((26a) and (26b)) and the nonlinear ((68a) and (68b)) models. The variances and covariances of errors, estimated by the residuals in these models, could be included as weights for different parts of objectives ((27) or (69)). Powerful software is able to construct these seemingly unrelated kinds of equations.
4.2. Extension to additional data sets The model can readily be extended to include additional data sets. Often encountered in practice is the case of many judges who estimate the same, or different, pairwise comparison matrices. Such a case can easily be described by linear or nonlinear objective functions. For example, if T l judges 01 = 1. . . . . T I) estimate alternatives by the first criterion, T2 judges (t 2 = 1. . . . . T2) by the second criterion, etc., till T,, judges (t,, = 1. . . . . T,,) estimate the alternatives by the ruth criterion and Q judges (q = 1. . . . . Q) estimate the criteria priorities themselves. These could be different judges, but assuming the existence of a unique vector 3' of all needed priorities, we can very easily write such a model with an NLS (or LS) objective that is an evident extension of (69):
NLS= ~
Y'~ ~
,(a(J't~)ik- y~j)/y~j))2+
j = l tj=l i,k=l
b~q/) q=l j,gffil
y/(J) iffil
Y'. y/(/)
---) min.
(70)
li=l
(70) differs from the previous, simpler (69) only in the presence of additional sums corresponding to information given by additional judges. The structure of priority ratios, constructed from the elements of the united priority vector y, is the same both in (69) and in (70). If the same judges each independently estimate all pairwise comparison matrices, there also exists another possibility, that is, to consider three-way matrices of pairwise comparisons where the third direction corresponds to the different judges. Four-way matrices can also be generated. For example, the fourth direction may correspond to judgements made at different times, or under different conditions. For evaluations using many-way matrices, see Lipovetsky and Tishler (1994), Lipovetsky (1994c).
4.3. A free parameter The range of applications of the SHM can be enlarged by including a free parameter ~o, indicating the significance of the various levels of the system under consideration. For example, in (69), or in (27), we can manipulate the relative weights of the alternatives and criteria:
NLS = (1 - ~)IIEII2 +
~I16112.
(71)
This multi-objective function can be tried with different values of the parameter ~p from the range [0, 1], corresponding to the importance of the different levels of decision making.
4.4. Linear and nonlinear estimations with missing data Linear and nonlinear estimations of the SHM can be carried out using the data in the upper triangle of pairwise judgement matrices, because only these data are given directly by judges. We can use the approach
S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
562
described in Lipovetsky (1994a) to evaluate the priorities by this method. The structure of SHM models tolerates the absence of all kinds of values in the matrices of pairwise comparisons, and even the complete absence of some of these matrices. The summation (27), or (69) and (70), for such cases is carried out simply on the items corresponding to the elements that are all present. 4.5• Fuzzified matrices
The SHM can be easily extended for fuzzified matrices of pairwise comparisons. Such a Fuzzy-SHM is described in Lipovetsky (1994b). In that work, several types of distribution of a random variable and its inverse were assessed, and Cauchy distribution was found to be the most appropriate description for fuzzified pairwise comparison matrices. Solutions by AHP and by SHM estimating approaches were modified for the evaluation of priorities using fuzzified data. 4.6. Intermediate levels
The SHM method can be applied rather well to the case with many intermediate levels of subcriteria. Let us describe the basic structure of the SHM for a system of n alternatives (i = 1. . . . . n), m subcriteria ( j = 1. . . . . m), and T criteria (t = 1. . . . . T), with the corresponding matrices of pairwise comparisons among alternatives A~j't) (of the nth order) for each j and t, among subcriteria B ~° (of the m the order) for each t, and among criteria C (of Tth order)• The global synthesized ~riorities 7r~ . . . . . 7r, for all alternatives can be obtained in the process: a(1,1)
•..
Or(re'l)
O~(1,1 )
• ..
O/(re'l)
~i,T, ... ~-.~, tU))
7"1"1 ,B"2
t~(l)
/'2
yl+...+ O~(1,1 )
• • •
O ,(rn'l)
•
(72)
"YT '
t3~')
where columns Ol ( j ' t ) correspond to the vectors of alternative priorities a} j'° related to the matrices A (j't), vectors /3mcontain priorities of subcriteria fl)t) related to the matrices B~t); and elements Tt . . . . . Yr are priorities of criteria related to the matrix C. Instead of (12) we have now the following matrix X of weighted priorities:
#(~,l)
...
#(,,,i)
I
I
#(~.r)
...
#(,.,T)
#1.,)
...
#~.1)
I
i
#,.T)
...
#-.n
X=
I ~.~")
...
~.~")
..•
I
(73)
I I
~. ~''~>
...
~.~'~)
where each element of this matrix corresponds to the related weighted priority coefficient from (72), i.e., ~i~j'') =- a~J'013~')%,
i = 1 . . . . . n, j = 1 . . . . . m t = 1 . . . . . T.
(74)
By estimating the values of (73) we can easily find all other needed priorities. For each subcriterion j and criterion t, the local priorities ct~j't) are proportional to (74). With normalization to one ((1 la) and (1 lb)) for each .
(75a)
i=1
~ ~,'J")= ~ f l ) t ) , t = Y,i=l j=l j
(75b)
S. Lipovetsky / European Journal o f Operational Research 93 (1996) 550-564
563
Thus, the subcriteria priorities [3Jt; of (75a) are proportional (up to the common constant 3,,) to the sums in the columns of (73) with each given t. Criteria priorities y, are proportional to the sums of all subcriteria (75b). The global priorities 7ri for all alternatives i = 1. . . . . n are defined by (72). Thus they are equal to the sums of the elements in the rows of (73): "n"i =
•
sri(j''), i = 1 . . . . . n.
(76)
j=l t=l
This is an evident generalization of (24) to the summing of the weighted local priorities by all m subcriteria and all T criteria. From (26a) and (26b), extended for all hierarchical levels under consideration, we obtain the objective which is a clear generalization of (27): [ ~ik
bk
t=l j = l i , k = l
+
E t,r=l
t=l j , f = t
c,T E E~:i (~'T) i
j
"
i
~i(~'') -*rain, i
(77)
j
where a~i't), b)5) and Cr,. are elements of the matrices A (J't), B (t) and C, respectively. With normalizing conditions of the sort of (28) or (53) for all coefficients ~i(j't) (73), we obtain the generalization of (29) or (54) with (77). These problems can in turn be reduced to the linear system of (38) or to the eigenproblem of (55) with a combined hyper-matrix S of order ( n m T ) X (nmT) to estimate the united vector ~, expressed by stacking the columns of (73). This matrix S could be constructed as Kronecker's sums and products of appropriate initial judgement matrices of three hierarchical levels. From (68a) and (68b), extended for this case of subcriteria and criteria, we can obtain the generalization of (69):
~'ik
- ~i(
) + E
t = l j = l i,k=l
+ ~ t,';= l
b~ -
t=l j,f=l
c,~-
~i(J'') i
~i(j'~) j
)
~i(i'')
~i(t'')
i
-)min.
(78)
In the same way we can consider complex systems with four or more levels of hierarchy.
5.
Summary
This paper describes a methodology for combining and estimating all local and criteria preferences in a matrix of weighted priorities. This method has proven to be convenient for considering synthesized priorities from a statistical point of view. Different extensions of this approach may be applied to hierarchical systems of many levels, judgements of many judges, multi-criteria objectives, fuzzified data and other applications. Statistical models produce conditional objective functions, minimization of which (using Kronecker's technique of sums and products of matrices) yields linear systems or eigenproblems for the estimation of aggregated priorities. Nonlinear models and nonlinear estimation can generate not only priorities but also their standard errors (which can be used for checking hypotheses on the global preferences). The numerical example indicates that the synthesized global priorities obtained using an optimizing approach may have an order of preferences that differs from the results of the classical AHP. So at least, if a decision-maker is not satisfied with the results
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S. Lipovetsky / European Journal of Operational Research 93 (1996) 550-564
obtained by AHP estimation, the problem can be reconsidered by applying the Synthetic Hierarchy Method optimizing approach. In general, the SHM promises to be very useful for the burgeoning field of theoretical investigation and for practical applications in numerous MCDM problems.
Acknowledgements The author wishes to thank David Mendels for editing the manuscript.
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