Group decision making using the analytic hierarchy process

Mathl. Compat. Modelling Vol. 17, No. 415, pp. 101-109, 1993 0695-7177193 $6.00 + 0.00

Printed in Great Britain. All rights reserved Copyright@ 1993 Pergamon Press Ltd

GROUP DECISION MAKING USING THE ANALYTIC HIERARCHY PROCESS

INDRANI BASAK

Pennsylvania State University, Altoona Campus

3000 Ivyside Park, Altoona, PA 16601-3760, U.S.A.

THOMAS SAATY

Joseph M. Katz Graduate School of Business, 322 Mervis Hall

University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

Abstract-Various methods of consensus of preference rankings of individuals are described in applying group decision making. Here, we discuss two different approaches, one deterministic and the other stochastic. The first is applicable to situations where the group is small, and the second applies to opinions of the population at large where one cannot deal with people on an individual basis. The dividing line between these two approaches is not made in stone and it is the choice of the decision maker to select one approach or the other. How does one synthesize the judgments of a group? In these approaches we address the issues of obtaining priority weights for a group of individuals. Importance ranking of the judges is considered in the deterministic case.

1. INTRODUCTION

In group decision making, aggregation of the preference rankings of individuals into a consensus ranking is the most important problem. Consensus is a process of general agreement on public issues. The question remains as to how many individuals must agree in order to carry the group. Decision for a group can be dictated by one or several people from within or outside the group, or by partial or total participation by the group. When the decision is made through participation, there are at least three known types of consensus-spontaneous, emergent, and manipulated. Spontaneous consensus is exhibited in traditional communities, with very few public issues, who change their view as a collective entity. Emergent consensus happens only in nontraditional societies which are relatively secular and urbanized. After all points of view have been considered, each individual weighs and judges the ideas and then draws a rational conclusion. Crystallization of these judgments forms the public opinion. If the emergent majority is sufficiently forceful, the minorities would adopt its view and emergent consensus arises with the viewpoint of the majority. Manipulated consensus exists in societies in which emergent consensus can theoretically occur, but general agreement depends on who controls the means or power of persuasion. The last two types of consensus generally prevail in modern societies and will be studied in the paper. We consider ways to amalgamate or synthesize group judgments and thereby obtain a public judgment. A basic notion underlying these types of consensus, which is practical throughout the world, is that human wisdom is worthy of aggregation in making a decision.

In the AHP, two fundamentally different approaches are discussed in Section 2. One of them is deterministic and the other statistical or stochastic. The participating individuals may be geographically close together to engage themselves in extended discussion so that all empirical data and theoretical ratiocination have been discussed, or they may be dispersed so that they do not have the scope for discussion over the issue. When a small group of individuals work closely together-interacting and influencing each other, the deterministic approach would be appropriate. We synthesize their judgments mathematically by the eigenvalue method. When a large number of geographically scattered individuals provide the judgments, inconsistency (variation) between individuals is much more important than the inconsistency of a single one of them. Thus, we need a statistical procedure to deal with variation among several people to surface a single value for the weights of the alternatives.

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102 I. BASAK, T. SAATY

In both situations one may examine how to divide the judges into homogeneous subgroups with significant sub-group differences. In the statistical approach to the AHP, we suggest a solution to this problem in Section 3. The judgment matrices are obtained for each group in cases where the groups are significantly different from each other. Otherwise, a single judgment matrix is developed for the judgments of all the individuals.

In the close group decision process, the problem is to find a mathematical method for aggregat- ing the information articulated by its members to determine what weight is most reasonable on the basis of the varying weights assigned by members of the group. Despite apparent differences, the problem is to find a way to ascertain a deeper level of consensus. We consider this problem of amalgamating total information in the group in Sections 4 and 5.

2. TWO DIFFERENT APPROACHES TO GROUP DECISIONS WITH THE AHP

In the literature of the AHP, one needs to distinguish between two different approaches men- tioned in the introduction. The eigenvalue method is a deterministic approach. In the pairwise comparison matrix A = (( eij)); i, j = 1,2, . . . , n, the reciprocal condition a;i aji = 1 is necessary because of the underlying mathematical justification. It derives the relative priority weights using the assumption that one can observe the preferences with certainty and the only uncertainty lies in the elicitation of these preferences which give rise to the inconsistency condition a<j ajk # &k. When equal importance is given to voters in a group, Aczel and Saaty [l] show that the geometric mean is the proper way for synthesizing the judgments given by the judges as reciprocal matrices.

Let the function f(ti, x2, . . , x,) for synthesizing the judgments given by n judges, satisfy the following:

(9 Separability condition (S):

(ii)

forallxr,t~,... , x, in an interval P of positive numbers, where g is a function mapping P onto a proper interval J, and o is a continuous, associative and cancellative operation. [(S) means that the influences of the individual judgments can be separated as above.] Unanimity condition (U):

f( x,z,...,z)=z for all x in P.

(iii)

[(U) means that if all individuals give the same judgment c, that judgment should also be

the synthesized judgment.] Homogeneity condition (H):

(iv)

f(uz1,uzz ,... ,uz,) = uf(21,+2, “‘, G&J where u > 0

and Xk,21zk (k = 1,2,... ,n) are all in P. [For ratio judgments, (H) means that if all individuals judge a ratio u times as large as another ratio, then the synthesized judgment

should also be u times as large.] Power conditions (Pr):

+:,z; ,... ,xP,)=f(11,22,...,x,)P.

[(Pz), for example, means that if the kth individual judges the length of a side of a square to be Xk, the synthesized judgment on the area of that square will be given by the square of the synthesized judgment on the length of its side.] Special case (R = P-i):

f ( 1 -,-,...,- 1 1 1 > =

Xl x2 2, f(~lJ2,*-Jn)

[(R) is of particular importance in ratio judgments. It means that the synthesized value of the reciprocal of the individual judgments should be the reciprocal of the synthesized value of the original judgments.]

Group decision making 103

Aczel and Saaty [l] proved the following theorem.

THEOREM 1. The general separable (S) synthesizing functions satisfying the unanimity (U) and homogeneity (H) conditions are the geometric mean and the root-mean-power. If, moreover, the reciprocal property (R) is assumed even for a single n-tuple (x1, x2, . . . , x,,) of the judgments of n individuals, where not all 21: are equal, then only the geometric mean satisfies all the above conditions.

However, in any rational consensus, those who know more should, accordingly, influence the consensus more strongly than those who are less knowledgeable. Some people are clearly wiser and more sensible in such matters than others, and their opinions should be given appropriately greater weight. For such unequal importance of voters, in (S) not all g’s are the same function. In place of (S), the weighted separability property (WS) is now:

f(Xl, 22,. . . ,Ga) =91(~1)092(~2)o~~~og,(~,).

[(WS) implies that not all judging individuals have the same weight when the judgments are synthesized and these different influences are reflected in the different functions (91, g2,. . . , g,,).]

In this situation, Aczel and Alsina [2] proved the following theorem.

THEOREM 2. The general weighted-separable (WS) synthesizing functions with the unanimity (U) and homogeneity (H) properties are the weighted geometric mean f(x1, x2,. . . ,x,) =

91 92 21 x2 . *.xg- and the weighted root-mean-powers

f(XlrX2,.. .,x,)= 1 \/

q1x:+qzx:+-+qnxln,

where q1 + q2 + -. .+q,, = 1, qk > 0 (k = 1,2 ,..., n), 7 # 0, but otherwise ql,q2 ,..., q,,, 7 are arbitrary constants. If f also has the reciprocal property (R) and for a single set of entries

(Xl, 22,. . * , x,) ofjudgments of n individuals, where not all xk are equal, then only the weighted geometric mean applies.

A deterministic approach is well established in AHP literature. On the other hand, a statistical approach has not yet been properly understood. In the rest of this article, we introduce the statistical approach in greater detail. For a large number of geographically dispersed people, the statistical approach is the one more likely to be needed. It is assumed that the group as a whole has true preference parameters to be estimated by multiple paired comparison input matrices. The difference between existing statistical methods and the method shown here is that in the former, individuals express their judgments as preferred/not preferred/tied [3,4]. In the AHP, further information is elicited on how far apart the alternatives are perceived to be by the judges. Since the AHP is in principle based upon the elicitation of ratio information, the corresponding statistical estimation method utilizes a multiplicative model with input data errors given in the following perturbation form:

where the error cij has some continuous distribution. Therefore, the probability that eij is equal to one (so that pairwise preferences aij are error-free) is equal to zero. This means that preferences subject to error are well suited for statistical analysis. In fact, the score aij assigned by a judge to the alternative Ti over the alternative Tj is thought of as the product of two components. The first, xi/rj is the average preference of Ti over Tj in the decision making body to which the judge belongs. The second, cij, represents the deviation of aij from this average preference. The set of scale values ~1, ~2,. . . ,x, is derived leaving the unsystematic (random) chance deviations to the error cij. Since the errors Cij have continuous distribution, c<j and fji also have continuous distributions. Therefore the probability that the product Eij cji is equal to 1 (SO that the matrix A is reciprocal, aij aji = 1) is equal to zero. This means that if all the off diagonal entries are collected and synthesized, the pairwise comparison matrix will not be reciprocal.

In equation (I), one would expect the statistical distribution of cij to be such that the probability density of cij has a maximum at Eij = 1 (in order to have an error-free estimate with

104 I. BASAK,T. SAATY

maximum probability) and/or the expected value of Eij to be equal to one (in order to have the expected value of the observed preference ratios equal to the true preference ratios). These two requirements contradict each other and cannot be satisfied simultaneously in (1). If cij has a maximum at 1, then the distribution of Cij on the positive real line must be skew-symmetric and therefore its mean or expected value cannot be attained at 1. One must choose between the foregoing two properties. It would not be correct to apply the statistical approach in the setting of the deterministic paradigm, because none of these two properties is satisfied. In the deterministic approach, only n (n - 1)/2 entries of the paired comparison matrix are observed and it is assumed that the comparison matrix is reciprocal. Under this restriction, the corresponding statistical approach fails to have these two properties. The reasons are the following.

RESULT 1. Let X be a continuous and reciprocally symmetric random variable. If the expected value E(Xk) (k y an real number) exists, E(X-“) also exists and E(X”) = E(X-‘) 1 1. For k # 0, E(Xk) = E(X-k) > 1.

RESULT 2. Let f(t) b e a differentiable and reciprocally symmetric probability density function. Then f’(1) = -f(l) and f(z) has no maximum at z = 1.

For the proof of Results 1 and 2, see [5, Remarks 5 and 91, respectively. In the statistical approach, the reciprocal constraint must be satisfied in the true (parameter

values) preference ratios. One may find it analytically easier and more tractable to collect both the entries aij and aji. This is mainly due to the fact that one allows for errors represented by Eij in (1) without coupling Eji and Eij. In that case one finds it convenient to use the maximum likelihood procedure. Also, in this approach it is not necessary for each single individual to provide all the entries of the paired comparison matrix to estimate the priority weights. While there are a maximum of n (n - 1)/2 entries to be collected, only a minimum of n - 1 comparisons are required to establish ratios between n entities. The redundancy in questioning, however, provides more information and the resulting relative priorities would be less sensitive to judgment errors. In a decision-making committee, this approach allows for the possibility of incomplete judgments where some members may abstain from giving their opinions.

3. HOMOGENEOUS SUBGROUPS

In the statistical approach to the AHP, we consider the problem of how to separate judgments into homogeneous subgroups with significant differences between these subgroups. Individual judgments within a homogeneous subgroup can be synthesized to give a single answer for that subgroup.

Given n alternatives, x; i = 1,2,. . . , n, let aij be a positive number, which indicates the

intensity of preference of 1; over Tj (on a ratio scale). The judgment matrix is given by A = (aij). It is technically simpler (though perhaps unrealistic) for one to investigate the case where aij and aji can assume any values. Later we show that the reciprocal relation is a special case of this.

Let Am = (ayjk) be the pairwise comparison judgment matrix where a$ is the kth observa-

tion in the uth group, Ic = 1,2,. . . , ru, u = 1,2,. . . , g, where g is the number of groups and r, is the number of judgment matrices in the uth group. For the purpose of testing preference agreement, we may use a variant of the model specified in (1) to facilitate the use of the statistical methods. Because our purpose for the present is only to determine whether the judgments of the group are homogenous or are significantly different, we simplify (1) by introducing the matrix

By) = (b$‘), where b$ = In ack. For this alternate model, consider the expression:

by’k=pij(U)+cGkI i#j; i,j=1,2 ,..., n, k=1,2 ,..., r,, ~=1,2 ,..., 9,

with

Pji(u) = -Pji(u), i < j,

E[c$] = 0, E[($“)*] = o;, E[c$ $1 = -u;, (2)

E[c;~ csk] = pU C+ = -E[c;~ ctk], E[E;~ c;;(“] = p,, c,” = -E[c;~ c;‘], E[cGk &] = 0,


where i # j # 1 # m. The relation pji(u) = -pji(u) of the parameters ensures that, ideally, the

reciprocal property is satisfied. Let X, denote the dispersion matrix of ~~jk, i # j. It is specified

in the equations given in (2). The (~jk, i # j, k = 1,2,. . . , r, are assumed to be distributed normally with zero mean and dispersion matrix Xc,. We used the lognormal distribution because

(i) of its simplicity, (ii) the logarithms of preferences assume values on the real number line, and

(iii) the specification of a distribution is not important as we are simply testing the homogeneity of the judges and not evaluating priority weights.

One determines the homogeneity of the judgments of the different groups by first forming a likelihood function of the logarithm of the observed sij’s for all the individuals. Then one forms the corresponding likelihood

by considering separate subgroups where r, is the number of judges in the uth group where b”’ and CL(“) are vectors of byjk and ~ij(u). For a specific subgroup to be homogeneous, the value of the

likelihood (Ii) derived from it must exceed the value of the likelihood (12) for all the individuals, where

bk, p, and X are defined analogously to buk, p(“), and X, and r = C,, r,. The null hypothesis for this test of homogeneity is Ho:

T!“) A-3 n!“’ - 7Tj

for all u = 1,2, . . . , g and for all i, j.

f

We accept a particular homogeneous subgroup to be statistically valid, if -2lnX exceeds a certain critical value of its statistical distribution under Ho, where -2 In X satisfies the x2 distribution asymptotically with (g - 1) (n” - n + 2) degrees of freedom and

We reject the hypothesis Ho at 95% level of significance if -2 In X > C, where P[-2 In > C] = 0.05 and accept it otherwise. For the theoretical derivation of this procedure the reader is referred to [6]. At the end, there may be agreement within each of the homogeneous subgroups. But if the

subgroups conflict with each other, then what do we do? One might note that consensus within each subgroup was arrived at in a formally identical manner. In this situation, if there is a forceful majority subgroup, one possibility is to follow the rule that the majority wins. Alternatively, one can aggregate judgments of the homogeneous subgroups weighted by the importance of these subgroups to surface an opinion for the entire group.

4. CONSENSUS PRIORITY WEIGHTS

When several people are involved and they work closely together, they usually justify their judgments in a reasoned debate. Often consensus may not be reached and one must synthesize the judgments. As indicated earlier, we may use the deterministic approach in which individual judgment matrices are synthesized using the geometric mean of the entries when all the voters in the group have equal importance; or, if the participating individuals are believed to have unequal importance, then a weighted synthesizing method would be appropriate. In either case, the eigenvalue method is used to find consensus priority weights of the alternatives in a certain level of the hierarchy. These priority weights are then aggregated for an overall judgment.


For a large number of geographically dispersed people participating in a decision process, one does not have the opportunity to interact closely and influence each other’s views. A statistical approach would be suitable in this case. Consider a typical situation in a certain level of the hierarchy. Suppose n elements Tl, Tz, . . . , T, are compared pairwise by a number of individuals in a decision-making committee. Let *i be the priority weight of Ti, pi 2 0 and xi ?Ti = 1 (ensuring determinacy). For a particular Ti and Tj, i # j, the preference of Ti over Tj is given by N independent judges. For the purpose of independence of the observations, we assume that each judge gives a judgment on just one preference, requiring a total of N n (n - 1) judges. In fact, we can do better than that. Each individual may give judgments on pairs with no common elements; for example, (Tl, T2) and (Ta, T4). This procedure guarantees the independence of observations, but entails the implicit assumption that individual differences among the judges are unsystematic and can be ignored. Following the model described in (l), we have

=i aijk = - fijk.

nj Only one set of scale values 7rl, 7r2, . . . , 7rn will be derived to represent the population of the judges leaving the unsystematic individual differences to the error values Eijk. In (3), the cijh’s are assumed to be independent and do not depend on the ri’s. As discussed in Section 2, cijk

cannot have both the properties that fijk has a mode at 6ijk = 1 and expected value equal to unity. Let us first choose the second property. In other words, we want the expected value of aijk, E[a;jk] to be equal to ri/nj or that the expected value of fijk, E[Cijk] to be equal to one. We assume that the cijk’s to be identically distributed according to Gamma(p,p) since these are positive random variables with expectation unity and the choice of the Gamma distribution for judgment matrices have been shown by Vargas [7] to be appropriate. One can use this assumption if the data fit the Gamma (p, p) distribution. Note that p is treated as a parameter and is estimated jointly with the parameters ?r = (rr,7r2, . . . , n,). We observe that the estimation of priority weights ~1,7rz, . . . , n,, does not depend on p. On the other hand, the estimate of p depends on the priority weights. The maximum likelihood estimates of rri, ~2, . . . , A, derived later can be used to obtain the maximum likelihood estimate r of p as a solution to the following eauation:

A

Q(r) - In(r) = 1+ & [z Tbijk-g 2 Taijk] j

where bijk = lnaijk, T=t(t - l), q(r) = digamma function = v. For the remainder of this case, we assume that the fijk’s are independently distributed according

to Gamma(r, r), where r is obtained from (4). A statistical maximum likelihood procedure is used to obtain estimated values of the ri. In it, the likelihood function is formed, based on

judgments for the input matrices. Maximization of the logarithm of the likelihood leads to the following system of equations (subject to the constraint xi ri = l), [8]

$ [hi(p) - pf si] = 0, where t

N N N N (5) hi(p) = p? 2 Caijk pj

k=l j#l

and Si = c c F, k=l j#l

which are then solved using the following iterative estimation steps. Each stage of the iteration is indexed by I<, Ii’ = 1,2, . . . . For each value of h’, a revised value of each pi is obtained. A stage is sub-divided into sub-stages indexed by u; u = (I< - 1) n,. . , , K n - 1. For each sub-stage, a new estimate p of ?r is obtained through a change of one element of p at a time. The (U + l)th sub-stage value P(~+‘) is obtained from the uth sub-stage value p t”) through replacement only of

the element pi”‘) for which i = u - (Ii - 1) n + 1. Finally, the iterative scheme is given as follows:

[ 1 ~u+l) 4 _ hi(P(“)). I J”) t

(6)


The solutions to the iterative scheme given in (6) converge monotonically to the solution of (5). Estimated variances of those solutions are obtained using large sample theory. It can be shown

that ~(~1-al),~(pz-~~),...,~(~~-77~) [ I

T has a singular normal distribution of

dimensionality (n - 1) in a space of n dimensions with mean vector zero and dispersion matrix X, given by

(7)

where

Xr = (UQ) = (Cij)-‘,

cii=2Nr(n-1)+2Nr(n-1)+2Nr 2

?Ti 2

Tn lri lrn'

2Nr(n-1) + 2Nr + 2Nr 2Nr Cij =

4 ---I

xi xn rj rn *i Tj

bT = (h, 62,. . . r&-l), where

n-1

bi = - c uij ; i= 1,2 )..., n- 1, j=l

i= 1,2 )...) n- 1,

i,j’l,2 ,...) n-l; i # j,

n-l n-l

andaec=x Cc,. i=l j=l

The estimated matrix of variances and covariances of the solutions tells us how good the estimated values of the priority weights are. The trace or the determinant of the matrix (whichever is used ss a criterion) of estimated variances-covariances can be used as the decisive criterion. The smaller the trace or determinant, the better are the estimates of priority weights.

5. THE SPECIAL CASE OF RECIPROCAL MATRICES

Although reciprocal matrices are used in the deterministic mathematical approach, they can also be used in the statistical approach. Maximum likelihood methods for reciprocal matrices are a special case of the foregoing. We have

ri aijk = - fijk,

Tj

where i < j, and as in the non-reciprocal case, we assume that cijk are distributed according to Gamma(6,6). Here, 6 is treated as a parameter and is estimated jointly with the parameters 7r= (rr,rrz,..., n,,). As before, estimation of the priority weights ?rr, IQ,. . . , ?r, does not depend on 6. On the other hand, estimation of 6 depends on the ri. The maximum likelihood estimate of 7r1,*2,..., rr,, derived later can be used to obtain the maximum likelihood estimate d of 6 as a solution to the following equation:

\Ir(d) - In(d) = 1+ & [~~bijk-~~~aijk]‘~~ln(~)’ (8)

where bijk = haijk, \k(d) = digamma function = q. From here onwards we assume that

6ijk are independently distributed according to Gamma(d,d), where d is obtained from (8). The maximum likelihood procedure is then applied to estimate ‘ITi. The maximum likelihood estimate p of w is obtained as the solution to the following system of equations (subject to the constraint C7ri = 1) [9]:

St - - !Ji(P) = 0 Pi!

for i < (n + ‘) 2 ’

where

gi(P) = N (n ‘,l - 22) + c 2 ajik i and si = c eaijkpj,

j<i k=l j>i k=l

(9)


h(p) -pfQi = 0, for i > (n + l) 2 ’

where

and

mi(p) -pf Ti = 0, for i - 1) -- 2 ’

where

mi(P) = Pf c g aijk pj (11)

and j>i k=l

Ti = c gajjk b. j<i k=l 3

These equations are solved using similar iterative steps as described in the non-reciprocal case. The iterative steps for the three cases are, respectively:

(12)

(13)

(14)

The solutions to the iterations (12)-(14) converge monotonically to the solutions of (9)-(ll), respectively. The estimated variances of the solutions are obtained using large sample theory.

[ fi(Pl - Al), mpz - R2)r.. . AqPn - nd]T has a singular normal distribution of dimen-

sionality n - 1 in a space of R dimensions with zero mean vector and dispersion matrix !Z:, given

by

where

ZZr = (aij) = (Cij)-‘,

Cii_ wd+w+ I Nd 7r” r: 7ri x, ’

Cij = Nd(n-l)+ Nd + Nd Nd

2 ---Y ri Xn

bT = @I$::...&&

rj rr, Ai Tj

where

bi = - no bij ; i= 1,2 ,“., n-l, j=l

i=1,2 ,,.., n-l,

i,j= 1,2 ,..., n-l; i # j,

n-l n-l

and ~0s = C C aij. i=l j=l

These estimated variances of the solutions tell us how good the estimated values are. Maximum likelihood estimates of the priority weights at each level of the hierarchy are combined

to yield composite estimates of the priority weights of the alternatives for the overall preference. For example, consider a two level hierarchy. Suppose n alternatives z, i = 1,2,. . . , R are compared with respect to m criteria Cj, j = 1,2,. . . , m. Let us denote the relative worths of the

n alternatives, when compared according to the jth criterion by zj where xi = (R’r , di, . . . ,T:)’

and the worths of the m criteria by 7r where (ar,7rz, . . . , T,,,)~, C’ 4 = 1 and Cy rj = 1. The over-all worth vector A* of the n alternatives when compared under m criteria is obtained by solving the equation:

7r* = Dir, (16)


where

* T 7r* = (XT, n;, . . . , fn) , D= (d,d ,..., A”‘), 7r=(7r1,q ,..., 7rm) T .

The maximum likelihood procedure generates the vectors p' , p2, . . . , pm, as well as the vector p. These vectors are combined in the composite worth vector p+ of the alternatives for the overall preference by solving the equation

P* Gp,

where p*, 6, and p are the estimated values of x*, D, and 1c in (16), respectively. One obtains

the variances of these estimated composite priority weights, to determine the accuracy of the estimates using the following lemma.

LEMMA. If the error terms associated with the criteria and alternatives as well as those corresponding to the different criteria are uncorrelated, then the covariance matrix of the asymptotic distribution of p* is given by

X0= (J $Y),

where

c;“j =cov(&p;)= fI:$(&+X;,+e &+;,

a=1 o=l p=1

for i, j = 1,2, . . , n - 1, where a; ‘s is obtained from (7) or (15) and c&‘s are obtained from (7)

or (15) with the vector x replaced by 7ra and n replaced by m. We have

n-1 n-l n-l

bO’ = (by, b;, . . . ,b;_l), whereby=-xc;, and U$ = C C Ui”. j=l i=l j=l

6. CONCLUDING REMARKS

Often, confusion arises in the literature when people cannot distinguish between two fundamentally different approaches. In this article, we have made an attempt to point out the distinctive features of these two approaches and their different areas of application. The deterministic approach, supported by mathematical proof, is well-established in the literature. Comparatively, the statistical approach is not understood and has not been explained properly. In the context of group decision making in the Analytic Hierarchy Process, we have discussed different ways to make use of these two approaches.

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J. Aczel and T.L. Saaty, Procedures for synthesizing ratio judgments, Journal of Mathematical Psychology

27,93-102 (1983). J. Aczel and C. Alsina, On synthesis of judgments, So&-Econ. Planning Sci. 20, 333-339 (1986). R.A. Bradley and M.E. Terry, Rank analysis of incomplete block designs, I. The method of paired comparisons, Biometrickn 39, 324-345 (1952). RR. Davidson, On extending the Bradley-Terry model to accommodate ties in paired comparisons experiments, Journal of the American Statistical Association 65, 317-328 (1970). J. Fichtner, Some thoughts about the mathematics of the Analytic Hierarchy Process, In Hochschule dcr

Bundeswehr, Miinchen, (1983). I. Basak, When to combine judgments and when not to in Analytic Hierarchy Process: A new method, Mathematical Comput. Modelling 10 (6), 1-14 (1988). L.G. Vargas, Reciprocal matrices with random coefficients, Mathematical Modelling 3, 69-81 (1982). I. Basak, Inference in pairwise comparison experiments based on ratio scales, Journal of Mathematical Psychology (1991) (to appear). I. Basak, Estimation of the multicriteria worths of the alternatives in a hierarchical structure of comparisons, Communications in Statistics-Theory and Methods 113 (lo), 3719-3738 (1989).

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Group decision making using the analytic hierarchy process