Mat/d Modelling, Vol. 9, No. 3-5, pp. 161-176, 1987
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THE ANALYTIC HIERARCHY PROCESS-WHAT IT IS
AND HOW IT IS USED
R. W. SAATY
4922 Ellsworth Avenue, Pittsburgh, PA 15213, U.S.A
Abstract-Here we introduce the Analytic Hierarchy Process as a method of measurement with ratio
scales and illustrate it with two examples. We then give the axioms and some of the central theoretical
underpinnings of the theory. Finally, we discuss some of the ideas relating to this process and its
ramifications. In this paper we give special emphasis to departure from consistency and its measurement
and to the use of absolute and relative measurement, providing examples and justification for rank
preservation and reversal in relative measurement.
1. INTRODUCTION
The Analytic Hierarchy Process (AHP) is a general theory of measurement. It is used to derive
ratio scales from both discrete and continuous paired comparisons. These comparisons may be
taken from actual measurements or from a fundamental scale which reflects the relative strength
of preferences and feelings. The AHP has a special concern with departure from consistency, its
measurement and on dependence within and between the groups of elements of its structure. It has
found its widest applications in multicriteria decision making, planning and resource allocation
and in conflict resolution [l-6]. In its general form the AHP is a nonlinear framework for carrying
out both deductive and inductive thinking without use of the syllogism by taking several factors
into consideration simultaneously and allowing for dependence and for feedback, and making
numerical tradeoffs to arrive at a synthesis or conclusion. T. L. Saaty developed the AHP in 1971-
1975 while at the Wharton School (University of Pennsylvania, Philadelphia, Pa).
This paper is written in an expository style to make it more accessible to the nonmathematician
interested in the subject. Our purpose is three-fold: first to introduce the AHP through two
hierarchically structured examples (in contrast with more elaborate feedbackknetwork structures);
next to give the axioms and some of the central theoretical underpinnings of the theory; and finally,
to discuss some of the ideas relating to the theory and its application that are most relevant in
using the AHP. In particular, we write about its generalizations and about where to use absolute
or relative measurement.
For a long time people have been concerned with the measurement of both physical and
psychological events. By physical we mean the realm of what is fashionably known as the tangibles
as it relates to some kind of objective reality outside the individual conducting the measurement.
By contrast, the psychological is the realm of the intangibles as it relates to subjective ideas and
beliefs of the individual about himself or herself and the world of experience. The question is
whether there is a coherent theory that can deal with both these worlds of reality without
compromising either. The AHP is a method that can be used to establish measures in both the
physical and social domains.
In using the AHP to model a problem one needs a hierarchic or a network structure to represent
that problem and pairwise comparisons to establish relations within the structure. In the discrete
case these comparisons lead to dominance matrices and in the continuous case to kernels of
Fredholm operators [7], from which ratio scales are derived in the form of principal eigenvectors,
or eigenfunctions, as the case may be. These matrices, or kernels, are positive and reciprocal, e.g.
aij = l/aji. In particular, special efforts have been made to characterize these matrices, and several
of the more theoretical papers in this issue address questions and offer new information about
them. Because of the need for a variety of judgments, there has also been considerable work done
to characterize the process of synthesizing diverse judgments [S].
161
162 R. W. SAATV
2. TWO EXAMPLES
In general a hierarchical model of some societal problem might be one that descends from a
focus (an overall objective), down to criteria, down further to subcriteria which are subdivisions of
the criteria and finally to the alternatives from which the choice is to be made. Our first example
is an application of the AHP developed by Hamllainen and Seppalainen [9] to solve a large-scale
socio-technical decision problem with intangible criteria in Finland.
The parliament of Finland was faced with making a decision about what type of power plant to
build. Members of parliament were concerned with how the new power plant would affect Finland’s
national economy; the health, safety and environment for Finnish citizens; and how political factors
such as Finnish relations with the U.S.S.R. affect the type of plant to be adopted. Their goal was
to build a power plant that would best serve the overall welfare of the nation. Each of the main
criteria was further decomposed into subcriteria, followed by the alternatives: the different kinds
of power plants. A simple hierarchy of the problem is shown in Fig. 1.
There has been extensive work on how to structure hierarchies for practical problems. Two
general types of hierarchies are the forward and the backward process hierarchies. All problems
have been found to fall into one or the other of these two categories. Planning combines them in
an iterative fashion [4]. The elements of a hierarchy are grouped in clusters according to
homogeneity (see Axiom 2 in Section 3) and a level may consist of one or several homogeneous
clusters. The elements in each level may be regarded as constraints, refinements or decompositions
of the elements above. In a “complete” hierarchy, as in the example of “choosing the best college”
given below, all the elements in one level have all the elements in the succeeding level as descendants.
In this case the levels are single homogeneous clusters. Otherwise a hierarchy is “incomplete”, as
in the Finnish energy example below.
Focus: DVEQALL WELFARE OF THE NATION
Criteria: NATIONAL HEALTH, SAFETY POLITICAL
ECONOMY ENVIRONMENT FACTORS
Sub- I
criteria: Cheap Natural
Electricity
Resources Indepen-
dence
Foreign
Trade
Unavoidable
Pollution
Central-
ization
Capital
Resources
Accidents &
Long-term
Risks
Political
Cooperat-
iveness
Alternatives:
NO BIG
POWER
PLANTS
COAL-FIRED
POWER
PLANT
NUCLEAR
POWER
PLANT
Fig. 1. Hierarchy of the Finnish energy decision
The AHP-what it is and how it is used 163
We can make a few observations about this hierarchy. Obviously, it is simple: more elements
could be added, at any level, and more levels. The question is, “How much should one include in
a hierarchy?” A general rule is that the hierarchy should be complex enough to capture the situation,
but small and nimble enough to be sensitive to changes. For the members of the Finnish Parliament
the hierarchy given above captured the degree of complexity that they could address as a political
unit.
Pairwise comparisons are fundamental in the use of the AHP. The members of parliament must
first establish priorities for their main criteria by judging them in pairs for their relative importance,
thus generating a pairwise comparison matrix. Judgments which are represented by numbers from
the fundamental scale below are used to make the comparisons. The number of judgments needed
for a particular matrix of order n, the number of elements being compared, is n(n - 1)/2 because
it is reciprocal and the diagonal elements are equal to unity. The paper by Harker [lo, this issue,
pp. 353-3601 gives conditions under which it is possible to use fewer judgments and still obtain
accurate results.
The next step is for the members of parliament to compare the subcriteria that belong to each
of the main criteria, thus constructing three more pairwise comparison matrices for level 3. Then
the three alternatives are compared with respect to each of the subcriteria, leading to nine pairwise
comparison matrices for level 4. The final step is to weight or synthesize the results to obtain the
final priorities of the three power plants.
We now give a second example with more details of how these operations are conducted.
The author has a teenage son who graduated from high school last year with good grades and
high SAT scores. We used an AHP model to select the college of his choice. He was accepted by
Swarthmore College, Northwestern University, the University of Michigan, Vanderbilt University
and Carnegie-Mellon University (two blocks from his home). Since the cost of each is about the
same, it was not included as a criterion in the hierarchy. He has an artistic temperament, writing
right-handed in a hooked position, which is said to prove that he is “right-brained”. The result of
the analysis was that he chose Northwestern University, whose reputation and academic environment
are slightly less preferred than that of Swarthmore College, because it appealed more to his nature.
At Northwestern, he selected a dormitory room facing Lake Michigan, combining a convivial
4-year vacation site with study.
To summarize his outlook: as far as LOCATION was concerned, the farther away from home
a school is perceived to be, the better (but not necessarily linearly); REPUTATION (how the school
is rated) was slightly more important than ACADEMICS (personalized attention, small classes vs
large ones); and AMBIENCE (how happy he felt at the place) was a little more important than
either ACADEMICS or LOCATION. The hierarchy is shown below in Fig. 2.
Table 1. The fundamental scale
Intensity of importance
on an absolute scale Definition Explanation
I Equal rmportance Two activities contribute equally to the
objective
3 Moderate importance of one over another Experience and judgment strongly favor one
activity over another
5 Essential or strong importance Experience and judgment strongly favor one
acttvity cwer another
7 Vtry strong importance An activity is strongly favored and its dominance
demonstrated in practice
9 Extreme importance The evidence favoring one activity over
another is of the highest possible order of
affirmation
24.8 Intermediate values between the two adjac- When compromise is needed
ent judgments
Reciprocals If activity i has one of the above numbers
assigned to It when compared with activity
j, then j has the reciprocal value when compared
with i
Ration& Ratios arising from the scale If consistency were to be forced by obtaining
n numerical values to span the matrix
1 the elements being compared are closer together than indicated by the scale, one can use the scale 1.1, 1.2, .,
ill finer. one can use an appropriate even liner refinement.
164 R. W. SAATV
What question should one ask?
On examining the matrices below, we note that a pair of elements (i,j) in a level of the hierarchy
are compared with respect to a parent element in the level immediately above as a common
property or criterion used to judge as to which one has it more and by how much..The typical
way to phrase a question to fill an entry in the matrix of comparisons is: when considering two
elements, i on the left side of the matrix andj on the top, which has the property more, or which
one satisfies the criterion more, i.e. which one is considered more important under that criterion
and how much more (using the fundamental scale values from Table l)? This gives us aij (oraji).
The reciprocal value is then automatically entered for the transpose.
The question asked in making a pairwise comparison can influence the judgments provided and
hence also the priorities. It must be made clear from the start what the focus of the hierarchy is
and how the elements in the second level either serve to fulfill that focus or are its consequence,
and so on down the hierarchy for each parent element and its descendants.
We have the following for the matrix of pairwise comparisons of the criteria with respect to the
overall focus:
FOCUS LOCATION AMBIENCE REPUTATION ACADEMICS
LOCATION 1 l/7 115 115
AMBIENCE 7 1 2 3
REPUTATION 5 l/2 1 1
ACADEMICS 5 l/3 1 1
In this matrix the first nontrivial comparison is (LOCATION, AMBIENCE). The question is:
“How much more is LOCATION preferred over AMBIENCE?” AMBIENCE is actually preferred
very strongly (7 times) over LOCATION, so the reciprocal value l/7 is entered in the (1,2) position.
The value 7 is automatically entered in the transpose position (2, I) for (AMBIENCE, LOCATION).
As another illustration, AMBIENCE is judged to be between equal and moderately more important
than REPUTATION and hence the value 2 (2 times) is entered in the (2,3) position with the
reciprocal (l/2) automatically entered in the (3,2) position and so on.
A matrix is said to be consistent if aijaj, = aik, Vi, j, k. Note that this matrix is inconsistent.
For example, (AMBIENCE, REPUTATION) = 2, while (REPUTATION, ACADEMICS) = 1,
therefore to be consistent (AMBIENCE, ACADEMICS) = (AMBIENCE, REPUTATION) x (REPUTATION,
ACADEMICS) = (2) x (1) = 2. But it has the value 3. Thus the value entered in
Focus: CHOOSE BEST COLLEGE
Criteria: LOCATION AMBIENCE REPUTATION ACADEMICS
Alternatives:
b w +J/ .$J 4
SWARTHMORE NORTHWr\TCRN ll.MI(‘tI. VANDERBILT CMU
Fig. 2. Hierarchy for choosing a college.
The AHP-what it is and how it is used 165
the (2,4) position is larger than it should be to be consistent, but we then enter the reciprocal value
l/3 in the (4,2) position and note that l/3 < l/2. The relevance of this observation is that while
one value exceeds the corresponding consistent value, its reciprocal is less than the reciprocal of
the consistent value, and hence there is a tendency to compensate. When a positive reciprocal
matrix of order n is consistent, the principal eigenvalue has the value n. When it is inconsistent,
the principal eigenvalue exceeds n and its departure from n serves as a measure of inconsistency
by forming a ratio (called the consistency ratio, C.R.) of this difference to the average of the
corresponding differences from n of the principal eigenvalues of a large number of matrices of
randomly chosen judgments (see Section 4).
The next step is to derive the scale of priorities (or weights). It has been shown that this scale is
obtained by solving for the principal eigenvector of the matrix and then normalizing the result.
This is called the local derived scale before weighting by the priority of its parent criterion (which
for the second-level elements is always equal to unity, the weight of the focus). After weighting, it
is called the global derived scale. It has also been shown that the principal eigenvector is the only
way to obtain the derived scale that makes use of all the dominance information given in the
matrix when the latter is inconsistent. In computing the principal eigenvector all possible
intransitivities and chains of intransitivity enter into the calculations as the matrix is raised to
powers [ 111.
We obtain for the vector of relative weights:
(LOCATION, AMBIENCE, REPUTATION, ACADEMICS) = (0.053, 0.491,0.238,0.213)
with an inconsistency ratio of 0.02 (see Section 4 for further details).
The next step is to set up matrices of paired comparisons for the schools in level 3 compared
with respect to the criteria in level 2. There are four such matrices. In the illustrative matrix, given
below, we compare SWARTHMORE, NORTHWESTERN, U. MICHIGAN, VANDERBILT and
CMU with respect to their perceived desirability according to LOCATION:
LOCATION SWARTH NORTHW U. MICH VANDERB CMU wt
SWARTH
NORTHW
U. MICH
VANDERB
CMU
1 l/4 l/3 l/3 7 0.115
4 1 2 3 7 0.402
3 l/2 1 3 6 0.283
3 l/3 l/3 1 4 0.163
l/7 l/7 l/6 l/4 1 0.037
CR. = 0.092
Again the local derived scale for the schools, for criterion (1) LOCATION, after solving for the
principal eigenvector, is listed in column 4 under weights:
(1) (SWARTH, NORTHW, U. MICH, VANDERB, CMU) = (0.115, 0.402, 0.283, 0.163, 0.037).
We now give the local derived scales for the other three criteria, (2) AMBIENCE, (3)
REPUTATION and (4) ACADEMICS, omitting the pairwise comparison matrices:
(2) (SWARTH, NORTHW, U. MICH, VANDERB, CMU) = (0.034, 0.539, 0.250, 0.121, 0.056);
(3) (SWARTH, NORTHW, U. MICH, VANDERB, CMU) = (0.521, 0.235, 0.147, 0.038, 0.059);
(4) (SWARTH, NORTHW, U. MICH, VANDERB, CMU) = (0.564, 0.209, 0.132, 0.040, 0.055).
The software package supporting the AHP, called Expert Choice, for the IBM PC, was used to
make these calculations and can guide the decision maker to improve his inconsistency if needed.
When the consistency ratio exceeds 0.10 appreciably the judgments often need reexamination. To
synthesize the overall priority scale we multiply as follows:
166
I
0.270
0.387
0.201
0.086
0.055
This gives a final priority of 0.387 for NORTHWESTERN, the most preferred school, with
SWARTHMORE as runner-up with a priority of 0.270. The overall consistency of these judgments
is obtained as a ratio of sums of weighted inconsistency indices to the corresponding sums of
weighted random indices (see Section 4).
In the next section we examine some of the mathematical foundations of the AHP.
3. THE AXIOMS OF THE AHP
Three principles guide one in problem solving using the AHP [ 121: decomposition, comparative
judgments and synthesis of priorities as embodied in the previous example.
The decomposition principle is applied by structuring a simple problem with the elements in a
level being independent from those in succeeding levels, working downward from the focus in the
top level, to criteria bearing on the focus in the second level, followed by subcriteria in the third
level, and so on, from the more general (and sometimes uncertain) to the more particular and
concrete. Saaty [ 121 makes a distinction between two types of dependence which he calls functional
and structural. The former is the familiar contextual dependence of elements on other elements in
performing their function, whereas the latter is the dependence of the priority of elements on the
priority and number of other elements. Absolute measurement, sometimes called scoring, is used
when it is desired to ignore such structural dependence among elements, while relative measurement
is used otherwise.
The principle of comparative judgments is applied to construct pairwise comparisons of the
relative importance of elements in some given level with respect to a shared criterion or property
in the level above, giving rise to the kind of matrix encountered above and its corresponding
principal eigenvector.
The third principle is that of synthesizing the priorities. In the AHP priorities are synthesized
from the second level down by multiplying local priorities by the priority of their corresponding
criterion in the level above and adding, for each element in a level according to the criteria it
affects. (The second-level elements are multiplied by unity, the weight of the single top-level goal).
This gives the composite or global priority of that element, which in turn is used to weight the
local priorities of the elements in the level below compared to each other with it as the criterion,
and so on to the bottom level.
Four axioms govern the AHP and utilize the notion of paired comparisons as a primitive.
Let ‘11be a finite set of n elements called alternatives. Let K be a set of properties or attributes with
respect to which elements in (11are compared. We will refer to the elements of a as criteria. A criterion
is a primitive.
We perform binary comparisons on the elements in ‘It according to a criterion in c. Let >c be a
binary relation on 9t representing “more preferred than” with respect to a criterion C in 6. Let -c
be the binary relation “indifferent to” with respect to a criterion C in B. Hence, given two elements
Ai, AjcA, either Ai >.Aj or Aj >=Ai or Ai -c Aj, VC ~6. We use Ai kc Aj to indicate more
preferred or indifferent. A given family of binary relations >c with respect to a criterion C in 6 is
a primitive.
Let ‘@be the set ofmappings from% x 9l to R+ (the set ofpositive reals). Letf’: K -+ $. Let Pc gf(C)
for CE K. P, assigns a positive real number to every pair (Ai, Aj)c‘91 x 2l.Let Pc(Ai, Aj) = UijE R ‘,
Ai, Aj E 91. For each C E K;,the triple (%!Ix VI, R +, PC) is a fundumentul or primitiw scale. A fundamental
scale is a mapping of objects to a numerical system.
The AHP-what it is and how it is used 167
DeJinition. For Ai, A,E(U and CE&
Ai >,Aj iff Pc(Ai, Aj) > 1,
Ai +=Aj iff P&A,, Aj) = 1.
If Ai > c Aj, we say that Ai dominates Aj with respect to C E Cs.Thus, PC represents the intensity or
strength of preference for one alternative over another.
Axiom 1 (the reciprocal axiom)
For all Ai, Aj~‘U and CE~,
Pc(Ai, Aj) = l/Pc(A,, Ai).
This axiom says that the comparison matrices we construct are formed of paired reciprocal
comparisons, for if one stone is judged to be five times heavier than another, then the other must
perforce be one-fifth as heavy as the first. It is this simple but powerful relationship that is the
basis of the AHP.
Let A = (aij) = (P&A,, Aj)) be the set of paired comparisons of the alternatives with respect to a
criterion C E 6. By the definition of PC and Axiom 1, A is a positive reciprocal matrix. The object is
to obtain a scale of relative dominance (or rank order) of the alternatives from the paired
comparisons given in A.
We will now show how to derive the relative dominance of a set of alternatives from a
pairwise comparison matrix A. Let RMM(“)be the set of (n x n) positive reciprocal matrices
A = (aij) = (PJA,, Ai)), VCEE. Let [0, 11” be the n-fold Cartesian product of [0, l] and let
W: &n, > [0, 11” for AE RMM(“),W(A) is an n-dimensional vector whose components lie in the
interval [O, 11. The triple (RMM(“),[0, l]“, W) is a derived scale. A derived scale is a mapping between
two numerical relational systems.
An important aspect of the AHP is the idea of consistency. If one has a scale for a property
possessed by some objects and measures that property in them, then their relative weights with
respect to that property are fixed. In this case there is no judgmental inconsistency (although if
one has a physical scale and applies it to objects in pairs and then derives the relative standing of
the objects on the scale from the pairwise comparison matrix, it is likely that inaccuracies will have
occurred in the act of applying the physical scale and again there would be inconsistency). But
when comparing with respect to a property for which there is no established scale or measure, we
are trying to derive a scale through comparing the objects two at a time. Since the objects may be
involved in more than one comparison and we have no standard scale, but are assigning relative
values as a matter of judgment, inconsistencies may well occur. In the AHP consistency is defined
in the following way, and we are able to measure inconsistency (see Section 4).
Dejinition. The mapping PC is said to be consistent iff
PcAAitAj)Pc(Aj,AA = PAA,,Ad, Vi, j, k.
Similarly, the matrix A is consistent iff aijajr = air, Vi, j, k.
We now turn to the hierarchic Axioms 2-4, and related definitions.
In a partially ordered set, we define x < y to mean that x < y and x # y; y is said to cover
(dominate) x. If x < y, then x < t < y is possible for no t. We use the notation x- = {yl x covers
y} and x’ = {y )y covers x}, for any element x in an ordered set.
Let H be a finite partially ordered set. Then H is a hierarchy if it satisfies the following conditions:
(a) there is a partition of H into sets L,, k = 1,. , h, for some h where L1 = {b}, b
is a single element;
(b) XEL, implies x-EL,,+~, k = l,...,h - 1;
(c) XEL, implies x+EL~-~, k = 2,..., h.
The notions of fundamental and derived scales can be extended to x E L, and x- E Lt+, ,
replacing C and A, respectively. The derived scale resulting from comparing the elements in x
168 R. W. SAATY
with respect to x is called a local derived scale or the local priorities for the elements in X-.
De@ition. Given a positive real number p > 1, a nonempty set x- c Lk+ 1 is said to be
p-homogeneous with respect to x E Lk if for every pair of elements y1 , yz E.x-, l/p d P&r, y2) < p.
In particular the reciprocal axiom implies that Pc(yi,yi) = 1.
Axiom 2 (the homogeneity axiom)
Given a hierarchy H, XEH and XEL~, x- c L,,, is p-homogeneous for k = 1,. , h - 1.
Homogeneity is essential for meaningful comparisons, as the mind cannot compare widely
disparate elements. For example, we cannot compare a grain of sand with an orange according to
size. When the disparity is great, elements should be placed in separate clusters of comparable size,
or in different levels altogether.
Given L,, L k+1 5 H, let us denote the local derived scale for YE x- and x E L, by $k+ ,(y/x),
k = 2,3,. , h - 1. Without loss of generality we may assume that tik+ r(y/x) = 1. Consider the
matrix IC/,JLk/Lk- ,) whose columns are local derived scales of elements in L, with respect to
elements in Lk _ 1.
Dejinition. A set A is said to be outer dependent on a set C if a fundamental scale can be defined
on A iyith respect to every C E6.
The process of relating elements (e.g. alternatives) in one level of the hierarchy according to the
elements of the next higher level (e.g. criteria) expresses the dependence (what is called outer
dependence) of the lower elements on the higher so that comparisons can be made between them.
The steps are repeated upward in the hierarchy through each pair of adjacent levels to the top
element, the focus or goal.
The elements in a level may also depend on one another with respect to a property in another
level. Inputtoutput of industries is an example of the idea of inner dependence, formalized as
follows.
Dejinition. Let U be outer dependent on K. The elements in 91 are said to be inner dependent with
respect to C E Q if for some A E ‘?I,PI is outer dependent on A.
Axiom 3
Let H be a hierarchy with levels L,, LX,..., L,. For each L,, k = 1, 2,_.., h - I:
(1) L!i+, is outer dependent on Lk;
(2) L!i+1 is not inner dependent with respect to all XE L,;
(3) L, is not outer dependent on Lk+,
Axiom 4 (the axiom ofexpectations)
C c H - L,, A=Lh.
This axiom is merely the statement that thoughtful individuals who have reasons for their beliefs
should make sure that their ideas are adequately represented in the model. All alternatives, criteria
and expectations (explicit and implicit) can be and should be represented in the hierarchy. This
axiom does not assume rationality. People are known at times to harbor irrational expectations
and such expectations can be accommodated.
Based on the concepts in Axiom 3 we can now develop a weighting function. For each XE H,
there is a suitable weighting function (whose nature depends on the phenomenon being hierarchically
structured):
w,: x + [0, l] such that c W,(Y) = I.
)‘EX
Note that h = 1 is the last level for which x- is not empty.
The sets Li are the levels of the hierarchy, and the function w, is the priority function of the
elements in one level with respect to the objective x. We observe that even if x- # Lk (for some
level Lk), w, may be defined for all of L, by setting it equal to zero for all elements in L, not in x.
The weighting function is one of the more significant contributions towards the application of
hierarchy theory.
The AHP--what it is and how it is used 169
Definition. A hierarchy is complete if, Vx c L,, x+ c Lt_,
We can state the central question:
Basic Problem. Given any element x E L,, and subset S c L,, (E < p), how do we
define a function w,,~: S + [0, l] which reflects the properties of the priority functions
on the levels L,, k = cc,.. . ,p - 1. Specifically, what is the function
wb.Lh. .Lh 4 [O, l]?
In less technical terms, this can be paraphrased thus:
Given a social (or economic) system with a major objective b, and the set L, of basic
activities, such that the system can be modeled as a hierarchy with largest element
b and lowest level L,. What are the priorities of the elements of any level and in
particular those of L,, with respect to b?
From the standpoint of optimization, to allocate a resource to the elements any interdependence
may take the form of input-output relations such as, for example, the interflow of products between
industries. A high priority industry may depend on flow of material from a low priority industry.
In an optimization framework, the priority of the elements enables one to define the objective
function to be maximized, and other hierarchies supply information regarding constraints, e.g.
input-output relations.
We now present a method to solve the Basic Problem. Assume that Y = {yi,. ..,y,,} c L, and
that X = {xi,. ..,x,,_+,} c LL+l. Without loss of generality we may assume that X = L,, 1, and
that there is an element z E Lk such that yi E z-. Then consider the priority functions
w,: Y + [O, l] and wyj:X + [0, l]j = 1,. . . ,mk.
Construct the priority function of the elements in X with respect to z, denoted w, w: X + [0, 11,by
W(Xi) = Fwy,(xi)wz(Yj)t i = l,...,mk+r
j= 1
It is obvious that this is no more than the process of weighting the influence of the element yj
on the priority of xi by multiplying it with the importance of xi with respect to z.
The algorithms involved will be simplified if one combines the w,j(xi) into a matrix B by setting
bij = w,j(xi). If one further sets wi = w(xi) and WI = w,(yj), then the above formula becomes
wi = zbijwJ, i = l,...,nk+,.
j=l
Thus, one may speak of the priority vector w and, indeed, of the priority matrix B of the (k + 1)th
level; this gives the final formulation,
w = Bw’.
The following theorem is easy to prove.
Theorem. Let H be a complete hierarchy with largest element b and h levels. Let BI, be the
priority matrix of the kth level, k = 1,. .. , h. If w’ is the priority vector of the pth level with respect
to some element z in the (p - 1)th level, then the priority vector w of the qth level (p < 4) with
respect to z is given by
w =BqB,_,...B,+Iw’.
Thus, the priority vector of the lowest level with respect to the element b is given by
w = BhB,_ 1. . . Bzb,
if L, has a single element, b, = 1. Otherwise, b, is a prescribed vector. We note that the pairwise
comparison process takes into consideration nonlinearities. Such nonlinearities are captured by
the composition weighting process.
170 R. W. SAATY
Network systems
Often alternatives depend on criteria and criteria on alternatives and there should be a cycle
connecting the two which is more accurately studied with the network feedback approach. The
AHP has been generalized to deal with feedback as shown below, although people generally prefer
to simplify and arrange their thinking in terms of a linear hierarchy even if the answers are only
approximate.
A network is a set of nodes (each of which consists of a set of elements) and a set of arcs which
indicate the order of interaction among the components. The priorities of the elements in each
node are components of the principal eigenvector of the matrix of pairwise comparisons of the
relative impact of these elements with respect to an element or node with which they interact. The
interaction is indicated by an arc of the network. All such eigenvectors define what is known as a
supermatrix of impact priorities. By weighting the eigenvectors corresponding to each component
by the priority of that component in the system, the supermatrix is transformed into a stochastic
matrix. The limiting impact priorities are obtained by computing large powers of this matrix.
Formally we have the following.
Dejnition. A partially ordered set S is a network system if
(a) there is a partition of S into sets C,, k = 1,. , s;
(b) there is an ordering on C,, k = 1,. . ,s such that x c CI, implies either x- or x’
is in CE, for some kj or both x- E Ck,, xc E Ckj for one or more kj;
(c) For each XES, there is a suitable weighting function
w,: x- -+ [0, l] such that 1 w,(y) = 1
YEXand
for Ck c S, k = 1,. , s, there is a weighting function
where C; = {Ch 1Ck covers C,}
We shall now turn to the calculation procedures for the weights and for the inconsistency index.
4. THE EIGENVECTOR SOLUTION FOR WEIGHTS AND CONSISTENCY
There is an infinite number of ways to derive the vector of priorities from the matrix (aij). But
emphasis on consistency leads to an eigenvalue formulation.
If aij represents the importance of alternative i over alternativej and ajk represents the importance
of alternative j over alternative k then aik, the importance of alternative i over alternative k, must
equal aijajk for the judgments to be consistent. If we do not have a scale at all, or do not have it
conveniently, as in the case of some measuring devices, we cannot give the precise values of \vi/wj
but only an estimate. Our problem becomes A’w’ = imaxw’, where I.,,, is the largest or principal
eigenvalue of A’ = (a,!j) the perturbed value of A = (aij) with aji = l/aij forced. To simplify the
notation we shall continue to write Aw = kmaxi, where A is the matrix of pairwise comparisons.
The solution is obtained by raising the matrix to a sufficiently large power then summing over
the rows and normalizing to obtain the priority vector w = (wl,. , w,). The process is stopped
when the difference between components of the priority vector obtained at the kth power and at
the (k + 1)th power is less than some predetermined small value.
An easy way to get an approximation to the priorities is to normalize the geometric means of
the rows. This result coincides with the eigenvector for n < 3. A second way to obtain an
approximation is by normalizing the elements in each column of the judgment matrix and then
averaging over each row.
We would like to caution the reader that for important applications one should only use the
eigenvector derivation procedure because approximations can lead to rank reversal in spite of the
The AHP---what it is and how it is used 171
closeness of the result to the eigenvector [13]. It is easy to prove that for an arbitrary estimate x
of the priority vector,
,lirn, & Akx = cw
““ax
where c is a positive constant and w is the principal eigenvector of A. This may be interpreted
roughly to say that if we begin with an estimate and operate on it successively by A/&,,,, to get
new estimates, the result converges to a constant multiple of the principal eigenvector.
A simple way to obtain the exact value (or an estimate) of &,,% when the exact value (or an
estimate) of w is available in normalized form is to add the columns of A and multiply the resulting
vector with the vector w. The resulting number is A,,,,, (or an estimate). This follows from
and
The problem is now, how good is the principal eigenvector estimate w? Note that if we obtain
w = (wi,. ., w,) by solving this problem, the matrix whose entries are wi/wj is a consistent matrix
which is our consistent estimate of the matrix A. The original matrix A itself need not be consistent.
In fact, the entries of A need not even be transitive; i.e. A, may be preferred to A2 and A2 to A,
but A, may be preferred to A,. What we would like is a measure of the error due to inconsistency.
It turns out that A is consistent iff A,,,,, = n and that we always have II,,, 2 n. This suggests using
i “ax - n as an index of departure from consistency. But
where Li, i = 1,. . . , n are the eigenvalues of A. We adopt the average value (A,,,,, - n)/(n - l), which
is the (negative) average of E.i, i = 2,. . ., n (some of which may be complex conjugates).
It is interesting to note that 2(1.,,, - n)/(n - 1) is the variance of the error incurred in estimating
aij. This can be shown by writing
aij = (Wi/Wj)Eij,Eij > 0 and eij = 1 + dij,dij > - 1,
and substituting in the expression for &,,ax. It is dij that concerns us as the error component and
its value 16ij(< 1 for an unbiased estimator. Normalizing the principal eigenvector yields a unique
estimate of a ratio scale underlying the judgments.
The C.I. of a matrix of comparisons is given by C.I. = (3,,,, - n)/(n - 1). The consistency ratio
(C.R.) is obtained by comparing the C.I. with the appropriate one of the following set of numbers,
each of which is an average random consistency index (R.I.) derived from a sample of size 500, of
a randomly generated reciprocal matrix using the scale l/9, l/8, . . . , 1, . .8, 9 to see if it is about
0.10 or less. If it is not less than 0.10, study the problem and revise the judgments:
n
Random
consistency
index (R.I.)
1 2 3 4 5 6 7 8 9 10
0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49
172 R. W. SAATV
Why tolerate 10% inconsistency? The priority of consistency to obtain a coherent explanation
of a set of facts must differ by an order of magnitude from the priority of inconsistency which is
an error in the measurement of consistency. Thus, on a scale from O-1, inconsistency should not
exceed 0.10 by very much. Note that the requirement of 10% should not be made much smaller
such as 1% or 0.1%. The reason is that inconsistency itself is important, for without it new
knowledge which changes preference order cannot be admitted. Assuming all knowledge to be
consistent contradicts experience which requires continued adjustment in understanding. Thus
the objective of developing a wide-ranging consistent framework depends on admitting some
inconsistency.
This also accounts for why the number of elements compared should be small. If the number of
elements is large, their relative priorities would be small and error can distort these priorities
considerably. If the number of items is small and the priorities are comparable a small error does
not affect the order of magnitude of the answers and hence the relative priorities would be about
the same. For this to happen, the items must be < 10 so their values on the whole would be > 10%
each and hence remain relatively unaffected by 1% error for example.
The consistency index (C.I.) for an entire hierarchy is defined by
where wij = 1 for j = 1, and nij+, is the number of elements of the (j + 1)th level with respect to
the ith criterion of the jth level.
Let ICJ be the number of elements of C; , and let w(kJ(h)be the priority of the impact of the hth
component on the kth component, i.e. wck)(,,)= w,,(C,,) or wcki:C,, -+ w(~)(,,,.
If we label the components of a system along lines similar to those we followed for a hierarchy,
and denote by wjk the limiting priority of the jth element in the kth component, we have
ICC1
cs= i g wjk c WCk)(h)~k(j,h),
k=l j=l /I=1
where pk(j,h) is the C.I. of the pairwise comparison matrix of the elements in the kth component
with respect to the jth element in the hth component.
5. ABSOLUTE AND RELATIVE MEASUREMENT AND STRUCTURAL
INFORMATION
The following is largely taken from a paper by Saaty [6]. Cognitive psychologists have recognized
for some time that there are two kinds of comparisons, absolute and relative. In absolute
comparisons alternatives are compared with a standard in one’s memory that has been developed
through experience; in relative comparisons alternatives are compared in pairs according to a
common attribute. The AHP has been used with both types of comparisons to derive ratio scales
of measurement. We call such scales absolute and relative measurement scales, respectively. Relative
measurement in the AHP is well-developed and its use has already been illustrated in the school
selection example in this paper. Here is a brief description of absolute measurement. Incidentally
the software package Expert Choice [14] also includes this method of measurement under the
name of “ratings”.
Absolute measurement (sometimes called scoring) is applied to rank the alternatives in terms of
the criteria or else in terms of ratings (or intensities) of the criteria; e.g. excellent (A), very good (B),
good (C), average (D), below average (E), poor and very poor (F). After setting priorities on the
criteria (or subcriteria, if there are any) pairwise comparisons are also performed on the ratings
themselves to set priorities for them under each criterion. Finally, alternatives are scored by
checking off their rating under each criterion and summing these ratings for all the criteria. This
produces a ratio scale score for the alternative. The scores thus obtained of the alternatives can be
normalized.
The AHP- what it is and how it is used 173
Using absolute measurement, no matter how many new alternatives are introduced, or old ones
deleted, the ranks of the alternatives cannot reverse. This idea has been used to rank cities in the
U.S.A. according to nine criteria as judged by six different people [6]. Another example of an
appropriate use for absolute measurement is that of schools admitting students. Most schools set
their criteria for admission independently from the performance of the current crop of students
seeking admission. Their priorities are then used to determine whether a given student meets the
standard set for qualification. In that case absolute measurement should be used to determine
which students qualify for admission.
Absolute measurement needs standards, often set by society for its convenience, and sometimes
having little to do with the values and objectives of the judge making the comparisons. In completely
new decision problems or in old problems where no standards have been established, we must
continue to use relative measurement comparing alternatives in pairs to identify the best. The
question now is: “What happens to rank when using relative measurement and alternatives are
added or deleted?”
When relative measurement is used to, for example, buy a car, even when the priorities of the
criteria are set in advance independently of the alternatives, the car that qualifies in the end depends
on the number of cars examined. Adding a new car to the collection being examined may cause
reversal in the rank of the original cars. This phenomenon can be accounted for by considering
the normalization operation as a structural criterion which has to do with information generated
in the measurement process. With relative measurement the priority of such a criterion changes
when new alternatives are added or old ones deleted and hence the priorities of the old alternatives,
which depend on all the criteria, including this structural criterion, would change and a different
rank order may occur among the old alternatives. The analogy can be made with any mathematical
model, e.g. linear programming, when a new variable or a new constraint is added. There is no
necessary relation between the solution of the new problem and the old one.
As it should, rank is unaffected when only one criterion is involved and the judgments are
consistent. More generally, one can show that with consistency, the rank order of two alternatives
is unaffected when the judgment values of one dominate those of the other in every pairwise
comparison matrix under the criteria. However, the final rank can change even when the judgments
are consistent when an alternative dominates another under one criterion but is dominated by it
under another. The following example illustrates the idea of rank reversal due to structural criteria.
In an investment decision let us use only two criteria, RETURN and LOW RISK to determine
where it is best to invest. Let us for simplicity assume that they are equally important so that their
priorities are 0.5 and 0.5. We first take the two alternatives TAX FREE BONDS (A) and
SECURITIES (B), pairwise compare them according to preference first under RETURN and then
under LOW RISK, and obtain their composite weights. We have:
;W RISK 1 4 l;2 Pi;;rTURN 1 ix ; Pi;;
The final ranks are
and
A:05 x 0.75 + 0.5 x 0.33 = 0.54
B: 0.5 x 0.25 + 0.5 x 0.67 = 0.46.
A is preferred to B. On observing that the priorities of the alternatives under each criterion are
obtained by adding the elements in either column and dividing by the total which is the
normalization factor, we may rewrite the above arithmetic operations to indicate that normalization
is applied to rescale the priorities of the criteria and use them to weight the priorities of the
alternatives before they are normalized. Thus on writing 0.75 = 3/4, 0.25 = l/4, 0.33 = l/3,
0.67 = 2/3 and noting that 4 is the normalization factor for RETURN and 3 is the normalization
factor for LOW RISK the above can be written as
and
A:(0.5 x l/4)3 + (0.5 x l/3)1 = 0.54
B: (0.5 x l/4)1 + (0.5 x l/3)2 = 0.46.
174 R. W. SAATV
In other words, normalization may be regarded as an operation that transforms the weights of the
criteria from the original scale of (0.5, 0.5) to the new scale (0.5/4,0.5/3) which when normalized
becomes
[0.125/(0.125 + 0.167),0.167/(0.125 + 0.167)] = (0.43,0.57),
which are the resealed priorities of the criteria. That is, relative measurement may be regarded as
an operation which always introduces a new criterion that operates on the existing criteria by
modifying their priorities. Earlier we called this criterion a structural criterion. It is known in
traditional decision theory that rank reversal can occur when a new criterion is introduced and
here we have encountered a fundamentally new kind of criterion that is always present when we
perform relative measurement. Now let us see how rank reversal occurs here by introducing the
new alternative SAVINGS ACCOUNTS (C) with the resulting paired comparisons shown in the
following two matrices:
L y
As before, but now without the details, we have
A:0.5 x 0.3 + 0.5 x 0.31 = (0.5 x l/10)3 + (0.5 x l/13)4 = 0.30,
and
B: 0.5 x 0.1 + 0.5 x 0.62 = (0.5 x l/10)1 + (0.5 x l/13)8 = 0.36
C:O.5 x 0.6 + 0.5 x 0.08 = (0.5 x l/10)6 + (0.5 x l/13)1 = 0.34.
Thus the presence of alternative C has caused rank reversal between A and B and now B is
preferred to C which is preferred to A. In this case we would have to ignore the first ranking and
use the second. Let us note that this process of rank reversal does not contradict any existing
proven fact and we have no historical precedent in some well-developed multicriteria theory to
check it against. In fact the only known theory which is concerned with rank reversal under even
a single criterion is utility theory. There when rank reversal occurs under one criterion the new
alternatives which cause it are called relevant, otherwise they are called irrelevant. The relative
measurement approach of the AHP requires that in order for elements to be comparable in pairs,
it is essential that they all be homogeneous, which means that they must be relevant to begin with.
There is no escaping the fact that to deal with myriads of intangibles we need relative measurement
and that when we use it rank reversal may occur. It is likely that rank reversal due to structural
criteria happens in real life without being recognized. In a recent unpublished work, T. Saaty has
provided the following principle with regard to rank preservation and reversal:
“There is greater uncertainty about rank optimality with increasing rate of change
of information.”
Group judgments?
When dealing with group judgments, Saaty has proposed that any rule to combine the judgments
of several individuals should also satisfy the reciprocal property. A proof that the geometric mean,
which makes no requirement on who should vote first, satisfies this condition was later generalized
in a paper by Aczel and Saaty [S] and by Aczel and Alsina [15, this issue, pp. 31 l-3201.
Group judgment differences can be resolved through a consistency check. When several people
propose radically different judgments in certain positions of the matrix these can be tested with
other judgments on which there is wide agreement by solving the problem separately for each
controversial judgment and measuring the consistency. The judgment yielding the highest consistency
The AHP-what it is and how it is used 175
in the overall problem is retained. The following consistency comparison for each individual’s
judgments with those of the scale vector w derived from group judgments has been proposed:
t bijwj/wi - n2 - 0.1.
i,j= 1
Probabilistic judgments have been studied extensively by Vargas [16]. In particular he showed
that when the judgments are given by a y-distribution the derived vector belongs to a Dirichlet
distribution with a /?-distribution of each component.
6. TOPICS FOR INVESTIGATION
There are a number of areas that need further investigation and others in which ground needs
to be broken. The reader is referred to the AHP literature for applications and theoretical
developments in most of these areas:
(1)
(2)
(31
(41
(51
(61
(7)
(81
(91
(10)
(111
(121
(131
(141
Generalization of the hierarchy and systems networks to manifolds.
Deeper and more extensive research on continuous judgments.
Test different group decision-making approaches on the same problem and
search for common elements. Develop A, B, C guidelines for group participation
in decision making.
Investigate the relationship of the principal eigenvector to the Weber-Fechner
power law.
Develop applications of the AHP in game theory, particularly with respect to
negotiation.
Investigate the relationship of the AHP and optimization. Can the general
optimization problem be solved using the AHP?
Implement psychological studies to show how people’s strength of feeling can
be adequately represented by numerical scales.
Study the sensitivity of priorities to the number of criteria and, more generally,
to the size of the hierarchy.
Sample opinions on how satisfied clients are with AHP outcomes.
Formulate more cases using the AHP in resource allocation, planning, costbenefit
analysis and conflict resolution.
Is there power in hierarchic formulation and judgments to make better
predictions? How can it be tested?
The AHP and risk analysis: put forth a definitive theory about the use of
scenarios in risk analysis.
Investigate the relationship between the AHP and artificial intelligence.
Develop communication and causal languages using the AHP.
REFERENCES
There are five books in English on the AHP, some of which have been translated into other languages,
as indicated in the references. There have been issues of two journals solely devoted to articles on the
AHP, this special issue of Mathematical Modelling and Volume 20, Number 6 (1986) of Socio-Economic
Planning Sciences.
F. Zahedi has published a comprehensive survey article on the AHP [17] with up-to-date references
on its literature. S. Xu (Tianjin University, China) has also published a complete reference list on the AHP
[18]. One book has been written on the subject by a Japanese author and two by Chinese authors.
The subject has also been presented in a chapter in the book by S. I. Gass, Decision Making, Models
and Algorithms, A First Course, (Wiley, New York, 1985) and in a chapter in the book by D. Anderson,
D. Sweeney and T. Williams, Quantitative Methodsfor Business, 3rd edn (West Publishing, St Paul, 1986).
There is a software program available, Expert Choice [14], that runs on the IBM PC, the IBM XT, the
IBM AT and compatible computers. It requires 256K memory and one double-sided disk drive. There is
also a version of Expert Choice available for the Japanese IBM personal computer (with the Kanji
keyboard).
1. T. L. Saaty, The Analytic Hierarchy Process. McGraw-Hill, New York (1980). (This book has been translated into
176 R. W. SAATY
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15
16.
17.
Chinese by S. Xu et al.; information is available from them at the Inst. of Systems Engineering, Tianjin Univ., Tianjin,
China.) (A translation into Russian by R. Vachnadze is currently underway.)
T. L. Saaty and L. G. Vargas, The Logic of Priorities, Applications in Bu,sines.., energy, Health, Transportation. KluwerNijhoff.
The Hague (1981).
T. L. Saaty, Decision Making for Leaders, Wadsworth, Belmont, Calif. (1982). (Also reprinted as a paperback and
available from the author at 322 Mervis Hall, Univ. Pittsburgh, Pittsburgh, PA 15260.) (Available in French under the
title Deader face a la complexite. Entreprise Moderne D’Edition, Paris, 1984).
T. L. Saaty and K. P. Kearns, Analytical Planning. Pergamon Press, Oxford (1985).
T. L. Saaty and J. Alexander, A New Logic for Conflict Resolution. In preparation.
T. L. Saaty, Absolute and relative measurement with the AHP. The most livable cities in the United States. Socio-econ.
Plum Sci. 20(6), 3277331 (1986).
T. L. Saaty and L. Vargas, Stimulussresponse with reciprocal kernels: the rise and fall of sensation. J. math. Sci. (1987).
J. Aczel and T. L. Saaty, Procedures for synthesizing ratio judgments. J. math. Psycho/. 27(l), 93-102 (1983).
R. P. Hamalainen and T. 0. Seppalainen. The analytic network process in energy policy planning. Socio-econ. Plum.
Sci. 20(6), 3999405 (1986).
P. T. Harker, Alternative modes of questioning in the Analytic Hierarchy Process. Math/ Modelling 9, 353-360 (1987).
T. L. Saaty, Inconsistency and rank preservation. J. math. Psychol. 28(2), 2055214 (1984).
T. L. Saaty, Axiomatic foundation of the Analytic Hierarchy Process. Mgmt Sci. 32(7), 841-855 (1986).
T. L. Saaty. Rank generation, preservation and reversal in the Analytic Hierarchy Process. Decis. Sci. (July 1987).
Expert Choice, software package. Decision Support Software, McLean, Va.
J. Aczil and C. Alsina, Synthesizing judgements: a functional equations approach. Math/ Modelling 9, 31 l-320 (1987).
L. Vargas, Reciprocal matrices with random coefficients. J. moth. Modelling 3, 69-81 (1982).
F. Zahedi, The Analytic Hierarchy Process-a survey of the method and its applications. Interfaces 16(4), 96-108
(1986).
18 S. Xu, References on the Analytic Hierarchy Process. Inst. of Systems Engineering, Tianjin Univ., Tianjin, China (1986).