Theory of the Analytic Hierarchy Process. Part 2.1
Highlights are given of the decision making theory the Analytic Hierarchy Process (AHP) and its generalization to dependence and feedback, the Analytic Network Process (ANP) both of which deal with the measurement of tangible and intangible criteria in relative terms by using paired comparisons. The...
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
2003
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irk-123456789-502592013-10-09T03:06:40Z Theory of the Analytic Hierarchy Process. Part 2.1 Saaty, T.L. Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Highlights are given of the decision making theory the Analytic Hierarchy Process (AHP) and its generalization to dependence and feedback, the Analytic Network Process (ANP) both of which deal with the measurement of tangible and intangible criteria in relative terms by using paired comparisons. The fundamental scale of absolute numbers for representing judgments is introduced and the principal right eigenvector is shown to be the necessary vector of priorities derived from the possibly inconsistent matrix of comparisons of homogeneous elements. A method of synthesis of priorities is proposed. Rank preservation and reversal are discussed. When independent, the alternatives can be rated one at a time with respect to the criteria using intensities. Negative priorities are also introduced. This paper will be followed soon by two other papers. Викладено теорію прийняття рішень — Аналітичний Ієрархічний Процес (АНР) та його узагальнення на випадок наявності залежності та зворотнього зв’язку — Аналітичний Мережевий Процес (ANP), в яких використовуються виміри відносної вагомості критеріїв за допомогою парних порівнянь. Уведено також фундаментальну шкалу абсолютних чисел для подання думок, а також показано, що головний правий власний вектор, який одержано з можливо неузгодженої матриці порівнянь однорідних елементів, є вектором пріоритетів. Запропоновано метод синтезу пріоритетів. Обговорені питання збереження та реверсу рангів. Якщо альтернативи є незалежними, то вони можуть бути оцінені по критеріям інтенсивності. Уведені також негативні пріоритети. Цю статтю буде продовжено двома іншими. Излагается теория принятия решений — Аналитический Иерархический Процесс (АНР) и его обобщение на случай зависимости и обратной связи — Аналитический Сетевой Процесс (ANP), в которых используются измерения относительной значимости критериев с помощью парных сравнений. Введена фундаментальная шкала абсолютных чисел для представления мнений, а также показано, что главный правый собственный вектор, полученный из возможно несогласованной матрицы сравнений однородных элементов, является вектором приоритетов. Предложен метод синтеза приоритетов. Рассмотрены вопросы сохранения и реверса рангов. Если альтернативы независимы, то они могут быть оценены по критериям интенсивности. Введены также отрицательные приоритеты. Эта статья будет продолжена двумя другими. 2003 Article Theory of the Analytic Hierarchy Process. Part 2.1 / T.L. Saaty // Систем. дослідж. та інформ. технології. — 2003. — № 1. — С. 48-71. — Бібліогр.: 4 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/50259 519.5 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах |
| spellingShingle |
Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах Saaty, T.L. Theory of the Analytic Hierarchy Process. Part 2.1 Системні дослідження та інформаційні технології |
| description |
Highlights are given of the decision making theory the Analytic Hierarchy Process (AHP) and its generalization to dependence and feedback, the Analytic Network Process (ANP) both of which deal with the measurement of tangible and intangible criteria in relative terms by using paired comparisons. The fundamental scale of absolute numbers for representing judgments is introduced and the principal right eigenvector is shown to be the necessary vector of priorities derived from the possibly inconsistent matrix of comparisons of homogeneous elements. A method of synthesis of priorities is proposed. Rank preservation and reversal are discussed. When independent, the alternatives can be rated one at a time with respect to the criteria using intensities. Negative priorities are also introduced. This paper will be followed soon by two other papers. |
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Article |
| author |
Saaty, T.L. |
| author_facet |
Saaty, T.L. |
| author_sort |
Saaty, T.L. |
| title |
Theory of the Analytic Hierarchy Process. Part 2.1 |
| title_short |
Theory of the Analytic Hierarchy Process. Part 2.1 |
| title_full |
Theory of the Analytic Hierarchy Process. Part 2.1 |
| title_fullStr |
Theory of the Analytic Hierarchy Process. Part 2.1 |
| title_full_unstemmed |
Theory of the Analytic Hierarchy Process. Part 2.1 |
| title_sort |
theory of the analytic hierarchy process. part 2.1 |
| publisher |
Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
| publishDate |
2003 |
| topic_facet |
Проблеми прийняття рішень і управління в економічних, технічних, екологічних і соціальних системах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/50259 |
| citation_txt |
Theory of the Analytic Hierarchy Process. Part 2.1 / T.L. Saaty // Систем. дослідж. та інформ. технології. — 2003. — № 1. — С. 48-71. — Бібліогр.: 4 назв. — англ. |
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Системні дослідження та інформаційні технології |
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© Thomas L. Saaty, 2003
48 ISSN 1681–6048 System Research & Information Technologies, 2003, № 1
TIДC
ПРОБЛЕМИ ПРИЙНЯТТЯ РІШЕНЬ І
УПРАВЛІННЯ В ЕКОНОМІЧНИХ, ТЕХНІЧНИХ,
ЕКОЛОГІЧНИХ І СОЦІАЛЬНИХ СИСТЕМАХ
УДК 519.5
THEORY OF THE ANALYTIC HIERARCHY PROCESS. PART 2.1
THOMAS L. SAATY
Highlights are given of the decision making theory the Analytic Hierarchy Process
(AHP) and its generalization to dependence and feedback, the Analytic Network
Process (ANP) both of which deal with the measurement of tangible and intangible
criteria in relative terms by using paired comparisons. The fundamental scale of ab-
solute numbers for representing judgments is introduced and the principal right ei-
genvector is shown to be the necessary vector of priorities derived from the possibly
inconsistent matrix of comparisons of homogeneous elements. A method of synthe-
sis of priorities is proposed. Rank preservation and reversal are discussed. When in-
dependent, the alternatives can be rated one at a time with respect to the criteria us-
ing intensities. Negative priorities are also introduced. This paper will be followed
soon by two other papers.
1. INTRODUCTION
Not long ago, people believed that the human mind is a completely unreliable in-
strument to perform measurement and that the only meaningful measurement is
obtained on a physical measurement scale like the meter and the kilogram in-
vented by some clever person. They did not think how the measurements came to
have meaning for people and that this meaning depends on people’s purpose each
time they obtain a reading on that scale. In the winter, ice may be a source of dis-
comfort but an ice drink in the summer can be a refreshing source of comfort. A
number has no meaning except that assigned it by someone’s mind. We may all
agree on the number we read on a physical scale, but not on what exactly that
number means to us in practical terms. We tend to parrot abstractions that define a
number but not how we use that number, which is ultimately more important for
our survival. Thus it is our subjective values that are essential for interpreting the
readings obtained through measurement. This interpretation depends on what one
has in mind at the time and different people may interpret the same reading dif-
ferently for the same situation depending on their goal. The reading may be called
objective but the interpretation and use are both subjective. In this sense subjec-
tivity is important because without it objectivity has no intrinsic meaning. If the
mind of an expert can produce measurement close to what we obtain through
measuring instruments, then it has greater power than the instruments in dealing
with complexity because it can also measure things for which we have no instru-
ment. What we have to do is examine the possibility and validity of this assump-
tion as critically as we can. It turns out that when we have knowledge and experi-
Theory of the Analytic Hierarchy Process. Part 2.1
Системні дослідження та інформаційні технології, 2003, № 1 49
ence, our minds are very good measuring instruments. This does not mean that we
should discard what we use in science, but rather use it to support and strengthen
what we are born to do with our minds.
The basic problem is that we need to measure intangibles of which there is a
near infinite number and we can only do it by making comparisons in relative
terms. Even if everything were measurable, we would still need to compare the
different types of measurement and how important they are to us in some decision
in order to make tradeoffs among them and reach a final answer. If we use tangi-
bles and their measurements we would need to normalize them to a common
frame of reference to reduce them to relative numbers and then weight and com-
bine them with intangibles. Combining normalized priorities of measurable quan-
tities with normalized relative measurement of non-measurable things needs ratio
scales because we can weight and add the outcomes.
The Analytic Hierarchy Process (AHP) is a theory of relative measurement
on absolute scales of both tangible and intangible criteria based on the judgment
of knowledgeable and expert people. How to measure intangibles is the main con-
cern of the mathematics of the AHP. In the end we must fit our entire world ex-
perience into our system of priorities if we are going to understand it. The AHP is
based on four axioms: (1) reciprocal judgments, (2) homogeneous elements, (3)
hierarchic or feedback dependent structure, and (4) rank order expectations. The
synthesis of the AHP combines multidimensional scales of measurement into a
single “unidimensional” scale of priorities. Decisions are determined by a single
number for the best outcome or by a vector of priorities that gives an ordering of
the different possible outcomes. We can also combine our judgments or our final
choices obtained from a group when we wish to cooperate to agree on a single
outcome.
The AHP has been mostly applied to multi-objective, multi-criteria and mul-
tiparty decisions because decision making needs this diversity. To make tradeoffs
among the many objectives and many criteria, the judgments that are usually
made in qualitative terms are expressed numerically. To do this, rather than sim-
ply assigning a score out of a person’s memory that appears reasonable, one must
make reciprocal pairwise comparisons in a carefully designed scientific way. In
paired comparisons the smaller or lesser element is used as the unit, and the larger
or greater element is estimated as a multiple of that unit with respect to the com-
mon property or criterion for which the comparisons are made. In this sense
measurement with judgments is made more scientifically than by assigning num-
bers more or less arbitrarily. Because human beings are limited in size and the
firings of their neurons are limited in intensity, it is clear that there is a limit on
their ability to compare the very small with the very large. It is precisely for this
reason that pairwise comparisons are made on elements or alternatives that are
close or homogeneous and the more separated they are, the more need there is to
put them in different groups and link these groups with a common element from
one group to an adjacent group of slightly greater or slightly smaller elements.
From all the paired comparisons, one derives a scale of relative values for
the priorities. Because of inconsistency among the judgments and more impor-
tantly because of the need for the invariance of priorities, it is mathematically
necessary to derive the priorities in the form of the principal eigenvector of the
matrix of paired comparisons.
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 50
We learn from making paired comparisons in the AHP that if A is 5 times
larger than B and B is 3 times larger than C , then A is 15 times larger than C
and A dominates C 15 times. That is different from A having 5 dollars more
than B and B having 3 dollars more than C implies that A has 8 dollars more
than C . Defining intensity along the arcs of a graph and raising the matrix to
powers measures the first kind of dominance precisely and never the second. It
has definite meaning and as we shall see below, in the limit is measured uniquely
by the principal eigenvector. The use of ratios to represent dominance leads to
this idea of dominance and can be verified by taking a matrix whose entries are
written as ratios, raising it to powers and checking that the resulting coefficients
give the desired dominance from any point to any other. There is a useful connec-
tion between what we do with dominance priorities in the AHP and what is done
with transition probabilities both of which use matrix algebra to find their an-
swers. Transitions between states are multiplied and added. To compose the pri-
orities for the alternatives of a decision with respect to different criteria, it is also
necessary that the priorities of the alternatives with respect to each criterion be
multiplied by the priority of that criterion and then added over all the criteria.
The rank of the alternatives can be affected by how many alternatives there
are and also by the quality or measurement of those alternatives. Both of these are
not intrinsic properties of any alternative, but can affect the importance of that
alternative. It may at first seem convenient to assume that alternatives are inde-
pendent of each other; but that is not always natural. For example as we increase
the number of copies of an alternative, it often loses (and sometimes increases) its
importance. For example, if gold were an alternative and gold is important, and if
we increase the amount of gold available until it is near infinite, gold gradually
loses its importance. No new criterion is added and no judgment is changed. Rela-
tive measurement takes quantity into consideration. We often need to consider
this kind of dependence known as structural dependence. When we add more al-
ternatives, the ranks among old ones may change and what was preferred to an-
other now because of the presence of new ones may no longer be preferred to the
other. Another example is that of a company that sells cars A and B . Car B is
better than car A but it costs more to make. It is more desirable all around for
people to buy car B but they buy A because it is cheaper. The company adver-
tises that it is going to make car C that is similar to B but much more expensive.
People are now observed more and more to buy car B . The company never
makes car C . This is a real life example from marketing. However, in some deci-
sion problems we may want to treat by fiat the alternatives of a decision as com-
pletely independent both in property and in number and quality and want to pre-
serve the ranks of existing alternatives when new ones are added or old ones
deleted. The AHP allows for both these possibilities. Actually, change in rank in
the presence of relevant alternatives is a fact of our world. It is also a fact that
when the number of irrelevant alternatives is very large, they can cause rank to
change. Viruses are irrelevant in most decisions but they can eventually cause the
death of all decision makers and make mockery of the decisions they thought
were so important. In essence reality is much more interdependent than we have
allowed for in our limited ways of thinking. Admittedly there are times when we
wish to preserve rank no matter what the situation may be. We need to allow for
Theory of the Analytic Hierarchy Process. Part 2.1
Системні дослідження та інформаційні технології, 2003, № 1 51
both in our decision theories and not take the simple way out as most theories do
at present.
Our story must begin with how paired comparisons are made, what scales of
numbers to use and what kind of scale to derive from them and how. The AHP
has three parts: what a person has to do; what the method has to do; and what the
decision process has to do to deliver “right” answers. The decision maker needs to:
1. Know his problem well and know how to structure it;
2. Put the more important criteria up in a hierarchy, the subcriteria below
that and the alternatives below that; or in a network make clusters of elements and
then connect these clusters and elements in them according to influence as in a
hierarchy with the arrow pointing in the opposite direction to how one determines
dominance of influence;
3. Provide judgments in answering two kinds of questions that we know in
reality that everyone has a talent to do:
a. With respect to a property or criterion which of two elements is more
dominant, and
b. With respect to a control criterion and a criterion in a subnet, which of
two elements contributes more to or influences the control criterion
more through that criterion in the subnet? This is a conditional kind of
question and very important to keep in mind.
Then there are three things to know about the AHP method of setting priorities.
1. We can derive the fundamental integer scale 1–9 of the AHP that we use
to express judgments on pairs of elements from stimulus-response theory, and
then validate it with numerous examples in practice whose measurements are
known. We can use the measurements themselves instead of judgments if we
want. We can connect measurement from the small to the very large through clus-
tering to extend the 1–9 scale indefinitely;
2. Use a pairwise positive reciprocal comparisons matrix and derive the
principal right eigenvector as the vector of priorities when the matrix is consis-
tent. We use perturbation arguments to show that we must still solve the principal
eigenvalue problem when it is reasonably but not greatly inconsistent. We also
show that we need the principal eigenvector to capture transitivity by raising the
matrix to powers. Finally because a hierarchy is a special case of a supermatrix
raised to powers to capture the transitivity of influences, raising it to powers in-
volves the arithmetic operations of multiplying and adding. In the limit we obtain
the principal eigenvector as a necessary condition to preserve the invariance of
the priorities. This means that if we use these priorities to weight the columns of
the matrix to derive priorities from it, we obtain the original priorities back;
3. In both a hierarchic and a network structure of a decision, the priorities
can be arranged as parts of its columns in a supermatrix [1, 2] whose clusters
must be weighted according to their influence on each of them; the weights thus
derived are used to weight the blocks of the supermatrix corresponding to the
components to make it stochastic so it can be raised to powers to capture the in-
teractions and obtain the priorities in the limit. There are two possibilities here
depending on the irreducibility/reducibility of the matrix. The first yields a unique
outcome while the second involves a cycle of limiting outcomes that are com-
bined into an average outcome.
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 52
Finally the process of decision-making requires us to analyze a decision ac-
cording to benefits (B), the good things that would result from taking the deci-
sion; opportunities (O), the potential good things that can result in the future from
taking the decision; costs (C), the pains and disappointments that would result
from taking the decision; and risks (R), the potential pains and disappointments
that can result from taking the decision. We then create control criteria and sub-
criteria or even a network of criteria under each and develop a subnet and its con-
nection for each control criterion. We then need to determine the best outcome for
each control criterion and combine the alternatives in what is known as the ideal
form for all the control criteria under each of BOCR. Then we note the best alter-
native under B and use it to think of benefits and then the best under O, which
may be different than the one under C, and use it to think of costs and so on. Fi-
nally we must rate these four with respect to strategic criteria using the absolute
mode of the AHP to obtain priority ratings for B, O, C, and R. We then normalize
and use these weights to combine the four vectors of outcomes under BOCR to
obtain an overall answer. We can form the ratio BO/CR which does not need the
BOCR ratings to obtain marginal overall outcomes. Alternatively:
1) we can use the ratings to weight and subtract the costs and risks from the
sum of the weighted benefits and opportunities, or
2) we can subtract the costs and the risks from one and weight and add eve-
rything, or
3) we can take the reciprocals of the costs and the risks, normalize the result
and weight and add them to the weighted benefits and costs.
2. PAIRWISE COMPARISONS
Assume that one is given n stones, nAA ,...,1 , with known weights nww ,...,1 , re-
spectively, and suppose that a matrix of pairwise ratios is formed whose rows
give the ratios of the weights of each stone with respect to all others. Thus one has
the equation:
wn
w
w
n
w
w
AA
w
w
w
w
w
w
w
w
A
A
Aw
nn
n
n
nn
n
n
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
11
1
1
1
1
1
1
,
where A has been multiplied on the right by the vector of weights w . The result
of this multiplication is wn . To recover the scale from the matrix of ratios, one
must solve the problem wnA = or 0)( =− wnA I (where I is identify matrix).
This is a system of homogeneous linear equations. It has a nontrivial solution if
and only if the determinant of InA− vanishes, that is, n is an eigenvalue of A .
Now A has unit rank since every row is a constant multiple of the first row. As a
result, all its eigenvalues except one are zero. The sum of the eigenvalues of a
matrix is equal to its trace, the sum of its diagonal elements, and in this case the
Theory of the Analytic Hierarchy Process. Part 2.1
Системні дослідження та інформаційні технології, 2003, № 1 53
trace of A is equal to n. Thus n is an eigenvalue of A , and one has a nontrivial
solution. The solution consists of positive entries and is unique to within a multi-
plicative constant.
To make w unique, we can normalize its entries by dividing by their sum.
Thus, given the comparison matrix, we can recover the scale. In this case, the so-
lution is any column of A normalized. Notice that in A the reciprocal property
ijji aa 1= holds; thus, also 1=iia . Another property of A is that it is consistent:
its entries satisfy the condition ijikjk aaa = . The entire matrix can be con-
structed from a set of n elements that form a chain across the rows and columns
of A .
In the general case, the precise value of ji ww cannot be given, but instead
only an estimate of it as a judgment. For the moment, consider an estimate of
these values by an expert whose judgments are small perturbations of the coeffi-
cients ji ww . This implies small perturbations of the eigenvalues.
Let us for generality call nAAA ,...,, 21 stimuli instead of stones. The quanti-
fied judgments on pairs of stimuli ji AA , , are represented by an nn× matrix
njiaA ij ,...,2,1,),( ==′ . The entries ija are defined by the following entry
rules.
Rule 1. If aaij = , then 0,/1 ≠= aaaij .
Rule 2. If iA is judged to be of equal relative intensity to jA then 1=ija ,
1=jia ; in particular, 1=iia for all i .
Thus the matrix A′ has the form
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=′
111
11
1
21
212
112
nn
n
n
aa
aa
aa
A .
Having recorded the quantified judgments on pairs of stimuli ( ji AA , ) as
numerical entries ija in the matrix A′ , the problem now is to assign to the n
stimuli nAAA ,...,, 21 a set of numerical weights nww ,...,1 that would "reflect the
recorded judgments." In order to do that, the vaguely formulated problem must
first be transformed into a precise mathematical one. This essential, and appar-
ently harmless, step is the most crucial one in any problem that requires the repre-
sentation of a real-life situation in terms of an abstract mathematical structure. It
is particularly crucial in the present problem where the representation involves a
number of transitions that are not immediately discernible. It appears, therefore,
desirable in the present problem to identify the major steps in the process of rep-
resentation and to make each step as explicit as possible to enable the potential
user to form his own judgment as to the meaning and value of the method in rela-
tion to his problem and his goal.
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 54
Why we must solve the principal eigenvalue problem in general has a simple
justification based on the idea of dominance among the elements represented by
the coefficients of the matrix. Dominance between two elements is obtained as the
normalized sum of paths of all lengths between them. All the paths of a given
length k are obtained by raising the matrix to the power k. According to Cesaro
summability, the limit of the Cesaro sum ∑
=∞→
N
k
k
N
AN
0
/1lim that represents the av-
erage of all order dominance up to N , is the same as the limit of the sequence of
the powers of the matrix i.e. k
k
A
∞→
lim . Now we know from Perron theory that this
sequence converges to a matrix all whose columns are identical and are propor-
tional to the principal right eigenvector of A .
Without the theory of Perron, the proof is somewhat more elaborate, but
more interesting because it is related to the amount of inconsistency one allows.
The following theorem assures us that a sufficiently small perturbation
0, >= ijij
j
i
ij w
w
a εε , of a consistent matrix (which we know has a simple princi-
pal eigenvalue n) gives rise to a simple eigenvalue problem for an inconsistent
matrix but does not guarantee that it is the principal eigenvalue.
Theorem (Existence). If λ is a simple eigenvalue of A , then for small
0>ε , there is an eigenvalue )(ελ of )(εA with power series expansion in ε :
…+++= )2(2)1()( λελελελ
and corresponding right and left eigenvectors )(εw and )(εv such that
…+++= )2(2)1()( wwww εεε
…+++= )2(2)1()( vvv ελεε
Our general problem takes the form:
cw
w
w
w
aa
aa
aa
wA
nnn
n
n
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=′ 2
1
21
212
112
111
11
1
.
We know that the homogeneous system of linear equations cwwA =′ has a
solution w if c is an eigenvalue of A′ . We need to show that the priorities we
seek should be given by solving the principal eigenvalue problem of A′ and not
by some other method. For example, the method of least squares (LSM) determines
a vector by minimizing the Frobenius norm of the difference between A and a posi-
tive rank one reciprocal matrix [ ]ji yy / :
∑
=>
−
n
ji
jiij
y
yya
1,
2
0
)/(min .
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The method of logarithmic least squares (LLSM) determines a vector by mini-
mizing the Frobenius norm of [ ])/(log ijij xxa :
2
1,0
)]/(log[logmin j
n
ji
iijx
xxa∑
=>
− .
Continuing with our quest for the principal right eigenvector, we need to
take a small diversion to discuss the principle of invariance of priorities that pro-
vide a second necessary condition for deriving priorities.
The concept of invariance is one of the most central in mathematics. An in-
variant is a mapping of a set of elements with an equivalence relation, into an-
other set of elements that is constant or unchanged with respect to the equivalence
relation of the original set. Actually one speaks of an invariant of an element of
the original set with respect to a mapping of that entire set. It should be clear that
no matter what method we use to derive the weights iw , we need to get them back
as proportional to the expression ∑
=
n
j
jij wa
1
ni ...,1= , that is, we must solve
i
n
j
jij wcwa =∑
=1
, ni ...,1= . Otherwise ∑
=
n
j
jij wa
1
, ni ,...,1= would yield another
set of different weights and they in turn can be used to form new expressions
∑
=
n
j
jij wa
1
, ni ,...,1= , and so on ad infinitum violating the need to have priorities
that are invariant. Unless we solve the principal eigenvalue problem, our effort to
derive priorities becomes meaningless.
We now show that the perturbed eigenvalue from the consistent case is the
principal eigenvalue of A′ . Our argument involves both left and right eigenvec-
tors of A′ . Two vectors ),...,( 1 nxxx = , ),...,( 1 nyyy = are orthogonal if their
scalar product nn yxyx ++ ...11 is equal to zero. It is known that any left eigenvec-
tor of a matrix corresponding to an eigenvalue is orthogonal to any right eigen-
vector corresponding to a different eigenvalue. This property is known as bi-
orthogonality.
Theorem: For a given positive matrix A , the only positive vector w and
only positive constant c that satisfy cwwA = , is a vector w that is a positive mul-
tiple of the principal eigenvector of A , and the only such c is the principal eigen-
value of A
Proof: We know that the right principal eigenvector and the principal eigen-
value satisfy our requirements. We also know that the algebraic multiplicity of the
principal eigenvalue is one, and that there is a positive left eigenvector of A (call
it z ) corresponding to the principal eigenvalue. Suppose there is a positive vector
y and a (necessarily positive) scalar d such that ydyA = . If d and c are not
equal, then by bi-orthogonality y is orthogonal to z , which is impossible since
both vectors are positive. If d and c are equal, then y and w re dependent since
c has algebraic multiplicity one, and y is a positive multiple of w . This completes
the proof.
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 56
Thus we see that both requirments of dominance and invariance lead us to
the principal right eigenvector. The problem now is how good is the estimate of
w . Notice that if w is obtained by solving this problem, the matrix whose entries
are ji ww / is a consistent matrix. It is a consistent estimate of the matrix A′ . The
matrix A′ itself need not be consistent. In fact, the entries of A′ need not even be
transitive; that is, 1A may be preferred to 2A and 2A to 3A but 3A may be pre-
ferred to 1A . What we would like is a measure of the error due to inconsistency. It
turns out that A′ is consistent if and only if n=maxλ and that we always have
n≥maxλ .
Remark. Solving the principal eigenvalue problem to obtain priorities is
equivalent to the following two optimization problems: Find 0>iw , ni ,...,1=
which
1) maximize
i
j
n
j
ij
n
i w
w
a
n ∑∑
== 11
1 or, in the simpler linear optimization setting;
2) maximize ∑∑
==
n
j
ij
n
j
j aw
11
, obtained by multiplying the sum of each
column j by its corresponding jw and summing over j, subject to 1
1
=∑
=
n
i
iw .
3. STIMULUS RESPONSE AND THE FUNDAMENTAL SCALE
To be able to perceive and sense objects in the environment our brains miniaturize
them within our system of neurons so that we have a proportional relationship
between what we perceive and what is out there. Without proportionality we can-
not coordinate our thinking with our actions with the accuracy needed to control
the environment. Proportionality with respect to a single stimulus requires that
our response to a proportionately amplified or attenuated stimulus we receive
from a source should be proportional to what our response would be to the origi-
nal value of that stimulus. If )(sw is our response to a stimulus of magnitude s ,
then the foregoing gives rise to the functional equation )()( swbsaw = . (This
equation can also be obtained as the necessary condition for solving the Fredholm
equation of the second kind
)()(),( max swdttwtsK
b
a
λ=∫
obtained as the continuous generalization of the discrete formulation
wwA maxλ= for obtaining priorities where instead of the positive reciprocal ma-
trix A in the principal eigenvalue problem, we have a positive kernel,
0),( >tsK , with 1),(),( =tsKtsK that is also consistent i.e. =),(),( utKtsK
),( usK= , for all ts, and u . The solution of this functional equation in the real
domain is given by
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⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
a
sPeCsw a
sb
log
log)( log
loglog
,
where P is a periodic function of period 1 and 1)0( =P . One of the simplest
such examples with asu log/log= is )2/(cos)( BuuP = for which 1)0( =P .
The logarithmic law of response to stimuli can be obtained as a first order
approximation to this solution through series expansions of the exponential and of
the cosine functions as:
321 log)()( CsCuPeCuv u +≈= −β ,
where 0,log >−= ββba . The expression on the right is known as the Weber-
Fechner law of logarithmic response 0,log ≠+= absaM to a stimulus of mag-
nitude s. This law was empirically established and tested in 1860 by Gustav
Theodor Fechner who used a law formulated by Ernest Heinrich Weber regarding
discrimination between two nearby values of a stimulus. We have now shown that
it can be derived that Fechner’s version can be derived by starting with a func-
tional equation for stimulus response.
The integer-valued scale of response used in making paired comparison
judgments can be derived from the logarithmic response function as follows. For
a given value of the stimulus, the magnitude of response remains the same until
the value of the stimulus is increased sufficiently large in proportion to the value
of the stimulus, thus preserving the proportionality of relative increase in stimulus
for it to be detectable for a new response. This suggests the idea of just noticeable
differences (jnd), well known in psychology. Thus, starting with a stimulus 0s
successive magnitudes of the new stimuli take the form:
)1(00
0
0
0001 rss
s
s
ssss +=
∆
+=∆+= ,
2
0
2
01112 )1()1( αsrsrssss +=+=∆+= ,
)...,2,1,0(01 === − nsss n
nn αα .
We consider the responses to these stimuli to be measured on a ratio scale
)0( =b . A typical response has the form niaM i
i ,...,1,log == α , or one after
another they have the form:
ααα log,...,log2,log 21 anMaMaM n === .
We take the ratios niMM i ,...,1,/ 1 = of these responses in which the first
is the smallest and serves as the unit of comparison, thus obtaining the integer
values n,...,2,1 of the fundamental scale of the AHP. It appears that numbers are
intrinsic to our ability to make comparisons, and that they were not an invention
by our primitive ancestors. We must be grateful to them for the discovery of the
symbolism. In a less mathematical vein, we note that we are able to distinguish
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 58
ordinally between high, medium and low at one level and for each of them in a
second level below that also distinguish between high, medium and low giving us
nine different categories. We assign the value one to (low, low) which is the
smallest and the value nine to (high, high) which is the highest, thus covering the
spectrum of possibilities between two levels, and giving the value nine for the top
of the paired comparisons scale as compared with the lowest value on the scale. In
fact we show later that because of increase in inconsistency when we compare
more than about 7 elements, we don’t need to keep in mind more than 27 ± ele-
ments. This was first conjectured by the psychologist George Miller in the 1950’s
and explained in the AHP in the 1970’s [4]. Finally, we note that the scale just
derived is attached to the importance we assign to judgments. If we have an exact
measurement such as 2.375 and want to use it as it is for our judgment without
attaching significance to it, we can use its entire value without approximation.
A person may not be schooled in the use of numbers but still have feelings,
judgments and understanding that enable him to make accurate comparisons
(equal, moderate, strong, very strong and extreme and compromises between
these intensities). Such judgments can be applied successfully to compare stimuli
that are not too disparate but homogeneous in magnitude. By homogeneous we
mean that they fall within specified bounds. The foregoing may be summarized as
in Table 1 to represent the fundamental scale for paired comparisons.
T a b l e 1 . The Fundamental Scale of Absolute Numbers
Intensity of
Importance Definition Explanation
1 Equal Importance Two activities contribute equally to
the objective
2 Weak
3 Moderate importance Experience and judgment slightly
favor one activity over another
4 Moderate plus
5 Strong importance Experience and judgment strongly
favor one activity over another
6 Strong plus
7 Very strong or
demonstrated importance
An activity is favored very strongly
over another; its dominance
demonstrated in practice
8 Very, very strong
9 Extreme importance
The evidence favoring one activity
over another is of the highest
possible order of affirmation
Reciprocals of
above
If activity i has one of the
above nonzero numbers
assigned to it when
compared with activity j,
then j has the reciprocal
value when compared with i
A reasonable assumption
Rationals Ratios arising from the scale
If consistency were to be forced by
obtaining n numerical values to
span the matrix
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In a judgment matrix A instead of assigning two numbers iw and jw (that
generally we do not know), as one does with tangibles, and forming the ratio
ji ww / we assign a single number drawn from the fundamental scale of absolute
numbers shown in Table 1 above to represent the ratio 1/)/( ji ww . It is a nearest
integer approximation to the ratio ji ww / . The ratio of two numbers from a ratio
scale (invariant under multiplication by a positive constant) is an absolute number
(invariant under the identity transformation). The derived scale will reveal what
iw and jw are. This is a central fact about the relative measurement approach. It
needs a fundamental scale to express numerically the relative dominance relationship.
Remark: The reciprocal property plays an important role in combining the
judgments of several individuals to obtain a judgment for a group. Judgments
must be combined so that the reciprocal of the synthesized judgments must be
equal to the syntheses of the reciprocals of these judgments. It has been proved
that the geometric mean is the unique way to do that. If the individuals are ex-
perts, they my not wish to combine their judgments but only their final outcome
from a hierarchy. In that case one takes the geometric mean of the final outcomes.
If the individuals have different priorities of importance their judgments (final
outcomes) are raised to the power of their priorities and then the geometric mean
is formed [3].
Validation Examples
Fig. 1 shows five areas to which we can apply the paired comparison process in a
matrix and use the 1 to 9 scale to test the validity of the procedure. The object is
to compare them in pairs to reproduce their relative size. First, we arrange the ob-
jects we wish to compare among themselves in a square matrix )( ijaA = as to
their dominance with respect to a common property which is area in this example.
The judgments are entered using the fundamental scale derived above to cover a
homogeneous range of values from 1–9. An element compared with itself is al-
ways assigned the value 1 so the main diagonal entries of the pairwise comparison
matrix are all 1. The numbers 3, 5, 7, and 9 correspond to the verbal judgments
“moderately more dominant”, “strongly more dominant”, “very strongly more
B
E
D
A
C
Fig. 1. Estimating the Relative Areas of Five Polygons
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 60
dominant”, and “extremely more dominant” (with 2, 4, 6, and 8 for compromise
between these values). Reciprocal values are automatically entered in the trans-
pose position. We are permitted to interpolate values between the integers, if desired.
In the absence of a computer, we can approximate the priorities derived from
the matrix by assuming that it is nearly consistent (because in a consistent matrix
any column gives the priority vector), normalize each column, and then take the
average of the first entries in the five columns, and then the average of the second
entries and so on thus obtaining an estimate of the relative areas.
The actual relative values of these areas are 47.0=A , 05.0=B , 24.0=C ,
14.0=D , and 09.0=E with which the answer may be compared. By comparing
more than two alternatives in a decision problem, one is able to obtain better val-
ues for the derived scale because of redundancy in the comparisons, which helps
improve the overall accuracy of the judgments.
Here is another example (one of many) which shows that the scale works
well on homogeneous elements of a real life problem. A matrix of paired com-
parison judgments is used to estimate relative drink consumption in the United
States. To make the comparisons, as shown in table 2 the types of drinks are listed
on the left and at the top, and judgment is made as to how strongly the consump-
tion of a drink on the left dominates that of a drink at the top. For example, when
coffee on the left is compared with wine at the top, it is thought that it is con-
sumed extremely more and a 9 is entered in the first row and second column posi-
tion. A 1/9 is automatically entered in the second row and first column position. If
the consumption of a drink on the left does not dominate that of a drink at the top,
the reciprocal value is entered. For example in comparing coffee and water in the
first row and eighth column position, water is consumed more than coffee slightly
and a 1/2 is entered. Correspondingly, a value of 2 is entered in the eighth row
and first column position. At the bottom of, we see that the derived values and the
actual values are close.
T a b l e 2 . Which Drink is Consumed More in the U.S.? An Example of
Estimation Using Judgments
Drink Consump-
tion in the U.S. Coffee Wine Tea Beer Sodas Milk Water
Coffee 1 9 5 2 1 1 1/2
Wine 1/9 1 1/3 1/9 1/9 1/9 1/9
Tea 1/5 2 1 1/3 1/4 1/3 1/9
Beer 1/2 9 3 1 1/2 1 1/3
Sodas 1 9 4 2 1 2 1/2
Milk 1 9 3 1 1/2 1 1/3
Water 2 9 9 3 2 3 1
The derived scale based on the judgments in the matrix is:
Coffee Wine Tea Beer Sodas Milk Water
0.177 0.019 0.042 0.116 0.190 0.129 0.327
with a consistency ratio of 0.022.
The actual consumption (from statistical sources) is:
0.180 0.010 0.040 0.120 0.180 0.140 0.330
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4. CLUSTERING AND HOMOGENEITY; USING PIVOTS TO EXTEND THE
SCALE FROM 1–9 TO 1 – ∞
We note that our ability to make accurate comparisons of widely disparate objects
on a common property is limited. We cannot compare with any reliability the very
small with the very large. However, we can do it in stages by comparing objects
of relatively close magnitudes and gradually increase their sizes until we reach the
desired object of large size (see example later). In this process we can think of
comparing several close or homogeneous objects for which we obtain a scale of
relative values, and then again pairwise compare the next set of larger objects that
includes for example the largest object from the previous already compared
collection, and then derive a scale for this second set. We then divide all the
measurements in the second set by the value of the common object and multiply
all the resulting values by the weight of the common element in the first set, thus
rendering the two sets to be measurable on the same scale and so on to a third
collection of the objects using a common object from the second set.
In Fig. 2 an unripe cherry tomato is eventually and indirectly compared with
a large watermelon by first comparing it with a small tomato and a lime, the lime
is then used again in a second cluster with a grapefruit and a honey dew where we
then divide by the weight of the lime and then multiply by its weight in the first
cluster, and then use the honey dew again in a third cluster and so on. In the end
we have a comparison of the unripe cherry tomato with the large watermelon and
would accordingly extended the scale from 1–9 to 1–721.
Such clustering is essential, and must be done separately for each criterion.
We should note that in most decision problems, there may be one or two levels of
clusters and conceivably it may go up to three or four adjacent ranges of homoge-
neous elements (Maslow put them in seven groupings of human needs). Very
roughly we have in decreasing order of importance:
1. Survival, health, security, family, friends and basic religious beliefs some
people were known to die for;
2. Career, education, productivity and lifestyle;
3. Political and social beliefs and contributions;
4. Beliefs, ideas, and things that are flexible and it does not matter exactly
how one advocates or uses them.
Nevertheless one needs them, such as learning to eat with a fork or a chop-
stick or with the fingers as many people do interchangeably. These categories can
be generalized to a group, a corporation, or a government. For very important
decisions, two categories may need to be considered. Note that the priorities in
two adjacent categories would be sufficiently different, one being an order of
magnitude smaller than the other, that in the synthesis, the priorities of the
elements in the smaller set have little effect on the decision. How some
undesirable elements can be compared among themselves and then related to
desirable ones thus going from negative to positive elements keeping the
measurement of the two types positive is done as in the BOCR discussion given
later.
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 62
0.07
0.28
0.65
Unripe Cherry
Tomato Small Green Tomato Lime
0.08
0.22
0.70
Lime
1
08.0
08.0 =
0.65Η1=0.65
Grapefruit
75.2
08.0
22.0 =
0.65Η2.75=1.79
Honeydew
75.8
08.0
70.0 =
0.65Η8.75=5.69
0.10
0.30
0.60
Honeydew
1
10.0
10.0 =
5.69Η1=5.69
Sugar Baby Water-
melon
3
10.0
30.0 =
5.69Η3=17.07
Oblong Watermelon
6
10.0
60.0 =
5.69Η6=34.14
This means that 34.14/.07.487.7 unripe cherry tomatoes are equal
to the oblong watermelon.
Fig. 2. Comparisons According to Volume
5. WHEN IS A POSITIVE RECIPROCAL MATRIX CONSISTENT?
In light of the foregoing, for the validity of the vector of priorities to describe re-
sponse, we need greater redundancy and therefore also a large number of com-
parisons. We now show that for the sake of being close to consistency we need to
make a small number, 2/)1( −nn , of comparisons. An expert may provide n-1
comparisons to fill one row or a spanning tree from which the matrix is consistent
and he priorities are easily obtained. So where is the optimum number?
Let us relate the psychological idea of the consistency of judgments and its
measurement, to a central concept in matrix theory and also to the size of our chan-
nel capacity to process information. It is the principal eigenvalue of a matrix of
paired comparisons.
Let ][ ijaA = be an n-by-n positive reciprocal matrix, so all 1=iia and
jiij aa /1= for all nji ,...,1, = . Let ][ iww = be the principal right eigenvector of
A , let ),...,(diag 1 nwwD = be the n-by-n diagonal matrix whose main diagonal
entries are the entries of w , and set ][]/[/ 1
ijijij wwaADDE γ==− . Then E is
similar to A and is a positive reciprocal matrix since == jijiji wwa /γ
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ijijij wwa γ/1)/( 1 == − . Moreover, all the row sums of E are equal to the principal
eigenvalue of A :
maxmax
1
//][/ λλε ====∑∑
=
iiiiijj ij
n
j
ij wwwAwwwa .
The computation
=++== ∑∑∑ ∑
≠
=== =
)()(
1,11 1
max ji
n
ji
ji
ij
n
i
ii
n
i
n
j
ijn εεεελ
22
1,
1 2/)()( nnnnn
n
ji
ji
ijij =−+≥++= ∑
≠
=
−εε
reveals that .max n≥λ Moreover, since 2/1 ≥+ xx for all 0>x , with equality if
and only if 1=x , we see that n=maxλ if and only if all 1=ijγ , which is equivalent
to having all jiij wwa /= .
The foregoing arguments show that a positive reciprocal matrix A has
n≥maxλ , with equality if and only if A is consistent. When A is consistent we
have AnA kk 1−= As our measure of deviation of A from consistency, we choose
the consistency index
.
1
max
−
−
≡
n
nλ
µ
We have seen that 0≥µ and 0=µ if and only if A is consistent. We can
say that as jiij wwa /,0 →→µ , or 1/ →= ijijij wwaε . These two desirable
properties explain the term “ n ” in the numerator of µ ; what about the term
“ 1−n ” in the denominator? Since trace nA =)( is the sum of all the eigenvalues
of A , if we denote the eigenvalues of A that are different from maxλ by
12 ,..., −nλλ , we see that ∑
=
+=
n
i
in
2
max λλ , so ∑
=
=−
n
i
in
2
max λλ and =µ
∑
=−
=
n
i
in 11
1 λ is the average of the non-principal eigenvalues of A .
It is an easy, but instructive, computation to show that 2max =λ for every 2-
by-2 positive reciprocal matrix:
⎥
⎦
⎤
⎢
⎣
⎡
− 1
1
1α
α
⎥
⎦
⎤
⎢
⎣
⎡
+
+
=⎥
⎦
⎤
⎢
⎣
⎡
+
+
−− 11 )1(
1
2
)1(
1
αα
α
αα
α
.
Thus, every 2-by-2 positive reciprocal matrix is consistent.
Thomas L. Saaty
ISSN 1681–6048 System Research & Information Technologies, 2003, № 1 64
Not every 3-by-3 positive reciprocal matrix is consistent, but in this case we
are fortunate to have again explicit formulas for the principal eigenvalue and ei-
genvector. For
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1/1/1
1/1
1
cb
ca
ba
A
we have 1
max 1 −++= ddλ , 3/1)/( bacd = and
⎟
⎠
⎞
⎜
⎝
⎛ ++=
d
cbdbdw 11 , ⎟
⎠
⎞
⎜
⎝
⎛ ++=
d
cbddcw 12 , ⎟
⎠
⎞
⎜
⎝
⎛ ++=
d
cbdw 113 .
Note that 3max =λ when 1=d or abc /= , which is true if and only if A is
consistent.
In order to get some feel for what the consistency index might be telling us
about a positive n-by-n reciprocal matrix A , consider the following simulation:
choose the entries of A above the main diagonal at random from the 17 values
{1/9, 1/8,…,1, 2,…,8, 9}. Then fill in the entries of A below the diagonal by tak-
ing reciprocals. Put ones down the main diagonal and compute the consistency
index. Do this 50000 times and take the average, which we call the random index.
Table 3 shows the values obtained from one set of such simulations and also their
first order differences, for matrices of size 1, 2,…,15.
T a b l e 3 . Random Index
Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
R.I. 0 0 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 1.52 1.54 1.56 1.58 1.59
First
Order
Differ-
ences
0 0.52 0.37 0.22 0.14 0.10 0.05 0.05 0.04 0.03 0.02 0.02 0.02 0.01
Fig. 3 below is a plot of the first two rows of Table 3. It shows the asymp-
totic nature of random inconsistency. We also learn that one should not compare
more than about seven elements because the inconsistency increases due to a cor-
rect mental limitation, I believe is a result of evolution, to deal with a few things
at a time [4].
Since it would be pointless to try to discern any priority ranking from a set of
random comparison judgments, we should probably be uncomfortable about pro-
ceeding unless the consistency index of a pairwise comparison matrix is very
much smaller than the corresponding random index value in Table 3. The consis-
tency ratio (C.R.) of a pairwise comparison matrix is the ratio of its consistency
index :, defined above, to the corresponding random index value in Table 3. As a
rule of thumb, we do not recommend proceeding if the consistency ratio is more
than about .10 for 4≥n . For 3=n , we recommend that the . .C R be less than
0.05. By way of further elaboration, inconsistency may be thought of as an ad-
justment needed to improve the consistency of the comparisons. But the adjust-
ment should not be as large as the judgment itself, nor so small that using it is of
no consequence. Thus inconsistency should be just one order of magnitude
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smaller. On a scale from zero to one, the overall inconsistency should be around
10%. The requirement of 10% cannot be made smaller such as 1% or 0.1% with-
out trivializing the impact of inconsistency. But inconsistency itself is important
because without it, new knowledge that changes preference cannot be admitted.
Assuming that all knowledge should be consistent contradicts experience that
requires continued revision of understanding.
If the . .C R is larger than desired, we do three things:
1) find the most inconsistent judgment in the matrix (for example, that
judgment for which ijijij wwa /=ε is largest);
2) determine the range of values to which that judgment can be changed
corresponding to which the inconsistency would be improved;
3) ask the judge to consider, if he can, change his judgment to a plausible
value in that range. If he is unwilling, we try with the second most inconsistent
judgment and so on. If no judgment is changed the decision is postponed until
better understanding of the stimuli is obtained.
Before proceeding further, the following observations may be useful for a
better understanding of the importance of the concept of a limit on our ability to
process information and also change in information. The quality of response to
stimuli is determined by three factors. Accuracy or validity, consistency, and effi-
ciency or amount of information generated. Our judgment is much more sensitive
and responsive to large perturbations. When we speak of perturbation, we have in
mind numerical change from consistent ratios obtained from priorities. The larger
the inconsistency and hence also the larger the perturbations in priorities, the
greater is our sensitivity to make changes in the numerical values assigned. Con-
versely, the smaller the inconsistency, the more difficult it is for us to know where
the best changes should be made to produce not only better consistency but also
better validity of the outcome. Once near consistency is attained, it becomes un-
certain which coefficients should be perturbed by small amounts to transform a
near consistent matrix to a consistent one. If such perturbations were forced, they
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Fig. 3. Plot of Random Inconsistency
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could be arbitrary and thus distort the validity of the derived priority vector in
representing the underlying decision.
The third row of Table 3 gives the differences between successive numbers
in the second row. Fig. 4 is a plot of these differences and shows the importance
of the number seven as a cutoff point beyond which the differences are less than
0.10 where we are not sufficiently sensitive to make accurate changes in judgment
on several elements simultaneously.
6. IN THE ANALYTIC HIERARCHY PROCESS ADDITIVE COMPOSITION IS
NECESSARY
Synthesis in the AHP involves weighting the priorities of elements compared with
respect to an element in the next higher level, called a parent element, by the pri-
ority of that element and adding over all such parents for each element in the
lower level. In the AHP we assume that the criteria are preferentially independent
(inner independent among themselves) and therefore use additive synthesis. This
process of weighting and adding is carried out from the top to the bottom of a hi-
erarchy. The result is a multilinear form involving sums of products of all the pri-
orities from the top to the bottom. Multilinear forms are important for represent-
ing the weights of all the factors involved in the prioritization process to capture
nonlinear effects as accurately as desired by expanding the hierarchy both later-
ally and in depth to include all the important factors considered to have an influ-
ence on the outcome. When the criteria are not independent, their weights are de-
rived as a function of the weights of the other criteria. The resulting weights are
used as if they are preferentially independent. Additive composition is needed to
make possible tradeoff among them. The non-additive component is already sub-
sumed in the way we obtain the weights of these criteria.
Let us consider the example of two criteria and three alternatives measured
in the same scale such as dollars. If the criteria are each assigned the value 1, then
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0 2 4 6 8 10 12 14 16
Fig. 4. Plot of First Differences in Random Inconsistency
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weighting and adding produces the correct dollar value. However, it does not if
the weights of the criteria are normalized, with each criterion having a weight of
0.5. Once the criteria are given in relative terms, so must the alternatives be and
then the criteria must be related according to importance. A criterion that meas-
ures values in pennies cannot be as important as another measured in thousands of
dollars. In this case, the only meaningful importance of a criterion is the ratio of
the total money for the alternatives under it to the total money for the alternatives
under both criteria. By using these weights for the criteria, rather than 0.5 and 0.5,
one obtains the correct final relative values for the alternatives. The AHP is a spe-
cial case of the Analytic Network Process. As we see shall see later, the ANP
automatically assigns the criteria the correct weights, if one only uses the normal-
ized values of the alternatives under each criterion and also the normalized values
for each alternative under all the criteria without any special attention to weight-
ing the criteria. Its operation of raising the supermatrix to the limit powers does
that. The ANP is based on additive synthesis. This is an important example that
shows that additive synthesis in both the AHP and ANP is necessary to obtain the
correct final result in relative terms when the weights of the criteria depend on the
weights of the alternatives.
An Investment Example
An individual has three alternate ways, 21, AA and 3A , of investing a sum of
money for the same period of time. There are two types of returns, 1C and 2C
(for example, capital appreciation and interest), as shown in Table 4. The question
is, which is the best investment to make in terms of actual dollars earned?
The returns for each investment are shown below. It is easy to calculate the
actual total cost for each house by simply adding the two numbers; the relative
cost is then obtained by normalizing as shown in Table 4 below. We will then
show that the ANP process of prioritizing the houses in terms of the criteria and
then the criteria in terms of the houses will give the same relative costs as the
arithmetic computation. This validates the ANP as an additive synthesis process
with a real example where measureable criteria are involved.
T a b l e 4 . Calculating Returns Arithmetically
Alternatives
Criterion C1
Unnormalized
weight = 1.0
Criterion C2
Unnormalized
weight = 1.0
Weighted Sum
Unnormalized
Normalized or
Relative values
A1 200 150 350 350/1300=0.269
A2 300 50 350 350/1300=0.269
A3 500 100 600 600/1300=0.462
Column totals 1000 300 1300 1
What is the relative importance of each criterion? We cannot simply assign
each criterion the value one because the values assigned to the criteria need to be
normalized yielding 0.5, 0.5. But normalization indicates relative importance.
Relative values require that criteria be examined as to their relative importance
with respect to each other. What is the relative importance of a criterion, or what
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numbers should the criteria be assigned that reflect their relative importance?
Weighting each criterion by the proportion of the resource under it, as shown in
Table 5, and multiplying and adding as in the additive synthesis of the AHP, we
get the same correct answer obtained in Table 5 using arithmetic. For criterion
1C we have ( ) ( ) ( )[ ] 1300100000105150500300200500300200 =+++++++
and for criterion 2C we have ( ) ( )[ ( +++++++ 5015050030020000150150
)] 1300300100 =+ . Here the criteria are normalized accordingly, and their
weights sum to one. We see that when the criteria are normalized, the alternatives
must also be normalized to get the right answer. For example if we look in Table
4 we have 350/1300 for the priority of alternative 1A . Now if we simply weight
and add the values for alternative 1A in Table 4 we get for its final relative value
( ) ( ) 1300350130030015013001000200 =+ as it should be from Table 4. How-
ever if we normalize as in Table 5 below we get ( )( ) +130010001000200
( )( ) 13003501300300300150 =+ . We see that if the priorities of the alternatives
are not normalized one does not get meaningful answers. Thus at least in this
case, normalization of the priorities of the alternatives is necessary when the pri-
orities of the criteria depend on the priorities of the alternatives.
T a b l e 5 . Normalized Criteria Weights and Normalized Alternative Weights
From Measurements in Same Scale (Additive Synthesis)
Alternatives
Criterion C1
Normalized weight =
1000/1300=0.7692
Criterion C2
Normalized weight =
300/1300=0.2308
Weighted Sum
A1 200/1000 150/300 350/1300 = 0.2692
A2 300/1000 50/300 350/1300 = 0.2692
A3 500/1000 100/300 600/1300 =0.4615
7. THE DISTRIBUTIVE AND IDEAL MODES
We have seen in the example of the last section that in order to obtain the correct
final relative values for the alternatives when measurements on a measurement
scale are given, it is essential that the priorities of the criteria be derived from the
priorities of the alternatives. This is followed by using the normalized weights of
the alternatives multiplied by the priorities of the criteria and summed over the
criteria to obtain the relative value for each alternative. Thus when the criteria
depend on the alternatives we need to normalize the values of the alternatives to
obtain the final result. This procedure is known as the distributive mode of the
AHP. It is also used in case of functional dependence of the criteria on the alter-
natives for example. Here the criteria are compared with respect to each alterna-
tive separately as in the ANP. For example one asks about a given individual: is
this individual better as a musician or as a teacher and how much better? The
dominant mode of synthesis in the ANP is the distributive mode.
When the criteria do not depend on the values of the alternatives we need to
derive their priorities by comparing them pairwise with each other with respect to
higher level criteria or the goal. It is a process of trading off one unit of one crite-
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rion against a unit of another, an ideal alternative from one against an ideal alter-
native from another. To determine the ideal the alternatives are divided by the
largest value among them for each criterion. In that case the process of weighting
and adding assigns each of the remaining alternatives a value that is proportionate
to the value 1 given to the highest rated alternative. In this way the alternatives
are weighted by the priorities of the criteria and summed to obtain the weights of
the alternatives. This is the ideal mode of the AHP. Thus the distributive mode is
essential for synthesizing the weights of alternatives with respect to tangible crite-
ria with the same scale of measurement into a single criterion for that scale and
then they are treated as intangibles and compared pairwise and combined with
other intangibles with the ideal mode. The dominant mode of synthesis in the
AHP where the criteria are independent from the alternatives is the ideal mode.
The standard mode for synthesizing in the ANP where criteria depend on alterna-
tives and also alternatives may depend on other alternatives is the distributive
mode.
8. RATING ALTERNATIVES ONE AT A TIME IN THE AHP
The AHP has a second way to derive priorities known as absolute measurement.
It involves making paired comparisons but the criteria just above the alternatives,
known as the covering criteria, are assigned intensities that vary in number and
type. For example they can simply be: high, medium and low; or they can be: ex-
cellent, very good, good, average, poor and very poor; or for experience: more
than 15 years, between 10 and 15, between 5 and 10 and less than 5 and so on.
These intensities are compared pairwise among themselves to obtain their priori-
ties as to importance, and they are then put in ideal form by dividing by the larg-
est value. Finally each alternative is assigned an intensity with its accompanying
priority (called rating) for each criterion. The priority of each intensity is
weighted by the priority of its criterion and summed over the weighted intensities
for each alternative to obtain that alternative’s final rating that also belong to a
ratio scale. It is often necessary to have categories of ratings for alternatives that
are widely disparate so that one can rate the alternatives correctly. Ratings are
useful when standards are established with which the alternatives must comply.
They are also useful when the number of alternatives n is very large to perform
pairwise comparisons on them for each criterion. In this case if the number of cri-
teria is c, the number of rating operations in rating the alternatives is cn, whereas
doing all the pairwise judgments involves 2/)1( −ncn comparisons.
9. NEGATIVE PRIORITIES IN THE AHP
Were one to subtract the values of the alternatives under a criterion, taken as
costs, from those under another, taken as gains, both measured in the same unit,
and then normalize the results, the outcome can again be duplicated in relative
terms. We total each of the two columns for the alternatives under the two crite-
ria, subtract the second total from the first. We then assign each criterion the sum
of the values of the alternatives under it divided by the absolute value of the fore-
going difference (treating a difference of zero as a special case in which benefits
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and costs have equal weights). Note that the criteria are normalized with respect
to differences rather than sums. We then weight by the priorities of the criteria
and add. The outcome, as in the example in Table 5, is identical to subtracting the
second value for each alternative from the first summing and then dividing by the
absolute value of the sum of these differences. Clearly the process requires that
we use the distributive and not the ideal mode. The following example speaks for
itself. The original values were the numerators of the two middle columns of Table 6.
In the absence of a unit, at first it is not obvious as how to combine positive
and negative numbers as
priorities. Thus it is not
possible for example to
compare the relative im-
portance of benefits with
the relative importance of
costs. Pleasure is not a
higher form of pain nor is
good a higher form of evil.
The good and the bad are
different but they are op-
posites of the same type or dimension of measurement. One way that has been
justifiably used in the AHP instead of negative priorities is to take reciprocals,
weight and then add them to other positive priorities.
From the requirement of dominance by using a unit in paired comparisons,
we know that we can only ask how much an element dominates another element
and not how much an element is dominated by another element. It is not meaning-
ful to do it the opposite way without first using the smaller element as the unit to
determine how many times larger is the more dominant element and then estimat-
ing the smaller one as a fraction of it. Thus not only does one ask how much more
important one element is than another according to benefits and opportunities, but
also how much more costly or risky one element is than another with respect to a
certain criterion.
We know that people can trade off the benefits of an alternative against its
costs in making a decision but they do not do it by a process of wholesale com-
parison. They do it with respect to their own satisfaction (strategic) criteria and by
rating the contribution of benefits and the costs of that alternative separately to the
fulfillment of those criteria. When there are several alternatives, one uses for each
of the benefits and costs that alternative with the largest composite ideal priority.
It may be that the best-ranked alternative under benefits differs from that under
costs, but in any case one uses the ideal alternative in developing the ratings for
the benefits. The same approach applies to the costs. The results obtained from
the ratings take the form of non-normalized priorities for the benefits and the
costs. Normalizing these rating values yields the desired priorities that enable us
to tradeoff the benefits and costs of all the alternatives. The example in the next
section will help clarify these ideas.
More generally, in many decision problems four kinds of concerns are con-
sidered: benefits, opportunities, costs and risks; which we abbreviate as BOCR.
The first two are advantageous and hence are positive and the second two are dis-
advantageous and are therefore negative [5]. We have sometimes justifiably kept
T a b l e 6 . Subtracting Priorities From Same
Scale Measurements
Alternatives
Criterion C1
Normalized
weight = 10/4
Criterion C2
Normalized
weight = 14/4
Weighted
difference
A1 3/10 8/14 –5/4
A2 2/10 1/14 1/4
A3 5/10 5/14 0
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the last two positive in a situation where the decision was already made for exam-
ple to buy a car, and low cost as determined by the normalized reciprocal costs of
the alternatives was seen as a benefit that is then weighted and added to the bene-
fits.
Another and a more accurate way to deal with BOCR is to realize that
through normalization of the principal eigenvector one obtains a dimensionless
set of numbers that belong to an absolute scale. It is known that absolute numbers
can be both positive and negative and hence it is not necessary to confine the
BOCR to being positive.
There are at least four ways to combine BOCR priorities with corresponding
normalized weights rcob ,,, obtained by rating B and then C and then O and
finally R separately. The first way is the traditional one in which weighting
amounts to multiplying by the same constant. They are:
BO/CR,
)/1()/1( RCOB rcob +++ ,
)1()1( RCOB −+−++ rcob ,
.R-C-OB rcob +
The four methods do not always yield the same best answer. One counter
example demonstrates the truth of this observation. The question now is how to
interpret these priorities and use them appropriately in different situations. The
first is a tradeoff between a unit of BO against a unit of CR, a unit of the desir-
able against a unit of the undesirable. The second is a sum of the advantages ob-
tained when committed to action with low values of the disadvantages (the lesser
of the evils) considered as good or positive. The third is more optimistic and con-
siders the residual or complementary value, the fact that «not all is bad» as a posi-
tive measure. The fourth and last simply subtracts the sum of the weighted «bads»
from the sum of the weighted «goods» and can give rise to negative priorities.
Below we give an example to illustrate these four ways of aggregating BOCR.
Which to use depends on the circumstances one faces. The first is used when re-
sources are limited.
This paper will be continued in Part 2.2.
REFERENCES
1. Saaty, Thomas L. The Analytic Network Process, RWS Publications, 4922 Ellsworth
Avenue, Pittsburgh, Pa. 15213, 2001.
2. Saaty, Rozann W. Decision Making in Complex Environments: The Analytic Net-
work Process (ANP) for Dependence and Feedback; a manual for the ANP Soft-
ware; Creative Decisions Foundation, 4922 Ellsworth Avenue, Pittsburgh,
Pa. 15213, 2002.
3. Saaty, Thomas L. Fundamentals of the Analytic Hierarchy Process, RWS Publica-
tions, 4922 Ellsworth Avenue, Pittsburgh, Pa. 15413, 2000.
4. Saaty Thomas L. and Mujgan S. Ozdemir. Why the Magic Number Seven Plus or
Minus Two, (Submitted for publication.)
Received 25.01.2003
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