Risk Analvsis, Vol. 7, No.

2, 1987

Decision Analysis

Risk-Its Priority and Probability: The Analytic

Hierarchy Process

Thomas L. Saaty'

Received Augusr 29, 1986; revised Junuaty 16, 1987

Risk estimation involves priorities and probabilities which are themselves a form of priority
of natural alternatives. This paper provides illustrations of how one can deal with risk and
uncertainty using the Analytic Hierarchy Process, a new approach to measurement by ratio
scales. The paper also includes a discussion of how to deal with risk strategies involving
interdependence. Particular emphasis is made on the siting of nuclear power plants.

KEY WORDS Risk; Uncertainty; Decision Making; Hierarchies; Low Probability Events.

1. INTRODUCTION ning. These are the aspects that must be emphasized

rather than the mathematical estimation of the prob-
A risk situation differs from benefit-cost analy- abilities of the things that can go wrong. Undertak-
sis as it involves potentially very high unacceptable ing risky activity is not a simple matter of courage
costs that no one expects to pay. Here, an individual and blind commitment.
or group put themselves in a position where things It has been known for a long time that we do
may not work out as expected, leading to losses far not have accurate and justifiable methods to make
exceeding the costs of the undertaking itself. What predictions for the future based on the past. But,
are these situations and which ones are worth the people blithely go on doing it because they do not
risk? Why do some people find it natural to take a know any better. At first, when people use it they
chance on risky endeavors while others do not? Risk recognize that it is not a sure way. The longer groups
analysis must, of necessity, involve different time of people use it and give excuses for it, the more
horizons over which the stability of the system is likely they are to become conditioned by their own
investigated. It can also involve different geographic habits and feel outraged if it is challenged, or feel
locations where the penalty can be minimized. Fi- even worse if it fails (as it often does). Instead of
nally, in case things go wrong, quick action with backing off to the weak original position, they en-
appropriate systemic controls may be followed to trench themselves in a defensive position. In the end,
minimize damage. Thus, risk analysis is not analo- misguidedness prevails with flimsy justification. It
gous to throwing dice in a game framework. Rather it may even be better to consult a medicine man or a
is a complex real world setting that can be favorably soothsayer than to use this extrapolative form of
managed with intelligence, creativity, and prior plan- fortune telling cloaked in the guise of science or
'University of Pittsburgh, Pittsburgh, PA 15260. statistics.
159
0272-4332/R7/0600-0159$0500/101987 Society far Risk Analysis
160

The problem is the same with risk analysis. is hard to apply to their problems, and leans too
Dressing a skeleton with fancy clothes does not give heavily on theoretical manipulations.
it life. Here again, the field suffers from elaborate Instead, we need a better understanding and
techniques purported to deal with risk and have representation of complexity and of the attitudes
fallen short in real life applications. Let us comment needed to estimate what goes on in that complexity,
on the state of nature and our ability to know it. and different potential scenarios that could result
A sufficient account of the possibility of dis- from a combination of the structures we use and the
order should be included in our representation of attitudes we assume. Additionally, we need a judg-
risk to capture what might happen. We should never ment about the entire system of thinking that we use
be smug and cocksure. This kind of suspicion is in this situation: whether it is destined to work, and
likely to be helpful against surprises. what alternatives we have to deal with the problem.
Another problem is that we are accustomed to
dealing with complexity one factor at a time, in a
chain like manner, when actually thmgs are interde-
pendent. We need a way to represent such interac-
tions and dependencies holistically in our structures
for handling risk. 2. PRIORITIES, PROBABILITIES AND
A third assumption, particularly by those who UNCERTAINTY
have a good mind for analysis, is that we can get
probabilities from our minds by pretending that we Priority is the relative intensity of what is im-
understand the situation and by doing reasoning portant to people. probability or likelihood is nature's
analogous to things we really do understand (as if priority at the occurrence level. Note that complexity
nature is a simple coin or a series of well-organized can be better represented through priority because
devices clearly interpretable in terms of probabilities). there are usually elements whose contribution or
Still worse, often our probabilities are not de- influence is not probabilistic, but nevertheless de-
termined experimentally because we cannot set up scribable in terms of priorities. For example, given a
experiments; we lack the knowledge even to simulate set of factors, one can ask which of them has greater
them for catastrophic events that we think (or hope) influence on the outcome, or which one has the
have a low probability of occurrence. It is a process higher priority influence. This lack of complete rep-
of self-hypnosis when we convince ourselves and resentation has been a serious flaw in risk analysis
others that we have the right probabilities because where everything is described in terms of probabili-
they appear to be so natural, given our sensibly ties alone.
sounding causal explanations. We should be espe- In modern risk analysis we need to determine
cially wary of such explanations and our proclivity to numerically not just what is likely to happen, but
estimate probabilities. also what is important and what is not important if it
A final point is that an old Babylonian habit does happen. One must consider all observed factors
remains active in our technical minds. We think that and then establish priorities in the two senses we
the world is a riddle, and like magicians belonging to mentioned above: importance and likelihood of oc-
a brotherhood with a tradition and a language, we currence.
can unravel it if we can only find the key. Our There are 2 types of uncertainty: (a) uncertainty
modem key is to resort to complex theoretical about the occurrence of events, and (b) uncertainty
manipulations, thinking that, if pursued with suffi- about the range of judgments used to express prefer-
cient diligence, the right combination will fall in ences. The first is beyond the control of the decision
place and we would have the answer. maker. The second is a consequence of the amount of
A case in point is the traditional approach to the information available to him and his understanding
disposition of an individual to risk, based on the of the problem.
concept of utility function and the rate of change of We have studied(') the second type of uncer-
its second derivative with respect to the first deriva- tainty experienced by the decision maker in makmg
tive. The calculation of the utility function involves pairwise comparisons. We assume that one can asso-
the probability of risk. People have complained that ciate each judgment with an interval of numerical
this kind of technique-driven approach, neat as it is, values from which a reciprocal matrix of interval
Risk Priority and Probability 161

values in each position is constructed. The problem taxed lower than short-term returns and that tax-free
then is to generate the possible solutions from such a investments, such as municipal bonds, would be rated
matrix by assuming various probability distributions. high. The fourth criterion, liquidity, included small
The approach has also been generalized to an entire transaction costs, ease of withdrawal, and the ability
hierarchy. The purpose of the analysis is to study the to make small additions. These second-level-clustered
likelihood that alternatives change rank and, in par- criteria could have been followed by a third level of
ticular, the likelihood that a high priority alternative subcriteria consisting of all the above itemized
could become lost due to uncertainty and change breakdowns, but, to be brief, we have gone directly
ranks with a lower priority one (and in bad cases, to the portfolio aggregate alternatives in the next
with a really low priority one). This approach has level.
great relevance to problems of uncertainty and hgh The third level consists of the following alterna-
risk. tives: the first, tax free bonds, are low return, low
risk securities with great tax advantage; the second,
securities, are tied to the treasury rate with medium
3. A DECISION ON INVESTMENT return, low risk, and relative liquidity; the third,
saving accounts, involve low return, but their liquid-
A recently retired person wanted to formulate ity and very low risk are their advantage; the fourth,
an investment portfolio tailored to his particular speculative stocks, involve high return and high risk
needs. Most of us can identify readily with this investments and include the options market; and the
individual's problem of grappling with risk, uncer- fifth, blue chip stocks and bonds, are the medium
tainty, and other abstract intangibles. He constructed return, low risk investments.
the following hierarchy (Fig. 1): The judgment phase of the analytic hierarchy
For simplicity, the second level contained only process (AHP) requires the following scale of ab-
four criteria which are in turn an aggregation of solute values (not ordinals) to express judgments in
many subcriteria that came to mind. The first crite- making paired comparisons: 1, equal; 3, moderate; 5,
rion, return, includes interest, dividends, and capital strong; 7, very strong; 9, extreme (2, 4, 6, 8 for
appreciation. Although capital appreciation and compromise- reciprocals for the inverse comparison,
short-term income are taxed differently, this dif- and decimal refinements between if it is desired to
ference is taken into account in another group. obtain a predetermined set of final priorities). The
Estimated annualized returns were used when judg- smaller of two elements being compared is consid-
ing the individual securities. The next criterion, low ered to be the unit and the larger one is assessed to
risk, included subcriteria involving diversification and be so many times more than it, using the intensity of
low-perceived risk and volatility of return. The third feeling and translating it to the numerical value. One
criterion, tax benefits, assumes that capital gains are can show that the derived scale from the compari-

Level 1: BEST CHOICE OF PORTFOLIO

Goal

Level 2:
Criteria

Level 3:
Alternatives

Fig. 1. Hierarchy for the best choice of a portfolio.

162

sons is insensitive to small changes (corresponding to Table 1.

uncertainty between two neighboring intensities) in Return Low risk Tax benefits Liquidity Weights
the values of the paired comparisons. If exact num- Return 1 1/5 3 3 ,214
bers for the comparisons are known, and their ratios Low risk 5 1 5 4 ,598
are used, the derived scale gives their relative value Tax benefits 1/3 1/5 1 1 ,090
Liquidity 1/3 1/4 1 1 ,097
back. However, even when exact numbers are avail-
able, one ordinarily needs to compare the value of
these numbers by interpreting their relative impor-
tance and then using that instead of the numbers
themselves which may have no universal meaning
(e.g., to a family of modest income, a $20,000 car but also to allocate resources appropriately in pro-
portion to the values.
may be much more expensive than a $10,000 car, but
to a military budget planner they may be about the The AHP does not insist on consistent judg-
same). ments and provides an index for measuring incon-
The next step after structuring the herarchy is sistency, both for each matrix of comparisons and for
to carry out comparative judgments using the above the entire herarchy. It is also possible to find out
scale. The elements in the second level are arranged where the most inconsistent judgments are and to
in a matrix and judgments are elicited as to the change them if desired to improve the consistency,
relative importance of each criterion when compared although t h s is not required. The inconsistency ratio
with every other criterion on that level. The question for the above matrix is 0.072 which is less than the
here is: for a best choice of portfolio for the particu- tolerance level of 0.100. The reason for using a
lar individual’s circumstances (whch give rise to h s tolerance level of 0.100 is this: although the mind is
judgments), which criterion does he consider more primarily concerned with constructing a consistent
important and how strongly more? We have the decision, it must allow a modicum of inconsistency in
following matrix (Table I.) for comparing the criteria order to admit new information, giving rise to change
with respect to the goal, and the weights derived in the old judgments. However, inconsistency is less
from it: important than consistency by one order of magni-
A criterion (X),represented on the left, is com- tude (the 10% tolerance range)(’).
pared with respect to a criterion ( Y ) , represented on Next, the alternatives are compared in pairs with
the top of the matrix. If X is more important than Y , respect to each criterion in separate matrices. The
then a numerical value greater than 1 is used in the resulting scales from the 4 matrices are each weighted
(X,Y ) position. If Y is more important than X,then or multiplied by the importance of the criteria just
the reciprocal of this value is used. The reciprocal of derived above, and then added to obtain the overall
whatever value is entered in the ( X , Y ) position is importance of the alternatives (Table 11.). We have:
automatically entered in the ( Y , X)position. This portfolio was compared with the investor’s
For example, in answering the question, “Which existing portfolio (the percentages shown in the last
is more important, return on the left, or low risk on column). Although at present the investor makes use
top?,” we believe that low risk is strongly more of the second and third investments appropriately.
important than return and, hence, enter the value the bulk of the portfolio is tied up in medium return,
1/5 in the first row, second column position and low risk corporate securities. He decided to sell a
automatically enter the value 5 in the second row, large part of these and reinvest in municipal bonds
first column position, and so on. and more speculative issues to satisfy his investment
The paired comparisons give rise to a ratio scale objectives.
of weights of the relative importance or priorities of
the criteria. Ratio scales are a strong class of num-
bers whose ratios remain the same when each of 4. THE SITING OF A NUCLEAR POWER PLANT
them is multiplied by a positive constant. For exam-
ple, the ratio of the weight of 2 objects measured The AHP provides a rational mechanism to
either in pounds or in kilograms is the same. Ratio guide the process of public policy decision towards a
scales make it possible not only to rank alternatives. more responsible, consistent approach to the man-
Risk Priority and Probability 163

Table U.
Return Low risk Tax benefits Liquidity
(.214) (.598) (.OW) (.097) Final priority Existing portfolio
A ,054 ,162 ,521 ,051 .16 .oo
B ,165 .338 ,057 .298 .27 .30
C ,057 ,344 ,057 .487 .27 .20
D ,497 ,034 ,124 .068 .15 .oo
E .227 .122 ,241 .096 .15 .50
Inconsistency .031 ,031 ,023 .025
Composite inconsistency .025

agement of risk. In no instance is this need more separate level and prioritized in terms of the intensity
acutely felt than in the case of a nuclear power plant cost factors as they impact on different sizes of the
siting decision, since it entails the full gamut of population.
claims and counter-claims of an environmental, tech- This is meant simply to serve as an example of
nological, social, and economic character. how such a hierarchy would be structured. If one
Technological risk involves two components: the wished to determine the priorities of various institu-
likelihood of an event occurring, and the nature of tional constraints (e.g., political and administrative
the distribution of its negative impact along various considerations), the alternate sites and technologies
dimensions. With regard to such impacts, they in- can be followed by a still lower level of feasible
clude: the character of the potential loss (e.g., whether implementation strategies. (For costs hierarchy, see
it is a monetary loss, a physical injury or death, or an Fig. 2).
environmental disability), the extent of the loss in Now another hierarchy of the “benefits” is de-
terms of its intensity and diffusion, and its timing veloped. An essential difference in the two hierarchies
(how immediate is the danger, what would be its would be the level 6 specific effects, which now lists
frequency, and so on). the positive aspects of the technology. This level is as
The AHP can be used to address these negative follows: specific desirable effects-cheap power,
elements within an overall cost-benefit framework. clean power, greater productivity, neat environment,
We would thus structure the nuclear power plant high technology, prestige, equity.
siting problem into two hierarchies, a costs hierarchy The prioritization process of costs and benefits
and a benefits hierarchy (see Figs. 2-6). would then be used to form benefit-cost ratios. It
Priorities are established on the subjective has been our experience that such ratios are most
evaluations (by the actors in levels 2 and 3 of Fig. 2) meaningful when familiar technologies that have been
of the losses resulting from building the proposed adopted and used by society for some time are
nuclear power plant. Priorities are also set on the included to serve as comparison yardsticks. AHP
actor’s objectives in level 4 of Fig. 2. To incorporate could also easily be used to perform sensitivity analy-
the kinds of losses involved, the cost factors are sis with regard to variations in the structure of
clustered according to whether they are economic, the problem and in the judgments. It is also a
physical, or social (see level 5 of Fig. 2). In yet means of explicitly measuring consistency of the
another level, each cluster is decomposed into an judgments.
itemized list of subfactors (level 6 of Fig. 2). Again, Figures 2 and 3 give the complete benefit-cost
the subfactors are decomposed according to uncer- analysis representation of the problem of siting
tainty by estimating the degree of their intensity: nuclear power plants.
high, medium, or low (level 7 of Fig. 2). Further, In the next section, we give a generalization of
each of these is decomposed into “size of population the hierarchic approach to risk problems involving
affected” categories, ranging from large to medium interdependence. The theory will be given without
to a small number of people (level 8 of Fig. 2). particular emphasis on risk but, based on the earlier
Finally, the alternative locations and nuclear applications, the reader will be able to see how it
technologies that may be used are identified in a applies.
Hierarchic Levels
Focus GNPa I
I
I
I
Actors Economic Gmups I Political Grouoa I
I I
I
Aclor Sub Groups

Concern for
Objectives
-- ~ropertyvalues
Earthquakes
and Acls of God

Clusters of Economic
Undesirable ElfeCls
Internal External
----------- I 1 r
I 1
SpeClflC
~~~~
Undesirable ElfeCls

Degree of Intensity I
High (H) I
Medium (M) I
LOW (L) I
I
Numbers of I
Individuals Afiecled I
Large Numbers (L) I
Moderale Numbers (M) I
Small Numbers (S)
I
I
Allernalive Locations
and Technologies ()'

[ Loss 01 Income
I

Q Q
This decision mruclure applled to all categoflea in all kvel 7 cstegones

Fig. 2. Costs hierarchy for siting of a nuclear power facility. Levels 1-5 are common with Costs
Clusters of Benetits (5) Economic Physical

Specific Desirable
Effects (6)
= 7
Cheap Power
Productivity Neat Environment High Technology Prestige

Degree of Intensity (7) 0

High (H) I
III
II
Medium (M)
Low (L)

Numbers of (8)
Individuals Affected
Large Numbers (L)
Moderate Numbers (M)
Small Numbers (S)

Alternative Locations (9)

and Technologies

1
1
I
Cheap Power I
I
el I I
I
I
I
I
I
I
I
I
I
I
I
I
I
I This decision Structure applied to all categories in all level 7 categories
I
I
I
166

system on another, in a sequential manner. It is also

23 .....
a convenient way to decompose a complex problem
in search of cause-effect explanations in steps which
form a linear chain. One result of this approach is to
Component assume the functional independence of an upper

+
part, component, or cluster from its lower parts. Ths
eIernent often does not imply its structural independence
from the lower parts which involve information on
..... the number of elements, their measurements, etc. But
there is a more general way to structure a problem
A Linear Hierarchy A Nonlinear Network
involving functional dependence. It allows for feed-
back between components, and is a network system
Fig. 4. A + B means that A dominates B or that B depends of which a hierarchy is a special case. In both

-
on A . hierarchies and networks, the elements within each
component may also be dependent on each other(3).
5. THE GENERAL CASE OF DEPENDENCE IN Figure 4 shows two drawings which depict the
HIERARCHIC DECISION THEORY structural difference between the two frameworks.
In the above figure, a loop means that there is
A hierarchy is a simple structure used to repre- inner dependence of elements within a component.
sent the simplest type of functional (contextual or A nonlinear network can be used to identify
semantic) dependence of one level or component of a relationships among components using one’s own

I OVERALL BENEFITS

1980-85

SOCIO-POLITICAL MACRO-ECONOMIC

TECHNOLOGY

-Savings -Non-Proliferation -Spinoffs

-Security and of Nuclear -National Prestige Professional
Reliability Armament -Preservation of -Unskilled Trades
-Regional Technology -Regulatory Aspects

I I -Regional -Skilled Trades

COAL SOLAR AND NUCLEAR SYNTHETIC

GEOTHERMAL FUELS
I
Risk Priority and Probability 167

I OVERALL COSTS
I

I 1980-85
J I 1985-90
I I 1993-95
1
1
COSTS TO ENVIRONMENTAL POLITICAL
GOVERNMENT COSTS COSTS
C1 C2 c3

I 1
I -%lid Waste
I
-National (Pubk Acceptance)
-Governmeni Financing of
Power Generation RLD -Accident Risk -InIernalional (Safe Guards)
Programs -SiS Approval Problems -Regional (Jurirdlction)
-Regulatory -Effluent and Health Problems
-Export Financing
-Markeling Support
-Domestic Financing

SYNTHETIC
COAL SOLAR AND NUCLEAR
FUELS
OEOTHERMAL

thoughts, rela ively free of rules. It is especially suited one criterion than another for that alternative?’; I ie
for modeling dependence relations. Such a network other set for comparing alternatives in terms of
approach makes it possible to represent and analyze criteria by answering the question, “Given the crite-
interactions and also to synthesize their mutual ef- rion, how much more important is one alternative
fects by a single logical procedure. than another for that criterion?’
For emphasis, we note again that in the nonlin- In this manner, components which depend on
ear network diagram or system with feedback (above), one another have impacts which appear in two
there are 2 kinds of dependence: that between com- b l o ~ k s ( ~The
, ~ ) .overall, or limiting, priorities are ob-
ponents (in a way to allow for feedback circuits), and tained from the following supermatrix of interac-
the interdependence within a component combined tions. If we denote the 4 criteria by C,, C,, C,, and
with feedback between components. We have called C, and the alternatives by A, B, and C, we have the
these, respectively, outer and inner dependence. stochastic supermatrix:
If the criteria cannot be compared with respect
to an overall objective because of lack of informa- C, C, C, C4 A B C
tion, they can be compared in terms of the alterna- 0 0 0 0 .6279 .6279 .6279
tives. The systems approach can then be used to 0 0 0 0 .0942 .0942 .0942
replace the hierarchic approach. 0 0 0 0 .2060 .2060 -2060
Briefly, the system prioritization approach be- 0 0 0 0 .0719 .0719 .0719
gins with what is known as a supermatrix of blocks .250 SO0 .556 .545 0 0 0
of interaction among components. Each column of a .333 .333 .286 .273 0 0 0
block is the eigenvector of priorities of the impact of .417 .167 .158 .182 0 0 0
a component on an element in the system. These
eigenvectors are obtained from individual matrices of As we shall see later, this is an irreducible im-
paired comparisons: one set for comparing criteria in primitive matrix. The limiting priorities of the criteria
terms of alternatives by answering the question, and of the alternatives-are obtained by solving the
“Given the alternative, how much more important is eigenvalue problem W w = w , and normalizing the
first 4 components of w for the priorities of the whole” which alternative is more “preferred“ or more
criteria and the last 3 for the priorities of the alterna- “likely.” For other levels, one has specific criteria or
tives. This gives, respectively, .6279, .0942, .2060, elements for making comparisons and need not
.0719 and .3578, .3190, .3232. In this case, the same preface the question with “on the average.”
results could have been obtained, respectively, as the
first 4 nonzero elements from any of the first 4
5.2. Hierarchies with Cycles
columns of:

lim
k+m
(w,)~, Power Generation Example with Uncertainty

and the last 3 nonzero elements from its last 3 There are problems in which the future must be
columns. factored into the decisions taken in the present. In
that case, one lays out different time horizons and
different criteria (or scenarios) likely to prevail dur-
5.1. Questions to Ask ing one or the other of these time periods. They are
essentially a discrete characterization of the situation.
The analytic hierarchy process seeks to elicit In a situation like this, one needs to set priorities on
judgments from people by asking the appropriate the criteria for each time period. One also needs to
question that would produce the intended answer. By set priorities for the time periods for each criterion
asking the wrong question one would obtain non- by entering a judgment as to the time during which
sensical results. It remains to determine whether there the criterion is most likely to prevail. As in the first
are right questions to ask, what they are, and how example, the resulting priority vectors are then en-
readily people can respond to them. A part of the tered as columns of a supermatrix, representing the
solution lies in experience obtained using the process. interactions of the two levels of the hierarchy. It has
The other part depends on our understanding of the the form:
types of problems for which one sets priorities. In
general, the situation may be normative or descrip-
tive. In the former case, the question for the pairwise
comparison should be formed in terms of what is
more preferred or desired in order to satisfy a certain The columns of the submatrix A,, correspond to
criterion, constraint, property, or scenario. In de- the priority vectors of the criteria in terms of each
scriptive situations, the question will seek judgments time period arranged in the proper order, and the
which identify the degree or extent that two alterna- columns of A,, correspond to the priority vectors of
tives have a certain property (e.g., of two stones the time periods in terms of the criteria; the matrix
which is heavier and how much; of two roses which A is column stochastic. Again, without details we
is more red and so on). More abstractly, 2 criteria give Figs. 5 and 6 to show the benefits and costs
may be compared as to a higher level criterion. hierarchies and a summary of overall benefits and
Which one is more likely to produce, embody, or costs of the alternatives, their ratios, and marginal
fulfill that criterion, or is closer to defining or bring- ratios for decision purposes. Here, the nuclear alter-
ing about the criterion. Generically, we ask which is native is favored. See Table 111 for the results.
more “important,” meaning a greater possessor of
the attribute. These 2 categories of questions relate to Table III. Results from the Power Generation Benefits and
2 types of hierarchies discussed in the AHP litera- Costs Hierarchies
ture: the forward (descriptive) and the backward Total
(prescriptive) hierarchies that are often combined benefits/ B~ - ‘1
through iteration into a process of improving the Benefits ~~
Costs ~
costs ~~
c, - c,
likely outcome towards the desired outcome. coal ,137 ,152 ,901 ,901
It should be noted here that in a hierarchy with Solar and
geothermal ,169 ,216 !782 -
a single top element or focus, comparison of the
Nuclear ,288 ,210 1.371 2.603
second level elements with respect to the focus sug- Syn-fuels .406 ,432 ,940 1.097
gests that one asks “on the average” or “on the
Risk Priority and Probability 169

An interesting application of the feedback con- Now for the formal definitions. The discussion
cept has been used in the analysis of terrorism. A below parallels the theory of Markov Chains as given
hierarchy was used whose bottom level of alterna- in Gantmacher@),and adapted for our purpose. If
tives is linked to its top level of criteria, giving rise to w i j is the impact priority of the i"' element on the j *
a cyclic hierarchy known as a holarchy. The super- element in the system, then:
matrix application was essential in deriving the prior-
ities for the courses of action to be

5.3. Systems with Feedback

Let us consider the general situation of a system

in which components affect each other. It is thus
desirable to obtain priorities for the impacts of the
elements on all other elements in the system. To do
this for each criterion, we must perform pairwise
comparisons for each component with respect to the The sum of the impact priorities along all possi-
elements in each other component with which it ble paths from a given element gives the priorities of
interacts. The resulting eigenvector of each matrix is an element. This amounts to raising the matrix W to
entered as part of a column of a supermatrix, along powers. The last expression is equivalent to W ( h + k )
whose side and at its top all the elements are listed to = WhWk.
obtain a measure of their interaction. Thus, all eigen- Given that the initial priority of the i"' element
vectors representing the impact of the elements of a is w;'), we have the following absolute priority of the
component on one element of another component j"' element in paths of length k # 0:
are arranged in a single column next to their corre-
sponding elements. In this manner, we fill out the Wjk) =c w p v y .
entire matrix. Next, all eigenvectors corresponding to i
a pair of interacting components are weighted by the
priorities of the component which creates the impact. The problem is to find the limiting impact prior-
These priorities are obtained from a separate pair- ity (LIP) matrix W" and the limiting absolute prior-
wise comparison matrix of the components. The ity (LAP) vector w m , as k + m . For a priority
judgments of that matrix are entered by answering system, we may also be interested in determining
the question: Given a component of the system, priorities for finite values of k (that does not present
which other component affects it most with respect problems of existence as does the limiting case). Of
to the given criterion and how strongly (i.e., Which is particular interest, is to determine when the LAP
the most important component affecting it)? The priority is independent of the initial priorities w;').
resulting weighted supermatrix has each column Such independence is called the ergodicity of the
adding to unity. One must make sure that the sum is system.
precisely equal to unity. The following is a classification of elements
We are interested in 2 types of priorities. Those useful in characterizing a system. The reader may
that give the influence or impact of one element on wish to go on to the actual discussion of existence
any other element in the system are known as the and construction of LIP and LAP solutions.
impact priorities. We are also interested in the ab- The element j can be reached from the element
solute priority of any element regardless of which i if for some integer k 2 1, w;;) > 0 where W k=
elements it influences. Generally, we seek limiting ( w ! ; ) ) . Here W k gives the k-reach of each element. A
values of the two kinds of priorities. Calculation of subset of elements C of a system is closed (opposite
these priorities shows where existing trends might definition to that for Markov chains) if w;;) = 0
lead if there is no change in preferences which affect whenever i is in C and j is not in C. It follows that
the priorities. By experimenting with the process of no element can be reached from any element not in
modifying priorities and noting their limiting trends, C. The subset C is minimal if it contains no proper
we may be able to steer a system towards a more closed subset of elements. A set of elements which
desired outcome. forms a minimal closed subset corresponds to what is
170

known as an irreducible matrix; the system or sub- If j is cyclic with cyclicity c > 1 then w $ ) = 0 if
system itself is called irreducible. A system is called k is not a multiple of c and wj,!”) + c / u as m ao;-+

decomposable if it has 2 or more closed sets. k = mc, m positive and c the largest integer for
If we initially start with the j I h element for some which k = mc holds.
fixed j and denote its first impact on itself in a path We had said earlier that reducibility and primi-
of length k 2 1 by f / ( k ) , we have = w;), tivity play an important role in proving the existence
$2) = wf)-f/(l)w;) . .. f.’U = WJ:k)--fi(l) ( k - 1)-
wJJ
... of LIP and LAP. We now give a few basic facts
- f J. ’ k - l ) w =T3r=lf,(k)
J
JJ . .and Am gives the cumulative relating these concepts which will be useful in the
impact of j on itself. The mean impact (of j on ensuing discussion.
itself) is given by uj = I3;pPOV,(”. A nonnegative irreducible matrix is primitive if
According to priority influence, we have (the it has a unique principal eigenvalue. If the matrix has
new terms introduced below are essential, as we are another eigenvalue with the same modulus as the
not dealing with time transitions): principal eigenvalue, it is called imprimitive. If the
1. If = 1, j is called an enduring (recurrent) principal eigenvalue has multiplicity greater than
element. Thus, an element is enduring if the sum of unity (equal to unity), but there are not other eigen-
its impact priorities on itself in a single step (by a values of the same modulus as the principal eigen-
loop), in two steps (through a cycle involving one value, then the matrix is called proper (regular).
other element), in three steps (involving two other A primitive matrix is always regular and, hence,
elements, etc.) is equal to unity. proper but not conversely (e.g., the identity matrix
2. If > 1, j is called transitory (transient). An which has unity as an eigenvalue of multiplicity
element j that is either enduring or transitory is equal to the order of the matrix). A matrix is proper
called cyclic (periodic) with cyclicity c, if uj has if, and only if, in the normal form, the isolated
values c, 2c, 3c, ... where c is the greatest integer blocks are primitive. For a regular matrix, the num-
greater than unity with this property (wi/k)= 0 where ber of isolated blocks is unity.
k is not divisible by c). An enduring element j for We note that if all the entries of W are positive,
which uj is infinite is called fading (null). An endur- we have a primitive matrix and the theorem on
ing element j that is neither cyclic nor fading (i.e., stochastic primitive matrices applies; both LIP and
uj > 00) is called sustaining (ergodic). For either a LAP exist. LIP and LAP are the same and are given
transitory or a fading element j , w $ ) + O for
every i. Actually w is any column of lim, -
by the solution of the eigenvalue problem: W w = w.

result is true if W is a primitive matrix.

Wk. The same
If one element in an irreducible subsystem is
cyclic with cyclicity c, all the elements in that subsys- In general, the nonnegative matrix W may have
tem are cyclic with cyclicity c. It is known that if j is some zeros. In that case, it is either an irreducible or
a sustaining element, then as k 03, ~ JJ( 6+)l / u j ; j
-+
a reducible matrix. If it is irreducible, then it is either
is a fading element if this number is zero and sus- primitive, in which case the above discussion applies,
taining if it is positive. Either all the elements of an or it is imprimitive. In the latter case, it has a number
irreducible subsystem are all transitory or all endur- c of eigenvalues (called the index of imprimitivity)
ing, and the system itself is called transitory or that are not equal to unity whose moduli are equal to
enduring, respectively. unity. This number plays an important role in the
Remark. The following expression always exists solution of the general case from which we can also
whether a system is irreducible or not. In the former obtain the solution to ths case. It is sufficient to
case, its values are known and are as indicated: point out here that W ,W 2 , .., . W C p 1are all not
proper and multiples of these matrices tend toward
m-1 periodic repetition. The system is cyclic with cycli-
0 if i and j are transitory
l / u if i and j are enduring city c.
m4m k=O
Remark. The system is acyclic, cyclic, irreduc-
All finite systems of elements must have at least ible, or reducible, depending on whether the corre-
one sustaining element which generates a closed irre- sponding matrix W is primitive, imprimitive, irre-
ducible subset of elements. Since the enduring ele- ducible, or reducible.
ments of a finite system are all sustaining, the block If W is nonnegative and reducible, then it is
(or component) thus generated is called sustaining. reduced to the normal form. If the isolated blocks are
Risk Priority and Probability 171

primitive, they are said to correspond to unessential

components. The system is by definition called proper -
and LIP and LAP exist(6). 0 0 . - 0. 0 0
Important Remark. When our column stochastic w,, 0 . . o. 0 0
matrix is reducible, its essential components drive the 0 w,,. . . o 0 0
. . . .
system since they are “sources” or impact-priority- w= .
diffusing components, as opposed to “sinks” or tran- . . . .
sition-probability-absorbing states of a Markov chain. . . . Wn-1.n-2 0 0
In any diagram, except for loops, arrows initiate -0 0 . . . o Wn. n - 1 I
from and nonterminate at such components.
The solution for LIP is given by:

where + ( A ) is the minimum polynomial of W and

#’(A) is its first derivative with respect to A. Each
- * w32w21, w n , n - 1
column W is a characteristic vector of W corre-
sponding to A,, = 1. If A,, = 1 is simple (i.e., W is
regular), +(A) may be replaced by A(A), the char-
I
wn, n - lwn - 1,n - 2
. * * I
w32, . . - wn, n - lwn - 1 , n - 2 , w n , n - 1
acteristic polynomial of W . LAP is obtained as w m for all k 2 n - 1. Each coefficient in the last row
= Wmw(o)if W is proper, and the eigenvector solu- gives the composite priority impact of the last com-
tion of Ww = w if W is regular. ponent on each of the remaining components. Note
Remark. One can show that the matrices of W that the principle of hierarchical composition ap-
corresponding to essential components are positive, pears in the ( n , l ) position as the impact of the nth
and those to priority impacts from essential to unes- component on the first. The nth component drives
sential components are also positive. Only impacts the hierarchy and is the counterpart of an absorbing
from unessential to unessential or from unessential to state in a Markov chain. It is a component of ele-
essential components are zero. ments which diffuse or are a source of priority im-
Finally, if not all isolated blocks are primitive, pacts. The essence of the above is summarized by the
then each has an index of imprimitivity, as we pointed Principle of Hierarchical Composition: the com-
out earlier. We consider the least common multiple posite vector of a hierarchy of n levels is the entry in
of these which is the cyclicity c of the system. Using the ( n , l ) position of Wk-’, k 2 n - 1.
the powers of W , LIP is given by: This discussion shows that the composition pro-
cess in a hierarchy, which is additive, conforms with
the general composition process of a system with
W = ( l / c ) ( l + W + .-wc-l)(wc)m feedback obtained by an alternative approach well-
known in classical mathematics.
=( l / c ) ( l - W C ) ( l - w)-’(wc)”,
6. CONCLUSIONS
and LAP is given by w = @w(O).Both W and w are
called the mean LIP and mean LAP, respectively. If The foregoing ideas have been applied in a
there is a single isolated block, then the mean LAP variety of situations involving risk and uncertainty in
are independent of the initial priorities and are investment and in corporate and governmental plan-
uniquely determined by the solution of: Ww = w . ning, including a study on terrorism. It is generally
This is precisely the case of an irreducible im- useful in such cases to carry out the evaluation over a
primitive system. Several applications have been made period of time with different groups and synthesize
to calculate priorities in a system with feedback. The the outcomes according to the group procedure of
calculations are long, but the foregoing theory has the Analytic Hierarchy Process. The reader may con-
been found very useful for the purpose. Let us note sult the many references cited(’**)for further ideas on
in closing that the supermatrix of a hierarchy has the the AHP and its applications.
172

REFERENCES 6. F.R.Gantmacher, The Theory of Matrices, Volume I I (Chelsea,

New York, 1960).
7. Shubo Xu, References on the Analytic Hierarchy Process (In-
1. T. L. Saaty and L. Vargas, “Uncertainty and Rank Order in stitute of Systems Engineering, Tianjin University, Tianjin,
the Analytic Hierarchy Process,” European Journal of Oper- China, June, 1986).
ations Research (1986). 8. F. Zahedi, “The Analytic Hierarchy Process-a Survey of the
2. Expert Choice Manual, Software Package Manual for the AHP Method and Its Applications,” Interfaces (forthcoming, 1986).
Software for the IBM PC (Decision Support Software, Mc- 9. W. Feller, A n Introduction to Probability Theory and Applica-
Lean, Virginia, 1986). tions (John Wiley and Sons,New York, 1950).
3. T. L. Saaty, The Analytic Hierarchy Process (McGraw-Hill, 10. D. L. Isaacson and R. W. Madsen, Markov Chains, Theory
New York, 1980). and Applications. Volume in Wiley Series in Probability and
4. T. L. Saaty and M. Takizawa, “Dependence and Indepen- Mathematical Statistics: Probability and Mathematics Section
dence: From Linear Hierarchies to Nonlinear Networks,” (John Wiley and Sons, New York, 1976).
European Journal of Operational Research 24421, pp. 229-231 11. J. G. Kemeny and J. L. Snell, Mathematical Models in the
(1986). Social Sciences (Blaisdell Publishing Company, New York,
5. J. P.BeMett and T. L. Saaty, Terrorism: patterns for negotia- 1962).
tions; three case studies through hierarchies and holarchies (A 12. T. L.Saaty, “Priorities in Systems with Feedback,” The Inter-
study for the Arms Control and Disarmament Agency, p. 208 national Journal of Systems, Measurement and Decisions 1,
1977). See also Facing Tomorrow’s Terrorism Incident Toda.v 24-38 (1981).
(U.S.Department of Justim, LEAA, Washington, D.C.,
28-31).