saaty how to make a decision the analytic hirearchy process

How to Make a Decision: The AnalyticHierarchy Process

THOMAS L. SAATY 322 Menus HallUniversity of PittsburghPittsburgh, Pennsylvania 15260

People make three general types of judgments to express im-portance, preference, or likelihood and use them to choose thebest among alternatives in the presence of environmental, so-cial, political, and other influences. They base these judgmentson knowledge in memory or from analyzing benefits, costs, andrisks. From past knowledge, we sometimes can develop stan-dards of excellence and poorness and use them to rate the alter-natives one at a time. This is useful in such repetitive situationsas student admissions and salary raises that must conform withestablished norms. Without norms one compares alternativesinstead of rating them. Comparisons must fall in an admissiblerange of consistency. The analytic hierarchy process (AHP) in-cludes both the rating and comparison methods. Rationality re-quires developing a reliable hierarchic structure or feedbacknetwork that includes criteria of various types of influence,stakeholders, and decision alternatives to determine the bestchoice.

Policy makers at all levels of decision facilitate their decision making. Through

making in organizations use multiple trade offs it clarifies the advantages andcriteria to analyze their complex problems. disadvantages of policy options under cir-Multicriteria thinking is used formally to cumstances of risk and uncertainty. It isCopyrishl n-' 1944, Thi> Insliluti^ i>i ManaBL-mfiit birn-iitL-s DECISION ANAl.YSrS—SYSThMS(11141 2l(l2/94/24l1(i/0()l9Sl)l 2S DECISION ANALYSIS—AI'I'LICATIONSThi-. pjpef H'J5 ri'ffrifd

INTERFACES 24: 6 November-December 1994 (pp. 19-43)

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also a tool vital to forming corporate strat-egies needed for effective competition.

Nearly all of us, in one way or another,have been brought up to believe that clear-headed logical thinking is our only sureway to face and solve problems. We alsobelieve that our feelings and our judg-ments must be subjected to the acid test ofdeductive thinking. But experience sug-gests that deductive thinking is not natural.Indeed, we have to practice, and for a longtime, before we can do it well. Since com-plex problems usually have many relatedfactors, traditional logical thinking leads tosequences of ideas that are so tangled thattheir interconnections are not readily dis-cerned.

The lack of a coherent procedure tomake decisions is especially troublesome

We have been brought up tobelieve that clear-headedlogical thinking is our onlysure way to solve problems.

when our intuition alone cannot help us todetermine which of several options is themost desirable, or the least objectionable,and neither logic nor intuition are of help.Therefore, we need a way to determinewhich objective outweighs another, bothin the near and long terms. Since we areconcerned with real-life problems we mustrecognize the necessity for trade-offs tobest serve the common interest. Therefore,this process should also allow for consen-sus building and compromise.

Individual knowledge and experienceare inadequate in making decisions con-cerning the welfare and quality of life for a

group. Participation and debate are neededboth among individuals and between thegroups affected. Here two aspects of groupdecision making have to be considered.The first is a rather minor complication,namely, the discussion and exchangewithin the group to reach some kind ofconsensus on the given problem. The sec-ond is of much greater difficulty. The ho-listic nature of the given problem necessi-tates that it be divided into smaller subject-matter areas within which different groupsof experts determine how each area affectsthe total problem. A large and complexproblem can rarely be decomposed simplyinto a number of smaller problems whosesolutions can be combined into an overallanswer. If this process is successful, onecan then reconstruct the initial questionand review the proposed solutions. A lastand often crucial disadvantage of manytraditional decision-making methods is thatthey require specialized expertise to designthe appropriate structure and then toembed the decision-making process in it,

A decision-making approach shouldhave the following characteristics;—Be simple in construct,—Be adaptable to both groups and indi-

viduals,—Be natural to our intuition and general

thinking,—^Encourage compromise and consensus

building, and—Not require inordinate specialization to

master and communicate [Saaty 1982].In addition, the details of the processesleading up to the decision-making processshould be easy to review.

At the core of the problems that ourmethod addresses is the need to assess the

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THE ANALYTIC HIERARCHY PROCESS

benefits, the costs, and the risks of the pro-posed solutions. We must answer suchquestions as the following: Which conse-quences weigh more heavily than others?Which aims are more important than oth-ers? What is likely to take place? Whatshould we plan for and how do we bring itabout? These and other questions demanda multicriteria logic. It has been demon-strated over and over by practitioners whouse the theory discussed in this paper thatmulticriteria logic gives different and oftenbetter answers to these questions than or-dinary logic and does it efficiently.

To make a decision one needs variouskinds of knowledge, information, andtechnical data. These concern— Details about the problem for which a

decision is needed,^The people or actors involved,—Their objectives and policies,—The influences affecting the outcomes,

and—The time horizons, scenarios, and con-

straints.The set of potential outcomes or alterna-

tives from which to choose are the essenceof decision making. In laying out theframework for making a decision, oneneeds to sort the elements into groupingsor clusters that have similar influences oreffects. One must also arrange them insome rational order to trace the outcome ofthese influences. Briefly, we see decisionmaking as a process that involves the fol-lowing steps:(1) Structure a problem with a model that

shows the problem's key elements andtheir relationships.

(2) Elicit judgments that reflect knowledge,feelings, or emotions.

(3) Represent those judgments with mean-ingful numbers.

(4) Use these numbers to calculate thepriorities of the elements of the hierar-chy.

(5) Synthesize these results to determinean overall outcome.

(6) Analyze sensitivity to changes in judg-ment [Saaty 1977].

The decision making process describedin this paper meets these criteria. I call itthe analytic hierarchy process (AHP). The

Deductive thinking is notnatural.

AHP is about breaking a problem downand then aggregating the solutions of allthe subproblems into a conclusion. It facili-tates decision making by organizing per-ceptions, feelings, judgments, and memo-ries into a framework that exhibits theforces that influence a decision. In the sim-ple and most common case, the forces arearranged from the more general and lesscontrollable to the more specific and con-trollable. The AHP is based on the innatehuman ability to make sound judgmentsabout small problems. It has been appliedin a variety of decisions and planning pro-jects in nearly 20 countries.

Here rationality is—Focusing on the goal of solving the

problem;—Knowing enough about a problem to de-

velop a complete structure of relationsand influences;

—Having enough knowledge and experi-ence and access to the knowledge andexperience of others to assess the prior-

November-December 1994 21

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ity of influence and dominance (impor-tance, preference, or likelihood to thegoal as appropriate) among the relationsin the structure;

—Allowing for differences in opinion withan ability to develop a best compromise.

How to Structure a HierarchyPerhaps the most creative part of deci-

sion making that has a significant effect onthe outcome is modeling the problem. Inthe AHP, a problem is structured as a hier-archy. This is then followed by a processof prioritization, which we describe in de-tail later. Prioritization involves elicitingjudgments in response to questions aboutthe dominance of one element over an-other when compared with respect to aproperty. The basic principle to follow increating this structure is always to see ifone can answer the following question:Can I compare the elements on a lowerlevel using some or all of the elements onthe next higher level as criteria or attri-butes of the lower level elements?

A useful way to proceed in structuring adecision is to come down from the goal asfar as one can by decomposing it into themost general and most easily controlledfactors. One can then go up from the alter-natives beginning with the simplest subcri-teria that they must satisfy and aggregatingthe subcriteria into generic higher level cri-teria until the levels of the two processesare linked in such a way as to make com-parison possible.

Here are some suggestions for an elabo-rate design of a hierarchy: (1) Identify theoverall goal. What are you trying to ac-complish? What is the main question? (2)Identify the subgoals of the overall goal. Ifrelevant, identify time horizons that affect

the decision. (3) Identify criteria that mustbe satisfied to fulfill the subgoals of theoverall goal. (4) Identify subcriteria undereach criterion. Note that criteria or subcri-teria may be specified in terms of ranges t)fvalues of parameters or in terms of verbalintensities such as high, medium, low. (5)Identify the actors involved. (6) Identifythe actors' goals. (7) Identify the actors'policies. (8) Identify options or outcomes.(9) For yes-no decisions, take the most pre-ferred outcome and compare the benefitsand costs of making the decision withthose of not making it. (10) Do a benefit/cost analysis using marginal values. Be- ,cause we are dealing with dominance hier-archies, ask which alternative yields thegreatest benefit; for costs, which alterna-tive costs the most, and for risks, which al-ternative is more risky.The Hospice Problem

Westmoreland County Hospital in West-ern Pennsylvania, like hospitals in manyother counties around the nation, has beenconcerned with the costs of the facilitiesand manpower involved in taking care ofterminally ill patients. Normally these pa-tients do not need as much medical atten-tion as do other patients. Those who bestutilize the limited resources in a hospitalare patients who require the medical atten-tion of its specialists and advanced tech-nology equipment—whose utilization de-pends on the demand of patients admittedinto the hospital. The terminally ill needmedical attention only episodically. Mostof the time such patients need psychologi-cal support. Such support is best given bythe patient's family, whose members areable to supply the love and care the pa-tients most need. For the mental health of

INTERFACES 24:6 22


the patient, home therapy is a benefit.From the medical standpoint, especiallyduring a crisis, the hospital provides agreater benefit. Most patients need thehelp of medical professionals only during acrisis. Some will also need equipment andsurgery. The planning association of thehospital wanted to develop alternativesand to choose the best one consideringvarious criteria from the standpoint of thepatient, the hospital, the community, andsociety at large. In this problem, we needto consider the costs and benefits of thedecision. Cost includes economic costs andall sorts of intangibles, such as inconve-nience and pain. Such disbenefits are notdirectly related to benefits as their mathe-matical inverses, because patients infinitelyprefer the benefits of good health to theseintangible disbenefits. To study the prob-lem, one needs to deal with benefits andwith costs separately.Approaching the Problem

1 met with representatives of the plan-ning association for several hours to decideon the best alternative. To make a decisionby considering benefits and costs, onemust first answer the question; In thisproblem, do the benefits justify the costs?If they do, then either the benefits are somuch more important than the costs thatthe decision is based simply on benefits, orthe two are so close in value that both thebenefits and the costs should be consid-ered. Then we use two hierarchies for thepurpose and make the choice by formingratios of the priorities of the alternatives(benefits ^,/costs c,) from them. One askswhich is most beneficial in the benefits hi-erarchy (Figure 1) and which is most costlyin the costs hierarchy (Figure 2). If the

benefits do not justify the costs, the costsalone determine the best alternative—thatwhich is the least costly. In this example,we decided that both benefits and costshad to be considered in separate hierar-chies. In a risk problem, a third hierarchyis used to determine the most desired alter-native with respect to all three: benefits,costs, and risks. In this problem, we as-sumed risk to be the same for all contin-gencies. Whereas for most decisions oneuses only a single hierarchy, we con-structed two hierarchies for the hospiceproblem, one for benefits or gains (whichmodel of hospice care yields the greaterbenefit) and one for costs or pains (whichmodel costs more).

The planning association thought theconcepts of benefits and costs were toogeneral to enable it to make a decision.Thus, the planners and I further subdi-vided each (benefits and costs) into de-tailed subcriteria to enable the group to de-velop alternatives and to evaluate the finerdistinctions the members perceived be-tween the three alternatives. The alterna-tives were to care for terminally ill patientsat the hospital, at home, or partly at thehospital and partly at home.

For each of the two hierarchies, benefitsand costs, the goal clearly had to be choos-ing the best hospice. We placed this goal atthe top of each hierarchy. Then the groupdiscussed and identified overall criteria foreach hierarchy; these criteria need not bethe same for the benefits as for the costs.

The two hierarchies are fairly clear andstraightforward in their description. Theydescend from the more genera! criteria inthe second level to secondary subcriteria inthe third level and then to tertiary subcri-


SAATY

QOM.

GENERALCRfTERIA

SECONDARYSUBCRITERIA

TERTIARYSUBCRITERIA

Choosing Best Hospice

Benefits Hierarchy

Physical0.16

Recipient Benefits0.64

Psycho-social0.44

Economic

nstrtutkmal Benefits0.26

Societal Benefits0.10

Psycho-social0.23

Economic0.03

Diroct care of- patientsD.D2

. Psiliativ« cars0.14

Reduced costs0.01

Improved*-productivitv

0.03

Vslunteersupport0.02

histvvoiking in' familiss0.06

Relief of post-- death distress

0.12

Emotional supporito family and patient0.21

Alleviation of guilt0.03

- - - (Each altemative model below is ccmnected to everv tertiary subcriterionl - - -

Publtcrty andpublic relations0.19Voluntootrecruitmant0.03

ProfessForialrecruitment andsupport0.06

Reduced lengthot stay0.006

Better udllzat'ion• of resources0,023

Increased finandalsupport fnam thecommunityaooi

Death as asocial issue0,02

Rehun^nization of. medical, professiona/

and heatm institutionsCOS

ALTERNATIVES MODEL 10.43

Unit of beds with Team•ivir>q home care (as in anospital or nursing homs)

MODEL 20.12

Mixed bad. contraciual home care(Parjy in hospital tor emergency

care and partiy in home when bettor- no nurses go to the ixiuse)

MODEL 30.45

Hospital ar>d home care sharecase marwgement (with visiting

nurses to tho i>ome; H extramelysick pationt goes to tha hospital)

Figure 1: To choose the best hospice plan, one constructs a hierarchy modeling the benefits tothe patient, to the institution, and to society. This is the benefits hierarchy of two separatehierarchies.

teria in tho fourth level on to the alterna-tives at the bottom or fifth level.

At the general criteria level, each of thehierarchies, benefits or costs, invoivedthree major interests. The decision shouldbenefit the recipient, the institution, andsociety as a whole, and their relative im-portance is the prime determinant as towhich outcome is more likely to be pre-ferred. We located these three elements onthe second level of the benefits hierarchy.As the decision would benefit each partydifferently and the importance of the bene-fits to each recipient affects the outcome,the group thought that it was important tospecify the types of benefit for the recipi-

ent and the institution. Recipients wantphysical, psycho-social and economic ben-efits, while the institution wants only psy-chosocial and economic benefits. We lo-cated these benefits in the third level of thehierarchy. Each of these in turn neededfurther decomposition into specific items interms of which the decision alternativescould be evaluated. For example, while therecipient measures economic benefits interms of reduced costs and improved pro-ductivity, the institution needed the morespecific measurements of reduced length ofstay, better utilization of resources, and in-creased financial support from the commu-nity. There was no reason to decompose

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GOAL

GENERALCRITERIA

SECONDARYSUBCRITERIA

TERTIARYSUBCRITERIA

Choosing Best Hospice

Costs Hierarchy

Communrty Costs0.14

Costs0.21

Societal Costs0.15

Capital0.03

Operating0.40

Educatton0.07

Bad debt0.15

Recrurtment0.06

Community0.01

Training staff0.06

Staff0.06

Volunteers0.01

(Each altemative model below is conrwcted to every node that has no further branch)

ALTERNATIVES MODEL 10.58

Unit of beds with teamgiving home care (as in ahospital or nursir^ horr>e)

MODEL 20.19

Mixed bed. contractual home care(Partly in !v>spftal for emergency

care and partly in horrve when better- no nurses go to the house)

MODEL 30.23

Hospital and home care sharecase menegement (with visiting

nurses to the home; (f uxtremelysick patient goes to the hosprtaO

Figure 2: To choose the best hospice plan, one constructs a hierarchy modeling the community,institutional, and societal costs. This is the costs hierarchy of two separate hierarchies.

the societal benefits into a third level sub-criteria, hence societal benefits connects di-rectly tu the fourth level. The group con-sidered three models for the decision alter-natives, and located them on the bottom orfifth level of the hierarchy: In Model 1, thehospital provided full care to the patients;In Model 2, the family cares for the patientat home, and the hospital provides onlyemergency treatment (no nurses go to thehouse); and in Model 3, the hospital andthe home share patient care (with visitingnurses going to the home).

In the costs hierarchy there were alsothree major interests in the second levelthat would incur costs or pains: commu-nity, institution, and society. In this deci-sion the costs incurred by the patient werenot included as a separate factor. Patientand family could be thought of as part ofthe community. We thought decomposi-

tion was necessary only for institutionalcosts. We included five such costs in thethird level: capital costs, operating costs,education costs, bad debt costs, and re-cruitment costs. Educational costs apply toeducating the community and training thestaff. Recruitment costs apply to staff andvolunteers. Since both the costs hierarchyand the benefits hierarchy concern thesame decision, they both have the same al-ternatives in their bottom levels, eventhough the costs hierarchy has fewerlevels.Judgments and Comparisons

A judgment or comparison is tbe numer-ical representation of a relationship be-tween two elements that share a commonparent. The set of all such judgments canbe represented in a square matrix in wbicbthe set of elements is compared with itself.Each judgment represents the dominance


SAATY

of an element in the column on the left

over an element in the row on top. It re-

flects the answers to two questions: Which

of the two elements is more important

with respect to a higher level criterion, and

how strongly, using the 1-9 scale shown in

Table 1 for the element on the left over the

element at the top of the matrix? If the ele-

ment on the left is less important than that

on the top of the matrix, we enter the re-

Intensity ofImportance Definition

ciprocal value in the corresponding posi-

tion in the matrix. It is important to note

that the lesser element is always used as

the unit and the greater one is estimated as

a multiple of that unit. From all the paired

comparisons we calculate the priorities and

exhibit them on the right of the matrix. For

a set of n elements in a matrix one needs

n{ii - l)/2 comparisons because there are

u I's on the diagonal for comparing ele-

Explanation

2, 4, 6, 8

Reciprocalsof above

Rationals

1.1-1,9

Equal Importance

Moderate importance

Strong importance

Very strt)ng or demonstratedimportance

Extreme importance

For compromise between theabove values

If activity i has one of the abovenonzero numbers assigned toit when compared withactivity /, then / has thereciprocal value whencompared with ('

Ratios arising from the scale

For tied activities

Two activities contribute equally tothe objective.

Experience and judgment slightlyfavor one activity t>ver another.

Experience and judgment stronglyfavor one activity over another.

An activity is favored very stronglyover another, its dominancedemonstrated in practice.

The evidence favoring one activityover another is of the highestpossible order of affirmation.

Sometimes one needs to interpolatea compromise judgmentnumerically because there is nogood word to describe it.

A comparison mandated by choosingthe smaller element as the unit toestimate the larger one as amultiple of that unit.

If consistency were to be forced byobtaining ;/ numerical values tospan the matrix.

When elements are close and nearlyindistinguishable; moderate is 1,3and extreme is 1,9,

Table 1: The fundamental scale is a scale of absolute numbers used to assign numerical valuesto judgments made by comparing two elements with the smaller element used as the unit andthe larger one assigned a value from this scale as a multiple of that unit.

INTERFACES 24:6 26


ments with themselves and of the remain-

ing judgments, half are reciprocals. Thus

we have {n^ — n)/2 judgments. In some

problems one may elicit only the minimum

of It — 1 judgments.

As usual with the AHP, in both the cost

and the benefits models, we compared the

criteria and subcriteria according to their

relative importance with respect to the par-

ent element in the adjacent upper level.

For example, in the first matrix of compari-

sons of the three benefits criteria with re-

spect to the goal of choosing the best hos-

pice alternative, recipient benefits are mod-

erately more important than institutional

benefits and are assigned the absolute

number 3 in the (1, 2) or first-row, second-

column position. Three signifies three

times more. The reciprocal value is auto-

matically entered in the (2, 1) position,

where institutional benefits on the left are

compared with recipient benefits at the

top. Similarly a 5, corresponding to strong

dominance or importance, is assigned to

recipient benefits over social benefits in the

(1, 3) position, and a 3, corresponding to

moderate dominance, is assigned to institu-

tional benefits over social benefits in the (2,

3) position with corresponding reciprocals

in the transpose positions of the matrix.

Judgments in a matrix may not be con-

sistent. In eliciting judgments, one makes

redundant comparisons to improve the va-

lidity of the answer, given that respon-

dents may be uncertain or may make poor

judgments in comparing some of the ele-

ments. Redundancy gives rise to multiple

comparisons of an element with other ele-

ments and hence to numerical inconsisten-

cies. For example, where we compare re-

cipient benefits with institutional benefits

and with societal benefits, we have the re-

spective judgments 3 and 5. Now \i x = 3y

and X = 5z then 3i/ = 52 or y ^ 5/32. If the

judges were consistent, institutional bene-

fits would be assigned the value 5/3 in-

stead of the 3 given in the matrix. Thus the

judgments are inconsistent. In fact, we are

not sure which judgments are more accu-

rate and which are the cause of the incon-

sistency. Inconsistency is inherent in the

judgment process. Inconsistency may be

considered a tolerable error in measure-

ment only when it is of a lower order of

magnitude (10 percent) than the actual

measurement itself; otherwise the inconsis-

tency would bias the result by a sizable er-

ror comparable to or exceeding the actual

measurement itself.

When the judgments are inconsistent,

the decision maker may not know where

the greatest inconsistency is. The AHP can

Choosing BestHospice

RecipientBenefit5

InstitutionalBenefits

SocialBenefits Priorities

Recipient BenefitsInstitutional BenefitsSocietal Benefits

C,R, - ,033

11/3

31

1/3

.64M.11

Table 2: The entries in this matrix respond to the question. Which criterion is more importantwith respect to choosing the best hospice alternative and how strongly?


SAATY

show one by one in sequential order whichjudgments are the most inconsistent, andalso suggests the value that best improvesconsistency. However, this recommenda-tion may not necessarily lead to a more ac-curate set of priorities that correspond tosome underlying preference of the decision

A decision-making approachshould be natural to ourintuition and general thinking.

makers. Greater consistency does not im-ply greater accuracy and one should goabout improving consistency {if one can begiven the available knowledge) by makingslight changes compatible with one s un-derstanding. If one cannot reach an accept-able level of consistency, one shouldgather more information or reexamine theframework of the hierarchy.

Under each matrix I have indicated aconsistency ratio (CR) comparing the in-consistency of the set of judgments in thatmatrix with what it would be if the judg-ments and the corresponding reciprocalswere taken at random from the scale. For a3-by-3 matrix this ratio should be aboutfive percent, for a 4-by-4 about eight per-cent, and for larger matrices, about 10 per-cent.

Priorities are numerical ranks measuredon a ratio scale. A ratio scale is a set ofpositive numbers whose ratios remain thesame if all the numbers are multiplied byan arbitrary positive number. An exampleis the scale used to measure weight. Theratio of these weights is the same inpounds and in kilograms. Here one scale isjust a constant multiple of the other. The

object of evaluation is to elicit judgmentsconcerning relative importance of the ele-ments of the hierarchy to create scales ofpriority of influence.

Because the benefits priorities of the al-ternatives at the bottom level belong to aratio scale and their costs priorities also be-long to a ratio scale, and since the productor quotient (but not the sum or the differ-ence) of two ratio scales is also a ratioscale, to derive the answer we divide thebenefits priority of each alternative by itscosts priority. We then choose the alterna-tive with the largest of these ratios. It isalso possible to allocate a resource propor-tionately among the alternatives.

I will explain how priorities are devel-oped from judgments and how they aresynthesized down the hierarchy by a pro-cess of weighting and adding to go fromlocal priorities derived from judgmentswith respect to a single criterion to globalpriorities derived from multiplication bythe priority of the criterion and overallpriorities derived by adding the globalpriorities of the same element. The localpriorities are listed on the right of eachmatrix. If the judgments are perfectly con-sistent, and hence CR = 0, we obtain thelocal priorities by adding the values ineach row and dividing by the sum of allthe judgments, or simply by normalizingthe judgments in any column, by dividingeach entry by the sum of the entries in thatcolumn. If the judgments are inconsistentbut have a tolerable level of inconsistency,we obtain the priorities by raising the ma-trix to large powers, which is known totake into consideration all intransitivitiesbetween the elements, such as those 1showed above between x, xj, and 2 [Saaty

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1994]. Again, we obtain the priorities from

this matrix by adding the judgment values

in each row and dividing by the sum of all

the judgments. To summarize, the global

priorities at the level immediately under

the goal are equal to the local priorities be-

cause the priority of the goal is equal to

one. The global priorities at the next level

are obtained by weighting the local priori-

ties of this level by the global priority at

the level immediately above and so on.

The overall priorities of the alternatives are

obtained by weighting the local priorities

by the global priorities of all the parent cri-

teria or subcriteria in terms of which they

are compared and then adding. (If an ele-

ment in a set is not comparable with the

others on some property and should be left

out, the local priorities can be augmented

by adding a zero in the appropriate posi-

tion.)

The process is repeated in all the matri-

ces by asking the appropriate dominance

or importance question. For example, for

the matrix comparing the subcriteria of the

parent criterion institutional benefits (Table

3), psychosocia! benefits are regarded as

very strongly more important than eco-

nomic benefits, and 7 is entered in the (1,

2) position and 1/7 in the (2, 1) position.

InstitutionalBenefits Psychosocial Economic Priorities

PsychosocialEconomics

C.R. = .000

11/7

.875

.125

Table 3: The entries in this matrix respond tothe question. Which subcriterion yields thegreater benefit with respect to institutionalbenefits and how strongly?

In comparing the three models for pa-

tient care, we asked members of the plan-

ning association which model they pre-

ferred with respect to each of the covering

or parent secondary criterion in level 3 or

with respect to the tertiary criteria in level

4. For example, for the subcriteritin direct

care (located on the left-most branch in the

benefits hierarchy), we obtained a matrix

of paired comparisons (Table 4) in which

Model 1 is preferred over Models 2 and 3

by 5 and 3 respectively, and Model 3 is

preferred by 3 over Model 2. The group

first made all the comparisons using se-

mantic terms for the fundamental scale

and then translated them to the corre-

sponding numbers.

For the costs hierarchy, I again illustrate

with three matrices. First the group com-

pared the three major cost criteria and pro-

vided judgments in response to the ques-

tion: which criterion is a more important

determinant of the cost of a hospice

model? Table 5 shows the judgments ob-

tained.

The group then compared the subcriteria

under institutional costs and obtained the

importance matrix shown in Table 6.

Finally we compared the three models to

find out which incurs the highest cost for

each criterion or subcriterion. Table 7

shows the results of comparing them with

respect to the costs of recruiting staff.

As shown in Table 8, we divided the bene-

fits priorities by the costs priorities for each

alternative to obtain the best alternative,

model 3, the one with the largest value for

the ratio.

Table 8 shows two ways or modes of

synthesizing the local priorities of the al-

ternatives using the global priorities of


SAATY

Direct Care ofPatient Model I Model I! Model III Priorities

Model I—Unit/Team

Model II—

Mixed/Home Care

Model HI—

CaseManagement

1

1/5

1/3

1/3

0.64

0.10

0.26

C.R. = .033

Table 4: The entries in this matrix respond to the question. Which model yields the greaterbenefit with respect to direct care of the patient and how strongly?

their parent criteria: The distributive mode

and the idea! mode. In the distributive

mode, the weights of the alternatives sum

to one. It is used when there is depen-

dence among the alternatives and a unit

priority is distributed among them. The

ideal mode is used to obtain the single best

alternative regardless of what other alter-

natives there are. In the ideal mode, the lo-

cal priorities of the alternatives are divided

by the largest value among them. This is

done for each criterion; for each criterion

one alternative becomes an ideal with

value one. In both modes, the local priori-

ties are weighted by the global priorities of

the parent criteria and synthesized and the

benefit-to-cost ratios formed. In this case,

both modes lead to the same outcome for

hospice, which is model 3. As we shall see

below, we need both modes to deal with

the effect of adding {or deleting) alterna-

tives on an already ranked set.

Model 3 has the largest ratio scale values

of benefits to costs in both the distributive

and ideal modes, and the hospital selected

it for treating terminal patients. This need

not always be the case. In this case, there

is dependence of the personnel resources

allocated to the three models because some

of these resources would be shifted based

on the decision. Therefore the distributive

mode is the appropriate method of synthe-

Choosing BestHospice (Costs) Community Institutional Societal Priorities

CommunityCosts

InstitutionalCosts

Societal Costs

1/5

11/5

0.14

0.710.14

C.R. = .000

Table 5: The entries in this matrix respond to the question, Which criterion is a greater deter-minant of cost with respect to the care method and how strongly?

INTERFACES 24:6 30


InstitutionalCosts

CapitalOperatingHdiitzation

Bad DebtRecruitment

C.R. = .08

Capital

17471

Operating

1/71

1/91/41/5

Education

1/49121

Bad Debt

1/74

1/21

1/3

Recruitment

15

131

Priorities

0.050,570,100.210.07

Table 6: The entries in this matrix respond to the question. Which criterion incurs greater in-stitutional costs and how strongly?

sis. If the alternatives were sufficiently dis-

tinct with no dependence in their defini-

tion, the ideal mode would be the way to

synthesize.

I also performed marginal analysis to de-

termine where the hospital should allocate

additional resources for the greatest mar-

ginal return. To perform marginal analysis,

I first ordered the alternatives by increas-

ing cost priorities and then formed the

benefit-to-cost ratios corresponding to the

smallest cost, followed by the ratios of the

differences of successive benefits to costs.

If this difference in benefits is negative, the

new alternative is dropped from considera-

tion and the process continued. The alter-

native with the largest marginal ratio is

then chosen. For the costs and correspond-

ing benefits from the synthesis rows in Ta-

ble 8 I obtained;

Costs:Benefits:

0.200.12

Marginal Ratios:

0.210.45

0.12

0.20

0.590.43

= 0.60

0.45 - 0.12

0.21 - 0.20

0.43 - 0.45

0.59 - 0.21

- 33

- -0.05

The third alternative is not a contender

for resources because its marginal return is

negative. The second alternative is best, in

fact, in addition to adopting the third

model, the hospital management chose the

Institutional Costsfor Recruiting Staff

Model I—Unit/Team

Model II—Mixed/Home Care

Model III—CaseManagement

Model 1

1

1/4

1/4

Model II

4

1

1

Model III

4

1

I

Priorities

.66

.17

.17

C.R. = .000

Table 7: The entries in this matrix respond to the question. Which model incurs greater costwith respect to institutional costs for recruiting staff and how strongly?


SAATY

Distributive Mode ell Mode

Bent-fits

Direct Care of PatientPalliative CareVolunlt'er SupportNetworking in FamiliesRelief of Post Death StressEmotional Support of Family

and PatientAlleviation of GuiltReduced Economic Costs for

PatientImproved ProductivityPublicitv and Public RelationsVolunteer RecruitmentProfessional Recruitment and

SupportReduced Length of StayBetter Utilization of ResourcesIncreased Monetary SupportDeath as a Social IssueRehumanization of Institutions

Synthesis

CostsCommunity CostsInstitutional Capital CostsInstitutional Operating Costsinstitutional Costs for Educating;

the CommunityInstitutional Costs tor Training

StaffInstitutional Bad DebtInstitutional Costs of Recruiting

StaffInstitutional Costs of Recruiting

VolunteersSocietal Costs

Synthesis

Benefit/Cost Ratio

Priorities

0,020.14002

0.060.12

0.210.03

0.010.030,!90.03

0.060.0060 0230.0010.020.08

0.140.030.40

0,01

0.060.15

0.03

0.010.15

Model 1

0.640.640.090.460.30

0.300,30

0,120.120.630.64

0.650,260.090,730.200.24

0 428

0 330.760.73

0.65

0.560.60

0.66

0.600.33

0.583

0.734

Model 2

0.100.100.170,220.08

0.080.08

0.650.270,080.10

0.230,100.220.080.200.14

0.121

0.330.090.08

0.24

0.320.20

0.17

0.200.33

0.192

0,630

Model 3

0.260.26

0.740.320.62

0.620,62

0,230.610.290.26

0.120.640-690,190.600.62

0.451

0.330.150.19

0.11

0.120.20

0.17

0.200,33

0,224

2.013

Model 1

1.0001.0000.1221.0000,484

0,4840.484

0.1850.1971.0001.000

1.0000.4060.1301.0000.3330.387

0.424

1.0001.0001.000

1,000

1.0001.000

1.000

1.0001.000

0.523

0.811

Model 2

0.1560.1560.2300.4780,129

0.1290.129

1.0000.4430.1270.156

0.3540.4060.1301.0000.3330,226

0.123

1,0000.1180.110

0.369

0.5710.333

0.258

0.3331.000

0.229

0,537

Model 3

0.4060.4061.0000.6961.000

1.0001.000

0.3541,0000.4600.406

0.1851.0001.0000,2601.0001.000

0.453

1.0000.1970.260

0.169

0.2140.333

0.258

0.3331.000

0.249

1,819

Table 8: The benefit/cost ratios of the three models given in the bottom row of the table areobtained for both the distributive and ideal modes. Here one multiplies each of the six col-umns of priorities of a model by the column of criteria weights on the left and adds to obtainthe synthesis of overall priorities, once for the benefits (top half of table) and once for the costs(bottom half of table) and forms the ratios of corresponding synthesis numbers to arrive at thebenefit/cost ratio (bottom row of table).

INTERFACES 24:6 32


second model of hospice care for furtherdevelopment.Absolute Measurement

Cognitive psychologists have recognizedfor some time that people are able to maketwo kinds of comparisons—absolute andrelative. In absolute comparisons, peoplecompare alfernatives with a standard intheir memory that they have developedthrough experience. In relative compari-sons, they compared alternatives in pairsaccording to a common attribute, as we didthroughout the hospice example.

People use absolute measurement(sometimes also called rating) to rank inde-pendent alternatives one at a time in termsof rating intensities for each of the criteria.An intensity is a range of variation of a cri-terion that enables one to distinguish thequality of an alternative for that criterion.An intensity may be expressed as a numer-ical range of values if the criterion is mea-surable or in qualitative terms. For exam-pic, if ranking students is the objective andone of the criteria on which they are to beranked is performance in mathematics, themathematics ratings might be: excellent,good, average, below average, poor; or, us-ing the usual school terminology, A, B, C,D, and F. Relative comparisons are firstused to set priorities on the ratings them-selves, if desired, one can fit a continuouscurve through the derived intensities. Thisconcept may go against our socialization.However, it is perfectly reasonable to askhow much an A is preferred to a B or to aC. Tbe judgment of bow much an A ispreferred to a B might be different underdifferent criteria. Perhaps for mathematicsan A is very strongly preferred to a B,while for physical education an A is only

moderately preferred to a B. So the end re-sult might be that the ratings are scaled dif-ferently. For example one could have thefollowing scale values for the ratings:

A

BC

D

E

MATH

0.500.300.150.040.01

PHYSICALEDUCATION

0.300.300.20,OM.0.10

The alternatives are then rated or tickedoff one at a time on the intensities.

I will illustrate absolute measurementwith an example. A firm evaluates its em-ployees for raises. The criteria are depend-ability, education, experience, and quality.Each criterion is subdivided into intensi-ties, standards, or subcriteria (Figure 3).The managers set priorities for the criteriaby comparing them in pairs. They thenpairwise compare the intensities accordingto priority with respect to tbeir parent cri-terion (as in Table 9) or with respect to asubcriterion if they are using a deeper hier-archy. The priorities of the intensities aredivided by the largest intensity for eachcriterion (second column of priorities inFigure 3). Table 9 shows a paired compari-son matrix of intensities with respect to de-pendability. The managers answer thequestion. Which intensity is more impor-tant and by how much with respect to de-pendability? The answer will depend onthe kind of job. "Outstanding" is muchmore preferred t)ver "above average" for asoldier guarding a nuclear missile sightthan for a waiter in a restaurant. Compari-son of intensities requires expert judgment


SAATY

GOAL

Dependability.4347

Outstanding(0.182) 1.000

Above Average(0.114) 0.626

Average(0.070) 0.385

Below Average(0.042) 0.231

Unsatisfactory(0.027) 0.148

Education.211A

- Doctorate(0.144) 1.000

Masters(0.071) 0.493

Bachelor(0.041) 0.285

H.S.(0.014) 0.097

Uneducated(0.007) 0.049

Experience.1755

Exceptional(0.086) 1.000

A Lot(0.050) 0.580

Average(0.023) 0.267

V A Little(0.010) 0.116

None(0.006) 0.070

Outstanding(0.056) 1.000

Above Average(0.029) 0.518

Average(0.018) 0.321

Beiow Average(0.006) 0.107

Unsatisfactory(0.003) 0.054

Figure 3: An evaluation hierarchy can be used lo rate employees.

in each problem and for each criteritm. Fi-nally, the managers rate each individual(Table 10) by assigning the intensity ratingthat applies to him or her under each crite-rion. The scores of these intensities areeach weighted by the priority of its crite-rion and summed to derive a total ratioscale score for the individual (shown onthe right of Table 10). These numbers be-

long to a ratio scale, and the managers cangive salary increases precisely in propor-tion to the ratios of these numbers. Adamsgets the highest score and Kesselman thelowest. This approach can be used when-ever it is possible to set priorities for inten-sities of criteria; people can usually do thiswhen they have sufficient experience witha given operation. This normative mode

Above BelowOutstanding Average Average Average Unsatisfactory Priorities

OutstandingAbove AverageAverageBelow AverageUnsatisfactory

CR, = 0,015

1.0

V21/31/4

1/5

2.01,01/2

1/31/4

3.02,01,01/21/3

4,03.02,01,01/2

5,04,03,02.01.0

0,4190,2630.1600.0970,062

Table 9: Ranking intensities: Which intensity is preferred most with respect to dependabilityand how strongly?

INTERFACES 24:6 34


1,2,3,4,5,6.7.

Adams, V,Becker, L.Hayat, F.Kesseiman, S.O'Shea, K,Peters, T,Tobias, K.

Dependability0.4347

OutstandingAverageAverageAbove AverageAverageAverageAbove Average

Education0.2774

BachelorBachelorMastersH.S.DoctorateDoctorateBachelor

Experience0,1775

A LittleA LittleA LotNoneA LotA LotAverage

Quality 0,1123

OutstandingOutstandingBelow AverageAbove AverageAbove AverageAverageAbove Average

Total

0.6460,3790,4180,3690,6050.5830,456

Table 10: Ranking alternatives. The priorities of the intensities for each criterion are dividedby the largest one and multiplied by the priority of the criterion. Each alternative is rated oneach criterion by assigning the appropriate intensity. The weighted intensities are added toyield the total on the right.

requires that alternatives be rated one byone without regard to how many theremay be and how high or low any of themrates on prior standards. Some corpora-tions have insisted that they no longertrust the normative standards of their ex-perts and that they prefer to make pairedcomparisons of their alternatives. Still,wht-n there is wide agreement on stan-dards, the absolute mode saves time in rat-ing a large number of alternatives.Homogeneity and Clustering

Think of the following situation: weneed to determine the relative size of ablueberry and a watermelon. Here, weneed a range greater than 1-9. Humanbeings have difficulty establishing appro-priate relationships when the ratios get be-yond 9. To resolve this human difficulty,we can use a method in which we clusterdifferent elements so we can rate themwithin a cluster and then rate them acrossthe clusters. We need to add other fruits tomake the comparison possible and thenform groups of comparable fruits. In thefirst group we include the blueberry, agrape, and a plum. In the second group weinclude the same plum, an apple, and a

grapefruit. In the third group we includethe same grapefruit, a melon, and the wa-termelon. The AHP requires reciprocalcomparisons of homogeneous elementswhose ratios do not differ by much on aproperty, hence the absolute scale 1-9;when the ratios are larger, one must clusterthe elements in different groups and use acommon element (pivot) that is the largestin one cluster and the smallest element inthe next cluster of the next higher order ofmagnitude. The weights of the elements inthe second group are divided by the prior-ity of the pivot in that group and thenmultiplied by the priority of the same pivotelement (whose value is generally differ-ent) from the first group, making themcomparable with the first group. The pro-cess is then continued. The AHP softwareprogram Expert Choice performs thesefunctions for the user. The reason for usingclusters of a few elements is to ensuregreater stability of the priorities in face ofinconsistent judgments. Comparing morethan two elements allows for redundancyand hence also for greater validity of real-world information. The AHP often usesseven elements and puts them in clusters if


SAATY

there are more. (Elaborate mathematicalderivations are given in the AHP to showthat the number of elements comparedshould not be too large in order to obtainpriorities with admissible consistency.)Problems with Analytic DecisionMaking

At this point you may wonder why wehave three different modes for establishingpriorities, the absolute measurement modeand the distributive and ideal modes ofrelative measurement. Isn t one enough?Let me explain why we need more thanone mode.

A major reason for having more thanone mode is concerned with this question.What happens to the synthesized ranks ofalternatives when new ones are added orold ones deleted? With consistent judg-ments, the original relative rank order can-not change under any single criterion, butit can under several criteria.

Assume that an individual has expressedpreference among a set of alternatives, andthat as a result, he or she has developed aranking for them. Can and should that in-dividual's preferences and the resultingrank order of the alternatives be affected ifalternatives are added to the set or deletedfrom it and if no criteria are added or de-leted, which would affect the weights ofthe old criteria? What if the added alterna-tives are copies or near copies of one or ofseveral of the original alternatives andtheir number is large? Rank reversal is anunpleasant property if it is caused by theaddition of truly irrelevant alternatives,However, the addition of alternatives mayjust reflect human nature: the straw thatbroke the camel's back was considered ir-relevant along with all those that went be-

fore it. Mathematically, the number andquality of newly added alternatives areknown to affect preference among theoriginal alternatives. Most people, unaidedby theory and computation, make each de-cision separately, and they are not veryconcerned with rank reversal unless theyare forced for some reason to refer to theirearlier conclusions. I think it is essentialto understand and deal with this phe-nomenon.An Example of Rank Reversal

Two products A and B are evaluated ac-cording to two equally important attributesP and Q as in the following matrices:

p

A

B

A

1

1/5

B

51

Priorities

0.830.17

Q

AB

B

1/31

Priorities

0.250.75

We obtain the following priorities: W= 0.542, Wj, - 0.458, and A is preferred toB.

A third product C is then introducedand compared with A and B as follows:

p

A

B

C

A

1

1/51

B

5

1

5

C

1

1/51

Priorities

0.4550.0900.455

Q

A

BC

A

1

31/2

B

1/3

1

1/6

C

2

6

1

Priorities

0.2220.6660.111

INTERFACES 24:6


Synthesis yields W^ ^ 0.338, W= 0.379, and W^ - 0.283, Here B is pre-ferred to A and there is rank reversal.

For a decision theory to have a lastingvalue, it must consider how people makedecisions naturally and assist them in orga-nizing their thinking to improve their deci-sions in that natural direction. Its assump-tions should be tied to evolution and not topresent day determinism. This is the fun-damental concept on which the AHP isbased. It was developed as a result of a de-cade of unsuccessful attempts to use nor-mative theories, with the assistance ofsome of the world's best minds, to dealwith negotiation and trade-off in the stra-tegic political and diplomatic arena at theArms Control and Disarmament Agency inthe Department of State, In the early1970s, I asked the question, how do ordi-nary people process information in theirminds in attempting to make a decisionand how do they express the strength oftheir judgments? The answer to this ques-tion led me to consider hierarchies andnetworks, paired comparisons, ratio scales,homogeneity and consistency, priorities,ranking, and the AHP.Resolution of the Rank PreservationIssue

Early developers of utility theory axiom-atically ruled that introducing alternatives,particularly irrelevant" ones, should notcause rank reversal [Luce and Raiffa 1957],A theory that rates alternatives one at atime, as in the absolute measurement sal-ary-raise example given above, assumesthe existence of past standards establishedby experts for every decision problem andwould thus assume that every decision canbe made by rating each alternative by itself

without regard to any other alternative andwould inexorably preserve rank. But if paststandards are inapplicable to new prob-lems and if experts are not sufficiently fa-miliar with the domain of a decision to es-tablish standards and the environmentchanges rapidly, an insistence on makingdecisions based on standards will onlyforce the organization to shift its effortsfrom solving the problem to updating itsstandards. For example, practitioners haveimprovised many techniques to relate stan-dards defined by utility functions in thecontext of a specific decision problem.Connecting theory to practice is importantbut often difficult. We need to distinguishbetween fixing the axioms of a decisiontheory to be followed strictly in all situa-tions and learning and revising in the pro-cess of making a decision. The rank preser-vation axioms of utility theory and theAHP parallel the axioms of the classicalfrequentist method of statistics andBayesian theory. Bayesian theory violatesthe axioms of statistics in updating predic-tion by including information from a pre-vious outcome, a process known as learn-ing. When we integrate learning with deci-sion making, we question some of thestatic basic axioms of utility theory.

The AHP avoids this kind of formula-tion and deals directly with paired compar-isons of the priority of importance, prefer-ence, or likelihood (probability) of pairs ofelements in terms of a common attribute orcriterion represented in the decision hierar-chy. We believe that this is the natural {butrefined) method that people followed inmaking decisions long before the develop-ment of utility functions and before theAHP was formally developed.


SAATY

The major objection raised against theAHP by practitioners of utility theory hasbeen this issue of rank reversal. The issuesof rank reversal and preference reversalhave been much debated in the literatureas a problem of utility theory [Grether andPlott 1979; Hershey and Schoemaker 1980;Pommerehne, Schneider, and Zweifel1982; Saaty 1994, Chapter 5; Tversky andSimonson 1993; Tversky, Slovic, andKahneman 1990).

Regularity is a condition of choice theorythat has to do with rank preservation. R.Corbin and A. Marley [1974] provide autility theory example of rank reversal. It"concerns a lady in a small town, whowishes to buy a hat. She enters the onlyhat store in town, and finds two hats, Aand B, that she likes equally well, and somight be considered equally likely to buy.However, now suppose that the sales clerkdiscovers a third hat, C, identical to B,Then the lady may well choose hat A forsure (rather than risk the possibility ofseeing someone wearing a hat just likehers), a result that contradicts regularity,"Utility theory has no clear analytical an-swer to this paradox nor to famous exam-ples having to do with phantom alterna-tives and with decoy alternatives that arisein the field of marketing [Saaty 1994].

Because of such examples, it is clear thatone cannot simply use one procedure forevery decision problem because that proce-dure would either preserve or not preserverank. Nor can one introduce new criteriathat indicate the dependence of the alter-natives on information from each new al-ternative that is added. In the AHP, this is-sue has been resolved by adding the idealmode to the normalization mode in relative

measurement. The ideal mode prevents analternative that is rated low or "irrelevant"on all the criteria from affecting the rankof higher rated alternatives.

In the AHP, we have one way to allowrank to change, (1) below, and two ways

The essence of the AHP is theuse of ratio scales in elaboratestructures to assess complexproblems.

to preserve rank, (2) and (3) below.(1) We can allow rank to reverse by usingthe distributive mode of the relative mea-surement approach of the AHP.(2) We can preserve rank in the case of ir-relevant alternatives by using the idealmode of the AHP relative measurementapproach.(3) We can preserve rank absolutely by us-ing the absolute measurement mode of theAHP.

As a recap, in relative measurement, weuse normalization by dividing by the sumof the priorities of the alternatives to de-fine the distributive mode. In this mode,we distribute the unit value assigned to thegoal of a decision proportionately amongthe alternatives through normalization.When we add a new alternative, it takes itsshare of the unit from the previously exist-ing alternatives. This mode allows for rankreversal because dependence exists amongthe alternatives, which is attributable tothe number of alternatives and to theirmeasurements values and which is ac-counted for through normalization. For ex-ample, multiple copies of an alternative

INTERFACES 24:6 38


can affect preference for that alternative insome decisions. We need to account forsuch dependence in allocating resources, invoting and in distributing resources amongthe alternatives.

In the ideal mode, we would simplycompare a new alternative with the ideal(with the weight of one), and it would fallbelow or above the ideal and could itselfbecome the ideal. As a result, an alterna-tive that falls far below the ideal on everycriterion cannot affect the rank of the bestchosen alternative. Using absolute mea-surement, we rate alternatives one at atime with respect to an ideal intensity oneach criterion, and this process cannot giverise to rank reversal,

I conducted an experiment involving64,000 hierarchies with priorities assignedrandomly to criteria and to alternatives totest the number of times the best choiceobtained by the distributive and idealmodes coincided with each other. It turnsout that the two methods yield the sametop alternative 92 percent of the time. I ob-tained similar results for the top two alter-natives [Saaty and Vargas 1993a].Decision Making in ComplexEnvironments

The AHP makes group decision makingpossible by aggregating judgments in away that satisfies the reciprocal relation incomparing two elements. It then takes thegeometric mean of the judgments. Whenthe group consists of experts, each worksout his or her own hierarchy, and the AHPcombines the outcomes by the geometricmean. If the experts are ranked accordingto their expertise in a separate hierarchy,we can raise their individual evaluations tothe power of their importance or expertise

priorities before taking the geometricmean. We have also used special question-naires to gather data in the AHP.

Practitioners have developed multicri-teria decision approaches largely aroundtechniques for generating scales for alter-natives. But I believe that making decisionsin real life situations depends on the depthand sophistication of the structures deci-sion makers use to represent a decision orprediction problem rather than simply onmanipulations^although they are also im-portant. It seems to me that decision mak-ing and prediction must go hand in hand ifa decision is to survive the test of theforces it may encounter [Saaty and Vargas1991]. If one understands the lastmg valueof a best decision, one will want to con-sider feedback structures with possible de-pendencies among all the elements. Thesewould require iterations with feedback todetermine the best outcome and the mostlikely to survive. I believe that ratio scalesare mathematically compelling for this pro-cess. The AHP is increasingly used for de-cisions with interdependencies (the hier-archic examples I have described are sim-ple special cases of such decisions). Idescribe applications of feedback in Chap-ter 8 of Saaty [1994] and in a book I amcurrently writing on applications of feed-back, I and my colleague Luis Vargasused the supermatrix feedback approach ofthe AHP in October 1992 to show that thewell-known Bayes theorem used in deci-sion making follows from feedback in theAHP, Furthermore, we have since shownthrough examples that some decisions withinterdependence can be treated by theAHP but not Bayes theorem [Saaty andVargas 1993bl.


SAATY

The essence of the AHP is the use of ra-tio scales in elaborate structures to assesscomplex problems. Ratio scales are thefundamental tool of the mind that peopleuse to understand magnitudes. The AHPwell fits the words of Thomas Paine in hisCommon Sense, "The more simple any-thing is, the less liable it is to be disorderedand the easier repaired when disordered."

In August 1993, Sarah Becker compileda list of what are now more than 1,000 pa-pers, books, reports, and dissertations writ-ten on the subject of AHP, an early ver-sion of which is included as a bibliographyin my 1994 book [Saaty 1994].The Benefits of Analytic DecisionMaking

Many excellent decision makers do notrely on a theory to make their decisions.Are their good decisions accidental, or arethere implicit logical principles that guidethe mind in the process of making a deci-sion, and are these principles complete andconsistent? I believe that there are suchprinciples, and that in thoughtful people,they work as formalized and described inthe analytic hierarchy process. Still aca-demics differ about how people shouldand should not make decisions. Experi-ments with people have shown that whatpeople do differs from the theoretical andnormative considerations the experts con-sider important. This may lead one to be-lieve that analytical decision making is oflittle value. But our experience and that ofmany others indicate the opposite.

Analytic decision making is of tremen-dous value, but it must be simple and ac-cessible to the lay user, and must have sci-entific justification of the highest order.Here are a few ideas about the benefits of

the descriptive analytical approach. First isthe morphological way of thoroughlymodeling the decision, inducing people tomake explicit their tacit knowledge. Thisleads people to organize and harmonizetheir different feelings and understanding.An agreed upon structure provides groundfor a complete multisided debate. Second,particularly in the framework of hierar-chies and feedback systems, the processpermits decision makers to use judgmentsand observations to surmise relations andstrengths of relations in the flow of inter-acting forces moving from the general tothe particular and to make predictions ofmost likely outcomes. Third, people areable to incorporate and trade off valuesand influences with greater accuracy of un-derstanding than they can using languagealone. Fourth, people are able to includejudgments that result from intuition andemotion as well as those that result fromlogic. Reasoning takes a long time to learn,and it is not a skill common to all people.By representing the strength of judgmentsnumerically and agreeing on a value, deci-sion-making groups do not need to partici-pate in prolonged argument. Finally, a for-mal approach allows people to make grad-ual and more thorough revisions and tocombine the conclusions of different peo-ple studying the same problem in differentplaces [Saaty and Alexander 1989]. Onecan also use such an approach to piece to-gether partial analyses of the componentsof a bigger problem, or to decompose alarger problem into its constituent parts.This is not an exhaustive list of the uses ofthe AHP, However, to deal with complex-ity we need rationality, and that is bestmanifested in the analytical approach.

INTERFACES 24:6 40


APPENDIXThf AHP has four axioms: (1) reciprocal

judgments, (2) homogeneous elements, (3)hierarchic or feedback dependent struc-ture, and (4) rank order expectations [Saaty1986].

Assume that one is given n stones, Ax,. . . , A,,, with known weights U',, . . . , U',,,respectively, and suppose that a matrix ofpairwise ratios is formed whose rows givethe ratios of the weights of each stone withrespect to all others. Thus one has theequation;

A.

Aw ^ \

A.

= mL\

where A has been multiplied on the rightby the vector of weights it'. The result ofthis multiplication is mv. Thus, to recoverthe scale from the matrix of ratios, onemust solve the problem ATV = nw or {A- n!)w = 0. This is a system of homoge-neous linear equations. It has a nontrivialsolution if and only if the determinant of A- nl vanishes, that is, n is an eigenvalue ofA. Now A has unit rank since every row isa constant multiple of the first row. Thusall its eigenvalues except one are zero. Thesum of the eigenvalues of a matrix is equalto its trace, the sum of its diagonal ele-ments, and in this case the trace of A isequal to n. Thus n is an eigenvalue of A,and one has a nontrivial solution. The so-lution consists of positive entries and isunique to within a multiplicative constant.

To make lo unique, one can normalize itsentries by dividing by their sum. Thus,

given the comparison matrix, one can re-cover the scale. In this case, the solution isany column of A normalized. Notice thatin A the reciprocal property a,, = I/a,,holds; thus, also fl,, = 1. Another propertyof A is that it is consistent: its entries sat-isfy the condition a,^ - a,i/a,,. Thus the en-tire matrix can be constructed from a set ofn elements which form a chain across therows and columns.

In the general case, the precise value ofIV Jw, cannot be given, but instead only anestimate of it as a judgment. For the mo-ment, consider an estimate of these valuesby an expert who is assumed to makesmall perturbations of the coefficients. Thisimplies small perturbations of the eigen-values. The problem now becomes A'lv'^ A,,,uvU'' where X,,,,j, is the largest eigen-value of A'. To simplify the notation, weshall continue to write Aw = \,,,a^w, whereA is the matrix of pairwise comparisons.The problem now is how good is the esti-mate of w. Notice that if w is obtained bysolving this problem, the matrix whose en-tries are wjw, is a consistent matrix. It is aconsistent estimate of the matrix A. A itselfneed not be consistent. In fact, the entriesof A need not even be transitive; that is. A]may be preferred to Aj and A2 to A^ but A^may be preferred to A,. What we wouldlike is a measure of the error due to incon-sistency. It turns out that A is consistent ifand only if X,,,«, ^ n and that we alwayshave \»,,,, > n.

Since small changes in a,, imply a smallchange in A,,,,n, the deviation of the latterfrom n is a deviation from consistency andcan be represented by (X,,,,,, - »))/(" ~~ I)-which is called the consistency index (C.I.).When the consistency has been calculated,the result is compared with those of thesame index of a randomly generated recip-rocal matrix from the scale 1 to 9, with re-ciprocals forced. This index is called therandom index (R.I.). Table 11 gives the or-der of the matrix (first row) and the aver-age R.I. (second row).


SAATY

1 2 10

Random Consistency Index (R.I.) 0 0 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49

Table 11: The order of the matrix (first row) and the average R. I. (second row).

The ratio of C.I. to the average R.I. forthe same order matrix is called the consis-tciJC]/ ratio {C.R). A consistency ratio of0.10 or less is positive evidence for in-formed judgment.

The relations fl,, - \/a,, and a,, = 1 arepreserved in these matrices to improveconsistency. The reason for this is that ifstone #1 is estimated to be k times heavierthan stone #2, one should require thatstone #2 be estimated to be 1/k times theweight of the first. If the consistency ratiois significantly small, the estimates are ac-cepted; otherwise, an attempt is made toimprove consistency by obtaining addi-tional information. What contributes to theconsistency of a judgment are (1) the ho-mogeneity of the elements in a group, thatis, not comparing a grain of sand with amountain; (2) the sparseness of elements inthe group, because an individual cannot

Figure 4: Five figures drawn with appropriatesize of area. The object is to compare them inpairs to reproduce their relative weights.

hold in mind simultaneously the relationsof many more than a few objects; and (3)the knowledge and care of the decisionmaker about the problem under study.

Figure 4 shows five areas to which wecan apply the paired comparison process ina matrix and use the 1-9 scale to test thevalidity of the procedure. We can approxi-mate the priorities in the matrix by assum-ing that it is consistent. We normalize eachcolumn and then take the average of thecorresponding entries in the columns.

The actual relative values of these areasare A - 0.47, B - 0.05, C = 0.24, D - 0.14,and E = 0.09 with which the answer maybe compared. By comparing more than twoalternatives in a decision problem, one isable to obtain better values for the derivedscale because of redundancy in the com-parisons, which helps improve the overallaccuracy of the judgments.

ReferencesCorbin, R. and Marley, A. A. J. 1974, "Random

utility models with equality: An apparent, butnot actual, generalization of random utilitymodels," journal of Mathematical Psi/chalog]/,Vol. 11, No. 3, pp. 274-293.

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Pommerehne, W. W.; Schneider, F.; andZweifel, P. 1982, "Economic theory of choiceand the preference reversal phenomenon: Areexammation," The American Economic Re-

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