What Sis the Analytic Hierarchy Process

8/11/2019 What Sis the Analytic Hierarchy Process

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WH T

IS

THE ANALYTIC

HIERARCHY PROCESS?

Thomas L

Saaty

Universi ty

of Pittsburgh

1.

Introduction

In our everyday l i f e we

must

constantly make choices

concerning

what tasks to do or not to do when to do

them and

whether to do them a t a l l .

Many

problems

such

as

buying the most cost effective home

computer

expansion

a car or house; choosing a

school

or a

career investing money,

deciding

on a

vacation

place or even

voting for

a poli t ica l

candidate

are

common

everyday

problems

in

personal

decision

making.

Other

problems

can

occur

in

business decisions such

as

buying equipment

marketing a

product

assigning management

personnel

deciding on

inventory

levels

or

the best source

for borrowing

funds.

There are

also

local

and national

governmental

decisions

l ike

whether

to act

or

not

to

act on an issue

such

as

building

a

bridge or

a

hospital how

to

allocate

funds

within

a department

or how

to

vote on a c i ty

council

issue.

All these are essent ia l ly problems

of

choice. In

addition

they

are

complex

problems

of

choice.

They also involve making a

logical

decision. The human mind is

not capable

of

considering

a l l

the factors

and their effects

simultaneously.

People

solve

these

problems

today

with

seat

of

the

pants

judgments

or

by

mathematical

models

based

on

assumptions

not readily

verif iable

that draw

conclusions that

may

not be clearly useful .

Typically

individuals make

these choices on

a

reactive

and

frequently unplanned basis with l i t t l e forethought of how the

decisions

t ie

together to

form one integrated plan. This whole

process of deciding what when,

and whether

to

do

cer tain tasks

is

the

crux

of th is process

of

set t ing pr ior i t ies .

The

pr ior i t ies may

be

long-term or short-term simple or

complex.

NATO AS Series,

Vol

F48

Mathematical Models for Decision Support

Edited

by

G. Mitra

Springer-Verlag Berlin Heidelberg 1988


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11

some organiza t ion s

needea. This

organiza t ion can be obta ined

through a

hiera rch ica l

representa t ion .

But

tha t s

not

a l l .

Judgments and

measurements have to

be included and

in tegrated.

A procedure which sa t i s f i e s these requirements s the Analyt ic

Hierarchy Process (AHP).

The mathematical

th inking

behind the

process s

based

on

l inear algebra. Unti l recent ly i t s

connect ion

to dec is ion making was not

adequately

studied. With

the

in t roduct ion

of home computers bas ic l inear algebra

problems can

be solved eas i ly

so

tha t

t s now possible to use

the

HP

on personal computers. The HP di f fe r s

from

convent ional

dec is ion analysis techniques by requi r ing

tha t

i t s

numerical approach to pr io r i t i e s

conform

with s c i en t i f i c

measurement.

By t h i s we

mean

tha t

i f

appropriate s c i en t i f i c

experiments are

car r ied out using the

scale of

the

HP for

paired comparisons, the sca le derived

from

these

should

yie ld

re la t ive

values

tha t are

the

same or close to what the

physical

law

underlying

the experiment d ic ta te s according to known

measurements

in

tha t

area.

The

Analyt ic Hierarchy Process

i s

of

pa r t i cu la r value

when

subjec t ive

abs t rac t or

nonquantif iable

c r i t e r i a

are involved in the decision.

With the HP we

have

a means

of

iden t i fy ing the

re levant

facts

and

the in te r re la t ionsh ips tha t ex i s t . Logic

plays

a role but

not

to the extent

of breaking down a

complex

problem and

determining re la t ionsh ips

through a

deduct ive

process.

For

example,

logic says tha t

i f

I prefer A

over Band

B

over

C

then I must prefer A over C.

This

s not

necessar i ly

so

(consider the example of soccer team A bea t ing soccer

team

B

soccer team B beat ing soccer team C and then C turning

around

and

beat ing A

and

not only

tha t

the

odds makers

may

well

have

given the

advantage to

C pr ior to the contes t based on the

overa l l records

of

a l l

three

teams) and the

HP allows

such

incons i s t enc ies in i t s framework.

A bas ic premise of the HP s

i t s

re l i ance on the

concept

tha t

much of what we consider

to

be knowledge ac tua l ly

perta ins

to

our in s t inc t ive sense of the way th ings

r ea l ly

are . This would

seem to agree

with

Descartes ' pos i t ion tha t the mind i t s e l f i s

the f i r s t knowable

pr inc ip le .

The

HP

therefore takes as i t s

premise

the idea tha t t

s

our concept ion of

rea l i ty

tha t

i s

cruc ia l and not our

convent ional

representat ions of

tha t

r e a l i t y by such means

as

s t a t i s t i c s e tc . With the

HP

t i s

possible

for

prac t i t ione rs to

ass ign

numerical values

to

what

are

essen t i a l ly

abs t rac t concepts and then deduce from these

values decis ions to apply in the global framework.

The

Analyt ic

Hierarchy Process s a dec is ion making model

tha t

a ids us in making decis ions

in

our complex

world.

I t i s a

th ree pa r t

process which

includes iden t i fy ing and

organizing

deci s ion

objec t ives

c r i t e r i a

cons t ra in t s and a l te rna t ives

in to a hierarchy; eva lua t ing pai rwise comparisons between the

re levant elements a t

each

level of the hierarchy; and the

synthesis

using the so lu t ion algori thm of the resu l ts of the

pai rwise

comparisons over

a l l

the

l eve ls .

Further,

the

algori thm

r esu l t

gives the

re la t ive

importance

of

a l te rna t ive

courses

of

act ion.


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To

summarize, the HP process has

e igh t

major uses . t

allows

the

dec is ion maker

to :

1) design a form tha t

represent s

a

complex

problem; 2)

measure

pr i o r i t i e s

and

choose

among

a l te rna t ives ; 3) measure cons is tency; 4) pred ic t ; 5)

formulate

a

cos t /benef i t analys i s ;

6) design forward/backward planning;

7) analyze conf l i c t

reso lu t ion ; 8) develop

resource a l loca t ion

from the cos t /benef i t analysis .

For

the

pairwise comparison

judgments

a sca le

of

1 to 9 i s

ut i l i zed . This i s not simply an assignment

of

numbers. The

re la t ive in tens i ty

of

the elements being

compared

with

respect

to a pa r t i cu la r proper ty

becomes

c r i t i c a l . The numbers

indicate

the s t rength of

preference for

one over the

other .

Ideal ly when

the pairwise comparison

process i s begun,

numerical

values should not be assigned, ra ther the comparat ive

st rengths

should be verbal ized

as indicated

in

the

t ab le below

of

the

fundamental

sca le

of re la t ive

importance

tha t i s the

basis

for

the

HP

judgments.

2. The Scale

Pairwise

comparisons are fundamental

in

the

use

of the AHP

We

must

f i r s t

es tab l ish pr io r i t i e s for the main c r i t e r i a by

judging them in pa i r s for t he i r re la t ive importance, thus

generat ing a

pairwise

comparison matrix .

Judgments

used to

make the comparisons are

represented

by

numbers

taken from the

fundamental

sca le

below. The number

of

judgments needed fQr a

par t icu la r matrix of order n, the

number

of elements being

compared, i s n n-1) /2

because

it

i s rec iproca l

and the diagonal

elements

are

equal to

uni ty .

There

are condi t ions

under

which

it

i s

poss ible to

use

fewer judgments and still

obtain

accurate

re su l t s . The comparisons are made by asking how much more

important the element on the

l e f t

of the matr ix i s perceived

to

be with

respect

to the proper ty in ques t ion than the

element

on

the top of the matr ix . I t i s important

to

formulate

the r igh t

quest ion

to

get the r igh t answer.

The

scale

given below

can be val idated for i t s super ior i ty over

any

other

assignment

of

numbers to judgments by taking one

of

the

two i l l u s t r a t i ve matr ices given

below

and inser t ing ins tead

of the numbers given numbers from another

scale

tha t i s not

simply

a small

per turba t ion

or

constant

mul t ip le of

our

sca le .

I t

wi l l

be found tha t the resu l t ing derived

sca le

i s markedly

d i f fe ren t and does not correspond to the known

resul t .


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112

T BLE

1

THE FUND MENT L SC LE

In t ens i ty

of

Importance

on an

Absolute

Scale Def in i t ion

1 Equal

importance

3 Moderate importance

of

one over another .

5 Essent ia l or

s t rong

importance

7

Very

s t rong

impor-

tance.

9

Extreme importance

2 4 6 8

In termedia te values

between

the

two ad-

j acent judgments

Reciprocals

Rat ionals

I f

a c t i v i t y i has

one of the

above

numbers

ass igned

to

it when

compared

with

a c t i v i t y j

then

j has

the

rec iproca l value

when

compared

with

i .

Rat ios ar i s ing

from

the sca le .

Explanat ion

Two

ac t i v i t i e s

cont r ibu te equal ly to

the objec t ive .

Experience

and judgment

s t rongly favor one

a c t i v i t y over another .

Experience

and

judgment

s t rongly favor one

a c t i v i t y over

another

n a c t i v i t y i s s t rong ly

favored and

i t s

dominance demonstrated

in prac t i ce .

The evidence favoring

one a c t i v i t y over

another

i s

of

the

h ighes t poss ib le order

of

aff i rmat ion.

When

compromise i s

needed

I f

consis tency were to

be

forced

by

obta in ing n

numerical

values to span

the

matr ix .

When the

elements being

compared are

c loser

t oge the r than

ind ica ted

by

the sca l e

one

can use the sca le

1 . 1

1.2

1.9.

I f

still f i ne r one can use

the

appropria te percentage

ref inement


5/13


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114

these express ions . Fina l ly , numerical sca le values must in tu rn

be

assoc ia ted

with these

verba l express ions t ha t

lead

to

meaningful outcomes and pa r t i cu l a r l y in

known s i tua t ions

can be

te s ted for

t he i r

accuracy. Small

changes

in the

words

(or the

numbers)

should lead to small

changes in

the derived

answer.

Fina l ly we use consistency arguments along

with the well-known

work

of Fechner in psychophysics

to der ive

and

subs t an t i a t e

the

sca le and i t s range.

In

1860 Fechner

increasing

s t imu l i .

considered a

sequence

of j us t

He denotes the

f i r s t

one by s .

o

j us t not iceable st imulus by

s = s + t,s = s

1 0 0 0

s ( l+r)

o

having used

Weber s law.

Simi la r ly

2 2

s = s

t,s

= s l+r) = s ( l+r)

=s

2 1 1 1 0 0

In general

n

s = s = s (n =

0,1 ,2 ,

n n-1 0

not iceable

The

next

Thus s t imul i of not iceable di f fe rences fol low sequent ia l ly in a

geometric progress ion . Fechner f e l t tha t the c o r r e s p o n d i ~ g

sensat ions

should follow each

other in

an

ari thmet ic

sequence

occurr ing a t the disc re te

points

a t which j us t

not iceable

di f fe rences

occur . But the

l a t t e r are obta ined when we solve for

n. We

have

n

=

( log s -

log

s ) / log

n

0

and

sensat ion

i s a l inea r funct ion of the logari thm of the

s t imulus .

Thus

i f M denotes the

sensa t ion and

s the s t imulus ,

the psychophys ica l law of Weber-Fechner i s given

by

M

= a

log

s + b , a ;

We

assume t ha t

the

s t imul i ar i se in making pairwise

comparisons

in te re s ted

of

ra t ios .

of r e l a t ive ly comparable a c t i v i t i e s . We are

in

responses

whose numberical values

are

in the form

Thus

b

=

0,

from

which

we

must

have

log

s

=

0

or

s

a

The next

a

=

1,

which i s

poss ib le

by ca l ib ra t ing

a

un i t

s t imulus .

not iceable

response i s due

to

the s t imulus

s s a a

1 0

This yie lds

a

response

log / log = 1. The

next

st imulus

i s

2

s

=

s a

2 a

which yie lds a response of 2. In

th i s

manner

we

sequence

1 , 2 , 3 , . . . .

For

the purpose

of

consis tency

obta in

the

we

place

the


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7

The ac tua l consumption

from

s t a t i s t i c a l sources) i s :

.180

.010 .040 .120 .180 .140 .330

In the second

example

an

ind iv idua l

gives

judgments as to the

r e l a t ive

amount

of

pro t e in in each food.

Prote in

in

Foods

A

B

C

D E F G

A:

Steak

9 9

6 4

5

B:

Potatoes

1/2

1/4

1/3 1/4

C:

Apples

1/3 1/3 1/5

1/9

D: Soybean

1/2

1/6

E:

Whole

Wheat

3

1/3

Bread

F: Tasty Cake

1/5

G

Fish

Here the

der ived

sca le and

ac tua l

values are :

Steak Potatoes Apples

Soybean

Whole Wheat Tasty Fish

Bread Cake

.345 .031 .030 .065 .124 .078 .328

.370 .040

.000

.070 .110

.090 .320

with a

consis tency r a t i o of .028.

3. Absolute

and

Rela t ive Measurement

cogni t ive psychologis t s [1] have

recognized for

some t ime

tha t t he re are two kinds of comparisons, absolu te

and

re la t ive .

In the

former

an a l t e rna t ive

s compared

with

a

s tandard

in

memory

developed through experience;

in

the l a t t e r a l t e rna t ives

are

compared in

pa i r s according to

a common a t t r i bu t e .

The HP

has been used to ca r ry out both types of comparisons r e su l t i ng

in ra t io sca les

of

measurement.

We

c a l l

the

sca les

derived

from

absolu te and r e l a t i v e comparisons re spec t ive ly abso lu te

and

r e l a t i v e measurement

sca les .

Both r e l a t ive

and

abso lu te

measurement are inc luded

in

the

IBM

PC compat ible sof tware

package Expert

Choice

[2] .

Let us

note

t ha t r e l a t ive

measurement i s

usua l ly

needed

to

compare c r i t e r i a

in

a l l

problems pa r t i cu l a r ly

when in tangib le

ones are involved. Absolute measurement i s appl ied to rank the

a l t e rna t i ves

in

te rms

of the c r i t e r i a or

ra ther in

te rms

of

ra t ings

o r i n t e n s i t i e s of the c r i t e r i a

such as

exce l l en t ,

very

good, good, average, below average, poor

and

very

poor .

After

se t t i ng p r i o r i t i e s

on

the c r i t e r i a or subc r i t e r i a ,

i f

t he re

are

some)

pairwise

comparisons

are

a lso

performed

on

the

ra t ings which may be d i f f e ren t fo r each c r i t e r i o n or

subcr i t e r ion .

n a l t e rna t ive

i s

evaluated , scored o r ranked

by i de n t i f y ing for each c r i t e r i o n o r

subcr i t e r ion , the re levant


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8

ra t ing

which

describes t ha t

a l t e rna t ive s

best .

Fina l ly the

weighted or global

pr i o r i t i e s of

the

ra t ings one

under each

cr i t e r ion corresponding to the

a l t e rna t ive are

added to

produce ra t io scale

score

for t ha t a l t e rna t ive .

I f

des i red

in the

end,

the scores

of

a l l the a l te rna t ives m y be

normalized to uni ty .

Absolute measurement needs s tandards , of ten se t

by

so ie ty for

convenience, and sometimes has l i t t l e

to

do

with

the values

and

object ives

of

judge making comparisons. In completely new

decis ion problems

or

old

problems where no s tandards

have

been

establ ished we must use re la t ive measurement to iden t i fy the

best one mong the a l te rna t ives by comparing them in pa irs .

I t

i s

c lea r tha t with absolute

measurement

there can be no

reversa l in the

rank

of the a l te rna t ives i f new a l te rna t ive

i s added or another one deleted.

This

i s

desirable when the

importance of the

c r i t e r i a

al though independent

from

the

a l te rna t ives

according

to function, meaning or context does not

depend

on

t he i r number

and

on

t he i r pr io r i t i e s as t does in

re la t ive measurement. In the l a t t e r i f for example, the

s tudents in ce r ta in school perform badly on in te l l igence

t e s t s the

pr io r i ty

of in te l l igence which i s

an important

cr i t e r ion used to judge students m y be resca led by dividing

by

the sum

of

the pai red comparison

value

of the students

t ransformation carr ied out through normalizat ion. The pr ior i ty

of

each other c r i t e r ion i s also

resca led according

to

the

performance

of

the students . Thus the

c r i t e r i a

weights are

affected

by the

weights

of

the

a l te rna t ives .

I t i s

worth noting t ha t

although

rank m y change when

using

r e l a t ive measurement

with

respect to several

c r i t e r i a

t

does

not change when only one cr i t e r ion i s used and

the judgments

are

cons is ten t .

I t can

never

happen tha t

an

apple which i s

more red

than

another apple

becomes

l ess red than tha t apple on

introducing

t h i rd

apple in the

comparisons.

I t would be

counte r- in tu i t ive were

t ha t to happen. However,

when

judging

apples on

several c r i t e r i a each

t ime new apple

i s

introduced,

c r i t e r ion

tha t

i s concerned

with

the number

of

apples being compared

changes

as

does

another

cr i t e r ion

concerned with the

ac tua l

comparisons

of

the apples. Such

c r i t e r i a are

ca l led

s t ruc tura l . The w y

they

par t ic ipa te in

generating the f ina l

weights

di f fe r s from the t r ad i t iona l

w y

in which the

other

c r i t e r i a ca l led

funct ional ,

do [4] .

Let us now i l l u s t r a t e both types of measurement in decis ion

making.

4. Examples of Rela t ive and

Absolute

Measurement

Rela t ive Measurement:Reagan s Decision to Veto the Highway Bi l l

Before President Reagan

vetoed the Highway

B i l l

high

budget

b i l l to

repa i r

roads

and provide jobs in the

economy in

the

U.S. ,

w

predicted

t ha t

he would

veto

t

by

const ruct ing

two

hiera rch ies one

to measure the benef i ts

and

the o ther

the

costs

of the

possible

a l te rna t ives of

the

decis ion and

taking

tha t with the highest benef i t

to cos t ra t io .

The publ ic

image


11/13


12/13

120

the

pr io r i t y of the

corresponding

c r i t e r i on

and

summed for each

a l t e rna t ive to obta in

the

overa l l

pr io r i t y

shown for

t ha t

a l t e rna t ive . For

example, in

the

benef i t s

hierarchy the

der ived sca les and the

weights

of the c r i t e r i a may be arranged

and composed as

fol lows.

P o l i t i c a l

Effic iency

Employ-

Convenience

Composite

Image

ment

Weights

.634)

.157) .152)

.057)

Modify

.89

.10 .10 .20 .607

Bi l l

Sign .11

.90 .90

.80 .393

Bi l l

Note

for

example

tha t

the

composite weight:

.607

= .89x .634 +.10 x .157 .10 x

.152

.20 x .057

A f ina l

comment

here

i s

tha t

the

HP has a more elaborate

framework

to deal with

dependence

within a level of a hierarchy

or between levels [3] but

we

wil l

not

go in to such de ta i l s

here.

Absolute Measurement:Employee Evaluat ion

The problem

i s to

evaluate

employee

performance. The

hierarchy

for the evaluat ion

and

the

pr io r i t i e s

derived

through

paired

comparisons i s shown below.

I t i s then

followed

by

ra t ing each

employee for

the

qua l i ty of

his

performance

under each

c r i t e r ion and

summing

the

resu l t ing

scores to obtain his

overa l l

ra t ing .

The

same approach can

be

used for student

admissions, giving sa la ry ra ises

etc . The hierarchy can

be

more

e labora te including subcr i te r ia followed by the

in tens i t i e s for expressing

qual i ty .


13/13

2

Goal:

Employee

Performance

Evaluation

Criter ia:

Intens

i t i es

Tech

nical

(

.061)

Excello

(

.604)

Abv.Avg.

( . 24S)

Average

(.10S)

Bel. Av.

( .046)

Alternatives:

Maturity

(.196)

Very

(

.731)

Accep.

(.188)

Immat.

( .081)

1) Mr. X Excell

Very

2) Ms. Y Average Very

3) Mr. Z Excell Immat.

Writing

sk i l l s

(

.043)

Excello

( .733)

Average

(

.199)

Poor

(

.068)

Average

Average

Average

Verbal

ski l l s

.071)

Excello

(.7S0)

Average

.171)

Poor

(

.078)

Timely

work

(.162)

Nofollup

.731)

OnTime

(.188)

Remind

( .081)

Potential

(personal)

(.466)

Great

(

.7

SO)

Average

.171)

Bel.Av.

(.078)

Excell.

OnTime Great

Average Nofollup Average

Excell.

Remind Great

Let

us

now

show

how to obtain

the

to ta l

score

for

Mr.

X

.061

x

.604 + .196

x

.731 + .043

x

.199 + .071

x

.7S0

+

.162

x .188

+

.466 x .7S0 =

.623

Similarly

the

scores

for

Ms. Y and Mr. Z can be shown

to

be

.369 and .478 respectively.

I t is clear

that

we can rank any number of candidates along

these l ines.

REFERENCES

1. Blumenthal, A.L., The Process of

Cognition,

Prentice-Hall,

Englewood Cliffs ,

1977.

2.

Expert

Choice, Software Package for IBM PC, Decision

Support

Software, 1300

Vincent Place,

McLean,

V 22101.

3.

Saaty,

Thomas L., The

Analytic Hierarchy

Process, McGraw-

Hil l ,

1981.

4.

Saaty, Thomas L., Rank

Generation,

Preservation,

and

Reversal

in

the

Analytic Hierarchy Decision

Process ,

Decision

Sciences, Vol. 18, No.2,

Spring

1987.

What Sis the Analytic Hierarchy Process

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