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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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
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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
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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 .
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8/11/2019 What Sis the Analytic Hierarchy Process
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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.