AHP Theory and MathAHP Theory and Math

NOMINAL SCALES

Invariant under one to one correspondence

Used to name or label objects

ORDINAL SCALES

Invariant under monotone transformations

Cannot be multiplied or added even if the numbers belong to the same scale

INTERVAL SCALES

Invariant under a linear transformation

ax + b a > 0 , b 0

Different scales cannot be multiplied but can be added if numbers belong to the same scale

RATIO SCALES

Invariant under a positive similarity transformation

ax a > 0 Different ratio scales can be multiplied. Numbers fom the same ratio scale can be added.

ABSOLUTE SCALES

Invariant under the identity transformation

Numbers in the same absolute scale can be both added and multiplied.

RELATIVE VISUAL BRIGHTNESS-I

C1 C2 C3 C4

C1 1 5 6 7

C2 1/5 1 4 6

C3 1/6 1/4 1 4

C4 1/7 1/6 1/4 1

RELATIVE VISUAL BRIGHTNESS -II

C1 C2 C3 C4

C1 1 4 6 7

C2 1/4 1 3 4

C3 1/6 1/3 1 2

C4 1/7 1/4 1/2 1

RELATIVE BRIGHTNESS EIGENVECTOR

I IIC1 .62 .63

C2 .23 .22

C3 .10 .09

C4 .05 .06

Square of Reciprocal Normalized normalized of previous Normalized

Distance distance distance column reciprocal

9 0.123 0.015 67 0.61 15 0.205 0.042 24 0.22 21 0.288 0.083 12 0.11 28 0.384 0.148 7 0.06

CHAIRS EXAMPLE

C1 C2 C3 C4

C1 E B(M-S) B(S-V) V

C2 - E M B(M-S)

C3 - - E B(E-M)

C4 - - - E

SCALE COMPARISON

(2) 1-5 1 2 2 3 3 4 4 5 5

(3) 1-7 1 2 2 3 4 5 6 6 7

(4) 1-9 1 2 3 4 5 6 7 8 9

(9) 1-18 1 4 6 8 10 12 14 16 18

(11) 1-90 1 20 30 40 50 60 70 80 90

(12) .9 1 .9 x 1-9 scale

(26)

(27) . . .

Scal

e

Equ

al

Bet

wee

n

Mod

erat

e

Bet

wee

n

Stro

ng

Bet

wee

n2

2/n

Ver

y St

rong

Bet

wee

n

Ext

rem

e

25.

21

25.1

22

25.2

23

25.3

24

98/n

98/1

98/8

1

1

Eigenvector for each scale(1) 0.451 0.261 0.169 0.119(2) 0.531 0.237 0.141 0.091(3) 0.577 0.222 0.125 0.077(4) 0.617 0.224 0.097 0.062(5) 0.659 0.213 0.083 0.044(6) 0.689 0.198 0.074 0.039(7) 0.702 0.199 0.066 0.034(8) 0.721 0.188 0.060 0.031(9) 0.732 0.185 0.057 0.026(10) 0.779 0.162 0.042 0.017(11) 0.886 0.098 0.014 0.003(12) 0.596 0.229 0.105 0.070(13) 0.545 0.238 0.124 0.094(14) 0.470 0.243 0.151 0.135(15) 0.352 0.236 0.191 0.221(16) 0.141 0.162 0.230 0.467(17) 0.340 0.260 0.212 0.187(18) 0.445 0.271 0.171 0.113(19) 0.513 0.266 0.142 0.078(20) 0.561 0.259 0.122 0.059(21) 0.431 0.260 0.172 0.137(22) 0.860 0.111 0.021 0.009(23) 0.953 0.043 0.003 0.001(24) 0.984 0.015 0.001 0.000(25) 0.995 0.005 0.000 0.000(26) 0.604 0.214 0.107 0.076(27) 0.531 0.233 0.134 0.102

0.608 0.219 0.111 0.062

What is the Ratio of Two Ratio Scale Numbers?

The ratio W i / Wj of two numbers W i and W j that belong to the same ratio scale a W a > 0 is a number that is not like W i and W j . It is not a ratio scale number. It is unit free.

It is an absolute number.It is invariant only under the identity transformation.

Example: The ratio of 6 kilograms of bananas and 2 kilograms of bananas is 3. The number 3 tells us that the first batch of bananas is 3 times heavier than the second. The number 3 is not measured in kilograms. It is a cardinal number. It would become meaningless if it were altered.

The Fundamental Scale

The fundamental scale of the AHP, being an estimate of two ratio scale numbers involved in paired comparisons, is itself an absolute scale of numbers. The smaller element in a comparison is taken as the unit, and one estimates how many times the dominant element is a multiple of that unit with respect to a common attribute, using a number from the fundamental scale.

The Derived Scale of the AHP

The scale derived from the paired comparisons in the AHP is a ratio scale w1,…, wn.. The comparisons themselves are based on the fundamental scale of absolute numbers. When normalized, each entry of the derived scale is divided by the sum

w1+…+ wn. . Because the sum of numbers from the same ratio scale is also a number from that scale, normalization of the wi means that the ratio of two ratio scale numbers is taken. It follows that the normalized scale is a scale of absolute numbers. It is only mean-

ingful to divide wi by one or by the sum of several such wi to obtain a meaningful absolute number. Thus the ideal mode in the AHP divides wi by the largest entry in the scale w1,…, wn.

The Composite Overall Scale in the AHP

Synthesis in the AHP produces a composite scale of absolutenumbers. It is obtained by multiplying an absolute number representing relative dominance with respect to a certaincriterion by another absolute number which is the relativeweight of that criterion. The result is an absolute number thatis then added to other such numbers to yield an overall composite scale of absolute relative dominance numbers.

This compounding of dominance is similar to compounding probabilities that are themselves absolute numbers that arerelative .

w = w a ijij

n

1 =j

max

1 = wi

n

1=i

...

...

...

...

1 n

1 1 1 1 n 1 1

n n 1 n n n n

A A

w w w w w wAAw n n w

w w w w w wA

12 1

12 2

1 2

1 ...

1/ 1 ...

1/ 1/ ... 1

n

n

n n

a a

a aA

a a

ija jiai

Let A1, A2,…, An, be a set of stimuli. The

quantified judgments on pairs of stimuli Ai, Aj, are

represented by an n‑by‑n matrix A = (aij), ij = 1,

2, . . ., n. The entries aij are defined by the

following entry rules. If aij = a, then aji = 1 /a, a 0. If Ai is judged to be of equal relative intensity

to Aj then aij = 1, aji = 1, in particular, aii = 1 for

all i.

Clearly in the first formula n is a simple eigenvalue and all other eigenvalues are equal to zero.

A forcing perurbation of eigenvalues theorem:

If is a simple eigenvalue of A, then for small > 0, there is an eigenvalue () of A() with power series expansion in :

()= + (1)+ 2 (2)+…

and corresponding right and left eigenvectors w () and v () such that w()= w+ w(1)+ 2 w(2)+…

v()= v+ v(1)+ 2 v(2)+…

Aw=nw

Aw=cw

Aw=maxw

How to go from

to

and then to

max .1

n

n

On the Measurement of Inconsistency

A positive reciprocal matrix A has with equality if and only if A is consistent. As our measure of deviation of A from consistency, we choose the consistency index

max n

so and is the average of

the non- principal eigenvalues of A.

n

iin

2max

2

1

1

n

iin

n

iin

2max

We know that and is zero if and only if A is consistent. Thus the numerator indicates departure from consistency. The term “n-1” in the denominator arises as follows: Since trace (A) = n is the sum of all the eigenvalues of A, if we denote the eigenvalues of A that are different from max by 2,…,n-1, we see that ,

0

w(s) = dt w(t)t)K(s, b

a

max

w(s)= t)w(t)dtK(s, b

a

w(s)= t)w(t)dtK(s, b

a 1 = w(s)dsb

a

KK((s,ts,t)) K K((t,st,s)) = = 1 1

KK((s,ts,t)) K K((t,ut,u))= K= K((s,us,u), ), for all for all s, t,s, t, and and uu

KK((s,ts,t))= k= k11((ss)) k k22((tt))

KK((s,ts,t))=k=k((ss))/k/k((tt))

k(s)ds

k(s) = w(s)

S

KK((as, atas, at)=)=aKaK((s,ts,t)=)=kk((asas))/k/k((atat))=a k=a k((ss))/k/k((tt))

ww((asas))=bw=bw((ss))where b=where b=a.a.

a

s P Ce = w(s) a

s b

log

loglog

loglog

v(u)=Cv(u)=C11 e e--u u PP((uu))

The periodic function is bounded and the negative exponential leads to an alternating series. Thus, to a first order approximation this leads to the Weber-Fechner law:

bsa log

r)+(1s= ss

s+ s= s+ s= s0

0000011

The Weber-Fechner law

20

201112 s)r+(1s = r)+(1s = s+s = s

2,...) 1, 0, = (n s = s = s n01-nn

s -s = n 0n

log

)log(log

0a b,+ s a = M log

MM0 0 = = aa log s log s00, M, M1 1 = = aa log log , M, M22

= 2= 2aa log log ,... , M,... , Mn n = n= naa log log ..

w)wv )-/(w A v( = w jjTjj11

Tj

n

2j=1

.WB...BB = W ,1+p1-qq

.WB...BB = W ,21-hh

Choosing the Best House

Price Remodeling Costs

Size(sq. ft.)

Style

200 300 500

150 50 100

300020005500

ColonialRanch

Split Level

ABC

Figure 1.Figure 1. Ranking Houses on Four CriteriaRanking Houses on Four CriteriaWe must first combine the economic factors so we have threecriteria measured on three different scales. Two of them are tangible and one is an intangible. The tangibles must be measuredin relative terms so they can be combined with the priorities ofthe intangible.

In relative terms, the normalized sums should be 350/1300 .269 350/1300 .269 600/1300 .462

200 + 150 = 350300 + 50 = 350500 + 100 = 600

Combining the two economic criteria into a single criterion

Choosing the Best House

Price Remodeling Size(sq. ft.)

Style

200/1000300/1000500/1000

150/30050/300100/300

300020005500

ColonialRanch

Split Level

ABC

Adding relative numbers does not give the relative value of the final outcome in dollars. We must weight the criteria first and use their priorities to weight and add and then we get the right answer. 200/1000 +150/300 350/1300 300/1000 + 50/300 350/1300 500/1000 +100/300 600/1300

Choosing the Best House

Price(1000/1300)

RemodelingCosts (300/1300)

Size(sq. ft.)

Style

200/1000300/1000500/1000

150/30050/300

100/300

300020005500

ColonialRanch

Split Level

ABC

Figure 2.Figure 2. Ranking Houses on Four CriteriaRanking Houses on Four Criteria

The criteria are assigned priorities equal to the ratio ofthe sum of the measurements of the alternatives under each to the total under both. Then multiplying and addingfor each alternative yields the correct relative outcome.

Figure 4. Combining the Two Costs through Figure 4. Combining the Two Costs through Additive or Multiplicative SynthesisAdditive or Multiplicative Synthesis

C h o o s in g th e B e s t H o u s e

E co n o m ic F a c to rs(co m b in in g P rice a n d

R e m o d e lin g C o s t)

S ize(sq . ft.) S ty le

3 5 0 /1 3 0 03 5 0 /1 3 0 06 0 0 /1 3 0 0

.2 6 9

.2 6 9

.4 6 2

3 0 0 0 /1 0 5 0 02 0 0 0 /1 0 5 0 05 5 0 0 /1 0 5 0 0

C o lo n ia lR a n ch

S p lit L e ve l

ABC

===

A d d itiveS yn th e s is

M u ltip lica tiveS yn th e s is

.2 5 6

.2 7 2

.4 7 2

Now the three criteria: Economic factors, size and style can be compared and synthesized as intangibles. We see that all criteriameasured on the same scale must first be combined as we did with the two economic factors.

x a=)a-x a( +1

x a +1 )x a( =)x ( = x = x

iiiii

iiiiia

ia

ia iii

loglogexplogexplogexp

x x x = ) x , ,x ,x( f qn

q2

q1n21

n21

x q + x q + x q = ) x , ,x ,x( f nn2211n21

where qwhere q11+q+q22+...+q+...+qnn=1, q=1, qkk>0 (k=1,2,...,n), >0 (k=1,2,...,n), > 0, but > 0, but

otherwise qotherwise q11,q,q22,...,q,...,qnn,, are arbitrary constants are arbitrary constants

11

m

iia

x ..., ,x (i)n

(i)1

x

1=i

m

..., ,x1=i

man

a1

ii

www

2www

www

= W

)j(ni

)j(ni

)j(ni

)j(i

)j(i2

)j(i2

)j(i1

)j(i1

)j(i1

ij

n ji

2

i

1

i

n j21

n j21

Acyclic CyclicIrreducible

max= 1 is a simpleroot

C other eigenvalues with modulus= 1 (they occur in conjugatepairs)

Reducible max= 1 is a

multiple rootC other eigenvalues with modulus= 1 (they occur in conjugatepairs)

Characterization of WCharacterization of W in Terms of in Terms of Eigenvalue Multiplicity.Eigenvalue Multiplicity.

IW000

W

000W0

0000W

00000

= W

1-nn,

2-n 1,-n

32

21

0W000

W

000W0

0000W

W0000

= W

1-nn,

2-n 1,-n

32

21

n1,

CompetitorsCustomer Group

TimeHorizon

IndirectCompetitors

Public Health

Traits

ContemporaryIssues

Marketing Mix

Burger King White Collar Blue Collar Student Family PrioritiesWhite collar 1 4 5 1 / 3 0.299Blue collar 1 / 4 1 4 1 / 3 0.138Student 1 / 5 1 / 4 1 1/ 7 0.051Family 3 3 7 1 0.512

Pairwise judgments of the Customer Group for Pairwise judgments of the Customer Group for Burger KingBurger King

COMPETITORS

CUSTOMER

GROUPS

MARKETINGMIX

CONTEMPORARY

ISSUES

PUBLIC

HEALTH

TRAITS

INDIRECTCOMPETIT

ORS

TIMEHORIZ

ON

Competitors 0.169 0.200 0.151 0.222 0.249 0.252 0.193CustomerGroup

0.186 0.180 0.222 0.175 0.252 0.178 0.391

MarketingMix

0.139 0.181 0.162 0.201 0.157 0.218 0.112 0.195

Contemporary 0.103 0.113 0.097 0.127 0.092Public Health 0.167 0.163 0.170 0.220 0.218Traits 0.074 0.113 0.071 0.101 0.088 0.109 0.083IndirectCompetitors

0.162 0.229 0.106 0.127 0.112 0.169 0.125 0.276

Time Horizon 0.064 0.138

Market ShareCompany Predicted % Actual %PIZZA 33.7 37.0CHICKEN 26.0 28.4MEXICAN 15.2 22.8SUBS 25.0 11.7

Predicted and Actual Market Shares for Indirect Predicted and Actual Market Shares for Indirect CompetitorsCompetitors

ij

ij

ij

in

i

i

IA

fAf

1

max = 1

1

i

( ) / ( ) which converges as

if 1 or is a complex root of 1.

j j i

j i j i

nk k

ii

x I A x xA k

n

k

nn

k

kn

knn

kk wawawawawaeA1

21

22111111 ......

If the eigenvalues are all different, the correspondent eigenvectors are linearly independent and we can write e=a1w1+…+anwn, and hence,