AHP Theory and Math

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AHP Theory and Math
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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 gt 0 , b ?
0 Different scales cannot be multiplied but can
be added if numbers belong to the same scale
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RATIO SCALES Invariant under a positive
similarity transformation
ax a gt 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.
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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
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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
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RELATIVE BRIGHTNESS EIGENVECTOR
I II C1 .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
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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
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SCALE COMPARISON
Moderate
Strong
Between
Very Strong
Between
Between
Between
Extreme
Equal
Scale
(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)
. . .
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Eigenvector for each scale (1) 0.451 0.261 0.16
9 0.119 (2) 0.531 0.237 0.141 0.091 (3) 0.57
7 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 0.2
59 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 0.04
3 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
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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 gt 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.
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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.
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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.
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The Composite Overall Scale in the AHP Synthesis
in the AHP produces a composite scale of
absolute numbers. It is obtained by multiplying
an absolute number representing relative
dominance with respect to a certain criterion by
another absolute number which is the
relative weight of that criterion. The result is
an absolute number that is 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 are relative .
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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.
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Awnw Awcw Aw?maxw
How to go from
to
and then to
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 ? gt 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)
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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
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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 ,
so and
is the average of the non-
principal eigenvalues of A.
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K(s,t) K(t,s) 1
K(s,t) K(t,u) K(s,u), for all s, t, and u
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K(s,t) k1(s) k2(t)
K(s,t)k(s)/k(t)
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K(as, at)aK(s,t)k(as)/k(at)a k(s)/k(t)
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w(as)bw(s)where b?a.
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v(u)C1 e-?u P(u)
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
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The Weber-Fechner law
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M0 a log s0, M1 a log ?, M2 2a
log ?,... , Mn na log ?.
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Figure 1. Ranking Houses on Four Criteria
We must first combine the economic factors so we
have three criteria measured on three different
scales. Two of them are tangible and one is an
intangible. The tangibles must be measured in
relative terms so they can be combined with the
priorities of the intangible.
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Combining the two economic criteria into a single
criterion
200 150 350 300 50 350 500 100 600
In relative terms, the normalized sums should be
350/1300 ?.269
350/1300 ?.269
600/1300 ?.462

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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
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Figure 2. Ranking Houses on Four Criteria
The criteria are assigned priorities equal to the
ratio of the sum of the measurements of the
alternatives under each to the total under both.
Then multiplying and adding for each alternative
yields the correct relative outcome.
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Figure 4. Combining the Two Costs through
Additive or Multiplicative Synthesis
Now the three criteria Economic factors, size
and style can be compared and synthesized as
intangibles. We see that all criteria measured on
the same scale must first be combined as we did
with the two economic factors.
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where q1q2...qn1, qkgt0 (k1,2,...,n), ? gt 0,
but otherwise q1,q2,...,qn,? are arbitrary
constants
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Characterization of W? in Terms of Eigenvalue
Multiplicity.
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Pairwise judgments of the Customer Group for
Burger King
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Predicted and Actual Market Shares for Indirect
Competitors
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If the eigenvalues are all different, the
correspondent eigenvectors are linearly
independent and we can write ea1w1anwn, and
hence,