Transcript Slide 1

A Study on the Compatibility
between Decision Vectors
Claudio Garuti
Universidad Federico Santa María, Chile
[email protected]
Valério Salomon
Sao Paulo State University, Brazil
[email protected]
June 6, 2010
1
A Study on the Compatibility
between Decision Vectors
1. Introduction
2. Compatibility between two vectors
3. Examples of compatibility indexes utilization
4. Concluding remarks
References
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1. Introduction
x = wD
x: decision vector, D: decision matrix,
w: weights for the criteria
Alternative
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Criterion 1 Criterion 2 … Criterion j
… Criterion n
1
d11
d12
…
d1j
…
d1n
2
d21
d22
…
d2j
…
d2n
…
…
…
…
…
…
…
i
di1
di2
…
dij
…
din
…
…
…
…
…
…
…
m
dm1
dm2
…
dmj
…
dmn
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1. Introduction
When we can say that x is a good decision vector?
“Garbage in, Garbage out”
Or else, if w and D were good,
then, x must also be good.
With AHP, the most widely used MCDM method
(Wallenius et al. 2008), we try to answer that
question with the consistency checking:
A w = lMAX w
m = (lMAX – n)/(n – 1) must be less than 10%.
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1. Introduction
An example of consistency checked
Relative electric consumption of household appliances
Alternatives
Electric range (A)
Refrigerator (B)
TV (C)
Dishwasher (D)
Iron (E)
Hair-dryer (F)
Radio (G)
A
1
1/2
1/5
1/8
1/7
1/9
1/9
B
2
1
1/4
1/5
1/5
1/7
1/9
C
D
E
F
5
8
7
9
4
5
5
7
1
2
5
6
1/2 1
4
9
1/5 1/4 1
5
1/6 1/9 1/5 1
1/8 1/9 1/9 1/5
G
9
9
8
9
9
5
1
w Actual
.393 .392
.261 .242
.131 .167
.110 .120
.061 .047
.028 .028
.016 .003
Consistency checking: m = 0.02, OK!
But, perhaps more important than that,
w and Actual are close to each other.
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1. Introduction
Decision vectors from applications
of three different MCDM methods
Alternative
AHP MACBETH MAUT
Hire more personnel
.15
24
.22
Keep the same personnel .37
64
.61
Outsource
.48
71
.72
Criteria like costs and quality from the service were considered.
Different MCDM methods applications result
on different cardinal decision vectors.
But, they result in the same ordinal decision vector.
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2. Compatibility between 2 vectors
When two vectors are close to each other, we can say
that they are compatible .
Saaty (2006) proposes a compatibility index, S, based
on the Hadamard Product.
Garuti (2007) proposes another compatibility index, G,
based on the inner product between two vectors.
Despite Compatibility is a new theme in MCDM, some
limitations of S has already been identified.
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2. Compatibility between 2 vectors
Saaty ‘s compatibility index, between x and y
1 T
S = 2 e A •BTe
n
When: aij = xi/xj, bij = yi/yj, e is a row-matrix with all
components equal to 1, and A • B is the cellwise
or Hadamard Product: aij • bij= aij bij
If x = y, then S = 1
If S > 1.1, then Saaty (2006) proposes that x and y
were considered as not compatible
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2. Compatibility between 2 vectors
The inner product
x ∙ y = |x| |y| cosa = S(xi yj)
x
y
a = 0° → total projection →
→ total vector similarity → total compatibility
x∙y=1
x
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a = 90° → no projection →
→ no vector similarity → total incompatibility
y
x∙y=0
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2. Compatibility between 2 vectors
Garuti‘s compatibility index, between x and y
min(xi , yi ) xi + yi
G=∑
max(xi , yi ) 2
If x = y, then G = 1
If G < .9, then Garuti (2007) proposes that x and y
were considered as not compatible
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3. Examples of S and G utilization
Relative electric consumption of household appliances
Alternatives
Electric range
Refrigerator
TV
Dishwasher
Iron
Hair-dryer
Radio
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w Actual
.393 .392
.261 .242
.131 .167
.110 .120
.061 .047
.028 .028
.016 .003
We have:
S = 1.455
G = .92
So w and Actual are:
• not compatible with S
• compatible with G
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3. Examples of S and G utilization
Normalizing the data from Slide 6
Alternative
AHP MACBETH MAUT
Hire more personnel
.15
.15
.14
Keep the same personnel .37
.40
.39
Outsource
.48
.45
.47
For AHP and MACBETH we have S = 1.003 and G = .94
For AHP and MAUT we have S = 1.002 and G = .96
For MACBETH and MAUT we have S = 1.002 and G = .96
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4. Concluding remarks
With the compatibility index, G, we can answer:
“When we can say that x is a good decision vector?”
Instead of S, G is a weighted index, since it uses the
average between vectors components or even the
weights for the criteria, w.
As Compatibility is a new theme in MCDM, more study
and applications will be necessary to prove this
theory in Decision Making.
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References
Garuti, C. Measuring compatibility (closeness) in weighted
environments: when close really means close? Int.
Symposium on AHP, 9, Vina del Mar, 2007.
Saaty, T. L. Fundamentals of decision making and priority
theory. 2 ed. Pittsburgh : RWS, 2006.
Wallenius, J., et al. Multiple criteria decision making,
multiattribute utility theory: recent accomplishments and
what lies ahead. Management Science, 7, 2008, Vol. 54, pp.
1336-1349.
Whitaker, R. 2007. Validation examples of the Analytic Hierarchy
Process and Analytic Network Process. Mathematical and
Computer Modelling. 2007, Vol. 46, pp. 840-859.
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A Study on the Compatibility
between Decision Vectors
Claudio Garuti thanks Sociedad
Argentina de Informática (SADIO) and
Universidad Federico Santa María
Valério Salomon thanks the Sao Paulo
Research Foundation (FAPESP) for
financial support
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