A Study on the Compatibility between Decision Vectors

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A Study on the Compatibility between Decision
Vectors

• Claudio Garuti
• Universidad Federico Santa María, Chile
• claudiogaruti_at_fulcrum.cl
•   Valério Salomon
• Sao Paulo State University, Brazilsalomon_at_feg.une
  sp.br

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A Study on the Compatibility between Decision
Vectors

• Introduction
• Compatibility between two vectors
• Examples of compatibility indexes utilization
• Concluding remarks
• References

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• Introduction
• What is Compatibility?
• What is Useful for?

Compatible (compatibilis) To have the right
proportion to joint or connect at the
same time with other (empathy) Under the
same decision problem, two compatible persons
should have close visions
What is useful for?

• To know when two metrics of decision are close
• To know how close are two different ways of
  thinking
• To know the closeness of two complex behavior
  patterns
• To know the degree of matching of two groups
  (sellers buyers)
• And many other issues.

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• Introduction
• What is Compatibility?
• What is Useful for?

Proximity Intensity M.Topology
O.Topology
Under the same decision problem, two compatible
persons should have close visions. But, what
means when says two compatible persons should
have close visions?
1.- It means that they should make the same
choice? Two candidates A, B for an
election Three people P1 choose candidate A, P2
P3 choice candidate B, P1 P2
regular people Intensity of preference 45-55
55-45 respectvely P3 extremist Intensity of
preference 5-95 Is really P2 more compatible
with P3 just because they make the same choice?
2.- Or, it means they should have similar metric
of decision In order topology metric of decision
means intensity of choice, (dominance degree of A
over B) So, compatibility is not related only
with order of choice, is something more complex,
more systemic, it is related with the intensity
of choices.
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• 1. Introduction
• Different Formulas to Assess Compatibility
• Hilbert formula (Hilberts index) C(A, B)
  Log Maxi(ai/bi)/Mini(ai/bi)
• Simple inner vector product (IVP) C(A, B)
  AB /n (Si ai x 1/bi )/n
• Hadamard product (Saatys index) C(A, B)
  ABt /n2 S iSj aij x 1/bij /n2
• Euclidian formula (normalized) C(A, B)
  SQRT(1/2 S i(aibi)2)
• Despite Compatibility is a new theme in MCDM,
  some limitations of these formulas has already
  been identified.

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2. A Compatibility Index G Garuti (2007) proposes
another compatibility index, G, starting on inner
product between two vectors, but based on the
physics view of vector relation Examples
WorkFd (Fd) x (projection of F-d) PVI
(VI) x (projection of V-I) Graphically
x

• 0 ? total projection ? ? total vector
  similarity ? total compatibility
• x y 1

y

• 90 ? no projection ? ? no vector similarity
  ? total incompatibility
• x y 0

x
y
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2. A Compatibility Index G Garutis compatibility
index, between x and y Projectioni x
Importancei (weight) If x y, then G 1 If G lt
.9, (or 1-Ggt10) then Garuti (2007) proposes that
x and y were considered as not compatible
vectors.
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3. Examples of compatibility index G Relative
electric consumption of household appliances
Alternatives A B C D E F G w Actual
Electric range (A) 1 2 5 8 7 9 9 .393 .392
Refrigerator (B) 1/2 1 4 5 5 7 9 .261 .242
TV (C) 1/5 1/4 1 2 5 6 8 .131 .167
Dishwasher (D) 1/8 1/5 1/2 1 4 9 9 .110 .120
Iron (E) 1/7 1/5 1/5 1/4 1 5 9 .061 .047
Hair-dryer (F) 1/9 1/7 1/6 1/9 1/5 1 5 .028 .028
Radio (G) 1/9 1/9 1/8 1/9 1/9 1/5 1 .016 .003
Consistency checking m 0.02, OK!
(compatibility needs consistency). w and Actual
seems to be close to each other, but are they
really close? How we can measure that closeness?
(weighted proximity problem).
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3. Examples of compatibility index G Relative
electric consumption of household appliances

• We have
• S 1.455 (45 gt 10)
• H 1.832 (83gt 10)
• IVP 1.63 (63gt 10)
• G 0.92 (8 lt 10)
• E0.0032 (0.3)
• So w and Actual are
• Not compatible by S, H, or IVP
• Compatible by G and E

Alternatives w Actual
Electric range .393 .392
Refrigerator .261 .242
TV .131 .167
Dishwasher .110 .120
Iron .061 .047
Hair-dryer .028 .028
Radio .016 .003
w and Actual are compatible vectors indeed and G
is the only one index that assess it correctly in
a weighted environment.
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3.- One more complex case Profiles Compatibility
w1
w9
w8
w7
w6
w5
w4
w3
w2
w10
Terminal criteria
1 0
SCALES OF INTENSITY
Patient X --------
GCIi Garutis General Compatibility Index
Between each Disease Profile and Patient
Profile IF G P 90 THEN the profiles are
compatibles. Note Higher compatibility represent
higher likelihood (or certainty) that patient
present that disease.
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• 4. Concluding remarks
• With the compatibility index G, we can answer
• When close really means close
• G is an index able to measure compatibility in
  weighted environment.
• G can assess if a specific metric is a good
  metric (compatible with actual metric (physical
  or economical)).
• G can establish if two complex profiles are
  aligned (for instance, degree of alignment
  between DO profiles).
• G can establish if two different people really
  have compatible point of views (compatible
  decision metric, very useful in conflict
  resolution).
• 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 to SADIO, Argentina and
  Universidad Federico Santa María, Chile
• claudiogaruti_at_fulcrum.cl
•   Valério Salomon thanks the Sao Paulo Research
  Foundation (FAPESP) for financial support