A Study on the Compatibility between Decision Vectors

Claudio GarutiUniversidad Federico Santa María, Chile

claudiogaruti@fulcrum.cl

Valério SalomonSao Paulo State University, Brazil

salomon@feg.unesp.br

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1. Introduction2. Compatibility between two vectors3. Examples of compatibility indexes utilization4. Concluding remarksReferences

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A Study on the Compatibility between Decision Vectors

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

1.1. To know when two metrics of decision are closeTo know when two metrics of decision are close2.2. To know how close are two different ways of thinking To know how close are two different ways of thinking 3.3. To know the closeness of two complex behavior patternsTo know the closeness of two complex behavior patterns4.4. To know the degree of matching of two groups To know the degree of matching of two groups (sellers & buyers)(sellers & buyers)……And many other issues.And many other issues.

1. IntroductionWhat is Compatibility?

What is Useful for?

What is useful for?

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. IntroductionWhat is Compatibility?

What is Useful for?

1.- It means that they should make the same choice? Two candidates: A, B for an electionThree people :P1: choose candidate A, P2 & P3: choice candidate B, P1 & P2: regular people: Intensity of preference= 45-55 & 55-45 respectvelyP3: extremist: Intensity of preference= 5-95Is really P2 more compatible with P3 just because they make the same choice?

2.- Or, it means they should have similar metric of decisionIn 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.

Proximity = Intensity

M.Topology O.Topology

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1. IntroductionDifferent Formulas to Assess Compatibility

1. Hilbert formula (Hilbert´s index): C(A, B) = Log {Maxi(ai/bi)/Mini(ai/bi)}

2. Simple inner vector product (IVP): C(A, B) = {A}•{B} /n = (i ai x

1/bi )/n

3. Hadamard product (Saaty`s index): C(A, B) = [A]•[B]t /n2 = [ij aij x 1/bij] /n2

4. Euclidian formula (normalized): C(A, B) = SQRT(1/2i(ai–bi)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 GGaruti (2007) proposes another compatibility index, G, starting on

inner product between two vectors, but based on the physics view of vector relation:Examples: Work=F•d = (Fd) x (projection of F-d); P=V•I= (VI) x (projection of V-I)

Graphically:

= 0° → total projection → → total vector similarity → total compatibility

x ∙ y = 1

= 90° → no projection → → no vector similarity → total incompatibility

x ∙ y = 0

x

y

xy

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2. A Compatibility Index GGaruti‘s compatibility index, between x and y

Projectioni x Importancei (weight)

If x = y, then G = 1

If G < .9, (or 1-G>10%) then Garuti (2007) proposes that x and y were considered as not compatible vectors.

∑2),max(

),min(

ii

ii

ii yx

yx

yxG

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Consistency checking: = 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).

Alternatives A B C D E F G w ActualElectric range (A) 1 2 5 8 7 9 9 .393 .392Refrigerator (B) 1/2 1 4 5 5 7 9 .261 .242

TV (C) 1/5 1/4 1 2 5 6 8 .131 .167Dishwasher (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 .047Hair-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

3. Examples of compatibility index GRelative electric consumption of household appliances

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3. Examples of compatibility index GRelative electric consumption of household appliances

We have:S = 1.455 (45% > 10%)H = 1.832 (83%> 10%)IVP =1.63 (63%> 10%)G = 0.92 (8% < 10%)E=0.0032 (0.3%)

So w and Actual are:•Not compatible by S, H, or IVP•Compatible by G and E

Alternatives w ActualElectric range .393 .392Refrigerator .261 .242

TV .131 .167Dishwasher .110 .120

Iron .061 .047Hair-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.

Disease D --------Disease E ---------

Disease G ---------Disease F ---------

fever pain anguish evolution cofe secretion allergy Tabbaco Diabetes HTA

Disease A --------Disease B --------Disease C --------

GCIGCIii = Garuti`s General Compatibility Index Between each Disease Profile and Patient Profile = Garuti`s General Compatibility Index Between each Disease Profile and Patient Profile

IF G IF G PP 90% THEN the profiles are compatibles.90% THEN the profiles are compatibles.

Note: Higher compatibility represent higher likelihood (or certainty) that patient present that disease.

SC

AL

ES

OF

SC

AL

ES

OF

INT

EN

SIT

YIN

TE

NS

ITY

Terminal criteria

w1 w9w8w7w6w5w4w3w2 w10

Compatibility Index G (closeness)10% (hepatitis)97% (irritable colon)73% (ulcer)15% (common flu)18% (H1N1)55% (colitis)…% (…..)

= Disease B (GCI=97%) (irritable colon)

3.- One more complex case: Profiles Compatibility3.- One more complex case: Profiles Compatibility

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4. Concluding remarksWith 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 D&O 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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ReferencesGaruti, 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.

A Study on the Compatibility between Decision Vectors

Claudio Garuti Thanks to SADIO, Argentina and Universidad Federico

Santa María, Chileclaudiogaruti@fulcrum.cl

Valério Salomon thanks the Sao Paulo Research Foundation (FAPESP) for

financial support

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