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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] A Study on the Compatibility between Decision Vectors. Introduction

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A study on the compatibility between decision vectors

A Study on the Compatibility between Decision Vectors

Claudio Garuti

Universidad Federico Santa María, Chile

[email protected]

Valério Salomon

Sao Paulo State University, [email protected]


A Study on the Compatibility between Decision Vectors

  • Introduction

  • Compatibility between two vectors

  • Examples of compatibility indexes utilization

  • Concluding remarks

    References


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.


Introduction

What is Compatibility?

What is Useful for?

Proximity=Intensity

M.TopologyO.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.


1. Introduction

Different Formulas to Assess Compatibility

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

Simple inner vector product (IVP): C(A, B) = {A}•{B} /n = (Si ai x 1/bi )/n

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

Euclidian formula (normalized): C(A, B) = SQRT(1/2 S i(ai–bi)2)

Despite Compatibility is a new theme in MCDM, some limitations of these formulashas already been identified.


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:Work=F•d = (Fd) x (projection of F-d); P=V•I= (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


2. A Compatibility Index G

Garuti‘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.


3. Examples of compatibility index G

Relative electric consumption of household appliances

Consistency checking: m = 0.02, OK! (compatibility needs consistency).

w and Actualseems to be close to each other, but are they really close?

How we can measure that closeness? (weighted proximity problem).


3. Examples of compatibility index G

Relative 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

  • w and Actual are compatible vectors indeed

    and G is the only one index that assess it correctly

    in a weighted environment.


    pain

    anguish

    evolution

    cofe

    secretion

    allergy

    Tabbaco

    HTA

    Diabetes

    fever

    Compatibility Index G (closeness)

    10% (hepatitis)

    97% (irritable colon)

    73% (ulcer)

    15% (common flu)

    18% (H1N1)

    55% (colitis)

    …% (…..)

    Disease A--------

    Disease B--------

    Disease D--------

    = Disease B (GCI=97%)

    (irritable colon)

    Disease E---------

    Disease C--------

    Disease F---------

    Disease G---------

    3.- One more complex case: Profiles Compatibility

    w2

    w3

    w4

    w5

    w6

    w7

    w8

    w9

    w10

    w1

    Terminal

    criteria

    1

    0

    SCALES OF

    INTENSITY

    Patient X--------

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

    IF G P90% THEN the profiles are compatibles.

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


    • 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 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.


    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.


    A study on the compatibility between decision vectors1

    A Study on the Compatibility between Decision Vectors

    Claudio Garuti Thanks to SADIO, Argentina and Universidad Federico Santa María, Chile

    [email protected]

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


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