1. A Study on the Compatibility between Decision Vectors Claudio Garuti Universidad Federico Santa María, Chile claudiogaruti@fulcrum.cl Valério Salomon Sao Paulo State University, Brazil salomon@feg.unesp.br salomon@feg.unesp.br June 6, 20101

2. 1.Introduction 2.Compatibility between two vectors 3.Examples of compatibility indexes utilization 4.Concluding remarks References June 6, 20102 A Study on the Compatibility between Decision Vectors

3. 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.To know when two metrics of decision are close 2.To know how close are two different ways of thinking 3.To know the closeness of two complex behavior patterns 4.To know the degree of matching of two groups (sellers & buyers) …And many other issues. 1.Introduction What is Compatibility? What is Useful for? What is useful for?

4. 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.Introduction What is Compatibility? What is Useful for? 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 d ecision 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

5. June 6, 20105 1. Introduction Different Formulas to Assess Compatibility 1.Hilbert formula (Hilbert´s index): C(A, B) = Log {Max i (a i /b i )/Min i (a i /b i )} 2.Simple inner vector product (IVP): C(A, B) = {A}{B} /n = (  i a i x 1/b i )/n 3.Hadamard product (Saaty`s index): C(A, B) = [A][B] t /n 2 = [  i  j a ij x 1/b ij ] /n 2 4.Euclidian formula (normalized): C(A, B) = SQRT(1/2  i (a i –b i ) 2 ) Despite Compatibility is a new theme in MCDM, some limitations of these formulas has already been identified.

6. June 6, 20106 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=Fd = (Fd) x (projection of F-d); P=VI= (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 x y

7. June 6, 20107 2. A Compatibility Index G Garuti‘s compatibility index, between x and y Projection i x Importance i (weight) If x = y, then G = 1 If G 10%) then Garuti (2007) proposes that x and y were considered as not compatible vectors.

8. June 6, 20108 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). AlternativesABCDEFGwActual Electric range (A)1258799.393.392 Refrigerator (B)1/2145579.261.242 TV (C)1/51/412568.131.167 Dishwasher (D)1/81/51/21499.110.120 Iron (E)1/71/5 1/4159.061.047 Hair-dryer (F)1/91/71/61/91/515.028 Radio (G)1/9 1/81/9 1/51.016.003 3. Examples of compatibility index G Relative electric consumption of household appliances

9. June 6, 20109 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 AlternativeswActual Electric range.393.392 Refrigerator.261.242 TV.131.167 Dishwasher.110.120 Iron.061.047 Hair-dryer.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.

10. Disease D -------- Disease E --------- Disease G --------- Disease F --------- fever painanguishevolutioncofesecretionallergyTabbacoDiabetes HTA Disease A -------- Disease B -------- Disease C -------- GCI i = Garuti`s 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. SCALES OF INTENSITY Terminal criteria w1 w9w8w7w6w5w4w3w2w10 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 Compatibility

11. June 6, 201011 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.

12. June 6, 201012 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.

13. A Study on the Compatibility between Decision Vectors Claudio Garuti Thanks to SADIO, Argentina and Universidad Federico Santa María, Chile claudiogaruti@fulcrum.cl Valério Salomon thanks the Sao Paulo Research Foundation (FAPESP) for financial support June 6, 201013