Comparing Rankings from using TODIM and a Fuzzy Expert System Valério A. P. Salomon Luís A. D. Rangel Sao Paulo State University (UNESP)Fluminense Federal University (UFF)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System2 Outline 1.Introduction 2.Theory background Correlation between ranks 3.Illustrative case Real Estate in Rio State 4.Discussion and conclusions Acknowledgments References

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System3 1. Introduction Multi-Criteria Decision Analysis (MCDA) methods [1] AHP, ANP, ELECTRE, MACBETH, MAUT, TOPSIS Decision problems Continuous (large number of alternative solutions, even, infinite) Discrete (small number of alternatives, perhaps, two) Choice, Sort, Ranking and Description [2]

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System4 1. Introduction Different MCDA methods may yield different results[11]: rank correlation [12] TODIM is an MCDA method developed to Ranking problems [13] Fuzzy Sets Theory (FST) was proposed to Classification problems [20] The use of FST in MCDA is slightly controversial [27]: FST may result in loss of information [28] Our aim is to prove that TODIM can provide a better solution than FST for Ranking problems

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System5 2. Theory background 2.1. Correlation between ranks Rank correlation coefficient [12] 2.2. TODIM method Prospect Theory [14] 2.3. Fuzzy expert systems If-Then rules [36], Mamdani model [39]

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System6 2. Theory background (Edmond-Mason coefficient)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System7 2. Theory background (examples)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System8 2. Theory background (TODIM’s value function)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System9 2. Theory background (TODIM elements) Matrix of evaluation: composed by the numerical evaluation for the alternatives regarding to all the criteria The matrix must be normalized, for each criterion Matrix of normalized alternatives: P = [p nm ] Number of criteria: m Number of alternatives: n Reference criterion, r, usually the highest weighted really Vector of weights: w = [w rc ] = w c /w r

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System10 2. Theory background (TODIM results) Dominance (Equation 3)  ( ,  j) =  (  i,  j) Overal value (Equation 3)  = (  (  i,  j) - min  (  i,  j)) / (max  (  i,  j) - min  (  i,  j))

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System11 2. Theory background (Fuzzy set)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System12 2. Theory background (Fuzzy expert system)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System13 3. Illustrative case (data) Volta Redonda is a city in the South of the State of Rio de Janeiro, Brazil. It has approximately 260,000 inhabitants. There are a large number of properties, residential and commercial, rented or available for rent. The major steel plant installed in the city in the 1940’s is a landmark of Brazilian industrialization.

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System14 3. Illustrative case (data) CriterionWeightNormalized weight Localization (C1)50.25 Construction area (C2)30.15 Construction quality (C3)20.10 State of conservation (C4)40.20 Garage spaces (C5)10.05 Rooms (C6)20.10 Attractions (C7)10.05 Security (C8)20.10

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System15 3. Illustrative case (matrix of evaluation) Residential propertyC1C2C3C4C5C6C7C8 A A A A A A A A A A A A A A A

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System16 3. Illustrative case (normalized matrix of evaluation) Residential propertyC1C2C3C4C5C6C7C8 A A A A A A A A A A A A A A A

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System17 3. Illustrative case (overall values without TODIM) Residential propertyOverall valueRank A A A A A A A A A A A A A A A

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System18 3. Illustrative case (TODIM application)  = 1 For C 1, p 11 < p 12, then  For C 2, p 12 > p 22, then  In Equation 3,  (A 1, A 2 )  In Equation 4,  0.644

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System19 3. Illustrative case (overall values with TODIM) Residential propertiesWithout TODIMWith TODIM A A A A A A A A A A A A A A A

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System20 3. Illustrative case (Fuzzy Expert System application) Fuzzy sets for Location (C1), Construction Quality (C3), State of Conservation (C4), Attractions (C7)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System21 3. Illustrative case (Fuzzy Expert System application) Fuzzy set for Construction area (C2)(Similar to C5, C6 and C8)

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System22 3. Illustrative case (Fuzzy Expert System application) Fuzzy rules Rule InputOutput LocationConstr. QualityState of conservationAttractionsEvaluation 1Bad 2 AverageBad 3 GoodBad... 79Good Bad 80Good AverageGood 81Good

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System23 3. Illustrative case (Fuzzy Expert System application) Residential propertyOverall valueRank A10 6 A20 6 A30 6 A40 6 A A60 6 A70 6 A80 6 A90 6 A100 6 A A120 6 A A A

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System24 4. Discussion and Conclusions Main contribution of this work: application of Fuzzy Expert System and its comparison with a TODIM application TODIM applied only with spreadsheets Fuzzy Expert System required specific software (fuzzyTECH.com) Sensitivity Analysis were conducted and did not affect the results TODIM application considered different weights for the criteria; Fuzzy Expert System considered the same weight (1/8 for all) Future research: compare TODIM with other techniques

Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System25 Acknowledgments Authors need to thank Prof. Dr. Luiz Flavio Autran Monteiro Gomes for valuable advises, comments, and suggestions This research has financial support from Brazilian Council for Scientific and Technological Development (Grant No. CNPQ /2011-8) Sao Paulo State Research Foundation (Grant No. FAPESP 2013/ )