1. 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@feg.unesp.br

2. 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

3. 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]

4. 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

5. 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]

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

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

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

9. 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

10. 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))

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

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

13. 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.

14. 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

15. Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System15 3. Illustrative case (matrix of evaluation) Residential propertyC1C2C3C4C5C6C7C8 A13290331640 A24180221420 A33347122510 A43124232540 A55360344911 A6289231510 A7185111401 A8580231601 A92121230600 A102120131510 A114280222731 A12190111520 A132160332611 A143320332821 A154180241611

16. Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System16 3. Illustrative case (normalized matrix of evaluation) Residential propertyC1C2C3C4C5C6C7C8 A10.0680.1030.1000.0750.0450.0690.1740 A20.0910.0640.0670.0500.0450.0460.0870 A30.0680.1230.0330.0500.0910.0570.0430 A40.0680.0440.0670.0750.0910.0570.1740 A50.1140.1270.100 0.1820.1030.0430.143 A60.0450.0310.0670.0750.0450.0570.0430 A70.0230.0300.0330.0250.0450.04600.143 A80.1140.0280.0670.0750.0450.06900.143 A90.0450.0430.0670.07500.06900 A100.0450.0420.0330.0750.0450.0570.0430 A110.0910.0990.0670.0500.0910.0800.1300.143 A120.0230.0320.0330.0250.0450.0570.0870 A130.0450.0570.1000.0750.0910.0690.0430.143 A140.0680.1130.1000.0750.0910.0920.0870.143 A150.0910.0640.0670.1000.0450.0690.0430.143

17. Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System17 3. Illustrative case (overall values without TODIM) Residential propertyOverall valueRank A10.3016 A20.24110 A30.2459 A40.2578 A50.4541 A60.19211 A70.15914 A80.3115 A90.18512 A100.18512 A110.3513 A120.12515 A130.2917 A140.3662 A150.3384

18. 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  -0.303 For C 2, p 12 > p 22, then  0.076... In Equation 3,  (A 1, A 2 )  0.017 In Equation 4,  0.644

19. Salomon & Rangel (2015)Comparing ranks from TODIM and Fuzzy Expert System19 3. Illustrative case (overall values with TODIM) Residential propertiesWithout TODIMWith TODIM A10.30160.6925 A20.241100.38610 A30.24590.3999 A40.25780.6207 A50.454111 A60.192110.28611 A70.15914015 A80.31150.4418 A90.185120.02014 A100.185120.21312 A110.35130.8583 A120.125150.10713 A130.29170.7194 A140.36620.9372 A150.33840.6736

20. 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)

21. 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)

22. 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

23. 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 A50.259 3 A60 6 A70 6 A80 6 A90 6 A100 6 A110.452 2 A120 6 A130.259 3 A140.741 1 A150.259 3

24. 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

25. 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 302692/2011-8) Sao Paulo State Research Foundation (Grant No. FAPESP 2013/03525-7)