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1. Criteria interactions in multiple criteria decision aiding: A Choquet formulation for the TODIM method

   Luiz Flavio Autran Monteiro Gomesa, Maria Augusta Soares Machadoa, Francisco Ferreira da Costab, Luis Alberto Duncan Rangelc aIbmec, Av. Presidente Wilson, 118, 11th floor, 20030-020, Rio de Janeiro, RJ, Brazil bPetroleo Brasileiro S.A., Av. República do Chile, 330, Torre Oeste, 33rd floor, 20031-170, Rio de Janeiro, RJ, Brazil cUFF-EEIMVR, Av. dos Trabalhadores, 420, Vila Santa Cecília,Volta Redonda, RJ, Brazil E-mail address: autran@ibmecrj.br
2. Our problem: modeling interactions between criteria in Multiple Criteria Decision Aiding (MCDA) B. Roy (2009): analyzed available aggregation models that have been conceived to take care of criteria interdependences; a special attention was then given to the use of the Choquet integral, by pointing out its possibilities and limitations S. Greco, V. Mousseau and R. Slowinski (2012): that problem has been tackled by Robust Ordinal Regression (ROR) S. Corrente, S. Greco and R. Slowinski (2012): ROR has also been applied to take into account imprecise evaluations in MCDA
3. In thispaper: we show how measures of criteria interaction can be computed for the TODIM method of MCDA TODIM (acronym for Interactive Multicriteria Decision Making, in Portuguese) is a method that relies on an additive value function built having as a basis the paradigm of prospect theory; it is based on nonlinear prospect theory as the shape of its value function is the same as the gains/losses function of prospect theory Although the original formulation of TODIM was based on the original prospect theory, that method has been extended towards a formulation based on Cumulative Prospect Theory (CPT), an evolution of that original prospect theory: see Tversky, A. and Kahneman, D. (1992) Advances in prospect theory, cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, 297-323
4. When introducing the CPT in 1992,Tversky & Kahneman defined the subjective value ν of an outcome x as a two-part power function of the form: α quantifies the curvature of the subjective value function for gains, β does for losses, and the parameter λ quantifies the loss aversion. For α, β < 1, the value function exhibits risk aversion over gains and risk seeking over losses. Furthermore, if λ, the loss-aversion coefficient, is greater than one, individuals are more sensitive to losses than gains.
5. TODIM (acronym for Interactive and Multicriteria Decision Making in Portuguese) GOMES,L.F.A.M.;MACHADO,M.A.S. and RANGEL,L.A.D. (2013) Behavioral multi-criteria decision analysis: the TODIM method with criteria interactions. Annals of Operations Research, available online in March 2013, http://dx.doi.org/10.1007/s10479-013-1345-0, 18 pp. FAN,Z.-P.; ZHANG, X.; CHEN, F.-D. and LIU, Y. (2013) Extended TODIM method for hybrid MADM problems. Knowledge-Based Systems, In press, http://dx.doi.org/10.1016/j.knosys.2012.12.014, 9 pp. MOSHKOVICH, H.M.; GOMES, L.F.A.M.; MECHITOV, A.I.and  RANGEL, L.A.D. (2012) Influenceofmodelsandscalesonthe ranking ofmultiattributealternatives. Pesquisa Operacional, v.32, n.3, pp. 523-542. KROHLING,R.A.; SOUZA, .T.M.de. (2012) CombiningProspectTheoryandFuzzyNumbers to Multi-criteriaDecisionMaking. Expert Systems with Applications, v. 39, Issue 13, p. 11487-11493. GOMES, L.F.A.M.; GONZÁLEZ, X.I. (2012) BehavioralMulti-CriteriaDecisionAnalysis: FurtherElaborationsonthe TODIM Method. Foundationsof Computing andDecisionSciences, v. 37, n. 1, p. 3-8. RANGEL, L.A.D.; GOMES, L.F.A.M.; CARDOSO, F. P. (2011) An application ofthe TODIM method to theevaluationofbroadband Internet plans. Pesquisa Operacional , v. 31, p. 235-249. MOSHKOVICH, H.; GOMES, L.F.A.M.; MECHITOV, A.I. (2011) Anintegratedmulticriteriadecision-making approach to real estateevaluation: case ofthe TODIM method. Pesquisa Operacional , v. 31, p. 3-20. GOMES, C.F.S.; GOMES, L.F.A.M.; MARANHÃO, F.J.C. (2010) Decisionanalysis for theexplorationofgas reserves: merging TODIM and THOR. Pesquisa Operacional , v. 30, p. 601-617. CHEN, F.-D.; ZHANG, X.; KANG F.; FAN, Z.-P. and CHEN, X. (2010) A Method for Interval Multiple Attribute Decision Making With Loss Aversion. 2010 International Conference of Information Science and Management Engineering, IEEE Computer Society, 453-456. GOMES, L. F. A. M. ; RANGEL, L.A.D. (2009) An Application of the TODIM Method to the Multicriteria Rental Evaluation of Residential Properties. European Journal of Operational Research, v. 193, p. 204-211. GOMES, L. F. A. M.; GOMES, C.F.S.; RANGEL, L.A.D. (2009) A comparativedecisionanalysiswith THOR and TODIM: rentalevaluation in Volta Redonda. Revista Tecnologia , v. 30, p. 7-11. GOMES, L. F. A. M. ; RANGEL, L.A.D.; MARANHÃO, F. J. C. (2009) Multicriteria Analysis of Natural Gas Destination in Brazil: An application of the TODIM Method. Mathematical and Computer Modelling, v. 50, p. 92-100 RANGEL, L.A.D.; GOMES, L.F.A.M. ; MOREIRA, R.A. (2009). Decision theory with multiple criteria: an aplication of ELECTRE IV and TODIM to SEBRAE/RJ. PesquisaOperacional, 29 (3), 577-590 NOBRE, F.F; TROTTA, L.T.F.; GOMES, L.F.A.M. (1999) Multi-criteria decision making – an approach to setting priorities in health care. Statistics in Medicine, v. 18, p. 3345-3354. GOMES, L.F.A.M.; LIMA, M.M.P.P. (1992) From Modelling Individual Preferences to Multicriteria Ranking of Discrete Alternatives: A Look at Prospect Theory and the Additive Difference Model. Foundations of Computing and Decision Sciences, v. 17, n. 3-4, p. 171-184. GOMES, L.F.A.M.; LIMA, M.M.P.P. (1991) TODIM: Basics and Application to Multicriteria Ranking of Projects with Environmental Impacts. Foundations of Computing and Decision Sciences, v. 16, n. 3, p. 113-127.
6. The computations in thispaper:the TODIM method is extended by applying the unipolarChoquet integral and we show how this extension allows computing measures of interaction between criteria From the original formulation of TODIM we compute the measure of relative dominance of each alternative Ai over another alternative Aj Through considering the fuzzy measures μ of interactions between criteria we can obtain the overall value of each alternative with no need of normalization If criteria are ordered we can then determine the fuzzy measures (i.e., the measures of criteria interactions)
7. The classical formulation of TODIM’s value function is indeed a particular case of the more general Tversky and Kahneman’s CPT where α=0.5 and λ=1/ θ A more general parametric form of the function c follows: [c (Ai,Aj) is the measure of value of alternative Ai as compared against alternative Aj according to criterion c]
8. The Choquet-extended TODIM method From the classical formulation of TODIM → Measure of relative dominance of each alternative Ai over another alternative Aj: Through consideringthefuzzymeasures μ of interactionsbetweencriteriawecanobtainthe overall value of eachalternativewith no need of normalization. This is accomplishedbyrewrittingtheequationabove: Where a: S →R, and I is the Choquet integral in relation to the fuzzy measure μ. [Choquet, G. (1953)Theory of capacities. Annales de l’ Institut Fourier, 5, p. 131-295; Grabisch, M. & Labreuche, C. (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Annals of Operations Research 175, 1, p. 247-290.]
9. Determining the fuzzymeasures Ifcriteriaare orderedwecanthen determine the fuzzymeasures (interactionsbetweencriteria) as follows: where kj are constants 9
10. An application case study: real estate valuation in Volta Redonda Our case study The case study is a valuation of residential properties carried out by real estate agents in the city of Volta Redonda, Brazil. Fifteen properties in different neighborhoods were analyzed as alternatives and a total of eight evaluation criteria were identified. A detailed description of the alternatives and criteria can be found in Gomes and Rangel (2009). The initial weights assigned to the criteria used to evaluate the properties were defined by decision makers (i.e., the real estate agents), assigning a number between 1 and 5 to each criterion where 1 would mean ‘least important’ and 5 would mean ‘most important’. The information is presented in a future table.
11. Alternatives: A1 – A house in an average location, with 290 m2 of constructed area, a high standard of finishing, in a good state of conservation, with one garage space, 6 rooms, a swimming pool, barbecue and other attractions, without a security system. A2 – A house in a good location, with 180 m2 of constructed area, an average standard of finishing, in an average state of conservation, with one garage space, 4 rooms, a backyard and terrace without a security system. A3 – A house in an average location, with 347 m2 of constructed area, a low standard of finishing, in an average state of conservation, two garage spaces, 5 rooms, a large backyard, without a security system. A4 – A house in an average location, with 124 m2 of constructed area, an average standard of finishing, in a good state of conservation, two garage spaces, 5 rooms, a fruit orchard, a swimming pool and barbecue, without security system. A5 – A house in an excellent location, with 360 m2 of constructed area, a high standard of finishing, in a very good state of conservation, four garage spaces, 9 rooms, a backyard and manned security boxes in the neighborhood streets. A6 – A house located between the periphery and the city center (periphery/average location) with 89 m2 of constructed area, an average standard of finishing, in a good state of conservation, with one garage space, 5 rooms, a backyard, without a security system. A7 – An apartment located in the periphery, with 85 m2 of constructed area, a low standard of finishing, in a bad state of conservation, one garage space, 4 rooms, a manned entrance hall with security. A8 – An apartment in an excellent location, with 80 m2 of constructed area, average standard of finishing, good state of conservation, with one garage space, 6 rooms, manned entrance hall with security. A9 – An apartment located between the periphery and the city center (periphery/average location), with 121 m2 of constructed area, an average standard of finishing, in a good state of conservation, no garage space, 6 rooms, without a security system. A10 – A house located between the periphery and the city center (periphery/average location), with 120 m2 of constructed area, a low standard of finishing, in a good state of conservation, with one garage space, 5 rooms, a large backyard, without a security system. A11 – A house in a good location, with 280 m2 of constructed area, an average standard of finishing, in an average state of conservation, with two garage spaces, 7 rooms, with an additional security system. A12 – An apartment located in the periphery, with 90 m2 of constructed area, a low standard of finishing, in a bad state of conservation, one garage space, 5 rooms, without additional security. A13 – An apartment located in the periphery in an average location, with 160 m2 of constructed area, a high standard of finishing, in a good state of conservation, two garage spaces, 6 rooms, with additional security features. A14 – An apartment in a good location, with 320 m2 of constructed area, high standard of finishing, in a good state of conservation, 2 garage spaces, 8 rooms, with in addition a security system. A15 – A house in a good location, with 180 m2 of constructed area, an average standard of finishing, in a very good state of conservation, one garage space, 6 rooms, with in addition a security system.
12. Computations Computations are performed in 4 steps: Step 1 - fuzzification of the scales of criteria in order to becomenon dimensional. In this presentation fuzzy triangular membership functions with null amplitude and mode equal to the original scale are used. Those fuzzy triangular membership functions are written as below: b,c,d are parameters. Parameters b and c locate the base of the triangle and parameter d locates the vertex.
13. Fuzzy triangular membership functions are graphically displayed as bellow: 16
14. Fuzzification of the scale for Localization (criterion 1) Fuzzification of the scale for Constructed area (criterion 2) 17
15. Fuzzification of the scale for Construction quality (criterion 3) Fuzzification of the scale for State of conservation (criterion 4) 18
16. Fuzzification of the scale for Garage (criterion 5) Fuzzification of the scale for Number of rooms (criterion 6) 19
17. Fuzzification of the scale for Attractions (criterion 7) Fuzzification of the scale for Security (criterion 8) 20
18. The evaluation matrix can now be rewritten after accomplishing the fuzzifications: 21
19. Step 2- determination of fuzzy measures Considering the order of criteria: C1>C4>C2>C3=C6=C8>C5=C7 We have the fuzzy measures to calculate the Choquet integral as: are fuzzy measures which are the weights for the group criteria. We have taken the highest value for μ1 because criterion 1 is the most important one. The other values are proportional or equal following the criteria order. This weighting is performed in a way such that the sum of all measures is equal to 1.0. 22
20. Step 3 - Computation of the Choquet integral 23
21. Some of the computed values of the Choquet integral are shown below as an example: 24
22. The calculations of the Choquet integral is the sum of all the values obtained for each column of the matrix. For the alternative A1, we have: For the alternative A2, we have: and so on. 25
23. Thus, we have: 26
24. Step 4 – Ranking of the alternatives With the values of the Choquet integral we obtain the ranking of the alternatives. This ranking is performed by ordering the obtained values of the Choquet integral. The ranking of the alternatives ordering is shown next: 27
25. Analysisof results Comparing these results with the classical TODIM method:
26. Among the properties that were evaluated, some were inserted as references, simply to assist in the analysis, as their rental values were already known. These properties are presented bellow: 29
27. Comparison of the ranking of alternatives according to the Choquet integral, the classical TODIM method and the known Monthly Rental Value (MRV) are presented next: The method using fuzzification and the Choquet integral misses in ordering A9 and A10(the rent of A10 is higher than that of A9) 30
28. Sensitivity Analysis The sensitivity analysis was performed by modifying the fuzzy measures by increasing and decreasing their values, and recalculating the Choquet integral. The results are presented below: 31
29. Comparing with real rental values: 32
30. The fuzzy measures used in the sensitivity analysis for the Choquet integral were: 33
31. Conclusions The use of the Choquet integral minimizes the calculations of the TODIM method since it is unnecessary to normalize the raw data Not only crisp values can be used but also interval data; this second situation would lead to using a fuzzy triangular number By using the Choquet integral more complex additive models can be used that allow for taking dependencies between criteria into consideration Suggestions for future research extending the TODIM method to situations when input data are not only precise, but also liable to be described by interval or by fuzzy numbers using the bipolar Choquet integral for taking more complex forms of interdependencies between criteria into consideration rental values will then be revised, by taking into consideration the existing values, aiming to use the bipolar Choquet integral and then check if the rank will change making use of Sugeno’s fuzzy inferential system in order to compare the obtained results against these computed by the Choquet-extend TODIM method; in particular, new concepts such as generalizations of Choquet integral as well as the bipolar CPT should be considered.
32. References 1. B. Roy. À propos de la signification des dépendances entre critères: quelle place et quels modes de prise en compte pour l'aide à la décision? RAIRO - Operations Research, Vol. 43 ( 2009) 255-275. 2. S. Greco, V. Mousseau and R. Slowinski. Robust Ordinal Regression for Value Functions Handling Interacting Criteria. CER 12-01, Février 2012. École Central Paris, Laboratoire Génie Industriel, (2012). 3. S. Corrente, S. Greco and R. Slowinski. Robust Ordinal Regression in case of Imprecise Evaluations. arXiv :1206.6317v1 [math.OC], 27 Jun (2012). 4. D. Kahneman and A.Tversky. The psychology of preferences, Scientific American, 246 (1982) 136-141. 5. D. Kahneman and A.Tversky. Choices, values, and frames, American Psychologist 39 (1984) 341-350. 6. D. Kahneman and A. Tversky. Prospect theory: an analysis of decision under risk, Econometrica, 47 (1979) 263-291. 7. A. Tversky and D. Kahneman. Advances in prospect theory, cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992) 297-323. 8. G. Choquet. Theory of capacities. Annales de l’ Institut Fourier, 5 (1953) 131-295. 9. M. Grabisch and C. Labreuche. A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Annals of Operations Research 175, 1 (2010) 247-290. 10. M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis. Tokyo Institute of Technology (1974). 11. S. Greco, B. Matarazzo and S. Giove. The Choquet integral with respect to a level dependent capacity. Fuzzy Sets and Systems 175, 1-35. Online version, doi: 10.1016/j.fss.2011.03.012 (2011). 12. S. Greco and F. Rindone. The bipolar Choquet integral representation. MPRA Paper No. 38957, posted 22. May 2012/18:29. Online at http://mpra.ub.uni-muenchen.de/38957/ (2011).
33. 謝謝Thankyou! Acknowledgements This research was partially supported by CNPq/Brazil through Research Project No. 310603/2009-9 and 302692/2011-8