Multicriteria Analysis with unknown preferences : An application of SMAA-2

1. Ubatuba, São Paulo, Brazil, August 2011 2011 Multicriteria Analysiswithunknownpreferences: An application of SMAA-2 Luiz Flávio Autran Monteiro Gomes, Ibmec/RJ, Av. Presidente Wilson, 118, Sala 1110, Centro, 20.030-020, Rio de Janeiro, RJ, Brazil, e-mail: autran@ibmecrj.br Luis Alberto Duncan Rangel, School of Metallurgical Industrial Engg. of Volta Redonda, Universidade Federal Fluminense, Av. dos Trabalhadores 420, Vila Santa Cecília, CEP 27255-125, Volta Redonda, RJ, Brazil, e-mail: duncan@metal.eeimvr.uff.br

2. Summary of thepresentationApplications of Multicriteria Analysis are normally dependent on preferences concerning alternatives for each criterion. They are also dependent on measures of importance of criteria. Either such preferences or measures are not available or are highly uncertain. Finnish researchers have developed in the last decade a family of analytical methods named SMAA. Methods belonging to this family are SMAA-1, SMAA-D, SMAA-O, SMAA-2, SMAA-3, SMAA-A, SMAA-TRI, Ref-SMAA and SMAA-P. Those consist in essence in formulating inverse problems in the weight space. Those problems lead to solving multidimensional integrals and can be approached by Monte Carlo simulation. In this article major concepts of SMAA methods are presented. A numerical application example of one of the most important among these methods, the SMAA-2 method, is shown. A future research development is foreseen.

3. The approach of Stochastic Multicriteria AcceptabilityAnalysis (SMAA): • In a multicriteria problem, when input data (such as preferences and weights) are notavailable, one can simulate them and then run a descriptive analysis • This descriptive analysis can lead to determining rank acceptability indices for each alternative; these indices serve to understand the variety of different preferences that support an alternative for achieving the highest rank or for achieving any particular rank • The information thus produced can be used in order to classify alternatives into more or less acceptable ones and those that are not acceptable at all • SMAA-2 assumes that each DM participating in the process has a preference structure represented by an individual vector of weights w and a real value utility function u (xi, w); the associated linear utility function would then look as follows:

4. How to solve theinverseproblem: • Theunknown input values are representedbystochasticvariablesijcorresponding to deterministicevaluationsgj(xi)withanassumedorestimatedjointprobabilitydistributionand a jointdensityfunctionfχ() in χ Rmxn • Onecanmake use of a uniformdistribution of weightswhen input information is totallyunknown: • Mostof thetime criteriavaluescanbetreated as independentstochasticvariableswhosejointdensity is expressed as a productsuch as • SMAA-2 thenallowsdetermining for eachalternativethe set of favorableweightsWi() computed as follows: • AnyweightwWi()makesthe general utility of xigreaterthanorequal to theutility of anyotheralternative. Allsubsequentanalyses are thenbasedontheproperties of these sets.

5. Through Monte Carlo simulation a number of indicescanbecomputed: The acceptability index describes the contribution from different evaluations to making a given alternative the most preferred among all alternatives: The central weight vector is the expected gravity center of the favorable weight space: The confidence factor is the probability that one alternative will reach the highest rank when the central weight vector and the central reference point are chosen: SMAA-2 contains most indices used in other methods of the SMAA famility, such asSMAA-1, SMAA-D, SMAA-O, SMAA-3, SMAA-A, SMAA-TRI, Ref-SMAA and SMAA-P

6. SMAA-2 in particular: • Therank of eachalternative is anintegerthat varies fromthehighest (= 1) to thelowest (= m) rank: • Byanalyzingthe set of stochasticweightsoneobtainsthefavorablerank as follows: • Thekeydescriptivemeasure of SMAA-2 is theacceptabilityindex: • Acceptabilityindices are in theinterval [0, 1], where 1 denotes thatthealternativewillneverreach a givenrank and 1 indicatesthatthealternativewillalwaysreachthatrank, regardless of thechosenweights

7. Some basicreferenceson SMAA methods: • Lahdelma, R.; Hokkanen, J. & Salminen, P. SMAA – stochastic multiobjective acceptability analysis. European Journal of Operational Research, 106: 137-143, 1998. • Lahdelma, R., Miettinen, K. & Salminen, P. Stochastis Multicriteria Acceptability Analysis using achievement functions, Technical Report 459, TUCS – Turku Center for Computer science, Available at http://www.tucs.fi, 2002. • Lahdelma, R.; Miettinen, K. & Salminen, P. Ordinal criteria in stochastic multicriteria acceptability analysis (SMAA), European Journal of Operational Research, 147(1): 117-127, 2003. • Lahdelma, R.; Miettinen, K. & Salminen, P. Reference point approach for multiple decision makers, European Journal of Operational Research, 164(3): 785-791, 2005. • Lahdelma, R. & Salminen, P. Prospect theory and stochastic multicriteria acceptability analysis (SMAA), Omega 37, 961-971, 2009. • Lahdelma R. & Salminen P. SMAA-2: stochastic multicriteria acceptability analysis for group decision making, Operations Research, v. 49, p. 444-454, 2001. • Lahdelma, R. & Salminen, P. Pseudo-criteria versus linear utility function in stochastic multi-criteria acceptability analysis, European Journal of Operational Research, 14: 454–469, 2002. • Lahdelma, R.; Salminen, P. & Hokkanen, J. Combining Stochastic Multiobjective Acceptability Analysis and DEA. In: D.K. Despotis & C. Zopounidis (eds.) Integrating Technology & Human Decisions: Global Bridges into the 21st Century, Athens: New Technologies Publications, 629-632, 1999. • Lahdelma, R.; Salminen, P. & Hokkanen, J. Locating a waste treatment facility by using stochastic multicriteria acceptability analysis with ordinal criteria, European Journal of Operational Research, 142:345–356, 2002.

8. Some basicreferenceson SMAA methods (cont.): • Tervonen, T. JSMAA: An open source software for SMAA computations. In: Proceedings of the 25th Mini EURO Conference on Uncertainty and Robustness in Planning and Decision Making (URPDM2010),C. Henggeler Antunes, D. Rios Insua & L.C. Dias (eds.), Coimbra, 2010. • Tervonen, T. & Figueira, J. R. A survey on stochastic multicriteria acceptability analysis methods, Journal of Multi-Criteria Decision Analysis, 15: 1–14, 2008. • Tervonen, T.; Hillege, H.; Buskens, E. & Postmus, D. A state-of-the-art multi-criteria model for drug benefit-risk analysis. SOM Research School Report No. 10001, University of Groningen, 2010. • Tervonen, T & Lahdelma, R. Implementing stochastic multicriteria acceptability analysis. European Journal of Operational Research, 178:2, 500-513, 2007. • Tervonen, T.; Lahdelma, R.; Almeida-Dias, J.; Figueira, J. R.; & Salminen, P. SMAA-TRI: a parameter stability analysis method for ELECTRE-TRI. In: Environmental Security in Harbours and Coastal Areas, G.A. Kiker & I. Linkov (eds), Berlin: Springer, 217-231, 2007. • Tervonen, T.; Lahdelma, R. & Salminen, P. A method for elicitating and combining group preferences for Stochastic Multicriteria Acceptability Analysis, Technical Report 638, TUCS – Turku Center for Computer Science, Available at http://www.tucs.fi, 2004.

9. Some predecessors of SMAA methods: • Bana e Costa, C.A. A multicriteria decision aid methodology to deal with conflicting situations on the weights. European Journal of Operational Research, 26: 22-34, 1986. • Charnetski, J.R. The multiple attribute problem with partial information: the expected value and comparative hypervolume methods, Ph.D. Thesis. Austin: The University of Texas at Austin, 1973. • Charnetski, J.R. & Soland, R.M. Statistical measures for linear functions on polytopes, Operations Research 24, 201-204, 1976. • Rietveld, P. Multiple objective decision methods and regional planning, Amsterdam: North-Holland, 1980. • Rietveld, P. & Ouwersloot, H. Ordinal data in multicriteria decision making, a stochastic dominance approach to siting nuclear power plants, European Journal of Operational Research 56: 249-262, 1992.

10. An application example: a freshgraduatewants to selectan MBA program • 5 alternatives and 4 criteria • Cost of investment • Additionalcosts (includingcost of living) • Expectedsalary (after 3 years) • Rank of thecourse in the Financial Times

12. Steps for applying SMAA-2: • Define theproblem in terms of criteria and alternatives • Choose one shape and scale for the utility function that DM must jointly use • Include all information available regarding weights • Compute ranks for acceptability indices for each alternative and for each rank • Compute the acceptability index, the central weight function, and the confidence factors • Present the results to the DM. Those must select which measure can be more useful, a or b, where a. ranks of acceptability and b. confidence factor. • DM must then decide if the procedure is complete or if it has to be reviewed

17. Our DM then decides to eliminatethe DUF alternative, becausethiscorresponds to theworstacceptabilityindex for thefirstrank A1 and because it hasthelowestconfidencelevel. Next, he decides to eliminatethe HKU alternative, becausethishastheworstvalues of acceptabilityindices for ranks A2, A3, and A4. He then decides to consider, in his final analysis, thealternatives EAESP, CBS, and LBS, becausethosepresentthehighestaveragevalues for theacceptabilityindices. Takingintoaccountthesethreealternativesonly and consideringthattheconfidencelevels are relativelylow for thesethreealternatives, in order to make a rationaldecisionthe DM mustevaluatethethe central weightvectors for each of them. The EAESP alternativehas central weightswiththelowestintercriteriavariation and thismakesthisalternative to bethemostattractive for theprevailingscenario. The EAESP alternativehasalsothebestvalues of acceptabilityindices for thefirst and secondranks (A1 and A2) as well as thehighestvalue of confidencelevelamongstallalternatives. Ranking alternativesbasedontheacceptabilityindiceswillbe as follows: Choosingthe overall bestalternative:

18. A future research development: • As a future research development we foresee developing and applying SMAA-P methods, i.e. inverse approaches to prospect theoretically founded multicriteria approaches AcknowledgementsResearch leading to this article was partially supported by the National Council for Scientific and Technological Development (CNPq) of Brazil through Processes No. 310603/2009-9 and 502711/2009-4 Thank you!