运用藤copula构造模型分析多组决策变量之间的相互依赖以及不确定性,创新地将SMAA方法与copula相依性分析结合在一起,得到更优良更有效的随机多准则可接受性分析方法。介绍基本的藤copula建模分析和SMAA的方法,展示两种方法结合使用的完整步骤,并在数据模拟分析实践中,通过对比新方法与原有的SMAA方法,得到在不同相互依赖结构下的表现结果,证明了新方法更普适、更准确的优点。
Stochastic multi-criteria acceptability analysis (SMAA) is a series of methods for multicriteria decision making. It is used to give decision opinions on the problem of uncertain or missing preferences of decision makers, the uncertainty of decision variables and incomplete or missing preference information can be represented by probability distribution. This method improves the past methods by considering inversely what specific parameters each decision outcome is determined by. However, the existing SMAA methods does not take into account the impact of the interdependence between these variables on the analysis results, or simply using a simple model such as a Gaussian distribution to describe these interdependencies, so that the analysis results do not have sufficient scientific basis. In order to fix this fault, this paper create an innovation by combining copula dependency analysis with stochastic multi-criteria acceptability analysis method, and uses vine copula modeling method to describe the uncertainty of decision variables and their interdependence, making the SMAA method complete and more effective. This paper introduces the basic methods of SMAA and vine copula separately, and gives the specific steps of combining the two methods. In simulation experiments, a comparative analysis is conducted between the original SMAA method and the method given in the paper to show their advantages and disadvantages under different dependency structures.
[1] Keeney R L, Raiffa H, Meyer R F. Decisions with multiple objectives:preferences and value trade-offs[M]. Cambridge, UK:Cambridge university press, 1993.
[2] Roy B. Decision aiding:major actors and the role of models[M]//Multicriteria methodoly for decision making. Boston, USA:Springer, 1996:237-267.
[3] Brans J P, Mareschal B. Promethee methods[M]//Multiple criteria decision analysis. New York, USA:Springer, 2005:163-186.
[4] Xu X Z. The SIR methods:a superiority and inferiority ranking method for multiple criteria decision making[J]. European Journal of Operational Research, 2001, 131(3):587-602. DOI:10.1016/S0377-2217(00)00101-6.
[5] Lahdelma R, Hokkanen J, Salminen P. SMAA-stochastic multiobjective acceptability analysis[J]. European Journal of Operational Research, 1998, 106(1):137-143. DOI:10.1016/S0377-2217(97)00163-X.
[6] Lahdelma R, Salminen P. SMAA-2:stochastic multicriteria acceptability analysis for group decision making[J]. Operations Research, 2001, 49(3):444-454. DOI:10.1287/opre.49.3.444.11220.
[7] Lahdelma R, Miettinen K, Salminen P. Ordinal criteria in stochastic multicriteria acceptability analysis (SMAA)[J]. European Journal of Operational Research, 2003, 147(1):117-127. DOI:10.1016/S0377-2217(02)00267-9.
[8] Yu W. ELECTRE TRI:aspects mthodologiques et manuel dutilisation[D]. Paris, France:Universit de Paris-Dauphine, 1992.
[9] Tervonen T, Lahdelma R, Dias J A, et al. SMAA-TRI:a parameter stability analysis method for ELECTRE-TRI[C]//Kiker G A, Linkov I. Environmental Security in Harbors and Coastal Areas. Berlin, Germany:Springer, 2007:217-231.
[10] Tervonen T, Figueira J R, Lahdelma R, et al. Towards robust ELECTRE Ⅲ with simulation:theory and software of SMAA-Ⅲ[J]. European Journal of Operational Research, under review, 2007.
[11] Lahdelma R, Salminen P. Stochastic multicriteria acceptability analysis using the data envelopment model[J]. European Journal of Operational Research, 2006, 170(1):241-252. DOI:10.1016/j.ejor.2004.07.040.
[12] Lahdelma R, Salminen P. Prospect theory and stochastic multicriteria acceptability analysis (SMAA)[J]. Omega, 2009, 37(5):961-971. DOI:10.1016/j.omega.2008.09.001.
[13] Corrente S, Figueira J R, Greco S. The SMAA-PROMETHEE method[J]. European Journal of Operational Research, 2014, 239(2):514-522. DOI:10.1016/j.ejor.2014.05.026.
[14] Tervonen T, Lahdelma R. Implementing stochastic multicriteria acceptability analysis[J]. European Journal of Operational Research, 2007, 178(2):500-513. DOI:10.1016/j.ejor.2005.12.037.
[15] Hokkanen J, Lahdelma R, Miettinen K, et al. Determining the implementation order of a general plan by using a multicriteria method[J]. Journal of Multi-Criteria Decision Analysis, 1998, 7(5):273-284. DOI:10.1002/(SICI)1099-1360(199809)7:5<273:AID-MCDA198>3.0.CO;2-1.
[16] Hokkanen J, Lahdelma R, Salminen P. A multiple criteria decision model for analyzing and choosing among different development patterns for the Helsinki cargo harbor[J]. Socio Economic Planning Sciences, 1999, 33(1):1-23. DOI:10.1016/S0038-0121(98)00007-X.
[17] Hokkanen J, Lahdelma R, Salminen P. Multicriteria decision support in a technology competition for cleaning polluted soil in Helsinki[J]. Journal of Environmental Management, 2000, 60(4):339-348. DOI:10.1006/jema.2000.0389.
[18] Lahdelma R, Salminen P, Hokkanen J. Locating a waste treatment facility by using stochastic multicriteria acceptability analysis with ordinal criteria[J]. European Journal of Operational Research, 2002, 142(2):345-356. DOI:10.1016/S0377-2217(01)00303-4.
[19] Tervonen T, Hakonen H, Lahdelma R. Elevator planning with stochastic multicriteria acceptability analysis[J]. Omega, 2008, 36(3):352-362. DOI:10.1016/j.omega.2006.04.017.
[20] Kangas J, Hokkanen J, Kangas A S, et al. Applying stochastic multicriteria acceptability analysis to forest ecosystem management with both cardinal and ordinal criteria[J]. Forest Science, 2003, 49(6):928-937.
[21] Xia Q, Hua Z S, Song S L. Location selection of distribution centers based on SMAA[J]. Lecture Notes in Electrical Engineering, 2014, 273(1):213-220. DOI:10.1007/978-3-642-40640-9-28.
[22] Zhu F L, Zhong P G, Wu Y N, et al. SMAA-based stochastic multi-criteria decision making for reservoir flood control operation[J]. Stochastic Environmental Research and Risk Assessment, 2017, 31(6):1485-1497. DOI:10.1007/s00477-016-1253-3.
[23] Lahdelma R, Makkonen S, Salminen P. Multivariate gaussian criteria in SMAA[J]. European Journal of Operational Research, 2006, 170(3):957-970. DOI:10.1016/j.ejor.2004.08.022.
[24] Genest C, Favre A C. Everything you always wanted to know about copula modeling but were afraid to ask[J]. Journal of Hydrologic Engineering, 2007, 12(4):347-368. DOI:10.1061/(asce)1084-0699(2007)12:4(347).
[25] Genest C, Rémillard B, Beaudoin D. Goodness-of-fit tests for copulas:a review and a power study[J]. Insurance Mathematics and Economics, 2009, 44(2):199-213. DOI:10.1016/j.insmatheco.2007.10.005.
[26] Joe H. Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependence parameters[J]. Lecture Notes-Monograph Series, 1996, 28:120-141.
[27] Bedford T, Cooke R M. Probability density decomposition for conditionally dependent random variables modeled by vines[J]. Annals of Mathematics and Artificial Intelligence, 2001, 32(1):245-268. DOI:10.1023/A:1016725902970.
[28] Aas K, Czado C, Frigessi A, et al. Pair-copula constructions of multiple dependence[J]. Insurance Mathematics and Economics, 2009, 44(2):182-198. DOI:10.1016/j.insmatheco.2007.02.001.
[29] Sklar A. Fonctions de répartition à n dimensions et leurs marges[J]. Publications de l'Institut de Statistique de l'Uni-versité de Paris, 1959(8):229-231.
[30] Genest C, Rivest L P. Statistical inference procedures for bivariate archimedean copulas[J]. Journal of the American Statistical Association, 1993, 88(423):1034-1043. DOI:10.1080/01621459.1993.10476372.
[31] Bedford T, Cooke R M. Vines-a new graphical model for dependent random variables[J]. The Annals of Statistics, 2002, 30(4):1031-1068. DOI:10.1214/AOS[WTB4]% 2F1031689016.
[32] 叶五一,郭人榛,缪柏其.基于R藤copula变点模型的金砖四国金融传染性与稳定性检验[J].中国科学技术大学学报, 2018, 48(8):655-666. DOI:10.3969/j.issn.0253-2778.2018.08.009.
[33] Diβmann J F. Statistical inference for regular vines and application[D]. Munich, Germany:Technische Universität München, 2010.
[34] Joe H, Kurowicka D. Dependence modeling:vine copula handbook[M]. Singapore:World Scientific, 2010. DOI:10.1142/7699.
