σ-µ efficiency analysis: A new methodology for evaluating units through composite indices

Greco, Salvatore and Ishizaka, Alessio and Tasiou, Menelaos and Torrisi, Gianpiero (2018): σ-µ efficiency analysis: A new methodology for evaluating units through composite indices.

Abstract

We propose a new methodology to employ composite indicators for performance analysis of units of interest using Stochastic Multiattribute Acceptability Analysis. We start evaluating each unit by means of weighted sums of their elementary indicators in the whole set of admissible weights. For each unit, we compute the mean, µ, and the standard deviation, σ, of its evaluations. Clearly, the former has to be maximized, while the latter has to be minimized as it denotes instability in the evaluations with respect to the variability of weights. We consider a unit to be Pareto-Koopmans efficient with respect to µ and σ if there is no convex combination of µ and σ of the rest of the units with a value of µ that is not smaller, and a value of σ that is not greater, with at least one strict inequality. The set of all Pareto-Koopmans efficient units constitutes the first Pareto-Koopmans frontier. By removing this set and computing the efficiency frontier for the rest of the units, one could obtain the second Pareto-Koopmans frontier. Analogously, the third, fourth and so on Pareto-Koopmans frontiers can be defined. This permits to assign each unit to one of this sequence of Pareto-Koopmans frontiers. We measure the efficiency of each unit not only with respect to the first Pareto-Koopmans frontier, as in the classic Data Envelopment Analysis, but also with respect to the rest of the frontiers, thus enhancing the explicative power of the proposed approach. To illustrate its potential, we apply it to a case study of world happiness based on the data of the homonymous report, annually produced by the United Nations’ Sustainable Development Solutions Network.

• Item Type: MPRA Paper
• Language: English
• Keywords: OR in societal problem analysis; Composite Indicators; Weighting; Sigma-Mu efficiency · Stochastic Multiattribute Acceptability Analysis · Data Envelopment Analysis.
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C43 - Index Numbers and Aggregation; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C44 - Operations Research ; Statistical Decision Theory; I - Health, Education, and Welfare > I3 - Welfare, Well-Being, and Poverty > I31 - General Welfare, Well-Being

References

Abberger, K., Graff, M., Siliverstovs, B., and Sturm, J.-E. (2017). Using rule-based updating procedures to improve the performance of composite indicators. Economic Modelling, 68:127–144.

Aertens, W., Kint, V., Van Orshoven, J., and Muys, B. (2011). Evaluation of modelling techniques for forest site productivity prediction in contrasting ecoregions using stochastic multicriteria acceptability analysis (SMAA). Environmental Modelling & Software, 26(7):929–937.

Ainslie, G. (2001). Breakdown of will. Cambridge, United Kingdom: Cambridge University Press.

Angilella, S., Corrente, S., and Greco, S. (2015). Stochastic multiobjective acceptability analysis for the choquet integral preference model and the scale construction problem. European Journal of Operational Research, 240(1):172–182.

Bandura, R. (2011). Composite Indicators and Rankings: Inventory 2011. Technical report, New York: Office of Development Studies, United Nations Development Programme (UNDP).

Becker, W., Saisana, M., Paruolo, P., and Vandecasteele, I. (2017). Weights and importance in composite indicators: Closing the gap. Ecological Indicators, 80:12–22.

Bewley, T. F. (2002). Knightian decision theory. part i. Decisions in Economics and Finance, 25(2):79–110.

Blackburn, D. W. and Ukhov, A. D. (2013). Individual vs. aggregate preferences: The case of a small fish in a big pond. Management Science, 59(2):470–484.

Booysen, F. (2002). An overview and evaluation of composite indices of development. Social Indicators Research, 59(2):115–151.

Boyd, S. and Vandenberghe, L. (2004). Convex optimization. Cambridge, United Kingdom: Cambridge university press.

Brans, J.-P., Vincke, P., and Mareschal, B. (1986). How to select and how to rank projects: The promethee method. European journal of operational research, 24(2):228–238.

Charnes, A. and Cooper, W. W. (1962). Programming with linear fractional functionals. Naval Research Logistics (NRL), 9(3-4):181–186.

Charnes, A., Cooper, W. W., Golany, B., Seiford, L., and Stutz, J. (1985). Foundations of data envelopment analysis for pareto-koopmans efficient empirical production functions. Journal of econometrics, 30(1-2):91–107.

Charnes, A., Cooper, W. W., and Rhodes, E. (1978a). Measuring the efficiency of decision making units. European journal of operational research, 2(6):429–444.

Charnes, A., Cooper, W. W., and Rhodes, E. (1978b). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6):429–444.

Chowdhury, S. and Squire, L. (2006). Setting weights for aggregate indices: An application to the commitment to development index and human development index. The Journal of Development Studies, 42(5):761–771.

Cooper, W. W., Seiford, L. M., and Zhu, J. (2011). Handbook on data envelopment analysis. International Series in Operations Research & Management Science. 2nd edition, Springer US.

Corrente, S., Figueira, J. R., and Greco, S. (2014). The smaa-promethee method. European Journal of Operational Research, 239(2):514–522.

Corrente, S., Figueira, J. R., Greco, S., and Słowinski, R. (2016a). A robust ranking method extending ´ electre iii to hierarchy of interacting criteria, imprecise weights and stochastic analysis. Omega.

Corrente, S., Greco, S., Kadzinski, M., and Słowinski, R. (2013). Robust ordinal regression in preference learning and ranking. Machine Learning, 93(2-3):381–422.

Corrente, S., Greco, S., Kadzinski, M., and Słowinski, R. (2016b). Inducing probability distributions ´ on the set of value functions by subjective stochastic ordinal regression. Knowledge-Based Systems, 112:26–36.

Costanza, R., Hart, M., Posner, S., and Talberth, J. (2009). Beyond GDP: The Need for New Measures of Progress. Pardee Center for the Study of the Longer-Range Future, Boston.

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE transactions on evolutionary computation, 6(2):182–197.

Decancq, K. and Lugo, M. A. (2013). Weights in Multidimensional Indices of Wellbeing: An Overview. Econometric Reviews, 32(1):7–34.

Decancq, K., Van Ootegem, L., and Verhofstadt, E. (2013). What if we voted on the weights of a multidimensional well-being index? an illustration with flemish data. Fiscal Studies, 34(3):315–332.

Doumpos, M., Gaganis, C., and Pasiouras, F. (2016). Bank Diversification and Overall Financial Strength: International Evidence. Financial Markets, Institutions & Instruments, 25(3):169–213.

Doumpos, M., Hasan, I., and Pasiouras, F. (2017). Bank overall financial strength: Islamic versus conventional banks. Economic Modelling, 64:513–523.

Durbach, I. (2009). On the estimation of a satisficing model of choice using stochastic multicriteria acceptability analysis. Omega, 37(3):497–509.

Elster, J. (1987). The Multiple self. Cambridge, United Kingdom: Cambridge University Press.

Elton, E. J., Gruber, M. J., Brown, S. J., and Goetzmann, W. N. (2009). Modern portfolio theory and investment analysis. John Wiley & Sons.

Figueira, J. R., Greco, S., Roy, B., and Slowinski, R. (2013). An Overview of ELECTRE Methods and their Recent Extensions. Journal of Multi-Criteria Decision Analysis, 20(1-2):61–85.

Figueira, J. R., Mousseau, V., and Roy, B. (2016). ELECTRE methods. In Greco, S., Ehrgott, M., and Figueira, J., editors, Multiple criteria decision analysis: State of the art surveys, pages 155–185. Springer.

Gan, X., Fernandez, I. C., Guo, J., Wilson, M., Zhao, Y., Zhou, B., and Wu, J. (2017). When to use what: Methods for weighting and aggregating sustainability indicators. Ecological Indicators, 81:491–502.

Gilboa, I., Maccheroni, F., Marinacci, M., and Schmeidler, D. (2010). Objective and subjective rationality in a multiple prior model. Econometrica, 78(2):755–770.

Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of mathematical economics, 18(2):141–153.

Greco, S., Ehrgott, M., and Figueira, J. (2016). Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research & Management Science. 2nd edition, New York: Springer.

Greco, S., Ishizaka, A., Matarazzo, B., and Torrisi, G. (2017a). Stochastic multi-attribute acceptability analysis (smaa): an application to the ranking of italian regions, Regional Studies, doi: 10.1080/00343404.2017.1347612, advance online publication.

Greco, S., Ishizaka, A., Tasiou, M., and Torrisi, G. (2017b). Measuring efficiency with multiple frontier data envelopment analysis, Mimeo.

Greco, S., Ishizaka, A., Tasiou, M., and Torrisi, G. (2018). On the methodological framework of composite indices: A review of the issues of weighting, aggregation and robustness, Social Indicators Research, doi: 10.1007/s11205-017-1832-9, advance online publication.

Greco, S., Mousseau, V., and Słowinski, R. (2008). Ordinal regression revisited: multiple criteria ranking ´ using a set of additive value functions. European Journal of Operational Research, 191(2):416–436.

Greco, S., Słowinski, R., Figueira, J., and Mousseau, V. (2010). Robust ordinal regression. ´ Trends in multiple criteria decision analysis, 142:241–283.

Grupp, H. and Schubert, T. (2010). Review and new evidence on composite innovation indicators for evaluating national performance. Research Policy, 39(1):67–78.

Hartley, J. E. and Hartley, J. E. (2002). The representative agent in macroeconomics. London, United Kingdom: Routledge.

Helliwell, J., Layard, R., and Sachs, J. (2012). World Happiness Report 2012. New York: Sustainable Development Solutions Network.

Helliwell, J., Layard, R., and Sachs, J. (2017). World Happiness Report 2017. New York: Sustainable Development Solutions Network.

Ishizaka, A. and Nemery, P. (2013). Multi-Criteria Decision Analysis: Methods and Software. Chichester, United Kingdom: John Wiley & Sons.

Karagiannis, G. (2017). On aggregate composite indicators. Journal of the Operational Research Society, 68(7):741–746.

Kirman, A. P. (1992). Whom or what does the representative individual represent? The Journal of Economic Perspectives, 6(2):117–136.

Kroll, C. and Delhey, J. (2013). A happy nation? opportunities and challenges of using subjective indicators in policymaking. Social Indicators Research, 114(1):13–28.

Lahdelma, R., Hokkanen, J., and Salminen, P. (1998). SMAA - Stochastic multiobjective acceptability analysis. European Journal of Operational Research, 106(1):137–143.

Lahdelma, R. and Salminen, P. (2001). SMAA-2 : Stochastic Multicriteria Acceptability Analysis for Group Decision Making. Operations Research, 49(3):444–454.

Lahdelma, R. and Salminen, P. (2009). Prospect theory and stochastic acceptability analysis (SMAA). Omega, 37(5):961–971.

Leskinen, P., Viitanen, J., Kangas, A., and Kangas, J. (2006). Alternatives to incorporate uncertainty and risk attitude in multicriteria evaluation of forest plans. Forest Science, 52(3):304–312.

Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1):77–91.

Mazziotta, M. and Pareto, A. (2016). On a Generalized Non-compensatory Composite Index for Measuring Socio-economic Phenomena. Social Indicators Research, 127(3):983–1003.

McClure, S. M., Laibson, D. I., Loewenstein, G., and Cohen, J. D. (2004). Separate neural systems value immediate and delayed monetary rewards. Science, 306(5695):503–507.

Menou, A., Benallou, A., Lahdelma, R., and Salminen, P. (2010). Decision support for centralizing cargo at a Moroccan airport hub using stochastic multicriteria acceptability analysis. European Journal of Operational Research, 204(3):621–629.

Mikuli´c, J., Koži´c, I., and Kreši´c, D. (2015). Weighting indicators of tourism sustainability: A critical note. Ecological Indicators, 48:312–314. Munda, G. (2005a). "Measuring sustainability": A multi-criterion framework. Environment, Development and Sustainability, 7(1):117–134.

Munda, G. (2005b). Multiple Criteria Decision Analysis and Sustainable Development. In Greco, S., Ehrgott, M., and Figueira, J., editors, Multiple Criteria Decision Analysis: State of the Art Surveys, pages 953–986.

OECD (2008). Handbook on Constructing Composite Indicators: Methodology and User Guide. OECD Publishing, Paris.

OECD (2016). Strategic Orientations of the Secretary-General for 2016 and beyond. Technical report, Meeting of the OECD Council at Ministerial Level. Retrieved from https://www.oecd.org/mcm/documents/strategic-orientations-of-the-secretary-general-2016.pdf.

Paruolo, P., Saisana, M., and Saltelli, A. (2013). Ratings and rankings: voodoo or science? Journal of the Royal Statistical Society: Series A (Statistics in Society), 176(3):609–634.

Pearson, K. (1901). LIII. On lines and planes of closest fit to systems of points in space. Philosophical Magazine Series 6, 2(11):559–572.

Ray, A. K. (2008). Measurement of social development: an international comparison. Social Indicators Research, 86(1):1–46.

Robinson, D. N. (1989). Aristotle’s psychology. New York: Columbia University Press.

Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15(3):234–281.

Saaty, T. L. (1980). The Analytic Hierarchy Process. McGraw-Hill, New York.

Saisana, M., Saltelli, A., and Tarantola, S. (2005). Uncertainty and sensitivity analysis techniques as tools for the quality assessment of composite indicators. Journal of the Royal Statistical Society. Series A: Statistics in Society, 168(2):307–323.

Saltelli, A. (2007). Composite indicators between analysis and advocacy. Social Indicators Research, 81(1):65–77.

Schelling, T. C. (1980). The intimate contest for self-command. The Public Interest, (60):94.

Sharpe, A. (2004). Literature Review of Frameworks for Macro-indicators. Centre for the Study of Living Standards, Ottawa.

Shin, D. C. and Johnson, D. M. (1978). Avowed happiness as an overall assessment of the quality of life. Social indicators research, 5(1-4):475–492.

Spearman, C. (1904). "General Intelligence", Objectively Determined and Measured. The American Journal of Psychology, 15(2):201–292.

Stiglitz, J., Sen, A. K., and Fitoussi, J.-P. (2009). The measurement of economic performance and social progress revisited: Reflections and Overview. Commission on the Measurement of Economic Performance and Social Progress, Paris.

Tervonen, T., Figueira, J., Lahdelma, R., Almeida Dias, J., and Salminen, P. (2009a). A stochastic method for robustness analysis in sorting problems. European Journal of Operational Research, 192(1):236– 242.

Tervonen, T., Figueira, J., Lahdelma, R., and Salminen, P. (2009b). SMAA-III: A simulation-based approach for sensitivity analysis of ELECTRE III. In Real-Time and Deliberative Decision Making, pages 241–253. Springer.

Tervonen, T. and Figueira, J. R. (2008). A survey on stochastic multicriteria acceptability analysis methods. Journal of Multi-Criteria Decision Analysis, 15(1-2):1–14.

Tervonen, T. and Lahdelma, R. (2007). Implementing stochastic multicriteria acceptability analysis. European Journal of Operational Research, 178(2):500–513.

Tervonen, T., Linkov, I., Figueira, J. R., Steevens, J., Chappel, M., and Merad, M. (2009c). Risk-based classification system of nanomaterials. Journal of Nanoparticle Research, 11(4):757–766.

Van Puyenbroeck, T. and Rogge, N. (2017). Geometric mean quantity index numbers with benefit-ofthe-doubt weights. European Journal of Operational Research, 256(3):1004–1014.

Yang, L. (2014). An Inventory of Composite Measures of Human Progress. Technical report, UNDP Human Development Report Office.