Portfolio optimization based on downside risk: a mean-semivariance ef¿cient frontier from Dow Jones blue chips

To create efficient funds appealing to a sector of bank clients, the objective of minimizing downside risk is relevant to managers of funds offered by the banks. In this paper, a case focusing on this objective is developed. More precisely, the scope and purpose of the paper is to apply the mean-semivariance efficient frontier model, which is a recent approach to portfolio selection of stocks when the investor is especially interested in the constrained minimization of downside risk measured by the portfolio semivariance. Concerning the opportunity set and observation period, the mean-semivariance efficient frontier model is applied to an actual case of portfolio choice from Dow Jones stocks with daily prices observed over the period 2005¿2009. From these daily prices, time series of returns (capital gains weekly computed) are obtained as a piece of basic information. Diversification constraints are established so that each portfolio weight cannot exceed 5 per cent. The results show significant differences between the portfolios obtained by mean-semivariance efficient frontier model and those portfolios of equal expected returns obtained by classical Markowitz mean-variance efficient frontier model. Precise comparisons between them are made, leading to the conclusion that the results are consistent with the objective of reflecting downside riskPla Santamaría, D.; Bravo Selles, M. (2013). Portfolio optimization based on downside risk: a mean-semivariance ef¿cient frontier from Dow Jones blue chips. Annals of Operations Research. 205(1):189-201. doi:10.1007/s10479-012-1243-xS1892012051Aouni, B. (2009). Multi-attribute portfolio selection: new perspectives. INFOR. Information Systems and Operational Research, 47(1), 1–4.Arenas, M., Bilbao, A., & Rodríguez, M. V. (2001). A fuzzy goal programming approach to portfolio selection. European Journal of Operational Research, 133, 287–297.Arrow, K. J. (1965). Aspects of the theory of risk-bearing. Helsinki: Academic Bookstore.Ballestero, E. (2005). Mean-semivariance efficient frontier: a downside risk model for portfolio selection. Applied Mathematical Finance, 12(1), 1–15.Ballestero, E., & Pla-Santamaria, D. (2004). Selecting portfolios for mutual funds. Omega, 32, 385–394.Ballestero, E., & Pla-Santamaria, D. (2005). Grading the performance of market indicators with utility benchmarks selected from Footsie: a 2000 case study. Applied Economics, 37, 2147–2160.Ballestero, E., Pérez-Gladish, B., Arenas-Parra, M., & Bilbao-Terol, A. (2009). Selecting portfolios given multiple Eurostoxx-based uncertainty scenarios: a stochastic goal programming approach from fuzzy betas. INFOR. Information Systems and Operational Research, 47(1), 59–70.Ben Abdelaziz, F., & Masri, H. (2005). Stochastic programming with fuzzy linear partial information on time series. European Journal of Operational Research, 162(3), 619–629.Ben Abdelaziz, F., Aouni, B., & El Fayedh, R. (2007). Multi-objective stochastic programming for portfolio selection. European Journal of Operational Research, 177(3), 1811–1823.Bermúdez, J. D., Segura, J. V., & Vercher, E. (2012). A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection. Fuzzy Sets and Systems, 188, 16–26.Bilbao, A., Arenas, M., Jiménez, M., Pérez- Gladish, B., & Rodríguez, M. V. (2006). An extension of Sharpe’s single-index model: portfolio selection with expert betas. Journal of the Operational Research Society, 57(12), 1442–1451.Chang, T. J., Yang, S. Ch., & Chang, K. J. (2009). Portfolio optimization problems in different risk measures using genetic algorithm. IEEE Intelligent Systems & Their Applications, 36, 10529–10537.Haugen, R. A. (1997). Modern investment theory. Upper Saddle River: Prentice-Hall.Huang, H. J., Tzeng, G. H., & Ong, C. S. (2006). A novel algorithm for uncertain portfolio selection. Applied Mathematics and Computation, 173(1), 350–359.Konno, H., Waki, H., & Yuuki, A. (2002). Portfolio optimization under lower partial risk measures. Asia-Pacific Financial Markets, 9, 127–140.Lin, C. M., Huang, J. J., Gen, M., & Tzeng, G. H. (2006). Recurrent neural network for dynamic portfolio selection. Applied Mathematics and Computation, 175(2), 1139–1146.Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7, 77–91.Ong, C. S., Huang, J. J., & Tzeng, G. H. (2005). A novel hybrid model for portfolio selection. Applied Mathematics and Computation, 169(2), 1195–1210.Pendaraki, K., Doumpos, M., & Zopounidis, C. (2004). Towards a goal programming methodology for constructing equity mutual fund portfolios. Journal of Asset Management, 4(6), 415–428.Pérez-Gladish, B., Jones, D. F., Tamiz, M., & Bilbao-Terol, A. (2007). An interactive three-stage model for mutual funds portfolio selection. Omega, 35(1), 75–88.Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32(1–2), 122–136.Sharpe, W. F. (1994). The Sharpe ratio. The Journal of Portfolio Management, 21(1), 49–58.Sortino, F. A., & Van der Meer, V. (1991). Downside risk. The Journal of Portfolio Management, 17(4), 27–31.Speranza, M. G. (1993). Linear programming model for portfolio optimization. Finance, 14, 107–123.Steuer, R., Qi, Y., & Hirschberger, M. (2005). Multiple objectives in portfolio selection. Journal of Financial Decision Making, 1(1), 5–20.Steuer, R., Qi, Y., & Hirschberger, M. (2007). Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection. Annals of Operations Research, 152, 297–317.Vercher, E., Bermúdez, J. D., & Segura, J. V. (2007). Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets and Systems, 158, 769–782

Modelos multicriterio para la selección de portafolios en la Bolsa de Madrid

En este trabajo se aplican modernas técnicas multicriterio para la selección de carteras, partiendo de información empírica muy amplia proveniente de la Bolsa de Madrid. En efecto, el número de títulos-valores considerados asciende a 104 durante un reciente período de cinco años, habiéndose recogido rendimientos por plusvalías, dividendos y ampliaciones de capital con periodicidad mensual a lo largo del periodo histórico. Basándose en este material estadístico, se han obtenido doce fronteras eficientes diversificadas, analizando sus peculiaridades. Estas fronteras se han diseñado de tal modo que cumplen en ellas las restricciones legales en cuanto a diversificación. Para seleccionar las carteras óptimas, se recurre a técnicas de bounding que se fundamentan en teoremas recientemente aparecidos en la literatura. Los objetivos del inversor son relevantes para optimizar los portafolios, teniendo en cuenta los coeficientes de aversión al riesgo, y más en general, las RMS entre rentabilidad y seguridad, de acuerdo con las preferencias inversoras. Los teoremas indicados permiten conseguir aproximaciones al óptimo cuando se carece de información completa sobre la función de utilidad. Este caso resulta especialmente importante, dadas las dificultades para especificar formas y parámetros de utilidad con respecto a fondos de inversión y otros inversores colectivos. Sin embargo, la fisonomía correspondiente al fondo de inversión y sus perfiles gestores dan lugar en la tesis a distintas alternativas de cartera, en casos tan diversos como las estrategias activas y la política buy & hold.Pla Santamaría, D. (2000). Modelos multicriterio para la selección de portafolios en la Bolsa de Madrid [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/5184Palanci

Extended Fuzzy Analytic Hierarchy Process (E-FAHP): A General Approach

[EN] Fuzzy analytic hierarchy process (FAHP) methodologies have witnessed a growing development from the late 1980s until now, and countless FAHP based applications have been published in many fields including economics, finance, environment or engineering. In this context, the FAHP methodologies have been generally restricted to fuzzy numbers with linear type of membership functions (triangular numbers-TN-and trapezoidal numbers-TrN). This paper proposes an extended FAHP model (E-FAHP) where pairwise fuzzy comparison matrices are represented by a special type of fuzzy numbers referred to as (m,n)-trapezoidal numbers (TrN (m,n)) with nonlinear membership functions. It is then demonstrated that there are a significant number of FAHP approaches that can be reduced to the proposed E-FAHP structure. A comparative analysis of E-FAHP and Mikhailov's model is illustrated with a case study showing that E-FAHP includes linear and nonlinear fuzzy numbers.Reig-Mullor, J.; Pla Santamaría, D.; Garcia-Bernabeu, A. (2020). Extended Fuzzy Analytic Hierarchy Process (E-FAHP): A General Approach. Mathematics. 8(11):1-14. https://doi.org/10.3390/math8112014S114811Chai, J., Liu, J. N. K., & Ngai, E. W. T. (2013). Application of decision-making techniques in supplier selection: A systematic review of literature. Expert Systems with Applications, 40(10), 3872-3885. doi:10.1016/j.eswa.2012.12.040Tavana, M., Zareinejad, M., Di Caprio, D., & Kaviani, M. A. (2016). An integrated intuitionistic fuzzy AHP and SWOT method for outsourcing reverse logistics. Applied Soft Computing, 40, 544-557. doi:10.1016/j.asoc.2015.12.005Medasani, S., Kim, J., & Krishnapuram, R. (1998). An overview of membership function generation techniques for pattern recognition. International Journal of Approximate Reasoning, 19(3-4), 391-417. doi:10.1016/s0888-613x(98)10017-8Medaglia, A. L., Fang, S.-C., Nuttle, H. L. W., & Wilson, J. R. (2002). An efficient and flexible mechanism for constructing membership functions. European Journal of Operational Research, 139(1), 84-95. doi:10.1016/s0377-2217(01)00157-6Mikhailov, L. (2003). Deriving priorities from fuzzy pairwise comparison judgements. Fuzzy Sets and Systems, 134(3), 365-385. doi:10.1016/s0165-0114(02)00383-4Appadoo, S. S. (2014). Possibilistic Fuzzy Net Present Value Model and Application. Mathematical Problems in Engineering, 2014, 1-11. doi:10.1155/2014/865968Mikhailov, L., & Tsvetinov, P. (2004). Evaluation of services using a fuzzy analytic hierarchy process. Applied Soft Computing, 5(1), 23-33. doi:10.1016/j.asoc.2004.04.001Hepu Deng. (1999). Multicriteria analysis with fuzzy pairwise comparison. FUZZ-IEEE’99. 1999 IEEE International Fuzzy Systems. Conference Proceedings (Cat. No.99CH36315). doi:10.1109/fuzzy.1999.793038Van Laarhoven, P. J. M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11(1-3), 229-241. doi:10.1016/s0165-0114(83)80082-7Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17(3), 233-247. doi:10.1016/0165-0114(85)90090-9Chang, D.-Y. (1996). Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research, 95(3), 649-655. doi:10.1016/0377-2217(95)00300-2Enea, M., & Piazza, T. (2004). Project Selection by Constrained Fuzzy AHP. Fuzzy Optimization and Decision Making, 3(1), 39-62. doi:10.1023/b:fodm.0000013071.63614.3dKrejčí, J., Pavlačka, O., & Talašová, J. (2016). A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic. Fuzzy Optimization and Decision Making, 16(1), 89-110. doi:10.1007/s10700-016-9241-0Cakir, O., & Canbolat, M. S. (2008). A web-based decision support system for multi-criteria inventory classification using fuzzy AHP methodology. Expert Systems with Applications, 35(3), 1367-1378. doi:10.1016/j.eswa.2007.08.041Isaai, M. T., Kanani, A., Tootoonchi, M., & Afzali, H. R. (2011). Intelligent timetable evaluation using fuzzy AHP. Expert Systems with Applications, 38(4), 3718-3723. doi:10.1016/j.eswa.2010.09.030Büyüközkan, G., & Güleryüz, S. (2016). A new integrated intuitionistic fuzzy group decision making approach for product development partner selection. Computers & Industrial Engineering, 102, 383-395. doi:10.1016/j.cie.2016.05.038Zheng, G., Zhu, N., Tian, Z., Chen, Y., & Sun, B. (2012). Application of a trapezoidal fuzzy AHP method for work safety evaluation and early warning rating of hot and humid environments. Safety Science, 50(2), 228-239. doi:10.1016/j.ssci.2011.08.042Calabrese, A., Costa, R., & Menichini, T. (2013). Using Fuzzy AHP to manage Intellectual Capital assets: An application to the ICT service industry. Expert Systems with Applications, 40(9), 3747-3755. doi:10.1016/j.eswa.2012.12.081Ishizaka, A., & Nguyen, N. H. (2013). Calibrated fuzzy AHP for current bank account selection. Expert Systems with Applications, 40(9), 3775-3783. doi:10.1016/j.eswa.2012.12.089Somsuk, N., & Laosirihongthong, T. (2014). A fuzzy AHP to prioritize enabling factors for strategic management of university business incubators: Resource-based view. Technological Forecasting and Social Change, 85, 198-210. doi:10.1016/j.techfore.2013.08.007Chan, H. K., Wang, X., & Raffoni, A. (2014). An integrated approach for green design: Life-cycle, fuzzy AHP and environmental management accounting. The British Accounting Review, 46(4), 344-360. doi:10.1016/j.bar.2014.10.004Tan, R. R., Aviso, K. B., Huelgas, A. P., & Promentilla, M. A. B. (2014). Fuzzy AHP approach to selection problems in process engineering involving quantitative and qualitative aspects. Process Safety and Environmental Protection, 92(5), 467-475. doi:10.1016/j.psep.2013.11.005Rezaei, J., Fahim, P. B. M., & Tavasszy, L. (2014). Supplier selection in the airline retail industry using a funnel methodology: Conjunctive screening method and fuzzy AHP. Expert Systems with Applications, 41(18), 8165-8179. doi:10.1016/j.eswa.2014.07.005Song, Z., Zhu, H., Jia, G., & He, C. (2014). Comprehensive evaluation on self-ignition risks of coal stockpiles using fuzzy AHP approaches. Journal of Loss Prevention in the Process Industries, 32, 78-94. doi:10.1016/j.jlp.2014.08.002Dong, M., Li, S., & Zhang, H. (2015). Approaches to group decision making with incomplete information based on power geometric operators and triangular fuzzy AHP. Expert Systems with Applications, 42(21), 7846-7857. doi:10.1016/j.eswa.2015.06.007Mangla, S. K., Kumar, P., & Barua, M. K. (2015). Risk analysis in green supply chain using fuzzy AHP approach: A case study. Resources, Conservation and Recycling, 104, 375-390. doi:10.1016/j.resconrec.2015.01.001Mosadeghi, R., Warnken, J., Tomlinson, R., & Mirfenderesk, H. (2015). Comparison of Fuzzy-AHP and AHP in a spatial multi-criteria decision making model for urban land-use planning. Computers, Environment and Urban Systems, 49, 54-65. doi:10.1016/j.compenvurbsys.2014.10.001Lupo, T. (2016). A fuzzy framework to evaluate service quality in the healthcare industry: An empirical case of public hospital service evaluation in Sicily. Applied Soft Computing, 40, 468-478. doi:10.1016/j.asoc.2015.12.010Tuljak-Suban, D., & Bajec, P. (2018). The Influence of Defuzzification Methods to Decision Support Systems Based on Fuzzy AHP with Scattered Comparison Matrix: Application to 3PLP Selection as a Case Study. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26(03), 475-491. doi:10.1142/s021848851850023xAkbar, M. A., Shameem, M., Mahmood, S., Alsanad, A., & Gumaei, A. (2020). Prioritization based Taxonomy of Cloud-based Outsource Software Development Challenges: Fuzzy AHP analysis. Applied Soft Computing, 95, 106557. doi:10.1016/j.asoc.2020.106557Jung, H. (2011). A fuzzy AHP–GP approach for integrated production-planning considering manufacturing partners. Expert Systems with Applications, 38(5), 5833-5840. doi:10.1016/j.eswa.2010.11.039Shaw, K., Shankar, R., Yadav, S. S., & Thakur, L. S. (2012). Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain. Expert Systems with Applications, 39(9), 8182-8192. doi:10.1016/j.eswa.2012.01.149Abdullah, L., & Zulkifli, N. (2015). Integration of fuzzy AHP and interval type-2 fuzzy DEMATEL: An application to human resource management. Expert Systems with Applications, 42(9), 4397-4409. doi:10.1016/j.eswa.2015.01.021Akkaya, G., Turanoğlu, B., & Öztaş, S. (2015). An integrated fuzzy AHP and fuzzy MOORA approach to the problem of industrial engineering sector choosing. Expert Systems with Applications, 42(24), 9565-9573. doi:10.1016/j.eswa.2015.07.061Kutlu, A. C., & Ekmekçioğlu, M. (2012). Fuzzy failure modes and effects analysis by using fuzzy TOPSIS-based fuzzy AHP. Expert Systems with Applications, 39(1), 61-67. doi:10.1016/j.eswa.2011.06.044Büyüközkan, G., & Çifçi, G. (2012). A combined fuzzy AHP and fuzzy TOPSIS based strategic analysis of electronic service quality in healthcare industry. Expert Systems with Applications, 39(3), 2341-2354. doi:10.1016/j.eswa.2011.08.061Taylan, O., Bafail, A. O., Abdulaal, R. M. S., & Kabli, M. R. (2014). Construction projects selection and risk assessment by fuzzy AHP and fuzzy TOPSIS methodologies. Applied Soft Computing, 17, 105-116. doi:10.1016/j.asoc.2014.01.003Patil, S. K., & Kant, R. (2014). A fuzzy AHP-TOPSIS framework for ranking the solutions of Knowledge Management adoption in Supply Chain to overcome its barriers. Expert Systems with Applications, 41(2), 679-693. doi:10.1016/j.eswa.2013.07.093Sun, L., Ma, J., Zhang, Y., Dong, H., & Hussain, F. K. (2016). Cloud-FuSeR: Fuzzy ontology and MCDM based cloud service selection. Future Generation Computer Systems, 57, 42-55. doi:10.1016/j.future.2015.11.025Ar, I. M., Erol, I., Peker, I., Ozdemir, A. I., Medeni, T. D., & Medeni, I. T. (2020). Evaluating the feasibility of blockchain in logistics operations: A decision framework. Expert Systems with Applications, 158, 113543. doi:10.1016/j.eswa.2020.113543Yalcin, N., Bayrakdaroglu, A., & Kahraman, C. (2012). Application of fuzzy multi-criteria decision making methods for financial performance evaluation of Turkish manufacturing industries. Expert Systems with Applications, 39(1), 350-364. doi:10.1016/j.eswa.2011.07.024Chang, S.-C., Tsai, P.-H., & Chang, S.-C. (2015). A hybrid fuzzy model for selecting and evaluating the e-book business model: A case study on Taiwan e-book firms. Applied Soft Computing, 34, 194-204. doi:10.1016/j.asoc.2015.05.011Li, N., & Zhao, H. (2016). Performance evaluation of eco-industrial thermal power plants by using fuzzy GRA-VIKOR and combination weighting techniques. Journal of Cleaner Production, 135, 169-183. doi:10.1016/j.jclepro.2016.06.113Mandic, K., Delibasic, B., Knezevic, S., & Benkovic, S. (2014). Analysis of the financial parameters of Serbian banks through the application of the fuzzy AHP and TOPSIS methods. Economic Modelling, 43, 30-37. doi:10.1016/j.econmod.2014.07.036Li, Y., Liu, X., & Chen, Y. (2012). Supplier selection using axiomatic fuzzy set and TOPSIS methodology in supply chain management. Fuzzy Optimization and Decision Making, 11(2), 147-176. doi:10.1007/s10700-012-9117-xKaya, Ö., Alemdar, K. D., & Çodur, M. Y. (2020). A novel two stage approach for electric taxis charging station site selection. Sustainable Cities and Society, 62, 102396. doi:10.1016/j.scs.2020.102396Chen, J.-F., Hsieh, H.-N., & Do, Q. H. (2015). Evaluating teaching performance based on fuzzy AHP and comprehensive evaluation approach. Applied Soft Computing, 28, 100-108. doi:10.1016/j.asoc.2014.11.050Javanbarg, M. B., Scawthorn, C., Kiyono, J., & Shahbodaghkhan, B. (2012). Fuzzy AHP-based multicriteria decision making systems using particle swarm optimization. Expert Systems with Applications, 39(1), 960-966. doi:10.1016/j.eswa.2011.07.095Che, Z. H., Wang, H. S., & Chuang, C.-L. (2010). A fuzzy AHP and DEA approach for making bank loan decisions for small and medium enterprises in Taiwan. Expert Systems with Applications, 37(10), 7189-7199. doi:10.1016/j.eswa.2010.04.010Krejčí, J. (2015). Additively reciprocal fuzzy pairwise comparison matrices and multiplicative fuzzy priorities. Soft Computing, 21(12), 3177-3192. doi:10.1007/s00500-015-2000-2Xu, Z., & Liao, H. (2014). Intuitionistic Fuzzy Analytic Hierarchy Process. IEEE Transactions on Fuzzy Systems, 22(4), 749-761. doi:10.1109/tfuzz.2013.2272585Mikhailov, L. (2000). A fuzzy programming method for deriving priorities in the analytic hierarchy process. Journal of the Operational Research Society, 51(3), 341-349. doi:10.1057/palgrave.jors.260089

On the use of multiple criteria distance indexes to find robust cash management policies

[EN] Cash management decision-making can be handled from a multiobjective perspective by optimizing not only cost but also risk. Nevertheless, choosing the best policies under a changing context is by no means straightforward. To this end, we rely on compromise programming to incorporate robustness as an additional goal to cost and risk within a multiobjective framework. As a result, we propose to calculate robustness as a multiple criteria distance index that is able to identify the best compromise policies in terms of cost and risk. Such policies are also robust to cash flow regime changes. We show its utility by transforming the Miller and Orr s cash management model into its robust counterpart using real data from an industrial company.Ministerio de Economia y Competitividad [grant number Collectiveware TIN2015-66863-C2-1-R], [grant number 2014 SGR 118]. Work partially funded by projects Collectiveware TIN2015-66863-C2-1-R (MINECO/FEDER) and 2014 SGR 118.Salas-Molina, F.; Rodriguez-Aguilar, JA.; Pla Santamaría, D. (2019). On the use of multiple criteria distance indexes to find robust cash management policies. INFOR Information Systems and Operational Research. 57(3):345-360. https://doi.org/10.1080/03155986.2017.1282291S34536057

A multi-objective approach to the cash management problem

[EN] Cash management is concerned with optimizing costs of short-term cash policies of a company. Different optimization models have been proposed in the literature whose focus has been only placed on a single objective, namely, on minimizing costs. However, cash managers may also be interested in risk associated to cash policies. In this paper, we propose a multi-objective cash management model based on compromise programming that allows cash managers to select the best policies, in terms of cost and risk, according to their risk preferences. The model is illustrated through several examples using real data from an industrial company, alternative cost scenarios and two different measures of risk. As a result, we provide cash managers with a new tool to allow them deciding on the level of risk to take in daily decision-making.Work partially funded by projects Collectiveware TIN2015-66863-C2-1-R (MINECO/FEDER) and 2014 SGR 118.Salas-Molina, F.; Pla Santamaría, D.; Rodriguez Aguilar, JA. (2016). A multi-objective approach to the cash management problem. Annals of Operations Research. 1-15. https://doi.org/10.1007/s10479-016-2359-1S115Baccarin, S. (2009). Optimal impulse control for a multidimensional cash management system with generalized cost functions. European Journal of Operational Research, 196(1), 198–206.Ballestero, E. (1998). Approximating the optimum portfolio for an investor with particular preferences. Journal of the Operational Research Society, 49(9), 998–1000.Ballestero, E. (2005). Mean-semivariance efficient frontier: A downside risk model for portfolio selection. Applied Mathematical Finance, 12(1), 1–15.Ballestero, E., & Pla-Santamaria, D. (2004). Selecting portfolios for mutual funds. Omega, 32(5), 385–394.Ballestero, E., & Romero, C. (1998). Multiple criteria decision making and its applications to economic problems. Berlin: Springer.Bates, T. W., Kahle, K. M., & Stulz, R. M. (2009). Why do US firms hold so much more cash than they used to? The Journal of Finance, 64(5), 1985–2021.Baumol, W. J. (1952). The transactions demand for cash: An inventory theoretic approach. The Quarterly Journal of Economics, 66(4), 545–556.Chen, X., & Simchi-Levi, D. (2009). A new approach for the stochastic cash balance problem with fixed costs. Probability in the Engineering and Informational Sciences, 23(04), 545–562.Constantinides, G. M., & Richard, S. F. (1978). Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time. Operations Research, 26(4), 620–636.da Costa Moraes, M. B., & Nagano, M. S. (2014). Evolutionary models in cash management policies with multiple assets. Economic Modelling, 39, 1–7.da Costa Moraes, M. B., Nagano, M.S., Sobreiro, V. A. (2015). Stochastic cash flow management models: A literature review since the 1980s. In P. Guarnieri (Ed.), Decision models in engineering and management. Decision engineering (pp. 11–28). Switzerland: Springer.Eppen, G. D., & Fama, E. F. (1969). Cash balance and simple dynamic portfolio problems with proportional costs. International Economic Review, 10(2), 119–133.Gao, H., Harford, J., & Li, K. (2013). Determinants of corporate cash policy: Insights from private firms. Journal of Financial Economics, 109(3), 623–639.Girgis, N. M. (1968). Optimal cash balance levels. Management Science, 15(3), 130–140.Gormley, F. M., & Meade, N. (2007). The utility of cash flow forecasts in the management of corporate cash balances. European Journal of Operational Research, 182(2), 923–935.Gregory, G. (1976). Cash flow models: A review. Omega, 4(6), 643–656.Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91.McNeil, A. J., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques and tools. Princeton: Princeton University Press.Melo, M. A., & Bilich, F. (2013). Expectancy balance model for cash flow. Journal of Economics and Finance, 37(2), 240–252.Miller, M. H., & Orr, D. (1966). A model of the demand for money by firms. The Quarterly Journal of Economics, 80(3), 413–435.Myers, S. C., & Brealey, R. A. (2003). Principles of corporate finance (7th ed.). NewYork: McGraw-Hill.Neave, E. H. (1970). The stochastic cash balance problem with fixed costs for increases and decreases. Management Science, 16(7), 472–490.Penttinen, M. J. (1991). Myopic and stationary solutions for stochastic cash balance problems. European Journal of Operational Research, 52(2), 155–166.Pinkowitz, L., Stulz, R. M., & Williamson, R. (2016). Do US firms hold more cash than foreign firms do? Review of Financial Studies, 29(2), 309–348.Pla-Santamaria, D., & Bravo, M. (2013). Portfolio optimization based on downside risk: A mean-semivariance efficient frontier from dow jones blue chips. Annals of Operations Research, 205(1), 189–201.Premachandra, I. (2004). A diffusion approximation model for managing cash in firms: An alternative approach to the Miller–Orr model. European Journal of Operational Research, 157(1), 218–226.Roijers, D. M., Vamplew, P., Whiteson, S., & Dazeley, R. (2013). A survey of multi-objective sequential decision-making. Journal of Artificial Intelligence Research, 48(1), 67–113.Ross, S. A., Westerfield, R., & Jordan, B. D. (2002). Fundamentals of corporate finance (6th ed.). NewYork: McGraw-Hill.Sharpe, W. F. (1966). Mutual fund performance. The Journal of Business, 39(1), 119–138.Sharpe, W. F. (1994). The sharpe ratio. The Journal of Portfolio Management, 21(1), 49–58.Srinivasan, V., & Kim, Y. H. (1986). Deterministic cash flow management: State of the art and research directions. Omega, 14(2), 145–166.Steuer, R. E., Qi, Y., & Hirschberger, M. (2007). Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection. Annals of Operations Research, 152(1), 297–317.Stone, B. K. (1972). The use of forecasts and smoothing in control-limit models for cash management. Financial Management, 1(1), 72–84.Whalen, E. L. (1966). A rationalization of the precautionary demand for cash. The Quarterly Journal of Economics, 80(2), 314–324.Yu, P. L. (2013). Multiple-criteria decision making: Concepts, techniques, and extensions. Berlin: Springer.Zeleny, M. (1982). Multiple criteria decision making. NewYork: McGraw-Hill.Zopounidis, C. (1999). Multicriteria decision aid in financial management. European Journal of Operational Research, 119(2), 404–415

Characterizing compromise solutions for investors with uncertain risk preferences

[EN] The optimum portfolio selection for an investor with particular preferences was proven to lie on the normalized efficient frontier between two bounds defined by the Ballestero (1998) bounding theorem. A deeper understanding is possible if the decision-maker is provided with visual and quantitative techniques. Here, we derive useful insights as a way to support investor's decision-making through: (i) a new theorem to assess balance of solutions; (ii) a procedure and a new plot to deal with discrete efficient frontiers and uncertain risk preferences; and (iii) two quality metrics useful to predict long-run performance of investors.Work partially funded by projects Collectiveware TIN2015-66863-C2-1-R (MINECO/FEDER) and 2014 SGR 118Salas-Molina, F.; Rodriguez-Aguilar, JA.; Pla Santamaría, D. (2019). Characterizing compromise solutions for investors with uncertain risk preferences. Operational Research. 19(3):661-677. https://doi.org/10.1007/s12351-017-0309-6S661677193Amiri M, Ekhtiari M, Yazdani M (2011) Nadir compromise programming: a model for optimization of multi-objective portfolio problem. Expert Syst Appl 38(6):7222–7226Ballestero E (1998) Approximating the optimum portfolio for an investor with particular preferences. J Oper Res Soc 49:998–1000Ballestero E (2007) Compromise programming: a utility-based linear-quadratic composite metric from the trade-off between achievement and balanced (non-corner) solutions. Eur J Oper Res 182(3):1369–1382Ballestero E, Pla-Santamaria D (2004) Selecting portfolios for mutual funds. Omega 32(5):385–394Ballestero E, Pla-Santamaria D, Garcia-Bernabeu A, Hilario A (2015) Portfolio selection by compromise programming. In: Ballestero E, Pérez-Gladish B, Garcia-Bernabeu A (eds) Socially responsible investment. A multi-criteria decision making approach, vol 219. Springer, Switzerland, pp 177–196Ballestero E, Romero C (1996) Portfolio selection: a compromise programming solution. J Oper Res Soc 47(11):1377–1386Ballestero E, Romero C (1998) Multiple criteria decision making and its applications to economic problems. Kluwer Academic Publishers, BerlinBilbao-Terol A, Pérez-Gladish B, Arenas-Parra M, Rodríguez-Uría MV (2006) Fuzzy compromise programming for portfolio selection. Appl Math Comput 173(1):251–264Bravo M, Ballestero E, Pla-Santamaria D (2012) Evaluating fund performance by compromise programming with linear-quadratic composite metric: an actual case on the caixabank in spain. J Multi-Criteria Decis Anal 19(5–6):247–255Ehrgott M, Klamroth K, Schwehm C (2004) An MCDM approach to portfolio optimization. Eur J Oper Res 155(3):752–770Fawcett T (2006) An introduction to ROC analysis. Pattern Recognit Lett 27(8):861–874Hernández-Orallo J, Flach P, Ferri C (2013) ROC curves in cost space. Mach Learn 93(1):71–91Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91Pla-Santamaria D, Bravo M (2013) Portfolio optimization based on downside risk: a mean-semivariance efficient frontier from dow jones blue chips. Ann Oper Res 205(1):189–201Ringuest JL (1992) Multiobjective optimization: behavioral and computational considerations. Springer Science & Business Media, BerlinSteuer RE, Qi Y, Hirschberger M (2007) Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection. Ann Oper Res 152(1):297–317Xidonas P, Mavrotas G, Krintas T, Psarras J, Zopounidis C (2012) Multicriteria portfolio management. Springer, BerlinYu P-L (1973) A class of solutions for group decision problems. Manag Sci 19(8):936–946Yu P-L (1985) Multiple criteria decision making: concepts, techniques and extensions. Plenum Press, BerlinZeleny M (1982) Multiple criteria decision making. McGraw-Hill, New Yor

An analytic derivation of the efficient frontier in biobjective cash management and its implications for policies

[EN] Cash managers who optimize returns and risk rely on biobjective optimization models to select the best policies according to their risk preferences. In the related portfolio selection problem, Merton (J Financ Quant Anal 7(4):1851¿1872, 1972) provided the first analytical derivation of the efficient frontier with all non-dominated return and risk combinations. This first proposal was later extended to account for three or more criteria by other authors. However, the cash management literature needs an analytical derivation of the efficient frontier to help cash managers evaluate the implications of selecting policies and risk measures. In this paper, we provide three analytic derivations of the efficient frontier determining a closed-form solution for the expected returns and risk relationship using three different risk measures. We study its main properties and its theoretical implications for policies. Using the variance of returns as a risk measure imposes limitations due to invertibility reasons.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Salas-Molina, F.; Pla Santamaría, D.; Rodriguez-Aguilar, JA. (2023). An analytic derivation of the efficient frontier in biobjective cash management and its implications for policies. Annals of Operations Research (Online). 328(2):1523-1536. https://doi.org/10.1007/s10479-023-05433-z152315363282Baumol, W. J. (1952). The transactions demand for cash: An inventory theoretic approach. The Quarterly Journal of Economics, 66(4), 545–556.Constantinides, G. M., & Richard, S. F. (1978). Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time. Operations Research, 26(4), 620–636.da Costa Moraes, M. B., Nagano, M. S., Sobreiro, V. A., et al. (2015). Stochastic cash flow management models: A literature review since the 1980s. In P. Guarnieri (Ed.), Decision Models in Engineering and Management (pp. 11–28). Berlin: Springer.Markowitz, H. (1952). The Portfolio selection. Journal of Finance, 7(1), 77–91.Merton, R. C. (1972). An analytic derivation of the efficient portfolio frontier. Journal of Financial and Quantitative Analysis, 7(4), 1851–1872.Miller, M. H., & Orr, D. (1966). A model of the demand for money by firms. The Quarterly Journal of Economics, 80(3), 413–435.Qi, Y. (2020). Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure. Journal of the Operational Research Society, 71(10), 1675–1690.Qi, Y. (2022). Classifying the minimum-variance surface of multiple-objective portfolio selection for capital asset pricing models. Annals of Operations Research, 311(2), 1203–1227.Qi, Y., & Li, X. (2020). On imposing ESG constraints of portfolio selection for sustainable investment and comparing the efficient frontiers in the weight space. SAGE Open, 10(4), 1–17.Qi, Y., & Steuer, R. E. (2020). On the analytical derivation of efficient sets in quad-and-higher criterion portfolio selection. Annals of Operations Research, 293(2), 521–538.Qi, Y., Steuer, R. E., & Wimmer, M. (2017). An analytical derivation of the efficient surface in portfolio selection with three criteria. Annals of Operations Research, 251(1–2), 161–177.Salas-Molina, F. (2019). Selecting the best risk measure in multiobjective cash management. International Transactions in Operational Research, 26(3), 929–945.Salas-Molina, F. (2020). Risk-sensitive control of cash management systems. Operational Research, 20(2), 1159–1176.Salas-Molina, F., Pla-Santamaria, D., & Rodríguez-Aguilar, J. A. (2018). Empowering cash managers through compromise programming. In H. Masri, B. Perez-Gladish, & C. Zopounidis (Eds.), Financial decision aid using multiple criteria (pp. 149–173). Berlin: Springer.Salas-Molina, F., Pla-Santamaria, D., & Rodriguez-Aguilar, J. A. (2018). A multi-objective approach to the cash management problem. Annals of Operations Research, 267(1), 515–529.Salas-Molina, F., Rodriguez-Aguilar, J. A., & Díaz-García, P. (2018). Selecting cash management models from a multiobjective perspective. Annals of Operations Research, 261(1), 275–288.Salas-Molina, F., Rodriguez-Aguilar, J. A., Pla-Santamaria, D., & García-Bernabeu, A. (2021). On the formal foundations of cash management systems. Operational Research, 21(2), 1081–1095.Salas-Molina, F., Rodríguez-Aguilar, J. A., & Guillen, M. (2023). A multidimensional review of the cash management problem. Financial Innovation, 9(67), 1–35.Savage, L. J. (1951). The theory of statistical decision. Journal of the American Statistical Association, 46(253), 55–67.Schroeder, P., & Kacem, I. (2019). Optimal cash management with uncertain, interrelated and bounded demands. Computers & Industrial Engineering, 133, 195–206.Schroeder, P., & Kacem, I. (2020). Competitive difference analysis of the cash management problem with uncertain demands. European Journal of Operational Research, 283(3), 1183–1192

Encompassing statistically unquantifiable randomness in goal programming: an application to portfolio selection

[EN] Random events make multiobjective programming solutions vulnerable to changes in input data. In many cases statistically quantifiable information on variability of relevant parameters may not be available for decision making. This situation gives rise to the problem of obtaining solutions based on subjective beliefs and a priori risk aversion to random changes. To solve this problem, we propose to replace the traditional weighted goal programming achievement function with a new function that considers the decision maker's perception of the randomness associated with implementing the solution through the use of a penalty term. This new function also implements the level of a priori risk aversion based around the decision maker's beliefs and perceptions. The proposed new formulation is illustrated by means of a variant of the mean absolute deviation portfolio selection model. As a result, difficulties imposed by the absence of statistical information about random events can be encompassed by a modification of the achievement function to pragmatically consider subjective beliefs.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. s This work is devoted to the memory of Professor Enrique Ballestero for his selfess dedication to it.Bravo Selles, M.; Jones, D.; Pla Santamaría, D.; Salas-Molina, F. (2022). Encompassing statistically unquantifiable randomness in goal programming: an application to portfolio selection. Operational Research (Online). 22(5):5685-5706. https://doi.org/10.1007/s12351-022-00713-156855706225Abdelaziz FB, Aouni B, El Fayedh R (2007) Multi-objective stochastic programming for portfolio selection. Eur J Oper Res 177(3):1811–1823Abdelaziz FB, El Fayedh R, Rao A (2009) A discrete stochastic goal program for portfolio selection: the case of united arab emirates equity market. INFOR Inf Syst Op Res 47(1):5–13Aouni B, La Torre D (2010) A generalized stochastic goal programming model. Appl Math Comput 215(12):4347–4357Aouni B, Ben Abdelaziz F, La Torre D (2012) The stochastic goal programming model: theory and applications. J Multi-Criteria Decis Anal 19(5–6):185–200Arrow KJ (1965) Aspects of the theory of risk-bearing. Academic Bookstore, HelsinkiBallestero E (1997) Utility functions: a compromise programming approach to specification and optimization. J Multi-Criteria Decis Anal 6(1):11–16Ballestero E (2001) Stochastic goal programming: a mean-variance approach. Eur J Op Res 131(3):476–481Ballestero E, Pla-Santamaria D (2004) Selecting portfolios for mutual funds. Omega 32(5):385–394Ballestero E, Romero C (1998) Multiple criteria decision making and its applications to economic problems. Kluwer Academic Publishers, DordrechtBallestero E, Bravo M, Pérez-Gladish B, Arenas-Parra M, Pla-Santamaria D (2012) Socially responsible investment: a multicriteria approach to portfolio selection combining ethical and financial objectives. Eur J Op Res 216(2):487–494Bhamra HS, Uppal R (2006) The role of risk aversion and intertemporal substitution in dynamic consumption-portfolio choice with recursive utility. J Econ Dyn Control 30(6):967–991Bilbao-Terol A, Jiménez M, Arenas-Parra M (2016) A group decision making model based on goal programming with fuzzy hierarchy: an application to regional forest planning. Ann Op Res 245(1–2):137–162Branke J, Deb K, Miettinen K, Slowiński R (2008) Multiobjective optimization: interactive and evolutionary approaches. Springer Science & Business Media, BerlinBravo M, Gonzalez I (2009) Applying stochastic goal programming: a case study on water use planning. Eur J Op Res 196(3):1123–1129Charnes A, Collomb B (1972) Optimal economic stabilization policy: linear goal-programming models. Soc-Econ Plan Sci 6:431–435Charnes A, Cooper WW (1957) Management models and industrial applications of linear programming. Manag Sci 4(1):38–91Charnes A, Cooper WW, Ferguson RO (1955) Optimal estimation of executive compensation by linear programming. Manag Sci 1(2):138–151Cheridito P, Summer C (2006) Utility maximization under increasing risk aversion in one-period models. Finance Stoch 10(1):147–158Choobineh M, Mohagheghi S (2016) A multi-objective optimization framework for energy and asset management in an industrial microgrid. J Clean Prod 139:1326–1338Debreu G (1960) Topological methods in cardinal utility theory. In: Mathematical Methods in the Social Sciences. Standford University Press, StandfordDíaz-Madroñero M, Mula J, Jiménez M (2014) Fuzzy goal programming for material requirements planning under uncertainty and integrity conditions. Int J Prod Res 52(23):6971–6988Elbasha EH (2005) Risk aversion and uncertainty in cost-effectiveness analysis: the expected-utility, moment-generating function approach. Health Econ 14(5):457–470Ewald CO, Yang Z (2008) Utility based pricing and exercising of real options under geometric mean reversion and risk aversion toward idiosyncratic risk. Math Methods Op Res 68(1):97–123Gass SI (1986) A process for determining priorities and weights for large-scale linear goal programmes. J Op Res Soc 37(8):779–785Ghahtarani A, Najafi AA (2013) Robust goal programming for multi-objective portfolio selection problem. Econ Model 33:588–592Gollier C (2001) The economics of risk and time. MIT press, CambridgeGonzález-Pachón J, Romero C (2016) Bentham, Marx and Rawls ethical principles: in search for a compromise. Omega 62:47–51González-Pachón J, Diaz-Balteiro L, Romero C (2019) A multi-criteria approach for assigning weights in voting systems. Soft Comput 23(17):8181–8186Grigoroudis E, Orfanoudaki E, Zopounidis C (2012) Strategic performance measurement in a healthcare organisation: a multiple criteria approach based on balanced scorecard. Omega 40(1):104–119Hanks RW, Weir JD, Lunday BJ (2017) Robust goal programming using different robustness echelons via norm-based and ellipsoidal uncertainty sets. Eur J Op Res 262(2):636–646Ignizio JP (1999) Illusions of optimality. Eng Optim 31(6):749–761Jiménez M, Bilbao-Terol A, Arenas-Parra M (2018) A model for solving incompatible fuzzy goal programming: an application to portfolio selection. Int Trans Op Res 25(3):887–912Johansson-Stenman O (2010) Risk aversion and expected utility of consumption over time. Games Econ Behav 68(1):208–219Jones D (2011) A practical weight sensitivity algorithm for goal and multiple objective programming. Eur J Op Res 213(1):238–245Jones D, Tamiz M (2010) Practical goal programming. Springer, New YorkKallberg JG, Ziemba WT (1983) Comparison of alternative utility functions in portfolio selection problems. Manag Sci 29(11):1257–1276Kihlstrom R (2009) Risk aversion and the elasticity of substitution in general dynamic portfolio theory: consistent planning by forward looking, expected utility maximizing investors. J Math Econ 45(9–10):634–663Kluyver T, Ragan-Kelley B, Pérez F, Granger BE, Bussonnier M, Frederic J, Kelley K, Hamrick JB, Grout J, Corlay S, et al (2016) Jupyter notebooks-a publishing format for reproducible computational workflows. In: ELPUB, pp. 87–90Konno H, Yamazaki H (1991) Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market. Manag Sci 37(5):519–531Kraft D (1988) A software package for sequential quadratic programming. Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt 28Krantz D, Luce D, Suppes P, Tversky A (1971) Foundations of measurement: geometrical, threshold, and probabilistic representations. Academic Press, New YorkKuchta D (2004) Robust goal programming. Control Cybern 33(3):501–510Langlais E (2005) Willingness to pay for risk reduction and risk aversion without the expected utility assumption. Theory Decis 59(1):43–50Markowitz H (1952) Portfolio selection. J Financ 7(1):77–91Masri H (2017) A multiple stochastic goal programming approach for the agent portfolio selection problem. Ann Op Res 251(1–2):179–192Matthews LR, Guzman YA, Floudas CA (2018) Generalized robust counterparts for constraints with bounded and unbounded uncertain parameters. Comput Chem Eng 116:451–467McCarl BA, Bessler DA (1989) Estimating an upper bound on the pratt risk a version coefficient when the utility function is unknown. Aust J Agric Econ 33:56Messaoudi L, Aouni B, Rebai A (2017) Fuzzy chance-constrained goal programming model for multi-attribute financial portfolio selection. Ann Op Res 251(1–2):193–204Miettinen K, Ruiz F, Wierzbicki AP (2008) Introduction to multiobjective optimization: interactive approaches. In: Multiobjective Optimization. Springer, Berlin, pp 27–57Muñoz MM, Ruiz F (2009) ISTMO: an interval reference point-based method for stochastic multiobjective programming problems. Eur J Op Res 197(1):25–35Muñoz MM, Luque M, Ruiz F (2010) Interest: a reference-point-based interactive procedure for stochastic multiobjective programming problems. OR Spectr 32(1):195–210Oliveira R, Zanella A, Camanho AS (2019) The assessment of corporate social responsibility: the construction of an industry ranking and identification of potential for improvement. Eur J Op Res 278(2):498–513Pratt JW (1964) Risk aversion in the small and in the large. Econometrica 32(1–2):122–136Romero C (1991) Handbook of critical issues in goal programming. Pergamon Press, OxfordSalas-Molina F, Rodríguez-Aguilar JA, Pla-Santamaria D (2018) Boundless multiobjective models for cash management. Eng Econ 63(4):363–381Schechter L (2007) Risk aversion and expected-utility theory: a calibration exercise. J Risk Uncertain 35(1):67–76Tamiz M, Jones D (1996) Goal programming and pareto efficiency. J Inf Optim Sci 17(2):291–307Tsionas MG (2019) Multi-objective optimization using statistical models. Eur J Op Res 276(1):364–378Woerheide W, Persson D (1993) An index of portfolio diversification. Financ Serv Rev 2(2):73–85Xu Y, Yeh CH (2012) An integrated approach to evaluation and planning of best practices. Omega 40(1):65–78Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–35

Monitoring multidimensional phenomena with a multicriteria composite performance interval approach

[EN] In the last two decades, the construction of composite indicators to measure and compare multidimensional phenomena in a broad spectrum of domains has increased considerably. Different methodological approaches are used to summarise huge datasets of information in a single figure. This paper proposes a new approach that consists in computing a multicriteria composite performance interval based on different aggregation rules. The suggested approach provides an additional layer of information as the performance interval displays a lower bound from a non-compensability perspective, and an upper bound allowing for full-compensability. The outstanding features of this proposal are: 1) a distance-based multicriteria technique is taken as the baseline to construct the multicriteria performance interval; 2) the aggregation of distances/separation measures is made using particular cases of Minkowski Lp metric; 3) the span of the multicriteria performance interval can be considered as a sign of the dimensions or indicators balance.Garcia-Bernabeu, A.; Hilario Caballero, A.; Pla Santamaría, D.; Salas-Molina, F. (2021). Monitoring multidimensional phenomena with a multicriteria composite performance interval approach. International Journal of Multicriteria Decision Making (Online). 8(4):368-385. https://doi.org/10.1504/IJMCDM.2021.120760S3683858

A multidimensional approach to rank fuzzy numbers based on the concept of magnitude

Ranking fuzzy numbers have become of growing importance in recent years, especially as decision-making is increasingly performed under greater uncertainty. In this paper, we extend the concept of magnitude to rank fuzzy numbers to a more general definition to increase in flexibility and generality. More precisely, we propose a multidimensional approach to rank fuzzy numbers considering alternative magnitude definitions with three novel features: multidimensionality, normalization, and a ranking based on a parametric distance function. A multidimensional magnitude definition allows us to consider multiple attributes to represent and rank fuzzy numbers. Normalization prevents meaningless comparison among attributes due to scaling problems, and the use of the parametric Minkowski distance function becomes a more general and flexible ranking approach. The main contribution of our multidimensional approach is the representation of a fuzzy number as a point in a $n$-dimensional normalized space of attributes in which the distance to the origin is the magnitude value. We illustrate our methodology and provide further insights into different normalization approaches and parameters through several numerical examples. Finally, we describe an application of our ranking approach to a multicriteria decision-making problem within an economic context in which the main goal is to rank a set of credit applicants considering different financial ratios used as evaluation criteria