Multi crteria decision making and its applications : a literature review

This paper presents current techniques used in Multi Criteria Decision Making (MCDM) and their applications. Two basic approaches for MCDM, namely Artificial Intelligence MCDM (AIMCDM) and Classical MCDM (CMCDM) are discussed and investigated. Recent articles from international journals related to MCDM are collected and analyzed to find which approach is more common than the other in MCDM. Also, which area these techniques are applied to. Those articles are appearing in journals for the year 2008 only. This paper provides evidence that currently, both AIMCDM and CMCDM are equally common in MCDM

NEUTROSOPHIC MULTI-OBJECTIVE LINEAR PROGRAMMING

For modeling imprecise and indeterminate data for multi-objective decision making, two different methods: neutrosophic multi-objective linear/non-linear programming neutrosophic goal programming, which have been very recently proposed in the literatuire. In many economic problems, the well-known probabilities or fuzzy solutions procedures are not suitable because they cannot deal the situation when indeterminacy inherently involves in the problem. In this case we propose a new concept in optimization problem under uncertainty and indeterminacy. It is an extension of fuzzy and intuitionistic fuzzy optimization in which the degrees of indeterminacy and falsity (rejection) of objectives and constraints are simultaneously considered together with the degrees of truth membership (satisfaction/acceptance). The drawbacks of the existing neutrosophic optimization models have been presented and new framework of multi-objective optimization in neutrosophic environment has been proposed. The essence of the proposed approach is that it is capable of dealing with indeterminacy and falsity simultaneously

Developing a meat supply chain network design using a multi-objective possibilistic programming approach

Purpose The purpose of this paper is to present a study in developing a cost-effective meat supply chain network design aiming to minimizing the total cost of transportation, the number of transportation vehicles and the delivery time of meat products. The developed model was also used for determining the optimum numbers and allocations of farms and abattoirs that need to be established and the optimal quantity flow of livestock from farms to abattoirs and meat products from abattoirs to retailers. Design/methodology/approach A multi-objective possibilistic programming model was formulated with a focus on minimizing the total cost of transportation, the number of transportation vehicles and the delivery time of meat products. Three sets of Pareto solutions were obtained using the three different solution methods. These methods are the LP-metrics method, the ɛ-constraint method and the weighted Tchebycheff method, respectively. The TOPSIS method was used for seeking a best Pareto solution as a trade-off decision when optimizing the three conflicting objectives. Findings A case study was also applied for examining the effectiveness and applicability of the developed multi-objective model and the proposed solution methods. The research concludes that the ɛ-constraint method has the superiority over the other two proposed methods as it offers a better solution outcome. Research limitations/implications This work addresses as interesting avenues for further research on exploring the delivery planner under different types of uncertainties and transportation means. Also, environmentalism has been increasingly becoming a significant global problem in the present century. Thus, the presented model could be extended to include the environmental aspects as an objective function. Practical implications The developed design methodology can be utilized for food supply chain designers. Also, it could be applied to realistic problems in the field of supply chain management. Originality/value The paper presents a methodology that can be used for tackling a multi-objective optimization problem of a meat supply chain network design. The proposed optimization method has the potential in solving the similar issue providing a compromising solution due to conflicting objectives in which each needs to be achieved toward an optimum outcome to survive in the competitive sector of food supply chains network. </jats:sec

Lexicographic Methods for Fuzzy Linear Programming

Fuzzy Linear Programming (FLP) has addressed the increasing complexity of real-world decision-making problems that arise in uncertain and ever-changing environments since its introduction in the 1970s. Built upon the Fuzzy Sets theory and classical Linear Programming (LP) theory, FLP encompasses an extensive area of theoretical research and algorithmic development. Unlike classical LP, there is not a unique model for the FLP problem, since fuzziness can appear in the model components in different ways. Hence, despite fifty years of research, new formulations of FLP problems and solution methods are still being proposed. Among the existing formulations, those using fuzzy numbers (FNs) as parameters and/or decision variables for handling inexactness and vagueness in data have experienced a remarkable development in recent years. Here, a long-standing issue has been how to deal with FN-valued objective functions and with constraints whose left- and right-hand sides are FNs. The main objective of this paper is to present an updated review of advances in this particular area. Consequently, the paper briefly examines well-known models and methods for FLP, and expands on methods for fuzzy single- and multi-objective LP that use lexicographic criteria for ranking FNs. A lexicographic approach to the fuzzy linear assignment (FLA) problem is discussed in detail due to the theoretical and practical relevance. For this case, computer codes are provided that can be used to reproduce results presented in the paper and for practical applications. The paper demonstrates that FLP that is focused on lexicographic methods is an active area with promising research lines and practical implications.Spanish Ministry of Economy and CompetitivenessEuropean Union (EU) TIN2017-86647-

A critical review of the approaches to optimization problems under uncertainty

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2001.Thesis (Master's) -- Bilkent University, 2001.Includes bibliographical references leaves 58-72.In this study, the issue of uncertainty in optimization problems is studied. First of all, the meaning and sources of uncertainty are explained and then possible ways of its representation are analyzed. About the modelling process, different approaches as sensitivity analysis, parametric programming, robust optimization, stochastic programming, fuzzy programming, multiobjective programming and imprecise optimization are presented with advantages and disadvantages from different perspectives. Some extensions of the concepts of imprecise optimization are also presented.Gürtuna, FilizM.S

Manufacturing cell formation in a fuzzy environment

The main objective of this study is to develop useful mathematical programming (FMP) models to solve cell formation (CF) problems in fuzzy environments. The dissertation was divided into three major parts. First, two mathematical programming models were developed to formulate the cell formation problems under consideration. The first model was a linear programming (LP) model for grouping parts and machines simultaneously into cells and solving the CF problem for dealing with exceptional elements (EEs). In second, a goal programming (GP) model to obtain a trade off between minimizing total cost of dealing with EEs and maximizing GE, a new similarity coefficient formula between parts also has been developed;In the second part, the fuzzy linear programming (FLP) methodology was applied to solve CF problems involving fuzzy situations. A new fuzzy operator, add-min, was proposed and its performances evaluated against the other six operators. Robustness and excellent performance in terms of clustering results and CPU executing time were verified for the FLP with the new operator. Fuzzy multiobjective linear programming (FMLP) then was used (1) to find the optimal trade-off between multiple goals in the proposed goal programming and (2) to compare the performance with the GP results. Numerical illustrations show that FMLP with the proposed operator performed much better than the GP did in terms of computational efficiency;Finally, an efficient heuristic genetic algorithm (HGA) was developed to solve all mathematical programming models, including the fuzzy models, presented in this dissertation. New heuristic crossover and mutation operators based on the special characteristics of CF were proposed to enhance computational performance. Our experiment showed that the proposed GA heuristic outperformed both the traditional GA approach and the mathematical programming models in terms of clustering results, computational time, and ease of use

Single machine scheduling problems with uncertain parameters and the OWA criterion

In this paper a class of single machine scheduling problems is discussed. It is assumed that job parameters, such as processing times, due dates, or weights are uncertain and their values are specified in the form of a discrete scenario set. The Ordered Weighted Averaging (OWA) aggregation operator is used to choose an optimal schedule. The OWA operator generalizes traditional criteria in decision making under uncertainty, such as the maximum, average, median or Hurwicz criterion. It also allows us to extend the robust approach to scheduling by taking into account various attitudes of decision makers towards the risk. In this paper a general framework for solving single machine scheduling problems with the OWA criterion is proposed and some positive and negative computational results for two basic single machine scheduling problems are provided

Computational studies of some fuzzy mathematical problems

In modelling and optimizing real world systems and processes, one usually ends up with a linear or nonlinear programming problem, namely maximizing one or more objective functions subject to a set of constraint equations or inequalities. For many cases, the constraints do not need to be satisfied exactly, and the coefficients involved in the model are imprecise in nature and have to be described by fuzzy numbers to reflect the real world nature. The resulting mathematical programming problem is referred to as a fuzzy mathematical programming problem.Over the past decades, a great deal of work has been conducted to study fuzzy mathematical programming problems and a large volume of results have been obtained. However, many issues have not been resolved. This research is thus undertaken to study two types of fuzzy mathematical programming problems. The first type of problems is fuzzy linear programming in which the objective function contains fuzzy numbers. To solve this type of problems, we firstly introduce the concept of fuzzy max order and non-dominated optimal solution to fuzzy mathematical programming problems within the framework of fuzzy mathematics. Then, based on the new concept introduced, various theorems are developed, which involve converting the fuzzy linear programming problem to a four objective linear programming problem of non-fuzzy members. The theoretical results and methods developed are then validated and their applications for solving fuzzy linear problems are demonstrated through examples.The second type of problems which we tackle in this research is fuzzy linear programming in which the constraint equations or inequalities contain fuzzy numbers. For this work, we first introduce a new concept, the α-fuzzy max order. Based on this concept, the general framework of an α-fuzzy max order method is developed for solving fuzzy linear programming problems with fuzzy parameters in the constraints. For the special cases in which the constraints consist of inequalities containing fuzzy numbers with isosceles triangle or trapezoidal membership functions, we prove that the feasible solution space can be determined by the respective 3n or 4n non-fuzzy inequalities. For the general cases in which the constraints contain fuzzy numbers with any other form of membership functions, robust numerical algorithms have been developed for the determination of the feasible solution space and the optimal solution to the fuzzy linear programming problem in which the constraints contain fuzzy parameters. Further, by using the results for both the first and second types of problems, general algorithms have also been developed for the general fuzzy linear programming problems in which both the objective function and the constraint inequalities contain fuzzy numbers with any forms of membership functions. Some examples are then presented to validate the theoretical results and the algorithms developed, and to demonstrate their applications

Multi-objective Optimization Proposal with Fuzzy Coefficients in both Constraints and Objective Functions.

Propuesta de optimización multiobjetivo con coeficientes difusos en restricciones y en funciones objetivo.AbstractFuzzy sets, and more specifically, fuzzy numbers can be a very suitable way to include uncertainty within the formulation and solution of linear problems with multiple goals. Goals in a decision problem do not need to be either maximized, or minimized, as in classical mathematical programming, but they are substituted by aspiration levels, and they need to be met in order to satisfy the decision-maker. Experience shows that it is easier for the decision-maker to formulate both objectives and constraints with fuzzy coefficients, rather than specify a defined quantity for the matrices A, b or g. This paper shows the versatility of a methodology that solves multi-objective linear problems, formulated with fuzzy coefficients. This conception becomes an alternative in contrast with the hard methodologies predominant in Operations Research, since the fuzzy approach allows the decision-maker to make uncertain assumptions both for the formulation and solution of optimization problems.Keywords: Fuzzy logic, multi-criteria analysis, triangular fuzzy numbers.ResumenLos conjuntos difusos y específicamente los números difusos constituyen una manera efectiva de incluir la incertidumbre en la formulación y solución de problemas lineales de optimización multiobjetivo. Las metas en un problema de decisión no necesitan ser maximizadas ni minimizadas, como ocurre en las herramientas clásicas de programación matemática, sino que se pueden sustituir por niveles de aspiración, las cuales constituyen las expectativas para un decisor. La experiencia demuestra que es más fácil para el decisor formular los objetivos y las restricciones en un problema con coeficientes difusos, en vez de simplemente especificar un número concreto en las matrices A, b ó g. Este artículo presenta la versatilidad de una formulación metodológica que permite resolver problemas multiobjetivo de tipo lineal, los cuales son formulados con coeficientes difusos. Esta concepción constituye una alternativa a las metodologías duras que dominan la investigación de operaciones, dado que la aproximación difusa permite que los decisores realicen presunciones inciertas en la formulación y solución en los problemas de optimización.Palabras Clave: Lógica difusa, análisis multiobjetivo, números triangulares difusos

Multiobjective Approach to Portfolio Optimization in the Light of the Credibility Theory

[EN] The present research proposes a novel methodology to solve the problems faced by investors who take into consideration different investment criteria in a fuzzy context. The approach extends the stochastic mean-variance model to a fuzzy multiobjective model where liquidity is considered to quantify portfolio's performance, apart from the usual metrics like return and risk. The uncertainty of the future returns and the future liquidity of the potential assets are modelled employing trapezoidal fuzzy numbers. The decision process of the proposed approach considers that portfolio selection is a multidimensional issue and also some realistic constraints applied by investors. Particularly, this approach optimizes the expected return, the risk and the expected liquidity of the portfolio, considering bound constraints and cardinality restrictions. As a result, an optimization problem for the constraint portfolio appears, which is solved by means of the NSGA-II algorithm. This study defines the credibilistic Sortino ratio and the credibilistic STARR ratio for selecting the optimal portfolio. An empirical study on the S&P100 index is included to show the performance of the model in practical applications. The results obtained demonstrate that the novel approach can beat the index in terms of return and risk in the analyzed period, from 2008 until 2018.García García, F.; González-Bueno, J.; Guijarro, F.; Oliver-Muncharaz, J.; Tamosiuniene, R. (2020). Multiobjective Approach to Portfolio Optimization in the Light of the Credibility Theory. Technological and Economic Development of Economy (Online). 26(6):1165-1186. https://doi.org/10.3846/tede.2020.13189S11651186266Acerbi, C., & Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487-1503. doi:10.1016/s0378-4266(02)00283-2Ahmed, A., Ali, R., Ejaz, A., & Ahmad, I. (2018). Sectoral integration and investment diversification opportunities: evidence from Colombo Stock Exchange. Entrepreneurship and Sustainability Issues, 5(3), 514-527. doi:10.9770/jesi.2018.5.3(8)Arenas Parra, M., Bilbao Terol, A., & Rodrı́guez Urı́a, M. V. (2001). A fuzzy goal programming approach to portfolio selection. European Journal of Operational Research, 133(2), 287-297. doi:10.1016/s0377-2217(00)00298-8Arribas, I., Espinós-Vañó, M. D., García, F., & Tamošiūnienė, R. (2019). Negative screening and sustainable portfolio diversification. Entrepreneurship and Sustainability Issues, 6(4), 1566-1586. doi:10.9770/jesi.2019.6.4(2)Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9(3), 203-228. doi:10.1111/1467-9965.00068Bawa, V. S. (1975). Optimal rules for ordering uncertain prospects. Journal of Financial Economics, 2(1), 95-121. doi:10.1016/0304-405x(75)90025-2Bermúdez, J. D., Segura, J. V., & Vercher, E. (2012). A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection. Fuzzy Sets and Systems, 188(1), 16-26. doi:10.1016/j.fss.2011.05.013Bezoui, M., Moulaï, M., Bounceur, A., & Euler, R. (2018). An iterative method for solving a bi-objective constrained portfolio optimization problem. Computational Optimization and Applications, 72(2), 479-498. doi:10.1007/s10589-018-0052-9Bi, T., Zhang, B., & Wu, H. (2013). Measuring Downside Risk Using High-Frequency Data: Realized Downside Risk Measure. Communications in Statistics - Simulation and Computation, 42(4), 741-754. doi:10.1080/03610918.2012.655826Carlsson, C., Fullér, R., & Majlender, P. (2002). A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems, 131(1), 13-21. doi:10.1016/s0165-0114(01)00251-2Chen, W., & Xu, W. (2018). A Hybrid Multiobjective Bat Algorithm for Fuzzy Portfolio Optimization with Real-World Constraints. International Journal of Fuzzy Systems, 21(1), 291-307. doi:10.1007/s40815-018-0533-0Choobineh, F., & Branting, D. (1986). A simple approximation for semivariance. European Journal of Operational Research, 27(3), 364-370. doi:10.1016/0377-2217(86)90332-2Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182-197. doi:10.1109/4235.996017Fang, Y., Lai, K. K., & Wang, S.-Y. (2006). Portfolio rebalancing model with transaction costs based on fuzzy decision theory. European Journal of Operational Research, 175(2), 879-893. doi:10.1016/j.ejor.2005.05.020Favre, L., & Galeano, J.-A. (2002). Mean-Modified Value-at-Risk Optimization with Hedge Funds. The Journal of Alternative Investments, 5(2), 21-25. doi:10.3905/jai.2002.319052García, F., González-Bueno, J., Guijarro, F., & Oliver, J. (2020). Forecasting the Environmental, Social, and Governance Rating of Firms by Using Corporate Financial Performance Variables: A Rough Set Approach. Sustainability, 12(8), 3324. doi:10.3390/su12083324García, González-Bueno, Oliver, & Riley. (2019). Selecting Socially Responsible Portfolios: A Fuzzy Multicriteria Approach. Sustainability, 11(9), 2496. doi:10.3390/su11092496García, F., González-Bueno, J., Oliver, J., & Tamošiūnienė, R. (2019). A CREDIBILISTIC MEAN-SEMIVARIANCE-PER PORTFOLIO SELECTION MODEL FOR LATIN AMERICA. Journal of Business Economics and Management, 20(2), 225-243. doi:10.3846/jbem.2019.8317García, F., Guijarro, F., & Moya, I. (2013). A MULTIOBJECTIVE MODEL FOR PASSIVE PORTFOLIO MANAGEMENT: AN APPLICATION ON THE S&P 100 INDEX. Journal of Business Economics and Management, 14(4), 758-775. doi:10.3846/16111699.2012.668859García, F., Guijarro, F., & Oliver, J. (2017). Index tracking optimization with cardinality constraint: a performance comparison of genetic algorithms and tabu search heuristics. Neural Computing and Applications, 30(8), 2625-2641. doi:10.1007/s00521-017-2882-2García, F., Guijarro, F., Oliver, J., & Tamošiūnienė, R. (2018). HYBRID FUZZY NEURAL NETWORK TO PREDICT PRICE DIRECTION IN THE GERMAN DAX-30 INDEX. Technological and Economic Development of Economy, 24(6), 2161-2178. doi:10.3846/tede.2018.6394Goel, A., Sharma, A., & Mehra, A. (2018). Index tracking and enhanced indexing using mixed conditional value-at-risk. Journal of Computational and Applied Mathematics, 335, 361-380. doi:10.1016/j.cam.2017.12.015González-Bueno, J. (2019). Optimización multiobjetivo para la selección de carteras a la luz de la teoría de la credibilidad. Una aplicación en el mercado integrado latinoamericano. Editorial Universidad Pontificia Bolivariana.Gupta, P., Inuiguchi, M., & Mehlawat, M. K. (2011). A hybrid approach for constructing suitable and optimal portfolios. Expert Systems with Applications, 38(5), 5620-5632. doi:10.1016/j.eswa.2010.10.073Gupta, P., Inuiguchi, M., Mehlawat, M. K., & Mittal, G. (2013). Multiobjective credibilistic portfolio selection model with fuzzy chance-constraints. Information Sciences, 229, 1-17. doi:10.1016/j.ins.2012.12.011Gupta, P., Mehlawat, M. K., Inuiguchi, M., & Chandra, S. (2014). Portfolio Optimization Using Credibility Theory. Studies in Fuzziness and Soft Computing, 127-160. doi:10.1007/978-3-642-54652-5_5Gupta, P., Mehlawat, M. K., Inuiguchi, M., & Chandra, S. (2014). Portfolio Optimization with Interval Coefficients. Studies in Fuzziness and Soft Computing, 33-59. doi:10.1007/978-3-642-54652-5_2Gupta, P., Mehlawat, M. K., Kumar, A., Yadav, S., & Aggarwal, A. (2020). A Credibilistic Fuzzy DEA Approach for Portfolio Efficiency Evaluation and Rebalancing Toward Benchmark Portfolios Using Positive and Negative Returns. International Journal of Fuzzy Systems, 22(3), 824-843. doi:10.1007/s40815-020-00801-4Gupta, P., Mehlawat, M. K., & Saxena, A. (2010). A hybrid approach to asset allocation with simultaneous consideration of suitability and optimality. Information Sciences, 180(11), 2264-2285. doi:10.1016/j.ins.2010.02.007Gupta, P., Mehlawat, M. K., Yadav, S., & Kumar, A. (2020). Intuitionistic fuzzy optimistic and pessimistic multi-period portfolio optimization models. Soft Computing, 24(16), 11931-11956. doi:10.1007/s00500-019-04639-3Gupta, P., Mittal, G., & Mehlawat, M. K. (2013). Expected value multiobjective portfolio rebalancing model with fuzzy parameters. Insurance: Mathematics and Economics, 52(2), 190-203. doi:10.1016/j.insmatheco.2012.12.002Heidari-Fathian, H., & Davari-Ardakani, H. (2019). Bi-objective optimization of a project selection and adjustment problem under risk controls. Journal of Modelling in Management, 15(1), 89-111. doi:10.1108/jm2-07-2018-0106Hilkevics, S., & Semakina, V. (2019). The classification and comparison of business ratios analysis methods. Insights into Regional Development, 1(1), 48-57. doi:10.9770/ird.2019.1.1(4)Huang, X. (2006). Fuzzy chance-constrained portfolio selection. Applied Mathematics and Computation, 177(2), 500-507. doi:10.1016/j.amc.2005.11.027Huang, X. (2008). Mean-semivariance models for fuzzy portfolio selection. Journal of Computational and Applied Mathematics, 217(1), 1-8. doi:10.1016/j.cam.2007.06.009Huang, X. (2009). A review of credibilistic portfolio selection. Fuzzy Optimization and Decision Making, 8(3), 263-281. doi:10.1007/s10700-009-9064-3Huang, X. (2010). Portfolio Analysis. Studies in Fuzziness and Soft Computing. doi:10.1007/978-3-642-11214-0Huang, X. (2017). A review of uncertain portfolio selection. Journal of Intelligent & Fuzzy Systems, 32(6), 4453-4465. doi:10.3233/jifs-169211Huang, X., & Di, H. (2016). Uncertain portfolio selection with background risk. Applied Mathematics and Computation, 276, 284-296. doi:10.1016/j.amc.2015.12.018Huang, X., & Wang, X. (2019). International portfolio optimization based on uncertainty theory. Optimization, 70(2), 225-249. doi:10.1080/02331934.2019.1705821Huang, X., & Yang, T. (2020). How does background risk affect portfolio choice: An analysis based on uncertain mean-variance model with background risk. Journal of Banking & Finance, 111, 105726. doi:10.1016/j.jbankfin.2019.105726Jalota, H., Thakur, M., & Mittal, G. (2017). Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework. Expert Systems with Applications, 71, 40-56. doi:10.1016/j.eswa.2016.11.014Jalota, H., Thakur, M., & Mittal, G. (2017). A credibilistic decision support system for portfolio optimization. Applied Soft Computing, 59, 512-528. doi:10.1016/j.asoc.2017.05.054Kaplan, P. D., & Alldredge, R. H. (1997). Semivariance in Risk-Based Index Construction. The Journal of Investing, 6(2), 82-87. doi:10.3905/joi.1997.408419Konno, H., & Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science, 37(5), 519-531. doi:10.1287/mnsc.37.5.519Li, B., Zhu, Y., Sun, Y., Aw, G., & Teo, K. L. (2018). Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint. Applied Mathematical Modelling, 56, 539-550. doi:10.1016/j.apm.2017.12.016Li, H.-Q., & Yi, Z.-H. (2019). Portfolio selection with coherent Investor’s expectations under uncertainty. Expert Systems with Applications, 133, 49-58. doi:10.1016/j.eswa.2019.05.008Li, X., & Qin, Z. (2014). Interval portfolio selection models within the framework of uncertainty theory. Economic Modelling, 41, 338-344. doi:10.1016/j.econmod.2014.05.036Liagkouras, K., & Metaxiotis, K. (2015). Efficient Portfolio Construction with the Use of Multiobjective Evolutionary Algorithms: Best Practices and Performance Metrics. International Journal of Information Technology & Decision Making, 14(03), 535-564. doi:10.1142/s0219622015300013Liu, B. (2004). Uncertainty Theory. Studies in Fuzziness and Soft Computing. doi:10.1007/978-3-540-39987-2Baoding Liu, & Yian-Kui Liu. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 10(4), 445-450. doi:10.1109/tfuzz.2002.800692Liu, N., Chen, Y., & Liu, Y. (2018). Optimizing portfolio selection problems under credibilistic CVaR criterion. Journal of Intelligent & Fuzzy Systems, 34(1), 335-347. doi:10.3233/jifs-171298Liu, Y.-J., & Zhang, W.-G. (2018). Multiperiod Fuzzy Portfolio Selection Optimization Model Based on Possibility Theory. International Journal of Information Technology & Decision Making, 17(03), 941-968. doi:10.1142/s0219622018500190Mansour, N., Cherif, M. S., & Abdelfattah, W. (2019). Multi-objective imprecise programming for financial portfolio selection with fuzzy returns. Expert Systems with Applications, 138, 112810. doi:10.1016/j.eswa.2019.07.027Markowitz, H. (1952). PORTFOLIO SELECTION*. The Journal of Finance, 7(1), 77-91. doi:10.1111/j.1540-6261.1952.tb01525.xMarkowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the Critical Line Algorithm. Annals of Operations Research, 45(1), 307-317. doi:10.1007/bf02282055Martin, R. D., Rachev, S. (Zari), & Siboulet, F. (2003). Phi-alpha optimal portfolios and extreme risk management. Wilmott, 2003(6), 70-83. doi:10.1002/wilm.42820030619Mehlawat, M. K. (2016). Credibilistic mean-entropy models for multi-period portfolio selection with multi-choice aspiration levels. Information Sciences, 345, 9-26. doi:10.1016/j.ins.2016.01.042Mehlawat, M. K., Gupta, P., Kumar, A., Yadav, S., & Aggarwal, A. (2020). Multiobjective Fuzzy Portfolio Performance Evaluation Using Data Envelopment Analysis Under Credibilistic Framework. IEEE Transactions on Fuzzy Systems, 28(11), 2726-2737. doi:10.1109/tfuzz.2020.2969406Mehralizade, R., Amini, M., Sadeghpour Gildeh, B., & Ahmadzade, H. (2020). Uncertain random portfolio selection based on risk curve. Soft Computing, 24(17), 13331-13345. doi:10.1007/s00500-020-04751-9Moeini, M. (2019). Solving the index tracking problem: a continuous optimization approach. Central European Journal of Operations Research. doi:10.1007/s10100-019-00633-0Narkunienė, J., & Ulbinaitė, A. (2018). Comparative analysis of company performance evaluation methods. Entrepreneurship and Sustainability Issues, 6(1), 125-138. doi:10.9770/jesi.2018.6.1(10)Palanikumar, K., Latha, B., Senthilkumar, V. S., & Karthikeyan, R. (2009). Multiple performance optimization in machining of GFRP composites by a PCD tool using non-dominated sorting genetic algorithm (NSGA-II). Metals and Materials International, 15(2), 249-258. doi:10.1007/s12540-009-0249-7Pflug, G. C. (2000). Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk. Probabilistic Constrained Optimization, 272-281. doi:10.1007/978-1-4757-3150-7_15Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21-41. doi:10.21314/jor.2000.038Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443-1471. doi:10.1016/s0378-4266(02)00271-6Rubio, A., Bermúdez, J. D., & Vercher, E. (2016). Forecasting portfolio returns using weighted fuzzy time series methods. International Journal of Approximate Reasoning, 75, 1-12. doi:10.1016/j.ijar.2016.03.007Saborido, R., Ruiz, A. B., Bermúdez, J. D., Vercher, E., & Luque, M. (2016). Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection. Applied Soft Computing, 39, 48-63. doi:10.1016/j.asoc.2015.11.005Sharpe, W. F. (1966). Mutual Fund Performance. The Journal of Business, 39(S1), 119. doi:10.1086/294846Sharpe, W. F. (1994). The Sharpe Ratio. The Journal of Portfolio Management, 21(1), 49-58. doi:10.3905/jpm.1994.409501Sortino, F. A., & Price, L. N. (1994). Performance Measurement in a Downside Risk Framework. The Journal of Investing, 3(3), 59-64. doi:10.3905/joi.3.3.59Srinivas, N., & Deb, K. (1994). Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms. Evolutionary Computation, 2(3), 221-248. doi:10.1162/evco.1994.2.3.221Vercher, E., & Bermúdez, J. D. (2012). Fuzzy Portfolio Selection Models: A Numerical Study. Financial Decision Making Using Computational Intelligence, 253-280. doi:10.1007/978-1-4614-3773-4_10Vercher, E., & Bermudez, J. D. (2013). A Possibilistic Mean-Downside Risk-Skewness Model for Efficient Portfolio Selection. IEEE Transactions on Fuzzy Systems, 21(3), 585-595. doi:10.1109/tfuzz.2012.2227487Vercher, E., & Bermúdez, J. D. (2015). Portfolio optimization using a credibility mean-absolute semi-deviation model. Expert Systems with Applications, 42(20), 7121-7131. doi:10.1016/j.eswa.2015.05.020Vercher, E., Bermúdez, J. D., & Segura, J. V. (2007). Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets and Systems, 158(7), 769-782. doi:10.1016/j.fss.2006.10.026Wang, S., & Zhu, S. (2002). Fuzzy Optimization and Decision Making, 1(4), 361-377. doi:10.1023/a:1020907229361Yue, W., & Wang, Y. (2017). A new fuzzy multi-objective higher order moment portfolio selection model for diversified portfolios. Physica A: Statistical Mechanics and its Applications, 465, 124-140. doi:10.1016/j.physa.2016.08.009Yue, W., Wang, Y., & Xuan, H. (2018). Fuzzy multi-objective portfolio model based on semi-variance–semi-absolute deviation risk measures. Soft Computing, 23(17), 8159-8179. doi:10.1007/s00500-018-3452-yZadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353. doi:10.1016/s0019-9958(65)90241-xZhai, J., & Bai, M. (2018). Mean-risk model for uncertain portfolio selection with background risk. Journal of Computational and Applied Mathematics, 330, 59-69. doi:10.1016/j.cam.2017.07.038Zhao, Z., Wang, H., Yang, X., & Xu, F. (2020). CVaR-cardinality enhanced indexation optimization with tunable short-selling constraints. Applied Economics Letters, 28(3), 201-207. doi:10.1080/13504851.2020.174015