Elsevier

Applied Soft Computing

Volume 85, December 2019, 105781
Applied Soft Computing

A polynomial goal programming approach for intuitionistic fuzzy portfolio optimization using entropy and higher moments

https://doi.org/10.1016/j.asoc.2019.105781Get rights and content

Highlights

  • Intuitionistic fuzzy multi-objective portfolio selection models are proposed.

  • A contingent and many realistic constraints are used in the optimization models.

  • Membership, non-membership and hesitancy degrees of the objectives are considered.

  • A polynomial goal programming approach is applied to aggregate the objectives.

  • Several customizable schemes as per decision-makers’ preferences are proposed.

Abstract

Although there are myriad works that deal with fuzzy portfolio optimization, there is a dearth of research that deals with the problem in an intuitionistic fuzzy environment. So, taking the route less travelled, we propose two intuitionistic fuzzy portfolio selection models for optimistic and pessimistic scenarios, respectively. For this purpose, we take into account four objectives, namely return, variance, skewness and entropy, along with some realistic constraints such as a cardinality constraint, a contingent constraint, and complete capital utilization. Also, short selling is prohibited. Another popular constraint, the “floor and ceiling” constraint, is presented and employed in the form of a flexible constraint by considering the confidence interval in the model. The membership and non-membership of the objectives are modelled using the extreme values of the four objectives. The proposed approach provides avenues for the inclusion and minimization of the hesitation degree into the decision making, thereby, resulting in a significantly better portfolio. The proposed model enables the decision makers to obtain a variety of results through the use of several schemes that can be postulated (customized) according to the decision makers’ preferences, and from which, results best suited to their preferences, can be chosen. A numerical illustration with eight different schemes is presented to demonstrate the virtues of the proposed model.

Introduction

A small amount of money, invested in the right channels, can contribute significantly to the growth of the economy as a whole. Also, apposite investment by an individual gives one some amount of financial control and power. Being a win–win situation, appropriate investment by individuals and organizations becomes imperative.

Since a plethora of alternatives is available, investors are always on the lookout for a propitious place to invest, and often choose conventional portfolios. Presently, there are two types of major investment areas: (i) the convenient one such as real estate, gold, and stock markets, and (ii) the smart one, which is constructing a portfolio. A portfolio usually consists of several classes of financial assets, such as equity shares, stocks, mutual funds, bonds and cash equivalents [1]. In simple terms, portfolio optimization is nothing but not “putting all eggs in one basket”. Mathematically, every portfolio has an associated risk and return, where returns come at the cost of a certain amount of risk. A calculated trade-off between return and risk, considering the preferences of the decision makers, forms the crux of portfolio optimization.

The generic portfolio optimization problem can be stated as thus: given a set of assets with historically determined individual returns and risks, the problem is to determine the set of assets to invest in, along with their proportions, so as to derive the maximum return at minimum risk. It can be argued that a linear relationship is expected to exist between return and risk; however, ‘constant returns to scale’ is an ideal case and rarely observed. Various intrinsic and extraneous factors affect the two key variables and therefore, determining a relationship between the two is a vast and fascinating area of research in itself.

Markowitz [2] was the scientist who introduced the modern portfolio selection theory. His pioneering work is regarded as a landmark in modern portfolio theory (MPT). It examines how the assets’ returns, risks, correlations, and diversification affect the portfolio return. The model proposed by Markowitz, popularly known as the mean–variance (MV) model, is a mean-risk bi-criteria optimization model, with the objective of obtaining an optimized portfolio, comprising the assets fetching the highest expected return for a pre-determined level of risk (measured by the standard deviation of their returns). The model is based on the idea that the return of a portfolio is the weighted linear combination of the returns of the constituent assets, and the portfolio risk is defined as the portfolio variance, or more appropriately, is a function of the correlations of the component assets. Among a given set of assets, Markowitz’s MV model seeks the optimal allocation of capital according to the investors’ return–risk preferences, by considering the first two moments about the rate of return of the portfolio. The optimal solutions of Markowitz’s portfolio selection problem can be determined using his “critical line method”, where the covariances between individual assets signify the importance of diversification in the portfolio. The overall risk can be reduced by selecting assets based not only on their mean and variance but also on their co-moments with other assets under consideration.

Today, the portfolio optimization problem is not limited to just mean and variance. Recently, several researchers have attempted various model simplification techniques or proposed different risk measures to expand Markowitz’s MV model. Several realistic constraints have also been imposed to make it more convincing and productive. Many researchers have attempted to consider practical aspects of portfolio management that often include constraints such as fixed and variable transaction costs, bounds on investment in each asset and cardinality constraints. The modification by Chang et al. in the portfolio selection problem by using a cardinality constraint is regarded as one of the major modifications in the portfolio optimization literature [3]. Soleimani et al. were the first to consider market sectors as a constraint in Markowitz’s MV model in addition to the cardinality constraint and minimization of transaction costs [4].

Although the addition of realistic constraints has further enriched Markowitz’s model, the inclusion of such constraints in the formulation results in a linear/nonlinear mixed integer programming problem, thereby greatly increasing the complexity. Consequently, the portfolio optimization problem turns out to be an NP-Hard problem. The basic assumptions of all the models are: (i) the return of each asset is normally distributed, (ii) the mean of the assets’ past return is the expected future return, (iii) variance is the considered risk measure associated with each asset, and (iv) covariance is the measure of joint risk of each pair of assets. However, all these assumptions do not comply with actual data as (i) the distributions of returns are often non-normal, and exhibit skewness and kurtosis, and (ii) using mean as an estimate of future returns conflicts with the dynamic behaviour of stock markets.

Most of the existing research primarily focuses on two main criteria; return and risk, and neglect the effects of higher moments as can be seen in Hjalmarsson and Manchev [5] and Kumar and Bhattacharya [6].

As per literature, the distribution of assets’ returns usually has a non-normal, asymmetric, leptokurtic and heavy-tailed nature [7], [8], [9], [10], highlighting the importance of higher order moments. Substantial research has been conducted with three- and four-moment frameworks; for example, Harvey et al. [11] used the mean–variance–skewness (MVS) framework with skewed normal distribution and proposed that higher order moments be incorporated in portfolio selection. Their research claims that this process can yield higher returns. Adcock [12] used quadratic programming for solving the MVS portfolio model by considering the multivariate extended skewed Student distribution. Konno and Suzuki [13] applied skewness in Markowitz’s model, highlighting its influence on assets’ rate of return and demonstrated that skewness strongly influences portfolio selection. Maringer et al. [14] considered the higher order moments in the classical Markowitz model and used stochastic algorithms to find the optimal portfolio. Doan et al. [15] examined the systematic skewness and kurtosis of Australian stock returns using higher order moments and demonstrated their significance in the decision-making process. Practically, investors prefer portfolios with a larger third order moment in the event of an equality of the first and second moments.

With the advancement in power of processors, the higher order moments models are becoming increasingly popular. However, these models come with their own set of serious issues, some prominent ones among them being: (a) errors in estimation may yield biased optimal portfolio weights which in turn may cause portfolios to produce “corner solutions” involving infeasible asset allocations, and (b) the resulting portfolio may have low diversity, with undesirable concentration of asset weights [16]. Though few assets may garner unexpected gains, low portfolio diversity may lead to an overall loss. To ensure diversification in the portfolio, in the recent past, several studies have employed entropy as an objective function (see, [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]). Recently, Li and Liu [28] proposed a new definition of credibility based fuzzy entropy that characterized the uncertainty caused due to deficiency of information resulting from the inability to predict the specified return values precisely. Since uncertainty spawns loss in a portfolio, we employ entropy to measure this risk degree of the portfolio. In the case where an investor is clueless about the portfolio return, he/she will have no preference regarding the fuzzy portfolio return values, therefore, resulting in portfolio return to be an equipossible fuzzy variable. Consequently, entropy of the portfolio return peaks, in other words the uncertainty of the portfolio reaches its maximum resulting in a most risky portfolio. Also, that in order to use entropy as a risk measure, we need not assume symmetrical membership functions of the asset returns.

Solving a multi-objective optimization problem to optimality is unfathomable due to the conflicting nature of the objectives. Complete attainment of higher priority objectives usually results in partial attainment of the ones with lower priorities. Thus, these problems require an aggregation procedure such as Goal Programming (GP) technique, a popular approach in portfolio selection [29], [30], [31]. Aouni et al. [30] provided an extensive literature review in this regard with both deterministic and precise as well as fuzzy and stochastic information.

Uncertainty is another key factor in a portfolio optimization model owing to imprecise, vague and uncertain data. In reality, financial information is uncertain rather than deterministic. To address this uncertainty, probability theory is widely used; however, it is incapable of analysing all types of uncertainty, including vague and ambiguous linguistic representations of data in financial markets. Consequently, using fuzzy logic is suggested [32].

Since the introduction of fuzzy set theory by Zadeh [33] in 1965, it has been recognized that these types of uncertainty can easily be captured using fuzzy sets. Zimmermann proposed fuzzy mathematical programming by applying the max–min operator introduced by Bellman and Zadeh [34] to a linear programming problem [35]. Katagiri and Ishii [36] were the first to use fuzzy theory in a portfolio selection problem.

Several GP variants, proposed for handling the uncertainty related to securities for portfolio selection, are usually based on probability and fuzzy theories. Aouni et al. [37] were the first to propose the formulation of a stochastic GP model that explicitly integrates the decision makers’ preferences through the concept of satisfaction functions, with normally distributed stochastic aspiration levels. Abdelaziz et al. [38], [39] presented a stochastic GP approach for generating a satisfactory portfolio assuming the equity returns to be non-normal and also suggested generating limited cases from an observed random distribution. Parra et al. [40] proposed a GP model based on investors’ preferences by expressing the aspiration levels (goals) using interval fuzzy numbers. Alinezhad et al. [41] applied the MINMAX fuzzy GP method to reformulate a fuzzy multi-objective portfolio problem into a fuzzy GP problem for a fuzzy allocated portfolio. The prevailing stochastic and fuzzy GP formulations for portfolio selection do not combine fuzziness and uncertainty simultaneously. Fuzzy–stochastic models are drawn upon to select assets to construct the best trade-off portfolio with combined fuzziness and randomness.

There are numerous implementations of fuzzy mathematical programming for portfolio selection problems in the literature. For example, Gupta et al. [42] proposed an expected value multi-objective portfolio rebalancing model with fuzzy parameters. Kocadagli et al. [43] proposed a novel fuzzy portfolio selection model using fuzzy goal programming techniques. Li et al. [44], using the definitions of possibilistic theory, proposed a model with background risk and solved it using a genetic algorithm. Mehlawat and Gupta [45] addressed the problem of portfolio selection with fuzzy parameters from the perspective of chance-constrained multi-objective programming. Parra et al. [40] applied a fuzzy GP approach to a portfolio selection problem with three objectives, namely, return, risk and liquidity. Fang et al. [46] proposed a fuzzy decision theory based portfolio rebalancing model with transaction costs taking into account the return, risk, and liquidity of the portfolio. Gupta et al. [47] proposed a “semi-absolute deviation portfolio selection model with five criteria, namely, short-term return, long-term return, dividend, risk, and liquidity”, and used fuzzy mathematical programming to solve it. Li et al. [32] proposed a MVS model for portfolio selection with fuzzy return. In another paper, Li et al. [48] introduced three MVS models for portfolio selection with fuzzy returns. With the help of empirical studies, they highlighted the asymmetric nature of portfolio returns and concluded that investors prefer a portfolio return with a larger degree of asymmetry in the event of equality of mean value and variance. Li and Liu [28] introduced the concept of fuzzy entropy for measuring the uncertainty of fuzzy variables. Qiu et al. [49] studied the fuzzy adaptive event triggered control problem containing unknown smooth functions and unmeasured states for a class of pure feedback nonlinear systems. Sun et al. [50] studied an adaptive fuzzy control problem with full state constraints for a class of non-triangular structural stochastic switched nonlinear systems. More recently, Mehlawat et al. [51] proposed a fuzzy multi-objective portfolio model involving higher moments based on data envelopment analysis and provided several schemes for investors having varied attitudes. More literature on fuzzy portfolio selection is available in the monograph by Gupta et al. [52].

To capture the inherent uncertainty in data, previous approaches proposed in the portfolio optimization literature used the fuzzy set theory. However, fuzzy set theory only captures the membership degree and ignores the non-membership and hesitation degrees. To exploit this limitation, we use the intuitionistic fuzzy set (IFS) theory. IFS theory, proposed by Atanassov in 1986 [53], [54] addresses both the degree of acceptance and the degree of rejection, thereby clearly illustrating the concept of “ambiguity”. We use the IFS theory to capture this ambiguity in the form of hesitation degree. This hesitation degree is efficiently used and minimized in the proposed approach to obtain significantly better portfolios. Recently, there have been a few pieces of research on portfolio selection in an intuitionistic fuzzy environment. For example, Chen et al. [55] proposed a “MVS fuzzy portfolio selection model based on intuitionistic fuzzy optimization” using an intuitionistic fuzzy min–max operator. Deng and Pan [56] proposed a multi-objective portfolio selection model based on intuitionistic fuzzy optimization. In addition to the intuitionistic fuzzy “min–max” operator method, they proposed a “max–min” operator method, as well as applied Compromise Programming (CP) and Nadir Compromise Programming (NCP) for portfolio optimization.

Apart from the above research, portfolio optimization in an intuitionistic fuzzy setting/environment is still generally uncharted territory. We have not come across any research that addresses portfolio selection in an intuitionistic fuzzy environment with hesitancy explicitly, which makes our proposed approach (model) a flag-bearer and a pioneer in this direction. So, taking the road less travelled, and inspired and motivated to fill this research gap, we propose in this paper an intuitionistic fuzzy portfolio selection (IFPS) model subject to several realistic constraints; the model gives the decision makers exclusive control over the membership, non-membership and hesitancy functions of the four objectives, namely mean (return), variance (risk), skewness and entropy, considered in the portfolio selection model. The model is solved using the polynomial goal programming technique that minimizes the underachievements of the membership functions and overachievements of the non-membership and hesitancy functions of the all four objectives, leading to significantly better results.

We propose a fuzzy mean–variance–skewness–entropy (MVSE) portfolio selection model with fuzzy assets returns. Using the possibility theory proposed by Carlsson and Fullér [57], we convert the fuzzy MVSE model into a crisp MVSE model. This crisp model is then solved as a single objective portfolio selection model to obtain the extreme (minimum and maximum) values of all the four objectives.

To define the IFPS model, we use the extreme values of the four objectives to construct their membership and non-membership functions, respectively. A parameter k is used to introduce the hesitancy in the proposed IFPS model. Based on the values taken (assumed) by the parameter k, the proposed model is further categorized into an optimistic IFPS model (k=1.1,1.3,,2.9) and a pessimistic IFPS model (k=0.1,0.2,,0.9). The four objectives of mean, variance, skewness and entropy are aggregated using the polynomial goal programming technique to construct new IFPS models wherein the membership, non-membership and hesitation functions of all the four objectives are considered as constraints. The underachievements of the membership functions and the overachievements of the non-membership and hesitation functions of all the objectives are minimized using the polynomial goal programming technique. In the ensuing polynomial goal programming objective, the underachievements of the membership functions and the overachievements of the non-membership and hesitation functions of the four objectives can be controlled using the parameter λi(i=1,2,,12). The parameter λi(i=1,2,,12) empowers the decision makers with an exclusive control over the IFPS models enabling them to tweak their preferences in order to obtain the best possible results. Using the parameter λi(i=1,2,,12), we devise eight different investment schemes for the decision makers to choose from, that are listed in Table 2. A real-world portfolio selection problem comprising twenty assets with fuzzy returns is presented, that demonstrates the virtues of the proposed model.

The portfolio selection problem in an intuitionistic fuzzy environment is still largely uncharted territory. We have not come across any research that addresses the problem in an intuitionistic fuzzy environment with hesitancy which makes our proposed approach a flag-bearer and a pioneer in this direction.

The proposed polynomial goal programming approach for the IFPS model introduces several interesting features. A comparison of the proposed model with others in the literature (on portfolio selection with higher moments, entropy and realistic constraints in general, and portfolio selection in an intuitionistic fuzzy environment with realistic constraints in particular) is presented in Table 1. Some novel contributions of the proposed approach are as follows:

  • 1.

    The proposed approach uses the IFS theory to capture uncertainty using the membership and non-membership degrees, thereby, providing avenues to include the hesitation degree in the decision making. This feature of the proposed study overshadows the previous fuzzy optimization approaches in the literature.

  • 2.

    In this paper, along with the traditional objectives of mean (return) and variance (risk), we consider the third order moment, viz. skewness, whose inclusion in the portfolio selection model has been proven to yield higher returns, and entropy to characterize the uncertainty caused due to deficiency of information resulting from the inability to predict the specified return values precisely.

  • 3.

    Several realistic constraints are incorporated in the proposed model, such as complete capital utilization, cardinality constraint and no short selling, to make the model more realistic. Apart from these traditional realistic constraints, another popular constraint, the “floor and ceiling constraint”, is employed as a flexible constraint by considering the confidence interval and perturbation in asset weights. Consequently, the flexible floor and ceiling constraint adjusts the proportion of the capital allocated, in accordance with the assets’ performance in relation to the objectives under consideration, so as to yield the best possible results. The flexibility that is introduced by the floor and ceiling constraint enables the solution to cross the thresholds of the bounds on the fraction of the capital that can be allocated to a particular asset. Additionally, there are times when the decision makers wish some particular assets, or their combinations, to be included in the portfolio to match their personal preferences. Keeping this in mind, one more constraint is proposed in the portfolio optimization model, a contingent constraint (or pre-assignment constraint), whose role is to incorporate these preferences. These realistic constraints enable our proposed model to demonstrate (mimic) the real market environment more effectively. The sheer number of objectives and constraints make the model more realistic and lifelike.

  • 4.

    Using the aforementioned objectives and constraints, we propose a fuzzy multi-objective portfolio selection model (that acts as a catalyst for the IFPS model) that maximizes the membership functions while simultaneously minimizing the non-membership and hesitancy functions of all the four objectives. The polynomial goal programming approach in addition to aggregating all the objective functions, provides the decision makers with exclusive control over the underachievements of the membership functions and overachievements of the non-membership and hesitation functions of all the four objectives, presented in the form of several investment schemes in this paper. These schemes enable the decision makers to fine-tune their aspirations regarding the objectives so that they can obtain the best possible results as per their preferences.

  • 5.

    Using the hesitation parameter, the proposed IFPS model is then used to present two different types of models: an optimistic IFPS model, and a pessimistic IFPS model, for optimistic investors and pessimistic investors, respectively.

  • 6.

    Each time the decision makers change their preferences for all the objectives in accordance with different schemes, the flexible floor and ceiling constraint modifies itself accordingly, to yield the best possible results, and the contingent constraint, in turn, includes the combination of assets in the portfolio that yields the best results (note that if no combination of assets provides the best results, all the given combinations are excluded from the portfolio), while also satisfying the cardinality constraint.

  • 7.

    The proposed model enables the decision makers to obtain a variety of results through the use of several schemes that can be postulated (custom-made) according to their preferences; they can then pick their desired results.

  • 8.

    The proposed IFPS model aided by the flexible floor and ceiling constraint and the contingent constraint, always works toward providing the best possible result which is also evident from the minimization of the underachievements of the membership functions and overachievements of the non-membership and hesitation functions. This feature of the proposed IFPS model makes it dynamic and intelligent.

Section 2 acquaints the readers with the preliminaries and basic concepts required to understand the proposed approach. The proposed multi-objective fuzzy IFPS models for optimistic and pessimistic scenarios are presented in Section 3; that are validated with the help of a numerical illustration in Section 4. The paper concludes with Section 5. All the results of the numerical illustration are provided in Appendix A Optimistic scenario, Appendix B Pessimistic scenario.

Section snippets

The concept of fuzzy set

A fuzzy set à in X, where X is the universe of discourse, is defined as Ã={(x,μÃ(x))|xX} with membership function μÃ:X[0,1].

Definition 1

[33]

A fuzzy set Ã=(a;α,β) with centre a, left spread α, and right spread β on R, is called a triangular fuzzy number (TFN) with membership function μÃ(x)=(xa+α)(α),aαxa,(a+βx)(β),axa+β,0,otherwise.

Definition 2

[33]

A ξlevel set of a fuzzy number à is defined as [Ã]ξ={xR|Ã(x)ξ} if ξ>0 and [Ã]ξ=cl{xR|Ã(x)>0} if ξ=0.

The concept of intuitionistic fuzzy set

An IFS à in X has the form Ã={(x,μÃ(x),νÃ(x))|xX}

Model description and notation

In this section, we discuss a fuzzy multi-objective portfolio selection model with n risky assets having fuzzy asset returns. The computation of assets’ mean, variance and skewness are based on possibility theory, whereas the entropy is computed by the credibility of Shannon’s function. For the convenience of the readers, we introduce the following notations that are used in the subsequent models.

Notation:

i:Index of the risky asset, i=1,2,,n
xi:Proportion of total investment invested in the ith

Numerical illustration

The proposed models are illustrated through a real-world portfolio selection problem. Historical data, corporate financial statements, market future information, and experts’ advice are used to consider a sample of twenty assets (n=20) from National Stock Exchange (NSE), India. The rates of return of the assets, Rei,i=1,2,,20 are characterized by TFNs, i.e., Rei=(ai;αi,βi) with centre ai, left spread αi and right spread βi, as presented in Table 3.

The cardinality (C) of the portfolio and the

Conclusions

This study proposed an IFPS model considering four objectives, namely mean (return), variance (risk), skewness and entropy, bounded by some realistic constraints such as complete capital utilization, cardinality and no short selling. Apart from these traditional constraints, two additional constraints are employed in the model: (i) a flexible floor and ceiling constraint and (ii) a contingent constraint. These two additional constraints mimic market conditions and investors’ preferences very

Declaration of Competing Interest

No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105781.

Acknowledgments

The author, Sanjay Yadav, is supported by the National Fellowship for Other Backward Classes (OBC) granted by University Grants Commission (UGC), New Delhi, India vide letter no. F./2016-17/NFO-2015-17-OBC-DEL-34358/(SA-III/Website). The author, Arun Kumar, is supported by the Rajiv Gandhi National Fellowship for SC Candidates granted by University Grants Commission (UGC), New Delhi, India vide letter no. F1-17.1/2015-16/RGNF-2015-17-SC-DEL-8966/(SA-III/Website).

References (58)

  • YueW. et al.

    A new fuzzy multi-objective higher order moment portfolio selection model for diversified portfolios

    Physica A

    (2017)
  • AouniB. et al.

    Portfolio management through the goal programming model: Current state-of-the-art

    European J. Oper. Res.

    (2014)
  • LiX. et al.

    Mean–variance-skewness model for portfolio selection with fuzzy returns

    European J. Oper. Res.

    (2010)
  • ZadehL.A.

    Fuzzy sets

    Inf. Control

    (1965)
  • ZimmermannH.J.

    Fuzzy programming and linear programming with multiple objective functions

    Fuzzy Sets and Systems

    (1978)
  • AouniB. et al.

    Decision-maker’s preferences modeling in the stochastic goal programming

    European J. Oper. Res.

    (2005)
  • AbdelazizF.B. et al.

    Multi-objective stochastic programming for portfolio selection

    European J. Oper. Res.

    (2007)
  • GuptaP. et al.

    Expected value multiobjective portfolio rebalancing model with fuzzy parameters

    Insurance Math. Econom.

    (2013)
  • LiT. et al.

    A fuzzy portfolio selection model with background risk

    Appl. Math. Comput.

    (2015)
  • FangY. et al.

    Portfolio rebalancing model with transaction costs based on fuzzy decision theory

    European J. Oper. Res.

    (2006)
  • GuptaP. et al.

    Asset portfolio optimization using fuzzy mathematical programming

    Inform. Sci.

    (2008)
  • MehlawatM.K. et al.

    Data envelopment analysis based fuzzy multi-objective portfolio selection model involving higher moments

    Inform. Sci.

    (2018)
  • AtanassovK.T.

    Intuitionistic fuzzy sets

    Fuzzy Sets Syst.

    (1986)
  • AtanassovK. et al.

    Interval valued intuitionistic fuzzy sets

    Fuzzy Sets and Systems

    (1989)
  • ChenG. et al.

    mean–variance–skewness fuzzy portfolio selection model based on intuitionistic fuzzy optimization

    Procedia Eng.

    (2011)
  • DengX. et al.

    The research and comparison of multi-objective portfolio based on intuitionistic fuzzy optimization

    Comput. Ind. Eng.

    (2018)
  • CarlssonC. et al.

    On possibilistic mean value and variance of fuzzy numbers

    Fuzzy Sets and Systems

    (2001)
  • SharpeW.F. et al.

    Portfolio Theory and Capital Markets, Vol. 217

    (1970)
  • MarkowitzH.

    Portfolio selection

    J. Finance

    (1952)
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