This article has been accepted for publication in a future issue of this journal, but has not been

fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2018.2842752, IEEE
Transactions on Fuzzy Systems

A multi-objective portfolio selection model with

fuzzy Value-at-Risk ratio
Bo Wang, You Li, Shuming Wang and Junzo Watada, Member, IEEE

Abstract—Considering nonstatistical uncertainties and/or in- on their confidence and knowledge about the future. Such
sufficient historical data in security return forecasts, fuzzy set experts’ opinion also involves empirical knowledge, rather
theory has been applied in the past decades to build portfolio than statistical information [4]. Generally, probability theory is
selection models. Meanwhile, various risk measurements such
as variance, entropy and Value-at-Risk have been proposed in used for analyzing a greater amount of data, while possibility
fuzzy environments to evaluate investment risks from differ- theory is used for the representation of events that involve
ent perspectives. Sharpe ratio, also known as the reward-to- linguistic knowledge. Therefore, it is reasonable to introduce
variability ratio which measures the risk premium per unit fuzzy set theory as an alternative tool to stochastic theory when
of the nonsystematic risk (asset deviation), has received great describing security future returns [5].
attention in modern portfolio theory. In this study, the Sharpe
ratio in fuzzy environments is first introduced, whereafter, a Based on the above facts, a lot of fuzzy portfolio selection
fuzzy Value-at-Risk ratio is proposed. Compared with Sharpe models have been proposed in recent years, e.g. fuzzy Value-
ratio, Value-at-Risk ratio is an index with dimensional knowledge at-Risk (VaR) [3], mean-variance [6], mean-semivariance [7],
which reflects the risk premium per unit of the systematic risk mean-entropy [8], mean-semiabsolute deviation [9] and mean-
(the greatest loss under a given confidence level). Based on the variance-skewness [10], which can be divided into two types
two ratios, a multi-objective model is built to evaluate their
joint impact on portfolio selection. Then the proposed model is in the light of the different risks measured. The first type
solved by a fuzzy simulation-based multi-objective particle swarm (T1) is to evaluate the nonsystematic risk (the risk that is
optimization algorithm, where the global best of each iteration specific to a firm) of a portfolio, which can be reduced by
is determined by an improved dominance times-based method. aspiring the diversification of capital allocations and includes
Finally, the algorithm superiority is justified via comparing most of the existing approaches such as mean-variance and
with existing solvers on benchmark problems, and the model
effectiveness is exemplified by using three case studies on portfolio mean-entropy. The second type (T2) measures the systematic
selection. risk (the risk caused by the fluctuation of the entire market)
of an investment e.g. VaR, which tells the largest loss of a
Index Terms—Fuzzy portfolio selection, Sharpe ratio, Value-
at-Risk ratio, multi-objective particle swarm optimization. portfolio under a given confidence level [11]. Nevertheless,
VaR cannot be eliminated via diversification as the VaR of a
portfolio is minimized if and only if all the capital has been
I. I NTRODUCTION allocated to the security with the lowest VaR.
Portfolio selection in finance seeks for optimal capital allo- Considering the different efficacy of the above risk measure-
cations to specific securities, so that an investment can maxi- ments, a number of researchers have applied more than one
mize the profit or minimize the risk. Inspired by Markowitz’s risk measurement to jointly evaluate portfolio performance,
pioneering work [1] in which a mean-variance portfolio selec- where the conflict among the different measurements were dis-
tion model was originally developed, many researchers have cussed. Based on the mean-variance structure, Jana et al. [12]
devoted themselves to this field in the past decades. proposed a fuzzy multi-objective portfolio selection model
Most of the existing studies treat security returns as random which takes entropy as an additional objective. Experimental
variables based on stochastic analysis of precise historical results show that the solutions obtained by the mean-variance
data. However, on one hand such precise data is not always model are often extremely concentrated on a few assets,
available, e.g. for an initial public offering (IPO) share. On the while the proposed method can generate solutions with a
other hand, the various inputs such as company performance, well diversified asset allocation. Similarly, Usta and Kan-
market forces of supply and demand, political factors that form tar [13] developed a multi-objective mean-variance-skewness-
the basis of future return forecasts are often assessed with entropy portfolio selection model, which indicates that an
some uncertainty [2]. This type of uncertainty is usually non- improvement on entropy reduces portfolio variance, but, at the
statistical and involves linguistic knowledge [3]. In addition, same time also decreases portfolio skewness. Recently, Kar
the forecasted returns need to be adjusted by experts based et al. [14] established a multi-objective uncertain portfolio
selection model by defining average return as expected value,
Bo Wang is with the School of Management and Engineering, Nanjing risk as variance and divergence among security returns as
University, Nanjing, China. You Li is with the School of Finance, Nanjing
University of Finance and Economics, Nanjing, China. Shuming Wang is cross-entropy. Then the model was solved by several methods
with the School of Economics and Management, University of Chinese such as weighted sum method, global criterion method and
Academy of Sciences, Beijing, China. Junzo Watada is with the Department evolutionary algorithms. Besides the above studies which
of Computer & Information Sciences, Universiti Teknologi PETRONAS,
Perak Darul Ridzuan, Malaysia. e-mail: (bowangsme@nju.edu.cn, liyoun- focus only on the T1 measurements, research efforts have been
j@126.com, wangshuming@ucas.ac.cn, junzo.watada@gmail.com) made as well to investigate the conflict between T1 and T2.

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Transactions on Fuzzy Systems

Alexander and Baptista [15] related VaR to mean-variance a multi-objective portfolio selection model (R-MOPSM) is
analysis and examined the economic implications of using constructed in this research which optimizes SR and VR
a mean-VaR model for portfolio selection. The case study simultaneously.
shows that the standard deviation of a portfolio is plausible Being a nonlinear multi-objective problem, R-MOPSM is
to increase when the risk measurement has been switched generally difficult to be solved. In the literature, a lot of
from variance to VaR. More recently, Brandtner [16] studied recent studies have been focused on the development of multi-
the portfolio selection under conditional VaR and spectral objective evolutionary algorithms. As a pioneer, Srinivas and
risk measurements, the performance of which was compared Deb developed a non-dominated sorting Genetic Algorithm
with that of the traditional mean-variance model. The analysis (NSGA) based on the fitness sharing of each individual [23].
reveals that conditional VaR finds non-diversification if a risk Nevertheless, the computational burden of NSGA is heavy
free asset exists, and only limited diversification without a risk and it ignores the effectiveness of elitism. Thus, Deb et al.
free asset, which could be a major drawback and should be further improved the above algorithm to NSGA-II by using a
aware of when replacing the traditional variance by conditional fast non-dominated sorting approach and a crowding distance
VaR. The readers may also refer to [17], [18] and [19] for measurement [24]. The experimental results on a number of
some other attempts on portfolio selection with different risk test problems justified the NSGA-II effectiveness. Inspired by
measurements. In summary, all of the above studies prove that the above success, researchers have endeavored to develop
the using of different risk measurements leads to inconsistent multi-objective particle swarm optimization (MOPSO) algo-
portfolio results, and it could be significant to investigate the rithms. Mostaghim and Teich proposed a σ-MOPSO algorithm
inherent conflict among the different measurements as well as which uses σ to represent the local guide of each particle [25].
their joint impact on portfolio selection. The algorithm can direct the particles move towards the
All of the above studies handle the expected return and optima, however, may lead to local convergence as the guides
risk separately, rather than analyzing how well the return used by σ-MOPSO are merely local optima and there is no
compensates an investor for the risk taken. In modern port- mechanism to improve the diversity of the final solution set.
folio theory, Sharpe ratio (SR), also known as the reward- Tripathi et al. introduced a TV-MOPSO algorithm where a
to-variability ratio has received great attention [20]. Recently, parameter named den is employed to determine the global
Nguyen et al. [21] introduced the SR in fuzzy environments best (Gbest) of each iteration, thus enhancing the diversity of
to build portfolio selection models. It is known that SR the final solutions [26]. Nevertheless, the den is calculated
evaluates investment risks by using standard deviation, which only according to the crowding distance sorting while NSGA-
belongs to the T1 measurements mentioned before, thus cannot II suggests to use both the non-dominated and the crowding
reflect the systematic risk of a portfolio. From the literature distance sorting. As a result, the elitism solutions obtained
as well as our experience [3], the T2 measurements (VaR from the non-dominated sorting which can improve the final
and conditional VaR) are not only systematic, but also more solution set are not considered in TV-MOPSO. Recently, a
acceptable for general investors. The reason is twofold. First, concept of dominance times (DT) was introduced in our
variance or entropy provides few information about how much previous study [19] to select Gbest, whereafter an IMOPSO
loss investors may suffer, while it is the loss of money that algorithm was developed. Basically, DT is a numerical value
concerns investors the most [22]. Therefore, compared with which counts the times that a non-dominated solution has
VaR, the T1 measurements are less sensitive to investors, thus dominated the others. Recent experiments show that the Pareto
introducing salient difficulties in assigning risk tolerance levels fronts obtained by IMOPSO are convergent and consist of
or estimating portfolio performance. Second, VaR produces ro- sufficient solutions, but, they cannot maintain an impressive
bust evaluations on investment risks, while its conservativeness diversity, which could be a main drawback of applying the
can be adjusted easily by setting different confidence levels. In existing DT-based Gbest selection.
this way, VaR provides investors with easy-to-adjust robustness Based on the above analysis, this study introduces a prac-
against investment risks. tical strategy to improve the Gbest selection of IMOPSO.
Considering the above advantages, a new index based on Then a fuzzy simulation-based multi-objective particle swarm
fuzzy VaR is introduced in this study, named VaR ratio (VR). optimization (FMOPSO) algorithm is developed, where fuzzy
Compared with SR, VR evaluates the risk premium per unit simulation techniques are used to calculate the SR and VR
of the greatest loss that an investment taken, i.e. reward-to- of portfolios with imprecise security returns and the improved
VaR. Theoretically, SR and VR are conflicting objectives as MOPSO is applied to solve the entire problem. The detailed
the former aims to reach the best trade-off between achieving knowledge of the improved Gbest selection is provided in
a high expected return and dispersing the capital to a basket Section V, while the FMOPSO effectiveness is justified on
of securities, while the latter pursuits an attractive trade-off ZDT problems [27] in Section VI.
between achieving a high expected return and centralizing The major contribution of this research includes: 1. R-
the capital on the-lowest-VaR security. Although a number MOPSM is built to investigate the trade-off between SR and
of existing studies have been focused on the conflict among VR when determining the portfolio composition; 2. FMOPSO
the original risk measurements such as variance and VaR, the is developed as an effective multi-objective solver to obtain
inherent nature between SR and VR has not been investigated. convergent and diversified Pareto fronts. The remainder of
Therefore, to investigate the the inherent conflict between the this paper is organized as follows. Section II briefly reviews
two ratios as well as their joint impact on portfolio selection, basic knowledge of fuzzy set theory, followed by the exist-

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Transactions on Fuzzy Systems

ing risk measurements in fuzzy portfolio selection. Section III. M OTIVATIONS

III introduces fuzzy SR and VR, whereafter the conflicting A. Fuzzy Sharpe ratio
nature between the two ratios and the different features with
In finance, SR measures the risk premium per unit of
the original risk measurements are discussed. In Section IV,
asset deviation, and the higher SR is, the better a portfolio
the mathematical model of R-MOPSM is given. Section V
performs. In what follows, the SR of an individual security
explains the improvements as well as the detailed knowledge
and a portfolio are given.
of FMOPSO. In Section VI, the superiority of FMOPSO is
Definition 3.1. Let ξi be a fuzzy variable that represents the
first justified on some benchmarks, then the effectiveness of
future return of security i, then its Sharpe ratio is defined as:
this research is exemplified by three case studies on portfolio
selection. Finally, Section VII summarizes our conclusions and E[ξi ] − Rf
future works. SR[ξi ] = , (7)
σ[ξi ]
II. P RELIMINARIES where SR[ξi ] denotes the fuzzy Sharpe ratio of security i, Rf
Recently, fuzzy set theory has been applied in various is the return on a benchmark asset e.g. a risk free rate, E[ξi ]
engineering problems to address the endogenous fuzzy uncer- and σ[ξi ] are the expected return and standard deviation of
tainty. In this section, the possibility, necessity and credibility ξi .
measurements of fuzzy variables are introduced, followed Definition 3.2. The Sharpe ratio of a portfolio that involves
by the concepts of the expected value operator and several n securities is:
existing risk measurements such as variance and Value-at- [ n ]
Risk. [ n ] E ∑ xi ξi − Rf
∑
Definition 2.1 ([28]). Let ξ be a fuzzy variable with mem- SR xi ξi = [ n
i=1
] , (8)
bership function µξ and r is a real number, the credibility ∑
i=1 σ xi ξi
function of an event ξ ≤ r is expressed as: i=1
1 where xi is the investment portion on security i.
Cr{ξ ≤ r} = [Pos{ξ ≤ r} + Nec{ξ ≤ r}] , (1)
2 SR uses standard deviation to measure the nonsystematic
where Pos{.} and Nec{.} represent the possibility and neces- risk, and the maximization of SR is to reach the best trade-off
sity measurements [28], defined as: between achieving a high return and dispersing the capital to
a basket of securities.
Pos{ξ ≤ r} = sup µξ (t), (2)
t≤r

Nec{ξ ≤ r} = 1 − sup µξ (t). B. Fuzzy VaR ratio

t>r
(3)
In contrast to SR, VR applies fuzzy VaR to calculate the
The credibility measurement is a self-dual set function, greatest loss of an investment under a given confidence level,
i.e., Cr{ξ ≤ r} = 1 − Cr{ξ > r}. Suppose ξ denotes a and the maximization of VR is to seek for the best trade-off
security future return, then Cr{ξ ≥ 0.2} = 0.9 means that the between achieving a high expected return and centralizing the
credibility of an event that in the future has a return of no less capital on the-lowest-VaR security. The VR of an individual
than 0.2, is 0.9 or 90%. security and a portfolio are given below.
Definition 2.2 ([29]). The expected value of ξ is calculated Definition 3.3. Let ξi be a fuzzy variable that represents the
as: future return of security i, then its VR is defined as:
∫ +∞ ∫ 0 E[ξi ] − Rf
E[ξ] = Cr{ξ ≥ r}dr − Cr{ξ ≤ r}dr. (4) VR[ξi ] = , (9)
VaR1−β [ξi ]
0 −∞
Definition 2.3 ([29]). The variance of ξ provides a mea- where VR[ξi ] denotes the fuzzy Value-at-Risk ratio of security
surement on the spread of the distribution around its expected i.
value. If ξ has a finite expected value, then its variance value Definition 3.4. The VR of a portfolio with n securities is:
is: [ ] [ n ]
2 [ n ] ∑
V[ξ] = E (ξ − E[ξ]) . (5) E xi ξi − Rf
∑
Definition 2.4 ([30]). Suppose that ξ is a fuzzy variable VR xi ξi = i=1
[ n ]. (10)
∑
which represents the loss of an investment, then the Value-at- i=1 VaR1−β x i ξi
i=1
Risk of ξ under confidence level (1 − β) is computed as:
VR of an investment measures the risk premium per unit
VaR1−β [ξ] = sup {λ|Cr(ξ ≥ λ) ≥ β} , (6) of the greatest loss. Obviously, the higher VR is, the better
where β ∈ (0, 1). this investment will be. Specifically, if VR< 0, the portfolio
Equation (6) tells that the greatest loss of the investment is not feasible as its return is lower than the risk free ratio.
under confidence level (1 − β) is λ. When 0 ≤ VR < 1, it is inadvisable to realize the investment
The above risk measurements have been applied to build as it suffers from a higher potential loss and a lower profit.
fuzzy portfolio selection models, the readers may refer VR=1 means that the risk premium amount equals that of the
to [3], [6] and [8] for the details. greatest loss. Then a portfolio is acceptable when VR> 1 as it

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Transactions on Fuzzy Systems

TABLE I
D IFFERENT FEATURES BETWEEN RISK MEASUREMENTS AND RATIOS ( THEORETICAL PERSPECTIVE )

Measurement Objective
[ n ] Realization Feature
∑
ER max E xi ξi Capital centralized on the highest-return security Centralized
[i=1 ]
∑n
SD min σ xi ξi Capital decentralized on a basket of securities Decentralized
[ i=1 ]
∑
n
E xi ξi −Rf
i=1
SR max [ ] Best trade-off between maximal ER and minimal SD Centralized vs. Decentralized
∑
n
σ xi ξi
[i=1n ]
∑
VaR min VaR xi ξi Capital centralized on the lowest-VaR security Centralized
[ i=1 ]
∑
n
E xi ξi −Rf
i=1
VR max [ ] Best trade-off between maximal ER and minimal VaR Centralized vs. Centralized
∑
n
VaR1−β xi ξi
i=1

produces a higher return and a lower loss. In real-applications, the portfolio which is relatively centralized on the-highest-
different VR levels could be assigned by investors according return security results in a higher VR value. Theoretically, VaR
to their personal risk attitudes. and VR are also different, because a security with a high (low)
Compared with SR, VR is a systematic index that provides expected return is generally accompanied by a high (low) VaR
dimensional knowledge, which is more acceptable for lay- value.
people to interpret portfolio performance. For example, in- The differences between the ratios and risk measurements
vestors know that maximizing SR can improve the portfolio, are further exemplified by using a simple case study. Con-
however, they cannot identify how much better an investment sidering the first two securities listed in Table VI of Section
becomes when SR increases, e.g. from 0.9 to 1.1. The reason VI, i.e. STV (-0.129, 0.091, 0.259) and CNEP (-0.280, 0.254,
is that the standard deviation used in SR is a non-dimensional 0.689), the portfolio decisions are made to optimize standard
risk measurement. By contrast, applying fuzzy VaR as a deviation, SR, VaR or VR respectively. The experimental
quantitative risk measurement, VR tells the risk premium per results are listed in Table II.
unit of the exact loss. In this manner, investors can easily
TABLE II
understand the performance of different portfolios. D IFFERENT FEATURES BETWEEN RISK MEASUREMENTS AND RATIOS
( APPLICATION PERSPECTIVE )
C. Different features between the ratios and original risk
measurements Objective Result Capital allocation Feature
min SD 0.102 STV:0.881, CNEP: 0.119 Decentralized
In this subsection, we first show the difference between the max SR 0.710 STV:0.487, CNEP: 0.513 Decentralized
original risk measurements and the corresponding ratios, then min VaR 0.082 STV:1.000, CNEP: 0.000 Centralized
the conflicting nature of SR and VR are explained. Especially, max VR 1.080 STV:0.000, CNEP: 1.000 Centralized
a simple case study is provided to explain the difference and
conflict from the application perspective.
The theoretical features of the expected return, risk measure- Table II expresses that the outcomes of standard deviation
ments and ratios are provided in Table I, where ER means the and SR are different: due to the less volatility of STV, a
expected return and SD denotes standard deviation. majority of the capital has been allocated to STV to achieve the
Table I shows that standard deviation aims to distribute the lowest standard deviation. Nevertheless, a trade-off between
capital to a basket of securities, and the more decentralized the expected return and standard deviation has been achieved
the portfolio is, the smaller standard deviation will be. By when we adopt SR as the measurement, thus changing the
contrast, SR aims to strike the best trade-off between capital portfolio composition as well as the investment result. Sim-
decentralization and centralization on the highest-return secu- ilarly, VaR and VR produce different outcomes as well: the
rity. Suppose that two portfolios are with the same standard VaR of STV is lower than that of CNEP, thus the whole capital
deviation, then the portfolio with a higher expected return has been distributed to the former to obtain a portfolio with
results in a higher SR value. From the theoretical perspective, the lowest VaR. However, the trade-off between the expected
standard deviation and SR are different, thus may produce return and VaR should be addressed when VR is treated as
different portfolio results. the measurement. Then Table II indicates that allocating the
In terms of VaR and VR, Table I illustrates that the former whole capital to CNEP provides the highest VR.
aims to centralize the capital on the security with the-lowest- The experimental results described above are consistent with
VaR, while the latter seeks for the best trade-off between the discussion on Table I, which jointly expose the difference
centralizing the capital on the-highest-return and the-lowest- between the risk measurements and ratios.
VaR securities. If two portfolios are with the same VaR, then In addition, the above knowledge also reveals the conflicting

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Transactions on Fuzzy Systems

nature between SR and VR: Table I shows that SR optimizes

the trade-off between capital centralization and decentral-  ∑
n

 xi ai − Rf
ization, while VR pursues an attractive trade-off between 


 max √i=1
different types of capital centralization. From the capital 


 ∑n
allocation perspective, the portfolio composition obtained by 
 xi σi2


maximizing SR should be more diversified than that of VR, 
 i=1

 ∑
as reflected in Table II. As mentioned before, a number of 

n
xi ai − Rf
existing studies have been focused on the conflict among i=1 (12)
the original risk measurements such as variance and VaR. 
 max ∑ n [ √ ]


However, the inherent nature between SR and VR has not been 
 x i ai + σ i ln (1/2β)

 i=1
investigated, which could be the primary motivation behinds 


 Subject to
this research. 
 ∑n




xi = 1

 i=1
xi ≥ 0.
IV. A MULTI - OBJECTIVE PORTFOLIO SELECTION MODEL

While β > 0.5, it equals:

This section provides the mathematical model as well as
some property analysis of R-MOPSM.
 ∑
n

 xi ai − Rf



 √
i=1

 max
A. Mathematical model 
 ∑n

 xi σi2
′ 

Suppose that pi is a fuzzy variable that denotes the fore- 
 i=1

 ∑
casted closing price of security i in the future, pi is the 

n
xi ai − Rf
closing price at present, then the return of security i is simply i=1 (13)
′
described by a fuzzy variable ξi as ξi = (pi − pi )/pi , where 
 max ∑ n { √ }

 −
the dividend is not considered in this study. 
 x i a i + σi ln [1/(2 2β)]

 i=1

 Subject to
The mathematical model of R-MOPSM is given as follows, 


 ∑n
which uses SR and VR to evaluate the investment perfor- 



xi = 1
mance: 
 i=1
xi ≥ 0.
 [ n ]

 ∑

 max SR x ξ
i i

 [i=1 ] Proof: The proof of the above theorem is similar to that

 ∑


n of the existing study [3], where the expected return, standard
 max VR xi ξi
i=1 deviation and VaR are calculated according to the credibility
(11)

 Subject to measurement of ξ. The readers may refer to [3] for the details.

 ∑


n

 xi = 1

 Theorem 2. Suppose that the security returns are indepen-
 i=1
xi ≥ 0, dent trapezoidal fuzzy variables with symmetrical distributions
denoted as ξi = (ai , bi , ci , di ), then for any xi , when β ≤ 0.5,
where the total capital is considered as unit 1, xi is the model (11) equals:
investment portion on security i in terms of percentage, SR[.]
and VR[.] represent the Sharpe ratio and VaR ratio of a 
portfolio.  ∑n ai + bi + ci + di

 xi − Rf

 4

 max √ i=1



 ∑n 3(ci − bi + αi )2 + αi2

 x
B. Some Properties of R-MOPSM 

i
24


i=1

 ∑
n a + b + c + di
In some special conditions when all the security returns  xi
i i i
− Rf
can be taken as fuzzy variables that follow the same type i=1 4 (14)

 max ∑ n
of distribution (such as trapezoidal, triangular or Gaussian 
 xi [(2β − 1)ai − 2βbi ]


distributions), the theorems below enable us to solve model 



i=1
(11) easily. 
 Subject to

 ∑n
Theorem 1. Suppose that the future returns of all can- 


 xi = 1
didate securities are independent Gaussian fuzzy variables 
 i=1
as ξi = FG(ai , σi2 ), the membership function is µξi (t) = xi ≥ 0.
Exp{−[(t − ai )/σi ]2 }, then for any xi , when β ≤ 0.5, model
(11) is equivalent to: When β > 0.5, it equals:

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∑
n
 ∑n a) Calculate all the possible combinations of xi ξi ,
 ai + bi + ci + di

 xi − Rf ζir
i=1

 4 i.e. select every of each ξi and compute:

 max √ i=1



 ∑n 3(ci − bi + αi )2 + αi2 oj = x1 ζ1r + ... + xi ζir + ... + xn ζnr , (18)

 xi

 24


i=1 where 0 ≤ r ≤ l, 1 ≤ j ≤ ln and the membership

 ∑n a + b + ci + di
 xi
i i
− Rf degree of oj is:
i=1 4 (15)

 max ∑ n µ(oj ) = µ(ζi1 ) ∧ ... ∧ µ(ζir ) ∧ ... ∧ µ(ζil ). (19)

 xi [(2β − 2)ci − (2β − 1)di ]



 b) Record all oj and the related membership degrees.


i=1

 Subject to If several oj values are the same, e.g. all equal t,

 ∑n

 then the one with the largest membership degree

 xi = 1

 i=1 is selected to construct the membership distribution
xi ≥ 0. ∑n
of xi ξi , that is:
Proof: The proof of the above theorem is similar to that i=1
of our previous work. The readers may refer to [3] for the µ(t) = sup µ(oj ), (20)
details.
In Theorem 2, when bi = ci , the trapezoidal fuzzy variables where j satisfies oj = t.
become triangular ones. Therefore, another theorem can be 4) Obtain the possibility and credibility measurements of
∑n
obtained without difficulty when all the returns are depicted xi ξi according to Equations (1), (2) and (3).
as independent triangular fuzzy variables. i=1
∑
n
Basically, the theorems given above can reduce the com- 5) Calculate the expected value and variance of xi ξi
putational burden of R-MOPSM when all the security returns i=1
by using Equations (4) and (5). The readers may refer
are independent and follow the same type of distribution.
to [10], [32] and [33] for the details. If compute fuzzy
∑n ∑
n
V. G ENERAL SOLUTION METHOD VaR, one only needs to replace xi ξi by − x i ξi .
i=1 i=1
Generally speaking, security returns could be fuzzy vari- It should be noticed that the selection of a larger l value
ables with different distributions and they may not always be ∑n
leads to a higher approximation of xi ξi , however, also
independent to each other. In this case, it is impossible to solve i=1
R-MOPSM by using the above theorems. Therefore, a fuzzy increases the computational burden exponentially. Therefore, it
simulation-based MOPSO algorithm is developed in this study is suggested that the l value should be set specially to different
to solve the proposed model in general situations. types of optimization problems.

A. Fuzzy simulation to calculate expected value, variance and B. PSO algorithm and Pareto optimal solution
VaR PSO algorithm uses particle collaborations to find optimal
Fuzzy simulation introduced by Liu [31] plays a pivotal role solutions within a possible space [34]. If the location of some
in the complicated calculations among fuzzy variables. The particle produces a better fitness value, then the others will
essence of fuzzy simulation is to approximate the membership adjust their positions to approach this one. Assume that S
distribution of the fuzzy variable ξi by a series of discrete particles search in a K-dimensional space for T iterations,
fuzzy vector ζ, so that the possibility and credibility measure- and the position of particle s is denoted as:
∑
n
ments of xi ξi can be approximately obtained. Suppose that Pis → (Pis,1 , ..., Pis,k , ..., Pis,K ) , (21)
i=1
ξi (i = 1, ..., n) represents a number of fuzzy variables with
∏n where 1 ≤ s ≤ S, 1 ≤ k ≤ K and T is a sufficient large
supports [Li , Ui ], where Li and Ui are the lower and upper integer. Then the velocity and position of each particle are
i=1
bounds of ξi , then the expected value, variance and VaR are updated as follows:
calculated as follows:
1) Divide each fuzzy variable ξi into l parts: vs,k = w · vs,k + c1 · Rand · (Pbests,k − Pis,k )
(22)
r +c2 · Rand · (Gbestt,k − Pis,k ),
ζir = Li + (Ui − Li ), (16)
l
Pis,k ← Pis,k + vs,k , (23)
where r and l are integers, and 0 ≤ r ≤ l.
2) Calculate the membership degree of each ζir , thus ap- where w is an inertia weight, c1 and c2 are learning rates, vs,k
proximating the membership function of ξi : means the velocity of particle s at dimension k with an upper
and lower bounds of Vmax, Vmin. Pbests,k is the personal
µξi → {µ(ζi1 ), ..., µ(ζir ), ..., µ(ζil )}. (17)
best that represents the best position of each particle itself in
∑
n the completed iterations, and Gbestt,k is the global best that
3) Simulate the membership distribution of x i ξi :
i=1 denotes the best position of all particles after t times iterations.

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The conventional PSO is a single-objective algorithm which to a clustering of particles to the elitism with a high DT, thus
cannot be employed directly to solve the proposed R-MOPSM improving the convergence of certain part of the final Pareto
(without invoking the weight method). In the literature, the front, however, ignores the spread of the solutions found.
concept of Pareto optimal solution has been applied in multi- To mitigate the above disadvantage, this study integrates
objective optimization. With regard to model (11), a solution a mutation strategy to improve the existing DT-based Gbest
X ∗ = (x∗1 , ..., x∗i , ..., x∗n ) is said to dominate another solution selection. When determining the Gbest of each iteration, an
X = (x1 , ..., xi , ..., xn ) if and only if Equation (24) is α value is first randomly generated in interval (0,1), then α
satisfied. is compared with a predefined threshold αL . If α ≤ αL , the
 [ n ] [ n ] Gbest selection remains the same as that of IMOPSO (Rule 1).
 ∑ ∗ ∑

 SR xi ξi ≥ SR x i ξi If α > αL , Gbest is determined by DT as well, however, from
[i=1 ] [i=1 ] (24) an opposite side i.e. the larger DT a non-dominated solution

 ∑n
∗
∑n
 VR xi ξi ≥ VR x i ξi , holds, the less possible it will be considered as Gbest (Rule
i=1 i=1 2). In this manner, an adequate trade-off between convergence
and X ∗ is treated as a Pareto optimal solution (or a non- and diversity can be obtained if an appropriate value has been
dominated solution) only when no other solution can dominate assigned to αL . Figure 2 depicts the flow chart of the mutation-
it. Then an aggregation of all Pareto optimal solutions forms based Gbest selection in FMOPSO, while its effectiveness is
the Pareto front. justified in Section VI.

C. FMOPSO algorithm
As mentioned before, the existing IMOPSO cannot always
obtain diversified Pareto fronts. Therefore, FMOPSO which
improves the Gbest selection of IMOPSO is developed in this
study as the solution of R-MOPSM. The details are explained
as follows.
1) An improved Gbest selection: Recent computation ex-
perience shows that the DT-based IMOPSO proposed in our
previous study may not obtain a diversified Pareto front. Figure
1 describes the IMOPSO performance on ZDT1-4, which
are two-objective minimization problems with 30 decision
variables.

1 1
ZDT1 ZDT2
0.8 0.8 Fig. 2. Flow chart of the Gbest selection in FMOPSO
0.6 0.6
F2

F2

0.4 0.4 2) Pbest selection and a strategy to mitigate local conver-

0.2 0.2 gence: In FMOPSO, the Pbest selection and the strategy to
0 0
mitigate local convergence are the same as that of IMPOSO.
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 The readers may refer to [19] and [35] for the details.
F1 F1
Based on all of the above knowledge, FMOPSO is devel-
1 1
ZDT3 ZDT4 oped as the solution of R-MOPSM, where fuzzy simulation is
0.8 used to calculate SR and VR, and the improved algorithm is
0.5
0.6 applied to solve the entire problem.
F2

F2

0
0.4
−0.5
0.2 VI. C OMPUTATIONAL S TUDY
−1 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 In this section, the superiority of FMOPSO is first justified
F1 F1
on ZDT1-4. Then the performance of this research is exem-
plified by solving three case studies on portfolio selection,
Fig. 1. Pareto fronts on ZDT1-4 obtained by IMOPSO where the trade-off between SR and VR is discussed, and the
superiority of FMOPSO is further justified via comparing with
Generally, the Pareto fronts in Figure 1 are convergent and the existing MOPSOs.
consist of a large amount of solutions, but, they cannot always
maintain an impressive diversity, which can be viewed in the
dashed squares. The reason is that in IMOPSO, the DT of A. Superiority of FMOPSO on ZDT1-4
each non-dominated solution is first counted, then the larger Using the mutation strategy, the first trail performance of
DT a solution holds, the more possible it will be selected as the FMOPSO on ZDT1-4 is provided in Figure 3 where αL is set
Gbest of the next iteration. Theoretically, this mechanism leads as 0.2 for ZDT1-3 and 0.5 for ZDT4, after testing.

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Transactions on Fuzzy Systems

value of F1 than F2, and as a result, the solutions with small F1

1 1
ZDT1 ZDT2 generally take high DT values. In this way, the using of a large
0.8 0.8
αL increases the chance that a non-dominated solution with
0.6 0.6 small F1 is selected as Gbest, thus improving the convergence
F2

F2
0.4 0.4 of the F1 minimization, but deteriorating the optimization on
0.2 0.2 F2. By contrast, a small αL raises the probability that a non-
0 0 dominated solution with small F2 is selected as Gbest, thus
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
F1 F1
improving the diversity of the final solution set, but, may lead
to a deterioration on the solution convergence.
1 1
ZDT3 ZDT4 According to the above analysis, it can be concluded that
0.8
0.5 an appropriate αL value, for example 0.2 in ZDT1 can be
0.6
used to realize a convergent and diversified Pareto front. In
F2

F2

0
0.4 addition, the αL value should be set specially to different types
−0.5
0.2 of optimization problems.
−1 0 The effectiveness of FMOPSO is further justified by com-
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
F1 F1 paring with IMOPSO, σ-MOPSO and TV-MOPSO. In the
comparisons, each algorithm was executed for 20 times,
Fig. 3. Pareto fronts on ZDT1-4 obtained by FMOPSO whereafter the statistical measurements such as the best, worst
and mean of generational distance (GD), spacing (SP) and
hypervolumn (HV) [36] are provided. Figures 1, 3, 5 and
Obviously, FMOPSO outperforms IMOPSO as the resulted 6 depict the Pareto fronts obtained by the first trail of each
Pareto fronts are highly convergent and more diversified than algorithm. Table III shows the GD, SP and HV comparisons a-
that of Figure 1. To investigate the impact of αL on Pareto mong the four algorithms, where the sample sizes of the actual
fronts, a series of experiments were implemented on ZDT1 Pareto fronts are all set as 5000 and the reference point when
with different αL values. Figure 4 depicts the experimental calculating HV is chosen as r = {1, 1}. Besides, the average
results. runtime costs (RT) of the 20 trails on each test problem are also
presented in Table III, while the particle number and iteration
times are all set as 50 and 800 respectively.
1 1
αL=0.0 αL=0.1
0.8 0.8

0.6 0.6 1 1
ZDT1 ZDT2
F2

F2

0.4 0.4 0.8 0.8

0.2 0.2 0.6 0.6

F2

F2

0 0 0.4 0.4
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
F1 F1 0.2 0.2

1 1 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
αL=0.5 αL=0.8
0.8 0.8 F1 F1

0.6 0.6 1 1
ZDT3 ZDT4
F2

F2

0.4 0.4 0.8

0.5

0.2 0.2 0.6

F2

F2

0
0 0 0.4
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 −0.5
F1 F1 0.2

−1 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fig. 4. Pareto fronts on ZDT1 obtained by different αL of FMOPSO F1 F1

Figure 4 together with the ZDT1 in Figures 1 and 3 (αL Fig. 5. Pareto fronts on ZDT1-4 obtained by σ-MOPSO
equals 1.0, 0.2 respectively) reveal the interactions between
αL and the corresponding Pareto front, i.e. the smaller αL First, Table III shows that all of the four algorithms produce
is (a high probability to select Gbest via Rule 2), the more convergent Pareto fronts as the corresponding GD values are
clustering the particles to the search space with small F2. absolutely small. Besides, the GD of FMOPSO and IMOPSO
Conversely, the larger αL is (a high probability to select Gbest is more attractive than that of σ-MOPSO and TV-MOPSO,
via Rule 1), the more clustering the particles to the search which proves that the DT-based Gbest selection can improve
space with small F1. This phenomenon can be interpreted via the convergence of the final solution set. Second, in view of
analyzing the feature of ZDT problem family, in which F1 is the SP and HV metrics, Table III together with Figures 1,
determined by only one variable while F2 is influenced by 29 3, 5 and 6 illustrate that FMOPSO basically outperforms the
variables. Theoretically, it is much easier to find the minimal others since its overall performance is the best. In addition,

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TABLE III
P ERFORMANCE COMPARISONS ON ZDT1-4: GD, SP, HV, RT( S ) 1 1
ZDT1 ZDT2
0.8 0.8
Algorithm ZDT1 ZDT2 ZDT3 ZDT4
FMOPSO 6.1E-5 7.4E-5 2.8E-4 7.2E-5 0.6 0.6

F2

F2
IMOPSO 7.1E-5 7.3E-5 2.5E-4 7.1E-5 0.4 0.4
Best σ-MOPSO 1.7E-4 5.5E-5 3.6E-4 4.3E-4
TV-MOPSO 8.6E-5 7.8E-5 3.1E-4 8.0E-5 0.2 0.2

FMOPSO 7.0E-5 7.8E-5 3.6E-4 8.0E-5 0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
IMOPSO 8.7E-5 8.3E-5 3.3E-4 8.2E-5 F1 F1
GD Worst σ-MOPSO 2.7E-4 1.3E-4 5.4E-4 4.8E-4
TV-MOPSO 1.2E-4 1.1E-4 4.4E-4 9.6E-5 1 1
ZDT3 ZDT4
FMOPSO 6.6E-5 7.6E-5 3.1E-4 7.7E-5 0.8
0.5
IMOPSO 8.2E-5 7.6E-5 2.9E-4 7.5E-5 0.6

F2

F2
Mean σ-MOPSO 2.1E-4 8.8E-5 4.2E-4 4.5E-4 0
TV-MOPSO 1.0E-4 9.1E-5 3.8E-4 8.9E-5 0.4
FMOPSO 3.3E-4 2.8E-4 5.9E-3 4.8E-4 −0.5
0.2
IMOPSO 1.6E-2 2.2E-2 8.2E-2 5.6E-3
Best σ-MOPSO 1.1E-3 1.7E-2 3.7E-2 1.8E-2 −1 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
TV-MOPSO 1.0E-3 4.2E-3 7.4E-3 2.8E-3 F1 F1
FMOPSO 1.1E-3 5.8E-4 9.3E-3 1.2E-3
IMOPSO 8.7E-2 1.3E-1 2.3E-1 1.0E-2
Fig. 6. Pareto fronts on ZDT1-4 obtained by TV-MOPSO
SP Worst σ-MOPSO 6.8E-3 3.8E-2 8.5E-2 3.2E-2
TV-MOPSO 2.0E-3 1.0E-2 2.2E-2 1.1E-2
FMOPSO 5.0E-4 4.1E-4 7.6E-3 7.6E-4
IMOPSO 3.1E-2 5.0E-2 1.5E-1 7.5E-3 obtained by experts’ opinion after observing the historical
Mean σ-MOPSO 2.2E-3 2.5E-2 6.8E-2 2.3E-2
TV-MOPSO 1.4E-3 6.7E-3 1.3E-2 5.3E-3
prices of 2011 [37].
FMOPSO 0.6612 0.3303 0.7782 0.6615 Based on the CP and FP given in Table IV, the future
IMOPSO 0.6489 0.3147 0.7467 0.6586 return of each security is calculated as (FP-CP)/CP (again, the
Best σ-MOPSO 0.6595 0.3246 0.7717 0.6582
TV-MOPSO 0.6604 0.3275 0.7739 0.6593 dividend is not considered in this case study), and the results
are listed in Table V.
FMOPSO 0.6583 0.3269 0.7701 0.6586
IMOPSO 0.6410 0.3103 0.7198 0.6521 2) Performance of R-MOPSM & FMOPSO: R-MOPSM is
HV Worst σ-MOPSO 0.6523 0.3211 0.7597 0.6509 applied to construct the portfolio selection problem, where
TV-MOPSO 0.6553 0.3205 0.7592 0.6513
Rf equals 0.05 and the confidence level of VaR is set as
FMOPSO 0.6601 0.3286 0.7739 0.6602 0.9, i.e. β = 0.1. Then FMOPSO is applied to solve the
IMOPSO 0.6455 0.3115 0.7360 0.6544
Mean σ-MOPSO 0.6568 0.3233 0.7682 0.6541 above problem, while its parameters are set according to
TV-MOPSO 0.6579 0.3239 0.7697 0.6550 existing literatures and our computational experience, as listed
FMOPSO 13.98 14.32 14.20 14.64 in Table VI. Specifically, the particle number and iteration
IMOPSO 13.42 14.22 14.09 14.59
RT σ-MOPSO 17.64 17.44 17.53 18.50 times follows the suggestion given in [38], the inertia weight
TV-MOPSO 19.25 18.12 18.23 19.97 and learning rates are set the same as the classical PSO, the
particle maximal/minimal velocity and αL are assigned after
performing a number of trails.
There are totally 45 Pareto optimal solutions found after
the negative SP and HV performance of IMOPSO also justifies 4.0 × 104 attempts, as shown in Figure 7. Table XII in the
that the mutation strategy used in FMOPSO is effective in im- Appendix lists the detailed knowledge of the optimal solutions,
proving the diversity of the non-dominated solutions. Finally, while Figure 8 depicts the capital allocation of four typical
the runtime costs listed in Table III indicate that the DT-based solutions i.e. 1, 14, 23 and 45. Specifically, solution 1 max-
FMOPSO and IMOPSO can reduce the computational burden imizes SR and solution 45 maximizes VR. Solutions 14 and
of the existing multi-objective algorithms. 23 are selected considering technical and visualized aspects
Based on the above facts, it is concluded that FMOPSO is respectively. From the technical aspect, after the execution of
more effective than the existing MOPSOs when solving ZDT FMOPSO, the DT value of solution 14 is the highest among
problems. the Pareto front (as shown in Table XII), which indicates that
the solution could be an elite as it dominates the most solutions
found in the search space. From the visualized aspect, solution
B. Case study 1 23 strikes an attractive trade-off between the numerical values
1) Problem description: In case study 1, a portfolio selec- of the two objectives, which would be more acceptable for
tion problem that includes 10 securities listed on the New York common investors with both SR and VR concerns.
Stock Exchange is solved. Consider a four-month investment Table XII together with Figure 7 show that SR and VR
problem at Dec 30, 2011. Table IV provides the abbreviations are conflicting objectives as the increasing of any one results
and closing prices (CP) at Dec 30, 2011 of the candidate in the decreasing of the other. Besides, the portfolio com-
securities, while the forecasted prices (FP) during the first 4 position obtained by optimizing SR is more diversified than
months of 2012 are triangular or Gaussian fuzzy variables that of VR. The reason is that VR aims to obtain a trade-

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10

TABLE IV
K NOWLEDGE OF THE 10 SECURITIES FROM N EW YORK S TOCK E XCHANGE

Security No. Symbol CP($) FP($) Security No. Symbol CP($) FP($)
1 STV 3.17 (2.76, 3.46, 3.99) 6 ACH 10.8 (9.62, 12.57, 13.68)
2 CNEP 2.05 (1.48, 2.54, 3.46) 7 CAST 6.12 (4.51, 5.65, 6.64)
3 CNTF 1.8 (1.25, 2.09, 2.63) 8 DANG 4.40 FN(5.28, 0.099)
4 CBEH 0.39 (0.21, 0.58, 0.73) 9 CNAM 0.28 FN(0.35, 0.124)
5 CHL 48.49 (45.05, 52.51, 55.99) 10 CBAK 3.15 FN(3.25, 0.015)

TABLE V
F UTURE RETURNS OF THE 10 SECURITIES

Security No. Symbol Return Expected value Security No. Symbol Return Expected value
1 STV (-0.129, 0.091, 0.259) 0.078 6 ACH (-0.109, 0.164, 0.267) 0.122
2 CNEP (-0.280, 0.254, 0.689) 0.231 7 CAST (-0.263, -0.077, 0.085) -0.006
3 CNTF (-0.306, 0.161, 0.461) 0.119 8 DANG FG(0.199, 0.099) 0.199
4 CBEH (-0.462, 0.487, 0.872) 0.346 9 CNAM FG(0.250, 0.124) 0.250
5 CHL (-0.071, 0.083, 0.148) 0.061 10 CBAK FG(0.032, 0.015) 0.032

2 0.4 0.4
Solution 1 Solution 14

Capital allocation

Capital allocation
VR=0.999 VR=1.262
0.3 SR=1.705 0.3 SR=1.603
1.8
0.2 0.2

1.6
0.1 0.1
Sharpe ratio

1.4 0 0
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
Security No. Security No.

1.2 0.8
Solution 23 1 Solution 45
Capital allocation

Capital allocation
VR=1.375 VR=1.742
0.6 0.8 SR=0.607
1 SR=1.374

0.4 0.6

0.8 0.4
0.2
0.2
0 0
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
VaR ratio Security No. Security No.

Fig. 7. Pareto front obtained by FMOPSO Fig. 8. Capital allocations of four typical solutions

brings different portfolio results and thus enriching investors’

off between two different types of capital centralization, i.e. decision region.
centralizing the capital on the-highest-return or the-lowest-
VaR security. In this case study, solution 45 distributes all TABLE VI
PARAMETER SETTING OF FMOPSO
the capital to security 6 as its VR is the highest, as shown in
Figure 8. Nevertheless, this behavior (allocating all the capital
Particle Iteration Vmax Vmin ω c1 c2 αL
to one security) increases the portfolio variability (variance) 50 800 1.2 -1.2 1.0 2 2 0.4
greatly, thus reducing the corresponding SR value. On the
other hand, SR pursues an adequate trade-off between capital
centralization and decentralization, i.e. centralizing the capital Besides, Table XII also illustrates that a non-dominated
on the-highest-return security and decentralizing the capital solution which involves certain trade-off between SR and VR
on a number of securities. In Figure 8, solution 1 allocates generally has a larger DT than that of the solutions who
the capital to a basket of securities to achieve a high SR extremely optimize either one. For example, the DT of solution
value, however, its VR is rather low as the investment portion 14 is 34728 which is obviously larger than that of solutions
on security 6 is relatively small. In addition, the knowledge 1 and 45. This phenomenon is caused by the random nature
involved in Table XII, Figures 7 and 8 also justifies that the of meta-heuristic algorithms, i.e. the randomness exists in the
proposed method can realize various trade-offs between SR particle initializations and iterations of FMOPSO, since it is
and VR, e.g. the portfolio composition of solutions 14 and much more difficult to find the maximal or minimal objective
23 is relatively more centralized than that of solution 1 and value than the others. Then the DT values listed in Table
obviously more decentralized than that of solution 45, which XII also justify the effectiveness of FMOPSO: by using the

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11

mutation-based Gbest selection, the probability that the non- in Table XII. Especially, the worst-case loss and the final in-
dominated solutions with small DT are selected as Gbest has vestment profit (i.e. the profit at April. 30) of the solutions are
been increased, thus improving the search ability of particles calculated. Figure 9 depicts their real market profit and worst-
in some extreme solution spaces such as those maximizing SR case performance, where the solution number is consistent
or VR. with that of Table XII.
Based on the above analysis, it can be concluded that SR
and VR are conflicting objectives which lead to different
1
portfolio compositions. And the proposed method is able to

Profit at April. 30
realize various trade-off between the two ratios, thus providing 0.8

investors with comprehensive decision support. 0.6

3) Effectiveness of VR-based R-MOPSM: Considering any 0.4

two solutions in Table XII, for example solutions 1 and 45. If 0.2
only use SR to evaluate portfolio performance, investors will 0
5 10 15 20 25 30 35 40 45
select solution 1 as the final decision, since its SR is higher. Solution No.
However, on one hand investors cannot realize how much
0.02

Worst−case performance
better solution 1 is when comparing with solution 45, i.e. they
cannot identify the exact benefit brought by the increment of 0

SR. On the other hand, SR ignores the robust estimation on −0.02

the investment risks caused by the fluctuation of the entire −0.04

market. −0.06
If use VR to evaluate solutions 1 and 45, then the latter is −0.08
better as its VR is higher. Besides, it should be noticed that 5 10 15 20 25 30 35 40 45
Solution No.
the information involved in VR is more sensitive to general
investors than that of SR, e.g. VR= 0.999 means that the risk
Fig. 9. Profit and worst-case performance of all the Pareto optimal solutions
premium of an investment is 0.999 ($) while the maximal loss
is taken as 1.0 ($). Rationally, solution 1 should not be made
as the investment decision because the risk premium is lower Figure 9 shows that with the increasing of the solution
than the loss suffered. Therefore, VR provides investors with number (i.e. the increasing of VR), the investment profit in
sensitive knowledge, and a mathematical model that combines general is decreasing while the worst-case performance has
SR with VR can provide investors with comprehensive deci- been improved. Although there exist some slight fluctuations,
sion support. the tendency is clear and in accordance with the comparison
In what follows, the effectiveness of R-MOPSM is further results of solutions 1, 14, 23 and 45. Besides, the fluctuations
justified by comparing the real market performance of the four in Figure 9 should be caused by the uncertainty of security
typical solutions mentioned before. To make the comparisons, future returns. Such fluctuations can hardly be eliminated as
the closing price after every half a month and the lowest the real market profit of each security can be any realization
price among the investment horizon of each security are first of the forecasted possibility (or probability) distribution.
collected from the real market, as shown in Table VII. Then the Based on the above facts, it is concluded that VR involves
profit of each security at time t is calculated as (CPt -CP)/CP, robust estimations on investment risks, and an adequate trade-
where CPt represents the closing price at t. The results are off between the return and risk can be realized in real-life
listed at the top of Table VIII. applications by using the proposed method. Moreover, the
According to the real market data, the performance of the tendency of the real market performance of all the Pareto
four solutions is compared at every time node, as shown at optimal solutions is obvious, which demonstrates that the pro-
the bottom of Table VIII. Experimental results illustrate that posed method is effective to handle multi-objective portfolio
solution 1 which maximizes SR brings the greatest investment optimization under uncertainties.
profit, however, its worst-case loss is also the highest. By 4) Algorithm comparisons: Finally, the effectiveness of
contrast, although the profit of solution 45 is the lowest, its FMOPSO is justified by comparing the performance of IMOP-
worst-case loss is 0. In terms of solutions 14 and 23 where SO, σ-MOPSO and TV-MOPSO in this case study. The Pareto
the trade-off between SR and VR exists, the investment profit optimal solutions obtained by each algorithm are depicted in
is acceptable when the corresponding loss has been reduced. Figure 10, while the metrics such as SP, HV and runtime costs
Especially, comparing solutions 14 and 23 with solution 1, are listed in Table IX. Especially, the reference point when
the average deteriorations in investment profit are 4.9% and calculating HV is chosen as r = {0, 0}.
21.2%, while the improvements on the worst-case loss are From the solution quality aspect, FMOPSO outperforms σ-
11.5% and 32.1% respectively, which are obviously higher. MOPSO and TV-MOPSO as the obtained Pareto front is more
That is, some solutions among the Pareto front can be applied diversified and produces a higher HV value, which proves
to reduce investment risks greatly when depressing the profit the effectiveness of the DT-based Gbest selection. In addition,
slightly. the comparison between FMOPSO and IMOPSO justifies the
Without loss of generality, we also investigate the real significance of the mutation strategy proposed in this research:
market performance of all the Pareto optimal solutions listed Figure 10 shows that the left-hand Pareto front of IMOPSO

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TABLE VII
R EAL MARKET DATA : HALF - MONTHLY CLOSING PRICES AND THE LOWEST PRICES DURING JAN . 1-A PRIL . 30, 2012 ($)

No. Symbol Jan 17 Jan 31 Feb 15 Feb 29 Mar 15 Mar 30 Apr 13 Apr 30 Worst
1 STV 3.52 3.57 3.77 3.98 3.58 3.47 3.31 3.52 3.25
2 CNEP 2.19 2.28 2.97 3.11 3.11 3.11 3.11 3.11 2.06
3 CNTF 1.80 2.06 2.43 1.70 1.50 1.49 1.40 1.51 1.31
4 CBEH 0.53 0.79 0.74 0.74 0.85 0.82 0.62 0.70 0.33
5 CHL 49.0 51.08 51.99 53.01 54.24 55.08 54.79 55.34 48.62
6 ACH 12.32 12.17 13.56 13.54 13.12 11.87 12.0 12.03 10.83
7 CAST 5.84 6.14 5.3 5.5 4.74 4.24 4.24 4.24 4.24
8 DANG 6.15 7.35 6.99 6.69 7.56 8.1 8.63 7.99 4.03
9 CNAM 0.41 0.58 0.57 0.69 0.54 0.5 0.79 0.69 0.24
10 CBAK 3.40 3.40 3.80 3.80 3.70 5.15 4.95 3.85 3.15

TABLE VIII
P ROFITABILITY OF THE 10 SECURITIES DURING JAN . 1-A PRIL . 30, 2012

No. Symbol Jan 17 Jan 31 Feb 15 Feb 29 Mar 15 Mar 30 April 13 April 30 Worst
1 STV +0.110 +0.126 +0.189 +0.256 +0.129 +0.095 +0.044 +0.110 +0.025
2 CNEP +0.068 +0.112 +0.449 +0.517 +0.517 +0.517 +0.517 +0.517 +0.005
3 CNTF 0 +0.144 +0.350 -0.056 -0.167 -0.172 -0.222 -0.161 -0.272
4 CBEH +0.359 +1.026 +0.897 +0.897 +1.179 +1.103 +0.590 +0.795 -0.154
5 CHL +0.011 +0.053 +0.072 +0.093 +0.119 +0.136 +0.130 +0.141 +0.003
6 ACH +0.141 +0.127 +0.256 +0.254 +0.215 +0.099 +0.111 +0.114 0
7 CAST -0.046 +0.003 -0.134 -0.101 -0.225 -0.307 -0.307 -0.307 -0.307
8 DANG +0.398 +0.670 +0.589 +0.589 +0.718 +0.841 +0.961 +0.816 -0.084
9 CNAM +0.464 +1.071 +1.036 +1.464 +0.929 +0.786 +1.821 +1.464 -0.143
10 CBAK +0.079 +0.079 +0.206 +0.206 +0.175 +0.635 +0.571 +0.222 0
Solution 1 Profit 0.307 0.599 0.623 0.723 0.641 0.601 0.858 0.745 -0.078
Solution 14 Profit 0.296 0.569 0.596 0.697 0.625 0.577 0.793 0.697 -0.069
Solution 23 Profit 0.249 0.476 0.559 0.701 0.526 0.415 0.721 0.621 -0.053
Solution 45 Profit 0.141 0.127 0.256 0.254 0.215 0.099 0.111 0.114 0.000

TABLE IX
2 P ERFORMANCE COMPARISONS ON CASE STUDY 1: SP, HV AND RT( S )
FMOPSO
IMOPSO
Metric FMOPSO IMOPSO σ-MOPSO TV-MOPSO
1.8 σ−MOPSO
TV−MOPSO SP 0.0203 0.0429 0.1251 0.0436
1 HV 2.6797 2.6431 2.4442 2.5001
1.6 RT 151.65 147.74 179.66 183.06
Sharpe ratio

1.4

2
FMOPSO.
1.2
From the runtime aspect, the four algorithms are practicable
1
to solve the multi-objective optimization (generally, due to the
less ambitious on runtime costs, it might not be a major issue
0.8
for solving portfolio selection problems around the execution
times listed in Table IX), and the DT-based FMOPSO and
IMOPSO can reduce more than 15% runtime costs of the other
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 two algorithms.
VaR ratio
Based on the above facts, it is concluded that FMOPSO out-
performs IMOPSO, σ-MOPSO and TV-MOPSO when solving
Fig. 10. Pareto fronts obtained by FMOPSO, IMOPSO, σ-MOPSO and
TV-MOPSO (Case study 1) the portfolio selection problem of this case study.

C. Case study 2
is as good as that of FMOPSO (sometimes even a little better In case study 2, a portfolio investment problem with 12
than FMOPSO, as depicted in Square 1). However, IMOPSO securities in different industries from the China Shanghai
performs poorly when VR is larger than 1.4, as shown in Stock Exchange is considered. The forecasted returns of
Square 2. By contrast, using the mutation strategy, although the securities are regarded to be asymmetric triangular and
there is a minor loss on the convergence of the left-hand Pareto trapezoidal fuzzy variables, as given in [39]. Then we solve
front, the overall search ability of FMOPSO has been greatly the portfolio selection problem by using the proposed method,
improved, i.e. an adequate trade-off between the convergence the experimental results of which are compared with that of
and diversity of the Pareto front can be realized by using the existing algorithms.

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Figure 11 depicts the Pareto fronts of R-MOPSM obtained

1.4
by each algorithm, while Table X provides the SP, HV
and runtime (RT) metrics. Again, the reference point when 1.3
calculating HV is chosen as r = {0, 0}. 1

1.2

Sharpe ratio
1.1
2

1.8 1

1
1.6 0.9
FMOPSO
Sharpe ratio

IMOPSO
0.8
1.4 σ−MOPSO
TV−MOPSO
2 0.7
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
1.2
VaR ratio
FMOPSO
IMOPSO
1 Fig. 12. Pareto fronts obtained by FMOPSO, IMOPSO, σ-MOPSO and
σ−MOPSO
TV−MOPSO
TV-MOPSO (Case study 3)

0.8
1 2 3 4 5 6 7 TABLE XI
VaR ratio P ERFORMANCE COMPARISONS ON CASE STUDY 3: SP, HV AND RT( S )

Fig. 11. Pareto fronts obtained by FMOPSO, IMOPSO, σ-MOPSO and Metric FMOPSO IMOPSO σ-MOPSO TV-MOPSO
TV-MOPSO (Case study 2) SP 0.0139 0.0209 0.0728 0.0400
HV 1.7397 1.6363 1.5872 1.6501
RT 203.23 194.55 248.22 252.47

TABLE X
P ERFORMANCE COMPARISONS ON CASE STUDY 2: SP, HV AND RT( S )

than the other algorithms as the SP and HV metrics are the

Metric FMOPSO IMOPSO σ-MOPSO TV-MOPSO
SP 0.0884 0.1362 0.1976 0.1849 best.
HV 11.3517 10.9193 9.4441 10.6660 In summary, the R-MOPSM and FMOPSO proposed in this
RT 165.12 160.33 195.27 200.14 research are practical to solve different types of multi-objective
portfolio selection problems.
The comparison results express that SR contradicts VR, and
FMOPSO achieves a more effective Pareto front than the other VII. C ONCLUSIONS
algorithms. In addition, the runtime cost given in Table X In this study, a fuzzy VaR ratio was introduced as an
also indicates that the computational burden of the existing index with dimensional knowledge when evaluating portfolio
multi-objective algorithms has been reduced by FMOPSO and performance. Then a multi-objective portfolio selection model
IMOPSO which apply DT to determine Gbest. Finally, among was built to capture the various trade-off between SR and VR.
the non-dominated solutions, investors can select a specific To generally solve the proposed model, a fuzzy simulation-
one that satisfies their requirements the best as the portfolio based multi-objective PSO algorithm was developed where the
decision. Gbest selection has been improved by integrating a mutation
strategy. Experimental results on three case studies show that
SR contradicts VR, and some solutions among the Pareto front
D. Case study 3 can reduce investment risks greatly when depressing the profit
To illustrate the effectiveness of the proposed method when slightly. Besides, the comparisons on several benchmarks as
there are a number of risky securities, we give another case well as the case studies justify that the proposed algorithm can
study based on existing works. Especially, the 10 Gaussian obtain convergent and diversified Pareto fronts when reducing
fuzzy returns listed in [40] and the 17 trapezoidal fuzzy more than 15% runtime cost of the existing approaches. In
returns (short-term) provided in [41] are used together, i.e. summary, the various non-dominated solutions achieved by
the number of the securities considered is 27. Then the the proposed method reveal the inherent conflict between SR
multi-objective portfolio selection problem is solved by using and VR, thus providing investors with comprehensive support
different algorithms, where Figure 12 depicts the approximate when determining the portfolio composition.
Pareto fronts and Table XI provides the performance metrics. Finally, several remarks on possible future research are
Similar to Case studies 1 and 2, the non-dominated solutions as follows. First, the development and application of some
listed in Figure 12 reveal the trade-off between SR and VR. other effective index such as Treynor ratio can be considered.
Besides, Table XI also justifies that FMOPSO is more effective Second, the proposed single-period model can be extended to

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TABLE XII
D ETAILED KNOWLEDGE OF PARETO OPTIMAL SOLUTIONS

No. VR SR DT Capital allocation

1 0.999 1.705 11475 (0.002 0.080 0.018 0.086 0.000 0.275 0.000 0.284 0.256 0.000)
2 1.001 1.705 11608 (0.000 0.143 0.028 0.070 0.000 0.227 0.000 0.279 0.253 0.000)
3 1.030 1.704 13821 (0.000 0.144 0.000 0.103 0.000 0.197 0.000 0.271 0.286 0.000)
4 1.062 1.697 16831 (0.000 0.137 0.000 0.065 0.000 0.287 0.000 0.276 0.236 0.000)
5 1.066 1.694 17205 (0.000 0.119 0.000 0.124 0.000 0.245 0.000 0.262 0.249 0.000)
6 1.085 1.686 19224 (0.000 0.078 0.000 0.063 0.000 0.353 0.000 0.269 0.237 0.000)
7 1.104 1.686 22272 (0.000 0.079 0.000 0.062 0.000 0.357 0.000 0.267 0.237 0.000)
8 1.138 1.680 25263 (0.000 0.082 0.000 0.073 0.000 0.360 0.000 0.273 0.240 0.000)
9 1.165 1.679 27201 (0.000 0.118 0.000 0.053 0.000 0.353 0.000 0.263 0.231 0.000)
10 1.184 1.675 29315 (0.001 0.130 0.000 0.084 0.000 0.222 0.000 0.276 0.287 0.000)
11 1.204 1.657 30866 (0.000 0.120 0.011 0.080 0.000 0.356 0.000 0.214 0.219 0.000)
12 1.233 1.645 32994 (0.000 0.138 0.000 0.050 0.000 0.338 0.000 0.276 0.198 0.000)
13 1.252 1.622 33273 (0.004 0.138 0.000 0.067 0.000 0.367 0.000 0.150 0.275 0.000)
14 1.262 1.603 34728 (0.000 0.074 0.000 0.101 0.000 0.341 0.000 0.258 0.225 0.000)
15 1.263 1.587 32705 (0.000 0.201 0.000 0.075 0.000 0.314 0.000 0.183 0.226 0.000)
16 1.283 1.585 33607 (0.000 0.229 0.000 0.097 0.000 0.255 0.000 0.200 0.212 0.006)
17 1.315 1.531 32442 (0.000 0.167 0.000 0.032 0.000 0.441 0.000 0.162 0.198 0.000)
18 1.332 1.514 32038 (0.002 0.122 0.000 0.065 0.000 0.394 0.000 0.095 0.311 0.012)
19 1.346 1.506 31996 (0.000 0.130 0.000 0.125 0.000 0.364 0.000 0.078 0.303 0.000)
20 1.353 1.495 31616 (0.005 0.172 0.001 0.117 0.000 0.313 0.000 0.075 0.317 0.000)
21 1.364 1.469 30490 (0.000 0.105 0.000 0.059 0.006 0.495 0.000 0.070 0.265 0.000)
22 1.371 1.397 26718 (0.000 0.102 0.000 0.047 0.000 0.544 0.000 0.051 0.255 0.000)
23 1.375 1.374 25354 (0.000 0.098 0.000 0.056 0.000 0.528 0.000 0.000 0.318 0.000)
24 1.380 1.373 25423 (0.000 0.012 0.000 0.052 0.000 0.523 0.000 0.098 0.314 0.000)
25 1.396 1.353 24444 (0.000 0.077 0.000 0.069 0.015 0.543 0.000 0.000 0.297 0.000)
26 1.417 1.300 21521 (0.002 0.011 0.000 0.101 0.000 0.521 0.000 0.010 0.353 0.001)
27 1.448 1.256 19210 (0.000 0.009 0.000 0.038 0.000 0.656 0.000 0.106 0.191 0.000)
28 1.470 1.225 19310 (0.000 0.015 0.000 0.039 0.000 0.591 0.000 0.026 0.329 0.000)
29 1.494 1.138 13399 (0.000 0.003 0.000 0.056 0.000 0.690 0.000 0.000 0.252 0.000)
30 1.518 1.117 12486 (0.000 0.000 0.000 0.051 0.000 0.708 0.000 0.008 0.232 0.000)
31 1.525 1.042 9408 (0.000 0.000 0.000 0.045 0.000 0.772 0.000 0.000 0.183 0.000)
32 1.545 1.018 8669 (0.000 0.037 0.000 0.003 0.000 0.772 0.000 0.000 0.188 0.000)
33 1.575 0.975 8602 (0.000 0.000 0.000 0.076 0.000 0.798 0.000 0.000 0.126 0.000)
34 1.582 0.915 4332 (0.000 0.000 0.000 0.050 0.000 0.827 0.000 0.000 0.123 0.000)
35 1.588 0.885 4327 (0.000 0.017 0.000 0.000 0.002 0.860 0.000 0.000 0.121 0.000)
36 1.607 0.852 4292 (0.000 0.000 0.000 0.000 0.000 0.854 0.000 0.000 0.146 0.000)
37 1.616 0.830 3860 (0.000 0.012 0.000 0.017 0.000 0.884 0.000 0.000 0.086 0.000)
38 1.627 0.784 2715 (0.000 0.009 0.000 0.016 0.000 0.915 0.000 0.000 0.060 0.000)
39 1.639 0.744 2459 (0.000 0.014 0.000 0.049 0.000 0.937 0.000 0.000 0.000 0.000)
40 1.670 0.732 2287 (0.000 0.000 0.000 0.032 0.000 0.943 0.000 0.000 0.025 0.000)
41 1.701 0.706 1997 (0.000 0.055 0.000 0.006 0.000 0.939 0.000 0.000 0.000 0.000)
42 1.723 0.644 1309 (0.004 0.016 0.000 0.000 0.000 0.980 0.000 0.000 0.000 0.000)
43 1.730 0.627 1137 (0.005 0.000 0.000 0.000 0.000 0.995 0.000 0.000 0.000 0.000)
44 1.733 0.615 1006 (0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000)
45 1.742 0.607 906 (0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000)

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