1. INTRODUCTION
Markowitz (1952) originaly presented mean variance singleperiod portfolio selection. In real world, the portfolio strategies are often multiperiod. The investor usually needs to rebalance his position from time to time. After Markowitz’s pioneering work, various researchers extended his work to multiperiod case. Among others, the representative works include Li and Ng (2000), who extend the originally meanvariance problem to dynamic discretetime setting. Li and Ng (2000) embed the original problem into a tractable auxiliary problem, then the analytical portfolio strategy and meanvariance efficient frontier are derived. The further study about dynamic extension of the Markowitz’s model includes, e.g., Zhu et al. (2004) discussed a generalized multiperiod meanvariance model with bankruptcy risk control, and showed that under the classical multiperiod meanvariance portfolio optimization model, the bankruptcies mostly occur in the early periods. Bielecki et al. (2005) study discrete time and continuoustime MV models, respectively. Cui et al. (2014) reformulated the multiperiod meanvariance model, and derived the optimal investment strategy and efficient frontier by using the meanfield approach. Note that all the portfolio models mentioned above do not allow cash additions or withdrawals. Cui et al. (2017) investigate analytically effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets. Li and Li (2012) investigate a multiperiod assetliability portfolio optimization problem which intends to control the probability of bankruptcy. Gao et al. (2015) considers the time cardinality constrained meanvariance dynamic portfolio selection problem in markets with correlated returns and in which the number of time periods to invest in risky assets is limited. Wu and Zeng (2015) studid a generalized multiperiod mean variance portfolio selection problem within the game theoretic framework for a definedcontribution pension scheme member.
All previous literatures are proposed on the basis on probability theory. They view the uncertainty associated with risky assets as random uncertainty. If using probabilistic approaches to handle the uncertainty of the financial markets, it is necessary to estimate the probability distribution of return for each risky asset. Due to the everchanging financial markets, the probability distribution of return of risky asset usually cannot be predicted accurately. Even though the probability distribution can be estimated, there is no guarantee that the return rates truly obey it. Actually, in real world, there are many nonrandom factors that affect the financial markets such as economics, policies, laws, regulations and people’s psychological factors. In particular, the influence of people’s psychological factors cannot be neglected in portfolio decisionmaking. In many cases, investors are usually provided with information which is characterized by vague linguistic descriptions such as high risk and low profit. Based on the facts mentioned above, to estimate the risky assets returns, it is necessary to take experts’knowledge and investors’subjective opinions into consideration. A possibility distribution can be regarded as an alternative to a probability distribution, in which the unquantifiable factors can be reflected easily. In fact, in many situations, it might be easier to estimate the possibility distributions of returns on risky assets than the corresponding probability distributions. In contrast to the probability theory, fuzzy set theory needs less information in the decision process. Thus, it is worthwhile to use fuzzy set theory to investigate the uncertainty of financial markets. Following the widely used fuzzy set theory in Zadeh (1999), researchers have used the fuzzy set theory to investigate portfolio selection problems under uncertain environment. Numerous models have been proposed by using different approaches. Li et al. (2010) proposed several fuzzy mean variance skewness models in fuzzy environment. Qin (2017) proposed a random fuzzy mean absolute deviation portfolio selection model. Some researchers extend the singleperiod fuzzy portfolio selection into multiperiod setting. By using experts’judgments, Sadjadi et al. (2011) formulated a fuzzy multiperiod portfolio selection model with different borrowing and lending rates. Zhang et al. (2012, 2014), and Liu et al. (2012, 2013) respectively proposed several kinds of multiperiod fuzzy portfolio selection models. Zhang and Zhang (2014) proposed a multiperiod mean absolute deviation fuzzy portfolio selection model with cardinality constraints.
Though possibility measure has been widely used in portfolio selection, it has limitation. One great limitation is that possibility measure is not selfdual. Using possibility measure which has no selfduality property, we can find that two fuzzy events with different occurring chances may have the same possibility value. These results are quite awkward and will confuse the decision maker. Thus, Huang (2008) presented meansemivariance model based on the selfdual credibility measure. Li et al. (2010) proposed crossentropy minimization credibilitic model and so forth, Zhang et al. (2014) proposed a credibilitic multiperiod meanvariance portfolio selection model. Mehlawat (2016) proposed a multiperiod mean entropy credibilitic portfolio selection model.
In many existing models are proposed on the framework of Markowitz’s mean variance with cardinality constraints, threshold constraints and so on. These constraints come from realworld practice where the administration of a portfolio made up of many assets is clearly not desirable because of transaction costs, complexity of management, or policy of the asset management companies. Because of its practical relevance, this cardinality constrained Markowitz model, and some variations have been intensively studied in the last decade. Especially from the computational viewpoint, some researchers proposed exact solution methods (i.e., Bertsimas and Shioda, 2009; Li et al., 2006; Shaw et al., 2008; Murray and Shek, 2012; Cesarone et al., 2013; Cui et al., 2013; Sun et al., 2013; Le Thi et al., 2009; Le The and Moeini, 2014). Since exact solution methods are able to solve only a fraction of practically useful cardinality constrained Markowitz models, many heuristic algorithms have also been proposed (ie., Fernández and Gómez, 2007; RuizTorrubiano and Suarez. 2010; Anagnostopoulos and Mamanis, 2011; WoodsideOriakhi et al., 2011; Deng et al., 2012). In these studies, it appears that the computational complexity for the solution of the cardinality constrained Markowitz model is much greater than the one required by the classical Markowitz model or by several other of its refinements. Indeed, the standard Markowitz model is a convex quadratic programming problem, while the cardinality constrained Markowitz model is a mixed integer quadratic programming problem that is a more difficult NPhard problem.
The contribution of this work is as follows. We propose a new credibilitic multiperiod mean semivariance portfolio selection model with borrowing constraints, transaction costs, threshold constraints and cardinality constraints. Using the fuzzy decisionmaking approach, the proposed model is transformed into a crisp mix integer dynamic optimization problem with path dependence. The proposed model is approximated to dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy, and is proved linearly convergent.
The rest of this paper is organized as follows. In Section 2, the definitions of the credibilitic mean, the credibilitic semivariance, and some properties are introduced. In Section 3, transaction costs, the threshold constraints and cardinality constraints are formulated into the multiperiod portfolio. A new credibilitic multiperiod portfolio selection model is constructed. The multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence. The proposed model is approximated to dynamic programming model. A discrete iteration method is proposed to solve it in Section 4. In Section 5, the comparison analysis of differently desired number of risky assets in the portfolio and different preference coefficients are given to illustrate the modeling idea and the effectiveness of the designed algorithm. Finally, some conclusions are given in Section 6.
2. PRELIMINARIES
Let ξ be a fuzzy variable with membership function μ, and u and x be real numbers. μ_{ξ}(x) represents the possibility distribution of ξ. For any given set B, the possibility and necessity measures of ξ ∈ B are, respectively, defined as (see Zadeh, 1999).(1)(2)
where Pos and Nec are the possibility and necessity measures, respectively.
Note that both possibility measure and necessity measure satisfy the properties of normality, nonnegativity and monotonicity. However, both of them are not selfdual. Based on the possibility measure and the necessity measure, Liu and Liu (2002) defined the credibility measure as follow, which is selfdual measure.
where Cr is selfdual measure and satisfies Cr{ξ ≤ r} + Cr{ξ ≥ r} = 1.
Definition 1 (Liu and Liu, 2002). Let x be a fuzzy variable, the expected value of x is defined as(4)
Theorem 1 (Liu and Liu, 2002) Let ξ be a fuzzy variable with finite expected value, and let λ and μ be two real numbers, then(5)
Theorem 2 (Liu and Liu, 2002) Let ξ and η be independent fuzzy variables with finite expected values, and let λ and μ be two real numbers, then(6)
Definition 2 (Huang, 2008) Let ξ be a fuzzy variable with finite expected value e, then the semivariance and credibilitic semivariance of ξ are defined as(7)(8)
where
Theorem 3 Let ξ be a fuzzy variable with finite expected value e，and let λ be any given nonnegative real number, then(9)
Proof. From Definition 2, it follows that
Thus, the proof of the Theorem 3 is ended. Furthermore, we can get the following form.
Thus, the credibilistic semivariance of ξ can be expressed as follows:(10)
Theorem 4 Let ξ and η be independent fuzzy variables with finite expected values. Then, for any nonnegative real numbers λ and μ, it holds that
Proof.
If ξ and η is independent fuzzy variables,
${\int}_{0}^{\infty}\text{Cr{}[{(\xi E(\xi ))}^{}][{(\mu E(\eta ))}^{}\ge r\text{}d}r}=0$
Then the Eq. (12) can be turned into
$SV(\lambda \xi +\mu \eta )={\lambda}^{\text{2}}SV(\xi )+{\mu}^{\text{2}}SV(\eta )$
Thus, the proof of the Theorem 4 is ended.
If ξ = (a, a, b) be a triangular fuzzy number, then its membership function μ_{ξ}(x) can be described as:(13)
From Eq. (3), the credibility of the event {ξ ≤ r} can be represented by
Theorem 5 Let ξ = (a, a, b) be a triangular fuzzy number, then the credibilistic expected value of ξ can be given by:(15)
Proof.
From the Definition 1, it follows that
$\begin{array}{l}E[\xi ]={\displaystyle {\int}_{0}^{\infty}Cr\{\xi \ge r\}dr}{\displaystyle {\int}_{\infty}^{0}Cr\{\xi \le r\}dr}\\ ={\displaystyle {\int}_{0}^{\infty}(1Cr\{\xi \le r\})dr}\\ \text{=}{\displaystyle {\int}_{0}^{a\alpha}1dr}+{\displaystyle {\int}_{a\alpha}^{a}(1\frac{ra+\alpha}{2\alpha})dr+{\displaystyle {\int}_{a}^{a+\beta}(1\frac{ra+\beta}{2\beta})dr}}+0\\ \text{=}\hspace{0.17em}a+\frac{\beta \alpha}{4}\end{array}$
Thus, the proof of the theorem is ended.
Theorem 6 Let ξ = (a, a, b) be a triangular fuzzy number. Then, semivariance of ξ can be obtained
Proof.
From the Definition 2, it follows that
From Eq. (14), it follows that
From Eq. (17) and Eq. (18), it follows that
Thus, the proof of the theorem is ended.
3. THE MULTIPERIOD PORTFOLIO SELECTION MODEL
Assume that there are n risky assets and one riskfree asset in financial market for trading. An investor wants to allocate his initial wealth W_{1} among n+1 assets at the beginning of period 1, and obtains the final wealth at the end of period T. He can reallocate his wealth among the n +1 assets at the beginning of each of the following T consecutive investment periods. Suppose that the return rates of the n risky assets at each period are denoted as triangular fuzzy variables. For the sake of description, let us first introduce the following notations. Let x_{i}_{0} be the initial investment proportion of risky asset i at period 0; x_{t} be the portfolio at period t, where ${x}_{t}=\left({x}_{1t},\hspace{0.17em}{x}_{2t},\hspace{0.17em}\cdots ,\hspace{0.17em}{x}_{nt}\right);\hspace{0.17em}{x}_{ft}$ be the investment proportion of riskfree asset at period t, where ${x}_{ft}=1{\displaystyle \sum _{i=1}^{n}{x}_{it}};\hspace{0.17em}{x}^{\beta}{}_{ft}$ be the lower bound of the investment proportion of riskfree asset at period t, where xft ³ xbft; Rit be the return of risky asset i at period t; r_{pt} be the return rate of the portfolio x_{t} at period t; r_{bt} be the borrowing rate of the riskfree asset at period t; r_{lt} be the lending rate of the riskfree asset at period t; l_{it} be the preset lower bound value of x_{it}; u_{it} be the preset upper bound value of x_{it}; r_{Nt} be the net return rate of the portfolio x_{t} at period t; W_{t} be the crisp form of the holding wealth at the beginning of period t; c_{it} be the unit transaction cost of risky asset i at period t; K be the desired number of risky assets in the portfolio at period t.
3.1. Return, Risk and Cardinality Constraints
In this section, we employ the credibilistic mean value of the return on the portfolio at each period to measure the return of portfolio. The risk on the return rate of portfolio at each period is quantified by the credibilistic semivariance. The cardinality constraints limit the number of risky assets to be held in an efficient portfolio. The return rate of security i at period t, R_{it} = (a_{it}, a _{it}, b_{it}), is triangular fuzzy variable for all i = 1, …, n and t = 1, … T.
Most of the brokerage houses provide the opportunity to make an acquisition on different assets by borrowing the money from the brokerage. Some researchers studied the borrowing constraints, for example, Sadjadi et al. (2011) proposed the fuzzy portfolio model with different rates for borrowing and lending. Deng and Li (2012) proposed a meanvariance fuzzy portfolio with borrowing constraints. Zhang and Zhang (2014), Zhang (2016), and Zhang (2017) proposed the fuzzy multiperiod portfolio model with different borrowing and lending rates. Considering the different rates for borrowing and lending, the credibilistic mean value of the portfolio x_{t} = (x_{1t}, x_{2t}, …, x_{nt})′ at period t (t = 1, …, T) can be expressed as(19)
when 1(r1t+…+rnt)³ 0, rft = rlt; when 1(r1t+…+rnt)£ 0, rft = rbt.
The transaction costs which are entailed by buying or selling assets to adjust the existing portfolio are one of the main concerns for portfolio managers. As mentioned by Arnott and Wagner (1990), and Yoshimoto (1996) ignoring the transaction costs may fail to obtain the efficient portfolio. So Güpınar and Rustem (2007), and Bertsimas and Pachamanova (2008) incorporated transaction costs into consideration to study the multiperiod portfolio selection problem. In this paper, the transaction costs are assumed Vshaped functions of differences between the t th period portfolio x_{t} = (x_{1t}, x_{2t}, …, x_{nt}) and the t1th period portfolio x_{t}_{1} = (x_{1t1}, x_{2t1}, …, x_{nt}_{1}). That’s to say, the transaction cost for asset i at period t (t = 1, …, T) can be expressed as(20)
Hence, the total transaction costs of the portfolio ${x}_{t}=\left({x}_{1t},\hspace{0.17em}{x}_{2t},\hspace{0.17em}\cdots ,\hspace{0.17em}{x}_{nt}\right)$ at period t (t = 1, … T) can be represented as(22)
The net return rate of the portfolio x_{t} at period t (t = 1, …, T) can be denoted as
The crisp form of the holding wealth at the beginning of the period t (t = 1, …, T) can be written as(23)
Derived from Eq. (11) and Eq. (16), the credibilistic semivariance of the portfolio x_{t} can be expressed as(24)
To formulate cardinality constraints into the multiperiod portfolio model, zeroone decision variables are added as:(25)
where $\sum _{i=1}^{n}{z}_{it}}\le K.$
Let l_{it} and u_{it} be the preset lower and upper bounds values of x_{it}, respectively, The threshold constraints of multiperiod portfolio selection can be expressed as(27)
Let the preset value be x^{b}_{ft}, where x^{b}_{ft} ≤ 0, the borrowing constraint of riskfree asset at period t is
For a rational investor, he/she wishes not only to maximize expected return but also to minimize the risk which is measured by the variance of the rate of return on a portfolio. So he/she must make a tradeoff between the two objectives. Let (1θ) and θ be the preference coefficients associated with criteria r_{pt} and SV_{t} (x_{t}) respectively. Then the investor attempts to maximize
Here the parameter θ sthe risk aversion factor of the investor. The greater the factor θ is, the more risk aversion the investor has. In this paper, we assume that 0≤θ≤1.
3.2. The Basic Multiperiod Portfolio Optimization Models
Assume that the investor wants to maximize his/her utility over the whole T periods investment. The multiperiod credibilitic portfolio selection problem with transaction costs, borrowing constraints, threshold constraints and cardinality constraints can be formulated as the following problem:
where constraint (a) denotes the wealth accumulation constraint; constraint (b) indicates that the investment proportion of riskfree asset at period t must exceed the given lower bound; constraint (c) represents the desired number of risky assets in the portfolio must not exceed the given value K; constraint (d) states threshold constraints of x_{it}.
4. SOLUTION ALGORITHM
In this section, the credibilitic multiperiod mean semivariance portfolio selection problem with real constraints will be approximated into a mix integer dynamic programming problem. A novel discrete iteration method will be proposed to solve the dynamic programming problem. The linear convergence of the method will be proved.
4.1. The Proposed Model Approximated to Dynamic Programming Problem
The subproblem of Model (29) at period t is as follows:
Let ${x}_{it1}=\overline{{x}_{it1}}$, where $\overline{{x}_{it1}}$ is preset value, $\sum _{i=1}^{n}\overline{{x}_{it1}}=\hspace{0.17em}}1{x}^{b}{}_{ft$, the Model (29) can be approximated into the following model:
The subproblem of Model (31) at period t is as follows:
Theorem 7. Let the optimal solution and objective function value of Model (30) and Model (32) respectively be x^{1*} and G(x^{1*}), x^{2*} and F(x^{2*}).
Then $G({x}^{1*})G({x}^{{2}^{*}})\le 4(1\theta )\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}(1{x}_{ft}^{b})$.
Proof. Because the feasible solution sets of the Model (30) and Model (32) are same, x^{1*} and x^{2*} are the feasible solutions of Model (30) and Model (32), respectively. Then, $G({x}^{1*})\ge G({x}^{2*})$ and
$\begin{array}{l}F({x}^{2*})\ge F({x}^{1*}),\hspace{0.17em}\text{that}\hspace{0.17em}\text{is}\\ G({x}^{1*})+F({x}^{2*})\ge G({x}^{2*})+F({x}^{1*})\end{array}$
then
The righthand side of Eq. (33) is(34)
Because ${x}_{it1}\ge 0,\hspace{0.17em}{x}_{it}\ge 0$,
$\begin{array}{l}2{\displaystyle \sum _{i=1}^{n}{c}_{it}\left{x}_{it1}{\overline{x}}_{it1}\right}\le 2{\displaystyle \sum _{i=1}^{n}{c}_{it}{x}_{it1}}+2{\displaystyle \sum _{i=1}^{n}{c}_{it}{\overline{x}}_{it1}}\\ =2\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}{\displaystyle \sum _{i=1}^{n}{x}_{it}}+2\underset{i=1}{\overset{n}{\mathrm{max}}}{\displaystyle \sum _{i=1}^{n}{\overline{x}}_{it1}}\\ \le 2(1\theta )\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}(1{x}_{ft}^{b})+2(1\theta )\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}(1{x}_{ft}^{b})\\ =4(1\theta )\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}(1{x}_{ft}^{b}).\end{array}$
So $G({x}^{1*})G({x}^{{2}^{*}})\le 4(1\theta )\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}(1{x}_{ft}^{b})$.
Because $\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}<<{r}_{it}$, where asset i∈ efficient asset set of portfolio, $4(1\theta )\underset{i=1}{\overset{n}{\mathrm{max}}}\left\{{c}_{it}\right\}(1{x}_{ft}^{b})$ is small, that $G({x}^{1*})G({x}^{{2}^{*}})$ is also small.
which ends the proof.
4.2. The Smallest and Biggest Value of State Variable at Every Period
In Model (31), investors can choose W_{t} between W_{t}^{min} and W_{t}^{max}. W_{t}^{min} and W_{t}^{max} can be respectively obtained as follows:
The investor considers to maximize the expected return of the portfolio at period t.
Let ${y}_{it}=\left{x}_{it}{\overline{x}}_{i(t1)}\right.$ Then the Model (35) can be turned into as follows.
x_{t}^{max*}(the optimal solution ${x}_{t}={\left({x}_{1t},\hspace{0.17em}{x}_{2t},\hspace{0.17em}\cdots ,\hspace{0.17em}{x}_{nt}\right)}^{\prime}$) can be obtained solving Model (36) by the CPLEX. r_{Nt}^{max} (the biggest of $\sum _{i=1}^{n}[{a}_{it}+\frac{{\beta}_{it}{\alpha}_{it}}{4}]{x}_{it}}+{r}_{ft}(1{\displaystyle \sum _{i=1}^{n}{x}_{it}}){\displaystyle \sum _{i=1}^{n}{c}_{it}{y}_{it}$) can also be obtained. Then, W_{t}_{+1}^{max} can be obtained as follows:(37)
where W_{1} is initial wealth, which is preset value.
The investor only considers to minimize the semivariance of the portfolio at period t, that is, the smallest value of the r_{Nt} can be obtained as follows:
x_{t}^{min*}(the optimal solution ${x}_{t}={\left({x}_{ft},\hspace{0.17em}{x}_{1t},\hspace{0.17em}{x}_{2t},\hspace{0.17em}\cdots ,\hspace{0.17em}{x}_{nt}\right)}^{\prime}$) can be obtained solving the Model (38) by the CPLEX. Simultaneously, r_{Nt}^{min} (the smallest of $\sum _{i=1}^{n}[{a}_{it}+\frac{{\beta}_{it}{\alpha}_{it}}{4}]{x}_{it}}\hspace{0.17em}+{r}_{ft}(1{\displaystyle \sum _{i=1}^{n}{x}_{it}}){\displaystyle \sum _{i=1}^{n}{c}_{it}\left{x}_{it}\overline{{x}_{it1}}\right$) is also obtained. Then, W_{t}_{+1}^{min} Figure 1. The multiperiod weighted digraph. can be obtained as follows:(39)
where W_{1} is initial wealth, which is preset value.
4.3. The Discrete Iteration Method
In this section, the maxplus algebra, which is proposed by Heidergott et al. (2006), will be used. The method solving the longest path of the following multiperiod weighted digraph will be proposed, and some definitions will be introduced first. Then, the method of finding the longest path of the multiperiod weighted digraph can be obtained. Finally, k+1th iteration method will be presented.
Definition 3. A semifield (or semiring) is a triplet A = {Q, ⊕, ⊗} consisting of nonempty set Q and two binary operations ⊕ and ⊗, called modiaddition and modimultiplication respectively, defined on Q, such that for all a_{1}, a_{2}, a_{3} in Q

(i) each of the operations ⊕ and ⊗ is commutative
${a}_{1}\oplus {a}_{2}={a}_{2}\oplus {a}_{1},{a}_{1}\otimes {a}_{2}={a}_{2}\otimes {a}_{1}$

(ii) each of the operations ⊕ and ⊗ is associative
$\begin{array}{l}\left({a}_{1}\oplus {a}_{2}\right)\oplus {a}_{3}={a}_{1}\oplus \left({a}_{2}\oplus {a}_{3}\right),\\ \left({a}_{1}\otimes {a}_{2}\right)\otimes {a}_{3}={a}_{1}\otimes \left({a}_{2}\otimes {a}_{3}\right)\end{array}$

(iii) the operation ⊗ is distributive with respect to the operation ⊕
${a}_{1}\otimes \left({a}_{2}\oplus {a}_{3}\right)={a}_{1}\otimes {a}_{2}\oplus {a}_{1}\otimes {a}_{3}$

(iv) there exists an element in Q which is the zero element, denoted by ξ, such that for all a in Q, we have
$\epsilon \oplus a=a,\hspace{0.17em}\epsilon \otimes a=\epsilon $
If there exists an element denoted by e in a semifield $A=\left\{Q,\hspace{0.17em}\oplus ,\otimes \right\}$, such that for all a in Q, we have
$e\otimes a=a$
then e is called an identity element of the semifield.
Definition 4.>a_{1}, a_{2}∈R, in semifield A = {R, max, +}, then ε = ∞ and e = 0.
Definition 5. Let us consider matrices ${A}_{n\times n}={\left({a}_{ij}\right)}_{n\times n},\hspace{0.17em}{B}_{n\times n}={\left({b}_{ij}\right)}_{n\times n},\hspace{0.17em}{a}_{ij},\hspace{0.17em}{b}_{ij}\in R$, in semifield A = {R, max, +}. Then $C=A\oplus B,\hspace{0.17em}{C}_{n\times n}={\left({c}_{ij}\right)}_{n\times n}$, where c_{ij} = max{a_{ij}, b_{ij}}.
Definition 6. Let us consider matrices ${A}_{n\times m}={\left({a}_{ij}\right)}_{n\times m},\hspace{0.17em}{B}_{m\times k}={\left({b}_{ij}\right)}_{m\times k},\hspace{0.17em}{a}_{ij},\hspace{0.17em}{b}_{ij}\in R$, in semifield A = {R, max, +}. Then $C=A\otimes B,\hspace{0.17em}{C}_{n\times k}={\left({c}_{ij}\right)}_{n\times k}$, where ${c}_{ij}=\underset{l=1}{\overset{m}{\mathrm{max}}}\{{a}_{il}+{b}_{lj}\}$.
The Model (31) is a mix integer dynamic programming problem with linear inequality constraints, the optimal solution can’t be obtained by the dynamic programming recursive relationship. In this section, a novel discrete iteration method is proposed. The method is as follows: Firstly, according to the network approach, discretizes the state variables and transforms the model into multiperiod weighted digraph. Secondly, uses the maxplus algebra to solve the largest path that is the admissible solution. Thirdly, based on the admissible solution, continues iterating until the two admissible solutions are real near. Finally, the method is proved linearly convergent.
The state variable W_{t} of the period t is discretized into four intervals of same widths from the smallest value to the biggest one. It means that there are five discrete values for the state variable in every period. In this way, Model (31) is transformed into a multiperiod weighted digraph as shown in Figure 1. The investment period, the value of the objective function of the period t and a discrete value of the state variable are respectively represented by the stage, the weight of the period t and the point of the multiperiod weighted digraph.
In this section, a discrete approximate iteration method will be proposed to solve the Model (31).
Step 1: The discrete state variables at period t (t = 2, …, T+1) can be obtained by discretizing the interval value of W^{max}  W^{min} into four equalities. That is
${W}_{it}={W}_{t}{}^{\mathrm{min}}+\frac{({W}_{t}{}^{\mathrm{max}}{W}_{t}{}^{\mathrm{min}})}{4}i,i=0,\hspace{0.17em}\cdots ,\hspace{0.17em}4$
Step 2: The weight of the arcs in Figure 1 can be obtained following three steps:
Step 2.1: The net expected return of the portfolio ${r}_{N1}\left(1,j\right),\hspace{0.17em}j=1,\hspace{0.17em}\cdots ,\hspace{0.17em}5$, can be obtained as follows:
Step 2.2: The net expected return of portfolio ${r}_{Nt}(j,\hspace{0.17em}k)$ at period t, j = 1, …, 5, k = 1, …, 5, can be obtained as follows:
Step 2.3: The weights F_{1}(1,j) and F_{t}(j, k) of the side at period t can be obtained as follows:
When ${r}_{Nt}(k,\hspace{0.17em}\hspace{0.17em}l)$ is known, the subproblem at period t of the Model (31) can be turned into(40)
Let ${y}_{it}=\left{x}_{it}{\overline{x}}_{i(t1)}\right$. Then the Model (41) can be turned into as follows.
${x}_{t}^{*}$ (the optimal solution ${x}_{t}={\left({x}_{ft},\hspace{0.17em}{x}_{1t},\hspace{0.17em}{x}_{2t},\hspace{0.17em}\cdots ,\hspace{0.17em}{x}_{nt}\right)}^{\prime}$) can be obtained solving the Model (41) by the CPLEX. Simultaneously, the objective function value F_{t}(j, k) can be obtained as follows:
$\begin{array}{c}{F}_{t}(j,\text{\hspace{0.17em}}k)=(1\theta )\left({\displaystyle \sum}_{i=1}^{n}[{a}_{it}+\frac{{\beta}_{it}{\alpha}_{it}}{4}]{x}_{it}^{*}{\displaystyle \sum}_{i=1}^{n}{c}_{it}{y}_{it}^{*}\right)\\ \theta \left({\displaystyle \sum}_{i=1}^{n}\left[\begin{array}{l}\frac{1}{6{\beta}_{it}}{(\frac{{\beta}_{it}{\alpha}_{it}}{4})}^{3}+\frac{1}{6{\alpha}_{it}}{(\frac{{\beta}_{it}+3{\alpha}_{it}}{4})}^{3}\hfill \\ \frac{1}{6{\alpha}_{it}}{(\frac{{\beta}_{it}{\alpha}_{it}}{4})}^{3}\hfill \end{array}\right]{({x}_{it}^{*})}^{2}\right)\end{array}$
Step 3: Calculation of the longest path of the multiperiod weighted digraph
According to Definition 6, the longest path F^{(1)} of the multiperiod weighted digraph of the first iteration can be obtained as follows:(42)
where ${F}_{1}^{(1)}\text{=}{({F}^{(1)}(1,\hspace{0.17em}\hspace{0.17em}j))}_{1\times 5},\hspace{0.17em}{F}_{\text{2}}^{(\text{1})}\text{=}{({F}_{\text{2}}^{(\text{1})}(i,\hspace{0.17em}j))}_{5\times 5},\hspace{0.17em}\cdots ,\hspace{0.17em}{F}_{T}^{(\text{1})}\text{=}{({F}_{T}^{(\text{1})}(i,j))}_{5\times 5}.$
Step 4: The discrete approximate iteration method k+1th iteration can be described as follows:
Let the longest path of the kth iteration be ${W}_{1}\to {W}_{{i}_{2}2}^{(k)}\to {W}_{{i}_{3}3}^{(k)}\to \cdots \to {W}_{{i}_{T+1}T+1}^{(k)}$. The optimal solutions of the longest path of Figure 1 are also feasible solutions of the multiperiod mean absolute deviation portfolio selection model. Based on $({W}_{1},\hspace{0.17em}{W}_{{i}_{2}2}^{(k)},\hspace{0.17em}{W}_{{i}_{3}3}^{(k)},\hspace{0.17em}\cdots ,\hspace{0.17em}{W}_{{i}_{T+1}T+1}^{(k)})$, the state variables from period 1 to period T are discretized into four intervals as following three steps.
Step 4.1:${W}_{2}^{\mathrm{min}}\hspace{0.17em}\text{and}\hspace{0.17em}{W}_{{i}_{2}2}^{\left(k\right)},\hspace{0.17em}{W}_{{i}_{2}2}^{\left(k\right)}\hspace{0.17em}\text{and}\hspace{0.17em}{W}_{2}^{\text{max}}$ are discretized into two same internals, respectively. The five discrete points of S2, i.e., ${W}_{2}^{\mathrm{min}},\hspace{0.17em}{W}_{22}^{(k+1)},\hspace{0.17em}{W}_{{i}_{1}2}^{(k+1)},\hspace{0.17em}{W}_{32}^{(k+1)},\hspace{0.17em}{W}_{2}^{\mathrm{max}}$ can be obtained.
Step 4.2: Based on $({W}_{{i}_{3}3}^{(k)},\hspace{0.17em}\cdots ,\hspace{0.17em}{W}_{{i}_{T+1}T+1}^{(k)})$, using the same method, the state variables from period 3 to period T + 1 are respectively discretized into the five points. The utility of period t can also be obtained.
Step 4.3: The longest path of the k+1th iteration F^{(k+1)} and another feasible solution can be obtained as follows:(43)
where ${F}_{1}^{(k+1)}\text{=}\hspace{0.17em}{({F}^{(k+1)}(1,\hspace{0.17em}j))}_{1\times 5},\hspace{0.17em}{F}_{\text{2}}^{(k+\text{1})}\text{=}\hspace{0.17em}{({F}_{\text{2}}^{(k+\text{1})}(i,\hspace{0.17em}j))}_{5\times 5},\cdots ,{F}_{T}^{(k+\text{1})}\text{=}{({F}_{T}^{(k+\text{1})}(i,j))}_{5\times 5}.$
If $\left{F}^{(k+1)}{F}^{(k)}\right\le {10}^{6},$ then the optimal solution of the longest path F(k+1) also is the approximately optimal solution of the Model (31). Otherwise turn Step 2.
4.4. Convergence and Complex Property of the Discrete Approximate Iteration Method
Theorem 8. The discrete approximate iteration method is convergent.
Proof. Let the longest path in period 1 be ${F}_{\text{1}}^{\mathrm{max}}(1,\hspace{0.17em}{j}_{2})$, and the longest path in period t be ${F}_{t}^{\mathrm{max}}({i}_{t},\hspace{0.17em}{j}_{t+1})$t = 2, …, T. Then the upper bound of the solution of Model (31) is
${F}_{1}^{\mathrm{max}}(1,\hspace{0.17em}{j}_{2})\times {F}_{2}^{\mathrm{max}}({i}_{2},\hspace{0.17em}{j}_{3})\times \cdots \times {F}_{T}^{\mathrm{max}}({i}_{T},\hspace{0.17em}{j}_{T+1})$
The longest path of the multiperiod weighted digraph of the kth iteration is obtained as follows:
where ${F}_{1}^{(k)}\text{=}\hspace{0.17em}{({F}^{(k)}(1,\hspace{0.17em}j))}_{1\times 5},\hspace{0.17em}{F}_{\text{2}}^{(k)}\text{=}\hspace{0.17em}{({F}_{\text{2}}^{(k)}(i,\hspace{0.17em}j))}_{5\times 5},\cdots ,\hspace{0.17em}{({F}_{T}^{(k)}(i,\hspace{0.17em}j))}_{5\times 5}.$.
Let the longest path of the k th iteration be ${W}_{1}\to {W}_{{i}_{2}2}^{(k)}\to {W}_{{i}_{3}3}^{(k)}\to \cdots \to {W}_{{i}_{T+1}T+1}^{(k)}$, Using the Step 4, the multiperiod weighted digraph of the k+1th iteration can be obtained. F^{(k+1)}, which is the value of the longest path of the multiperiod weighted digraph of the k+1th iteration, can be obtained by the Eq. (44). So ${F}^{(k+1)}\ge {F}^{(k)}$. The solution is becoming bigger and bigger. Because the solutions of Model (31) are increasing sets and there is an upper bound of the solution of Model (31). Then, the discrete approximate iteration method is convergent.
Thus, the proof of the Theorem 8 is ended.
5. NUMERICAL EXAMPLE
In this section, a numerical example is given to express the idea of the proposed model. Assume that an investor chooses thirty stocks from Shanghai Stock Exchange for his investment. The stocks codes are respectively S_{1}, …, S_{30}. He intends to make five periods of investment with initial wealth W_{1} = 1 and his wealth can be adjusted at the beginning of each period. He/she assumes that the returns, risk and turnover rates of the thirty stocks at each period are represented as trapezoidal fuzzy numbers. We collect historical data of them from April 2006 to March 2017 and set every three months as a period to handle the historical data. By using the simple estimation method in Vercher et al. (2007) to handle their historical data, the triangular possibility distributions of the return rates of assets at each period can be obtained as shown in Appendix.
Suppose that the unit transaction costs of assets of the two periods investment take the same value c_{it} = 0.003 (i = 1, …, 30; t = 1, …, 5), the desired number of risky assets in the portfolio K = 0, …, 9 at period t, t = 1, …, 5, the lower bound of the investment proportion of riskfree asset x^{b}_{ft} = 0.5, preference coefficient θ = 0, 0.1, …, 1, the borrowing rate of the riskfree asset r_{bt} = 0.017, the lending rate of the riskfree asset r_{lt} = 0.009, t = 1, …, 5, the lower l_{it} = 0.05 and upper bound constraints uit = 0.2 (i = 1, …, 30; t = 1, …, 5), x_{i}_{0} = 0(i = 1, …, 30).
As illustrations, the following numerical examples are given to show the effectiveness of the proposed model and the discrete approximate iteration method. The algorithm was programmed by C++ language and run on a personal computer with Pentium Dual CPU, 4GHz and 8GB RAM.
By using the discrete iteration method to solve the Model (31), the corresponding results can be obtained as follows.
If θ = 0.5, K = 6, the optimal solution will be obtained as the Table 1.
When θ = 0.5, K = 6, the optimal investment strategy at period 1 is x_{11} = 0.2, x_{131} = 0.2, x_{171} = 0.2, x_{191} = 0.2, x_{241} = 0.1, x_{281} = 0.2, x_{f}_{1} = 0.2 and being the rest of variables equal to zero, which means investor should allocate his initial wealth on asset 1, asset 13, asset 17, asset 19, asset 24 , asset 28, riskfree asset and otherwise asset by the proportions of 20%, 20%, 30%, 20%, 20% , 20%, 20% and being the rest of variables equal to zero among the thirty stocks, respectively. From Table 1, the optimal investment strategy at period 2, period 3, period 4 and period 5 can also be obtained. In this case, the available terminal wealth is 2.0396.
If θ = 0.5, K = 8, the optimal solution will be obtained as the Table 2.
When q = 0.5, K = 8, the available terminal wealth is 2.2268.
To display the influence of K on the optimal solution of multiperiod, its value is set as 6 and 8, respectively, and the Model (28) for portfolio decisionmaking will be used afterwards. After using the discrete approximate iteration method, the corresponding optimal investment strategies can be obtained as shown in Table 1 and Table 2. From Table 1 and Table 2, it can be seen that most of assets of the optimal solutions of K = 6 and K = 8 are same. There are six assets in period 1, i.e. asset 1, asset 13, asset 17, asset 19, asset 24, asset 28; there are six assets in period 2, i.e. asset 1, asset 12, asset 13, asset 15, asset 17, asset 18; there are six assets in period 3, i.e. asset 1, asset 12, asset 13, asset 15, asset 17, asset 18; there are six assets in period 4, i.e. asset 1, asset 12, asset 13, asset 15, asset 17, asset 18; there are six assets in period 5, i.e. asset 1, asset 12, asset 13, asset 15, asset 17, asset 18.
When θ = 0.5, K = 0, 1, …, 9, the optimal terminal wealth can be obtained as shown in Table 3.
In the used data sets, the same optimal solutions with the K = 8 is suitable for K ≥ 9. The experiments in this paper correspond to the values of K in the interval [0, 9]. In Table 3, the terminal wealth also becomes larger, when the preset the desired number of assets in the portfolio become larger, which reflects the influence of the desired number of assets on portfolio selection.
When K = 7, θ = 0, 0.1, …, 0.9, 1, the optimal terminal wealth can be obtained as shown in Table 4.
In the used data sets, the experiments in this paper correspond to the values of θ in the interval [0, 1]. In Table 4, the terminal wealth is same, when 0 ≤ θ ≤ 0.6; the terminal wealth becomes smaller, when preference coefficient θ which 0.6 < θ ≤ 1, become larger, which reflects the influence of preference coefficient θ on portfolio selection.
6. CONCLUSIONS
In this paper, we consider the multiperiod portfolio selection problem in fuzzy environment. We use the credibilitic mean value and the semivariance to measure the return and the risk of the multiperiod portfolio, respectively. A new multiperiod portfolio optimization model with transaction cost, borrowing constraints, threshold constraints and cardinality constraints is proposed. In the proposed model, the cardinality constraints are added as a constraint to control the number of assets to be held in an efficient portfolio. Based on the credibilitic theories, the proposed model is transformed into a crisp mix integer nonlinear dynamic optimization. Because of the transaction cost and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence. The proposed model is approximated to dynamic programming model. A novel discrete iteration method is designed to obtain the optimal portfolio strategy, and is proved linearly convergent. Finally, an example is given to illustrate the behavior of the proposed model and the designed algorithm using real data from the Shanghai Stock Exchange.