# A Hybrid Intelligent Algorithm for Optimal Birandom Portfolio Selection Problems

Authors: Qi Li; Guo-Hua Cao; Dan Shan
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Birandom portfolio selection problems have been well developed and widely applied in recent years. To solve these problems better, this paper designs a new hybrid intelligent algorithm which combines the improved LGMS-FOA algorithm with birandom simulation. Since all the existing algorithms solving these problems are based on genetic algorithm and birandom simulation, some comparisons between the new hybrid intelligent algorithm and the existing algorithms are given in terms of numerical experiments, which demonstrate that the new hybrid intelligent algorithm is more effective and precise when the numbers of the objective function computations are the same.


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## Body

## 1. Introduction

Portfolio theory was initially put forward by Markowitz [1] and has received great development since then. The main content of it is to study how to allocate one’s capital to a large number of securities and the study mainly focuses on three aspects: the first is how to estimate the security return; the second is how to build portfolio models; the third is how to design efficient algorithms to solve these models. Many scholars have made great contributions in this field.In the early literatures, security return was assumed to obey normal distribution. However, the following researches manifested that the assumption of normal distribution did not accord with the facts. Therefore, a lot of new distributions were used to describe the security return [2–9]. Among them, birandom distribution received great attention and development by some scholars [7–9] and literature [9] demonstrated that birandom distribution could reflect the features of securities’ technical patterns and the investors’ heterogeneity.When the security return followed birandom distribution, the existing literatures established several birandom portfolio models and designed the corresponding hybrid intelligent algorithms [7–9]. However, all these algorithms were based on genetic algorithm (GA) and existed some common shortcomings, such as low accuracy and inferior local search ability. To solve these birandom portfolio models effectively, this paper designs a new hybrid intelligent algorithm which integrates the improved LGMS-FOA algorithm and birandom simulation. The experimental results show that the new algorithm is more efficient.The rest of this paper is organized as follows: Section2 recalls some basic concepts about birandom theory; Section 3 provides an overview of birandom portfolio models; Section 4 presents a new hybrid intelligent algorithm which integrates the improved LGMS-FOA algorithm and birandom simulation; Section 5 provides numerical examples to test the effectiveness of the new hybrid intelligent algorithm; finally, a brief summary about this paper is given.


## 2. Birandom Theory

Definition 1.
A birandom variableε is a mapping from a probability space (Ω,A,Pr) to a collection of random variables such that, for any Borel subset B of the real line R, the induced function Pr(ε(w)∈B) is a measurable function with respect to w [7].Example 2.
LetΩ=(ω1,ω2) (ω1 represents security rising and ω2 represents security falling) and Pr(ω1)=a and Pr(ω2)=b. Assume that ε is a function on (Ω,A,Pr) as follows:
(1)ε(ω)={ε1ifω=ω1ε2ifω=ω2,
where ε1and ε2follow uniform distribution; then ε is a birandom variable according to Definition 1.Definition 3.
Letε=(ε1,ε2,…,εn) be a birandom vector on (Ω,A,Pr), and let f:Rn→Rm be a vector-valued Borel measurable function. Then the primitive chance of birandom event characterized by f(ε)≤0 is a function from (0,1] to [0,1], defined as [8]
(2)Ch{f(ε)≤0}(α)=sup⁡{β∣Pr{ω∈Ω∣Pr{f(ε(ω))≤0}≥β}≥α}.Definition 4.
Letε be a birandom variable defined on the probability space (Ω,A,Pr). Then the expected value of birandom variable ε is defined as
(3)E(ε)=∫0∞Pr{ω∈Ω∣E[ε(ω)]≥t}dt-∫-∞0Pr{ω∈Ω∣E[ε(ω)]≤t}dt,
provided that at least one of the above two integrals is finite [8].


## 3. Birandom Portfolio Models

In this section, we provide an overview of birandom portfolio models. Letxi represent the investment proportion in security i and let ki denote the return of the ith security for i=1,2,…,n, respectively. In particular, ki follows birandom distribution for i=1,2,…,n. Depending on different measures of investment profit and risk, birandom portfolio models are divided into different types.


### 3.1. Birandom Safety-First Model

For each portfoliox=(x1,x2,…,xn), the chance that return of total asset (RTA) is no less than a disaster level is used to represent investment risk, which is given by
(4)Ch{x1k1+x2k2+⋯+xnkn≥m}(β1)≥a.The chance that RTA is no less than some value is regarded to measure investment profit, which is represented by(5)Ch{x1k1+x2k2+⋯+xnkn≥t}(β2),
where m and t represent the disaster level and profit, respectively. β1, β2, and a represent the corresponding confidence level.Using (4) and (5), we obtained the following Birandom safety-first model, which was proposed in [9]:
(6)max⁡Ch{x1k1+x2k2+⋯+xnkn≥t}(β2)s.t.Ch{x1k1+x2k2+⋯+xnkn≥m}(β1)≥aiiiiiiii∑i=1nxi=1iiiiiiiixi≥0i=1,2,…,n.


### 3.2. Birandom Expected Value Model

In this model, investment risk was represented by (4) and the expect value of RTA was used to measure investment profit which was given by
(7)E[x1k1+x2k2+⋯+xnkn].Based on (4) and (7), Birandom expect value model was proposed in [8], which was formulated as follows:
(8)max⁡E[x1k1+x2k2+⋯+xnkn]s.t.Ch{x1k1+x2k2+⋯+xnkn≥m}(β1)≥as.t.∑i=1nxi=1s.t.xi≥0i=1,2,…,n.


### 3.3. Birandom Chance-Constrained Model

Literature [7] proposed birandom chance-constrained model, which was listed as below:
(9)max⁡Ch{x1k1+x2k2+⋯+xnkn≥t}(β2)s.t.∑i=1nxi=1s.t.xi≥0i=1,2,…,n.Remark 5.
The parameters of models (6), (8), and (9) have the same economic meaning.Remark 6.
For more details on these models, please refer to the corresponding literatures.


## 3.1. Birandom Safety-First Model

For each portfoliox=(x1,x2,…,xn), the chance that return of total asset (RTA) is no less than a disaster level is used to represent investment risk, which is given by
(4)Ch{x1k1+x2k2+⋯+xnkn≥m}(β1)≥a.The chance that RTA is no less than some value is regarded to measure investment profit, which is represented by(5)Ch{x1k1+x2k2+⋯+xnkn≥t}(β2),
where m and t represent the disaster level and profit, respectively. β1, β2, and a represent the corresponding confidence level.Using (4) and (5), we obtained the following Birandom safety-first model, which was proposed in [9]:
(6)max⁡Ch{x1k1+x2k2+⋯+xnkn≥t}(β2)s.t.Ch{x1k1+x2k2+⋯+xnkn≥m}(β1)≥aiiiiiiii∑i=1nxi=1iiiiiiiixi≥0i=1,2,…,n.


## 3.2. Birandom Expected Value Model

In this model, investment risk was represented by (4) and the expect value of RTA was used to measure investment profit which was given by
(7)E[x1k1+x2k2+⋯+xnkn].Based on (4) and (7), Birandom expect value model was proposed in [8], which was formulated as follows:
(8)max⁡E[x1k1+x2k2+⋯+xnkn]s.t.Ch{x1k1+x2k2+⋯+xnkn≥m}(β1)≥as.t.∑i=1nxi=1s.t.xi≥0i=1,2,…,n.


## 3.3. Birandom Chance-Constrained Model

Literature [7] proposed birandom chance-constrained model, which was listed as below:
(9)max⁡Ch{x1k1+x2k2+⋯+xnkn≥t}(β2)s.t.∑i=1nxi=1s.t.xi≥0i=1,2,…,n.Remark 5.
The parameters of models (6), (8), and (9) have the same economic meaning.Remark 6.
For more details on these models, please refer to the corresponding literatures.


## 4. Hybrid Intelligent Algorithm

In this section, a new hybrid intelligent algorithm is designed to solve models (6), (8), and (9), where the improved LGMS-FOA algorithm and birandom simulation are used.


### 4.1. Birandom Simulation

Because of the uncertainty of birandom variable, we should use birandom simulation [10] to solve equations (4), (5), and (7).Algorithm 1 (birandom simulation for (4)).
Step
1. Set l=1.
Step
2. Generate w1,w2,…,wN from Ω according to the probability Pr. Consider(10)wi={y1,y2,…,yn},i=1,2,…,N.
Step
3. Compute the probability
(11)δi=Pr{x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)≥m},hhhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,N,
by stochastic simulation.
Step
4. Set N* as the integer part of β1N.
Step
5. Return the N*th largest element δ* in {δ1,δ2,…,δN}.
Step
6. If δ*≥a, then l=l*1; else l=l*0.
Step
7. l=1 means that the solution is feasible; l=0 means that the solution is infeasible.Algorithm 2 (birandom simulation for (5)).
Step
1. Generate w1,w2,…,wN from Ω according to the probability Pr. Consider (12)wi={y1,y2,…,yn},i=1,2,…,N.
Step
2. Compute the probability
(13)δi=Pr{x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)≥t},hhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,N,
by stochastic simulation.
Step
3. Set N* as the integer part of β2N.
Step
4. Return the N*th largest element δ* in {δ1,δ2,…,δN}.Algorithm 3 (birandom simulation for (7)).
Step
1. Set e=0.
Step
2. Generate w1,w2,…,wN from Ω according to the probability Pr. Consider(14)wi={y1,y2,…,yn},i=1,2,…,N.
Step
3. Compute the probability
(15)δi=E[x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)],hhhhhhhhhhhhhhhhhhhhhhhii=1,2,…,N,
by stochastic simulation.
Step
4. Consider(16)e⟵e+E[x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)].
Step
5. Repeat the second to the fourth steps for N times.
Step
6. Consider(17)E[x1k1+x2k2+⋯+xnkn]=eN.


### 4.2. The Improved LGMS-FOA

Recently, optimization algorithms are widely used in many fields [11–14]. Fruit fly optimization algorithm (FOA) is a new stochastic optimization technique proposed in 2012 [11] and has received rapid developments in recent years. The principle of FOA is based on the food finding behavior of fruit fly swarm. The food finding process was shown in Figure 1 and has two steps: first, it smells food and flies to that direction; then, it uses vision to find food.Figure 1
Food finding process of fruit fly swarm.To enhance the search ability of FOA, LGMS-FOA was proposed and had been proved to be an effective algorithm by numerical testing [12]. However, LMGS-FOA in the literature [12] is only applied to solve unconstrained optimization problems. To solve birandom portfolio models, the improved LGMS-FOA is proposed by means of combination of LGMS-FOA with penalty function method. Without loss of generality, we take model (6), for example, to introduce the improved LGMS-FOA. The steps of it are listed as follows.Algorithm 4 (the improved LGMS-FOA).
Step
1. Parameters initialization.
The parameters include the number of iteration (maxgen), the population number (sizepop), the searching scope (r), the searching coefficient (d), the primary weight (w0), and the weight adjustment coefficient (ρ).
Step
2. Initial fruit fly swarm location.
The fruit fly swarm locationx=(x1,x2,…,xn) is randomly generated in the interval [-r,r] and the searching scope becomes bigger with the increase of r.
Step
3. Give the random direction and distance for food finding of an individual fruit fly. Consider(18)xP=x+w×d×(2×rand(1,n)-1),hhhhhhhhhhhhP=1,2,,…,sizepop;w=w0×ρgen.
Step
4. Generate fruit fly swarm that fits the constraint ∑i=1nyi=1.
Step
4.1. A solution yp=(y1,y2,…,yn) is represented by the individual fruit fly xp=(x1,x2,…,xn), where the relation between yp and xp is formulated as follows:
(19)yi=xix1+x2+⋯+xn,i=1,2,…,n,
which ensures that ∑i=1nyi=1.
Step
4.2. If the element of yp is less than zero, repeat Step 4.1 until the number of individuals is sizepop.
Step
5. Calculate the smell concentration by inputting yp into objective function.
Step
5.1. If yp=(y1,y2,…,yn) satisfies
(20)Ch{y1k1+y2k2+⋯+ynkn≥m}(β1)≥a,
then let
(21)Smelli=Ch{y1k1+y2k2+⋯+ynkn≥t}(β2).
Step
5.2. If yp=(y1,y2,…,yn) does not satisfy
(22)Ch{y1k1+y2k2+⋯+ynkn≥m}(β1)≥a,
then use penalty function method [13] and let
(23)Smelli=Ch{y1k1+y2k2+⋯+ynkn≥t}(β2)-b,
where b is a given positive integer.
Step
6. Find out the optimal individual among fruit fly swarm. Consider(24)[bestsmellbestindex]=max⁡(Smell).
Step
7. Keep the optimal objective function value and the corresponding x coordinate; then the fruit fly swarm flies towards that location:
(25)Smellbest=bestsmellx=x(bestinedx).
Step
8. Repeat Step 2 to Step 7 until termination condition is satisfied.


### 4.3. Hybrid Intelligent Algorithm

Through integrating the improved LGMS-FOA algorithm and birandom simulation, a new hybrid intelligent algorithm is built and the steps are listed as below.Algorithm 5 (hybrid intelligent algorithm).
m
Step
1. Initialize N fruit fly individuals that satisfy the constraints.
Step
2. Calculate the objective function values for all fruit fly individuals by birandom simulation.
Step
3. Find out the optimal objective value.
Step
4. Keep the optimal objective value and the corresponding location of fruit fly individual.
Step
5. Repeat Step 2 to Step 5, until termination condition is satisfied.


## 4.1. Birandom Simulation

Because of the uncertainty of birandom variable, we should use birandom simulation [10] to solve equations (4), (5), and (7).Algorithm 1 (birandom simulation for (4)).
Step
1. Set l=1.
Step
2. Generate w1,w2,…,wN from Ω according to the probability Pr. Consider(10)wi={y1,y2,…,yn},i=1,2,…,N.
Step
3. Compute the probability
(11)δi=Pr{x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)≥m},hhhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,N,
by stochastic simulation.
Step
4. Set N* as the integer part of β1N.
Step
5. Return the N*th largest element δ* in {δ1,δ2,…,δN}.
Step
6. If δ*≥a, then l=l*1; else l=l*0.
Step
7. l=1 means that the solution is feasible; l=0 means that the solution is infeasible.Algorithm 2 (birandom simulation for (5)).
Step
1. Generate w1,w2,…,wN from Ω according to the probability Pr. Consider (12)wi={y1,y2,…,yn},i=1,2,…,N.
Step
2. Compute the probability
(13)δi=Pr{x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)≥t},hhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,N,
by stochastic simulation.
Step
3. Set N* as the integer part of β2N.
Step
4. Return the N*th largest element δ* in {δ1,δ2,…,δN}.Algorithm 3 (birandom simulation for (7)).
Step
1. Set e=0.
Step
2. Generate w1,w2,…,wN from Ω according to the probability Pr. Consider(14)wi={y1,y2,…,yn},i=1,2,…,N.
Step
3. Compute the probability
(15)δi=E[x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)],hhhhhhhhhhhhhhhhhhhhhhhii=1,2,…,N,
by stochastic simulation.
Step
4. Consider(16)e⟵e+E[x1k1(y1)+x2k2(y2)+⋯+xnkn(yn)].
Step
5. Repeat the second to the fourth steps for N times.
Step
6. Consider(17)E[x1k1+x2k2+⋯+xnkn]=eN.


## 4.2. The Improved LGMS-FOA

Recently, optimization algorithms are widely used in many fields [11–14]. Fruit fly optimization algorithm (FOA) is a new stochastic optimization technique proposed in 2012 [11] and has received rapid developments in recent years. The principle of FOA is based on the food finding behavior of fruit fly swarm. The food finding process was shown in Figure 1 and has two steps: first, it smells food and flies to that direction; then, it uses vision to find food.Figure 1
Food finding process of fruit fly swarm.To enhance the search ability of FOA, LGMS-FOA was proposed and had been proved to be an effective algorithm by numerical testing [12]. However, LMGS-FOA in the literature [12] is only applied to solve unconstrained optimization problems. To solve birandom portfolio models, the improved LGMS-FOA is proposed by means of combination of LGMS-FOA with penalty function method. Without loss of generality, we take model (6), for example, to introduce the improved LGMS-FOA. The steps of it are listed as follows.Algorithm 4 (the improved LGMS-FOA).
Step
1. Parameters initialization.
The parameters include the number of iteration (maxgen), the population number (sizepop), the searching scope (r), the searching coefficient (d), the primary weight (w0), and the weight adjustment coefficient (ρ).
Step
2. Initial fruit fly swarm location.
The fruit fly swarm locationx=(x1,x2,…,xn) is randomly generated in the interval [-r,r] and the searching scope becomes bigger with the increase of r.
Step
3. Give the random direction and distance for food finding of an individual fruit fly. Consider(18)xP=x+w×d×(2×rand(1,n)-1),hhhhhhhhhhhhP=1,2,,…,sizepop;w=w0×ρgen.
Step
4. Generate fruit fly swarm that fits the constraint ∑i=1nyi=1.
Step
4.1. A solution yp=(y1,y2,…,yn) is represented by the individual fruit fly xp=(x1,x2,…,xn), where the relation between yp and xp is formulated as follows:
(19)yi=xix1+x2+⋯+xn,i=1,2,…,n,
which ensures that ∑i=1nyi=1.
Step
4.2. If the element of yp is less than zero, repeat Step 4.1 until the number of individuals is sizepop.
Step
5. Calculate the smell concentration by inputting yp into objective function.
Step
5.1. If yp=(y1,y2,…,yn) satisfies
(20)Ch{y1k1+y2k2+⋯+ynkn≥m}(β1)≥a,
then let
(21)Smelli=Ch{y1k1+y2k2+⋯+ynkn≥t}(β2).
Step
5.2. If yp=(y1,y2,…,yn) does not satisfy
(22)Ch{y1k1+y2k2+⋯+ynkn≥m}(β1)≥a,
then use penalty function method [13] and let
(23)Smelli=Ch{y1k1+y2k2+⋯+ynkn≥t}(β2)-b,
where b is a given positive integer.
Step
6. Find out the optimal individual among fruit fly swarm. Consider(24)[bestsmellbestindex]=max⁡(Smell).
Step
7. Keep the optimal objective function value and the corresponding x coordinate; then the fruit fly swarm flies towards that location:
(25)Smellbest=bestsmellx=x(bestinedx).
Step
8. Repeat Step 2 to Step 7 until termination condition is satisfied.


## 4.3. Hybrid Intelligent Algorithm

Through integrating the improved LGMS-FOA algorithm and birandom simulation, a new hybrid intelligent algorithm is built and the steps are listed as below.Algorithm 5 (hybrid intelligent algorithm).
m
Step
1. Initialize N fruit fly individuals that satisfy the constraints.
Step
2. Calculate the objective function values for all fruit fly individuals by birandom simulation.
Step
3. Find out the optimal objective value.
Step
4. Keep the optimal objective value and the corresponding location of fruit fly individual.
Step
5. Repeat Step 2 to Step 5, until termination condition is satisfied.


## 5. Numerical Experiments

To test the effectiveness of the new hybrid intelligent algorithm, we compared it with the existing algorithm [7–9] through numerical examples.


### 5.1. Experimental Setup

Assume that the investor selects three securities represented byk1, k2, and k3. The security return follows the birandom distribution in Example 2 and the corresponding parameters of each security are shown in Table 1. Besides, the investor takes t=0.04, m=-0.06, β1=0.7, β2=0.5, and a=0.6; then, models (6), (8), and (9) satisfying the above parameters are changed into
(26)max⁡Ch{x1k1+x2k2+x3k3≥0.04}(0.5)s.t.Ch{x1k1+x2k2+x3k3≥-0.06}(0.7)≥0.6s.t.x1+x2+x3=1s.t.xi≥0i=1,2,3,(27)max⁡E[x1k1+x2k2+x3k3]s.t.Ch{x1k1+x2k2+x3k3≥-0.06}(0.7)≥0.6s.t.∑i=13xi=1s.t.xi≥0i=1,2,3,(28)max⁡Ch{x1k1+x2k2+x3k3≥0.04}(0.5)s.t.∑i=13xi=1s.t.xi≥0i=1,2,3.Table 1
The distributions of securitiesk1, k2, and k3.
Parameters
k
1
k
2
k
3
a
i (i=1,2,3)
3/4
1/2
1/3
b
i (i=1,2,3)
1/4
1/2
2/3
ε
i
1 (i=1,2,3)
U [0, 7%]
U [0, 9%]
U [0, 11%]
ε
i
2 (i=1,2,3)
U [−9%, 0]
U [−11%, 0]
U [−13%, 0]The parameters of Algorithm5 are set up as follows.(1)
The parameter of birandom simulation isN=2000.(2)
The parameters of the improved LGMS-FOA aremaxgen=40, sizepop=30, r=40, d=20, w0=1, and ρ=0.8.To show the efficiency of Algorithm5, we choose two sets of parameters and the corresponding hybrid intelligent algorithms in [7, 8] which are named as Algorithms 6 and 7.The parameters of Algorithm 6 are listed as below.(1)
The parameter of birandom simulation isN=2000.(2)
The parameters of GA are as follows: the iteration number is 40, the population is 30, the mutation probability is 0.6, and the crossover probability is 0.1.The parameters of Algorithm 7 are given as follows.(1)
The parameter of birandom simulation isN=2000.(2)
The parameters of GA are as follows: the iteration number is 40, the population is 30, the mutation probability is 0.8, and the crossover probability is 0.05.Remark 5.
For Algorithms5 to 7, the numbers of objective function computations are the same and this can ensure the fairness of comparisons.


### 5.2. Experimental Results

Compute models from (26) to (28) by using Algorithms from 5 to 7, respectively, and repeat the experiments for 20 times. The average values of the optimal results are shown in Tables 2, 3, and 4.Table 2
The optimal results of model (26).
Model
Algorithm
Optimal solution
Objectivefunction
x
1
x
2
x
3
Model (26)
Algorithm5
0.9996
0.0003
0.0001
0.4230
Model (26)
Algorithm  6
0.6854
0.2304
0.0842
0.0425
Model (26)
Algorithm  7
0.1562
0.6832
0.1606
0.0805Table 3
The optimal results of model (27).
Model
Algorithm
Optimal solution
Objectivefunction
x
1
x
2
x
3
Model (27)
Algorithm5
0.9832
0.0043
0.0125
0.0159
Model (27)
Algorithm  6
0.6818
0.2948
0.0234
0.0071
Model (27)
Algorithm  7
0.6057
0.2634
0.1309
0.0062Table 4
The optimal results of model (28).
Model
Algorithm
Optimal solution
Objectivefunction
x
1
x
2
x
3
Model (28)
Algorithm5
0.0005
0.9987
0.0008
0.5367
Model (28)
Algorithm  6
0.0908
0.8339
0.0753
0.3382
Model (28)
Algorithm  7
0.1221
0.8703
0.0076
0.4052From Tables2 to 4, it can be found that the final searching quality of Algorithm 5 is better than Algorithms 6 and 7. So it can be concluded that the new hybrid intelligent algorithm is more efficient and precise than the existing algorithms when the numbers of objective function computations are the same.


## 5.1. Experimental Setup

Assume that the investor selects three securities represented byk1, k2, and k3. The security return follows the birandom distribution in Example 2 and the corresponding parameters of each security are shown in Table 1. Besides, the investor takes t=0.04, m=-0.06, β1=0.7, β2=0.5, and a=0.6; then, models (6), (8), and (9) satisfying the above parameters are changed into
(26)max⁡Ch{x1k1+x2k2+x3k3≥0.04}(0.5)s.t.Ch{x1k1+x2k2+x3k3≥-0.06}(0.7)≥0.6s.t.x1+x2+x3=1s.t.xi≥0i=1,2,3,(27)max⁡E[x1k1+x2k2+x3k3]s.t.Ch{x1k1+x2k2+x3k3≥-0.06}(0.7)≥0.6s.t.∑i=13xi=1s.t.xi≥0i=1,2,3,(28)max⁡Ch{x1k1+x2k2+x3k3≥0.04}(0.5)s.t.∑i=13xi=1s.t.xi≥0i=1,2,3.Table 1
The distributions of securitiesk1, k2, and k3.
Parameters
k
1
k
2
k
3
a
i (i=1,2,3)
3/4
1/2
1/3
b
i (i=1,2,3)
1/4
1/2
2/3
ε
i
1 (i=1,2,3)
U [0, 7%]
U [0, 9%]
U [0, 11%]
ε
i
2 (i=1,2,3)
U [−9%, 0]
U [−11%, 0]
U [−13%, 0]The parameters of Algorithm5 are set up as follows.(1)
The parameter of birandom simulation isN=2000.(2)
The parameters of the improved LGMS-FOA aremaxgen=40, sizepop=30, r=40, d=20, w0=1, and ρ=0.8.To show the efficiency of Algorithm5, we choose two sets of parameters and the corresponding hybrid intelligent algorithms in [7, 8] which are named as Algorithms 6 and 7.The parameters of Algorithm 6 are listed as below.(1)
The parameter of birandom simulation isN=2000.(2)
The parameters of GA are as follows: the iteration number is 40, the population is 30, the mutation probability is 0.6, and the crossover probability is 0.1.The parameters of Algorithm 7 are given as follows.(1)
The parameter of birandom simulation isN=2000.(2)
The parameters of GA are as follows: the iteration number is 40, the population is 30, the mutation probability is 0.8, and the crossover probability is 0.05.Remark 5.
For Algorithms5 to 7, the numbers of objective function computations are the same and this can ensure the fairness of comparisons.


## 5.2. Experimental Results

Compute models from (26) to (28) by using Algorithms from 5 to 7, respectively, and repeat the experiments for 20 times. The average values of the optimal results are shown in Tables 2, 3, and 4.Table 2
The optimal results of model (26).
Model
Algorithm
Optimal solution
Objectivefunction
x
1
x
2
x
3
Model (26)
Algorithm5
0.9996
0.0003
0.0001
0.4230
Model (26)
Algorithm  6
0.6854
0.2304
0.0842
0.0425
Model (26)
Algorithm  7
0.1562
0.6832
0.1606
0.0805Table 3
The optimal results of model (27).
Model
Algorithm
Optimal solution
Objectivefunction
x
1
x
2
x
3
Model (27)
Algorithm5
0.9832
0.0043
0.0125
0.0159
Model (27)
Algorithm  6
0.6818
0.2948
0.0234
0.0071
Model (27)
Algorithm  7
0.6057
0.2634
0.1309
0.0062Table 4
The optimal results of model (28).
Model
Algorithm
Optimal solution
Objectivefunction
x
1
x
2
x
3
Model (28)
Algorithm5
0.0005
0.9987
0.0008
0.5367
Model (28)
Algorithm  6
0.0908
0.8339
0.0753
0.3382
Model (28)
Algorithm  7
0.1221
0.8703
0.0076
0.4052From Tables2 to 4, it can be found that the final searching quality of Algorithm 5 is better than Algorithms 6 and 7. So it can be concluded that the new hybrid intelligent algorithm is more efficient and precise than the existing algorithms when the numbers of objective function computations are the same.


## 6. Conclusion

To solve birandom portfolio selection problems better, this paper designs a new hybrid intelligent algorithm which integrates the improved LGMS-FOA algorithm and birandom simulation. Comparisons between the new hybrid intelligent algorithm and the existing algorithms show that the new hybrid intelligent algorithm is more effective and precise when the numbers of the objective function computations are the same.



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*Source: 213518-2014-06-16.xml*