该项目负责人是永乐高ylg888888蚁文洁，项目成员有刘憬、刘佳、喻迎。本项目使用Matlab软件编程，对群体智能算法的效果进行测试，并选择合适的算法解决多目标集成供应链问题模型。构建考虑多种因素的多目标集成供应链数学模型。对细菌觅食算法、头脑风暴算法等群体智能算法进行改进，找到更适合应用于解决该数学模型的优化算法来解决集成供应链问题。通过设计严谨且符合现实情况的实验，比较目前已有的各种优化算法并就其算法特点进行讨论。

本项目创新点：在对集成多级供应链的研究中，同时考虑多原材料采购、多产品生产，并与多供应商、多生产商、多分销商结合,同时考虑提前期对完工时间的影响。在模型目标函数方面,从只考虑最小化总成本转变为同时考虑最小化总成本和最大化客户满意度(即最小化产品缺货率),构建了一个更贴近现实供应链模型。针对该模型，在标准细菌觅食优化算法的基础上进行算法结构改进，学习交流机制及非控制解选择机制改进，构建适用于来解决此类问题的新的多目标细菌觅食优化算法。通过构建模型及优化求解，有利于多级供应链中的成员以正确的数量、质量和恰当的方式，在正确的时间和地点进行产品的生产、加工和分销，以最低的成本提高供应链系统的服务水准, 从而提高企业的竞争力。这不仅丰富了群体智能计算理论，并驱动了集成供应链模型在实际中的应用。

项目的预期成果为完成两篇学术论文。实际成果发表三篇学术论文，《Niu B., Yi W.J.（蚁文洁）, Tan L.J, Liu J（刘佳）., Li Y, Wang H.. Multi-objective Comprehensive Learning Bacterial Foraging Optimization for Portfolio Problem. Lecture Notes in Computer Science. 2017, 10386: 69-76.（EI: 20173204021034）》。《Wang H, Liu J(刘佳）, Yi W.J.（蚁文洁）, Niu B, Jaejong Baek: An Improved Brain Storm Optimization with Learning Strategy. Lecture Notes in Computer Science. 2017:511-518. (EI: 20173204020699)》。刘憬《基于改进多目标细菌觅食算法的集成供应链问题优化求解》。

本项目论据充足，材料新颖，具有实践意义。符合自主创新基金项目结项要求。

Multi-objective Comprehensive Learning Bacterial Foraging Optimization for Portfolio Problem

Ben Niu^1,2*, Wenjie Yi¹, Lijing Tan^3*, Jia Liu¹, Ya Li⁴, Hong Wang^1,5*

1College of Management, Shenzhen University, Shenzhen, 518060, China

² School of Computing, Information, and Decision Systems Engineering, Arizona State University, Tempe, USA

3Department of Business Management, Shenzhen Institute of Information Technology,

Shenzhen, 518172, China

⁴School of Computer and Information Science, Southwest University,

Chongqing, 400715, China.

⁵Department of Mechanical Engineering, the Hong Kong Polytechnic University, Hong Kong

【Abstract】 Multi-objective portfolio optimization (PO) problem is always converted into a single objective problem by using the weighted method, which is sensitive to the pareto optimal front and requires that decision makers must have previous experience about the preference for weights. Based on multi-objective comprehensive learning bacterial foraging optimization (MOCLBFO), this paper proposes an algorithm which is specially designed for multi-objective PO problem. The corresponding coding strategy which considers each particle as a feasible solution is also given. In order to test the validity of the algorithm, multi-objective comprehensive learning particle swarm optimization (MOCLPSO) is chosen as the competing algorithm. Comparative experimental tests on ten assets PO problem demonstrate that MOCLBFO is able to find a more well-distributed Pareto set.

【Keywords】 Multi-objective problem, Comprehensive learning strategy, Bacterial foraging optimization, Portfolio optimization, Pareto solutions

1 Introduction

Portfolio optimization (PO) is the process of arranging the proportion of investment in different assets with the minimum risk to obtain the maximum profit. Many scholars have modified the original portfolio model proposed by Markowiz [1] by adding more constraints, such as no short sales [2,3], the transaction costs [3,4]. As the number of constraints increase, the difficulty to deal with such a multi-objective PO model is also increasing. As a result, traditional mathematical methods cannot solve it well. Therefore, scholars begin to use swarm intelligent algorithms, including particle swarm optimization (PSO) [5,6], bacterial foraging optimization (BFO) and its variants [3], and others.

Our proposed multi-objective comprehensive learning bacterial foraging optimization (MOCLBFO) [7] has been successfully applied to environmental/economic dispatch problem (EED) [7]. Based on MOCLBFO, this paper proposes a specific method for solving multi-objective PO model with two conflicting objectives and two constraints [8].

The rest of the article is organized as follows: Section 2 provides a brief description of BFO and MOCLBFO. Section 3 presents the multi-objective PO model and the related computational steps. The results and analyses of the experiment are shown and discussed in Section 4. Finally, the concluding remarks are provided in Section 5.

2 Multi-objective Comprehensive Learning Bacterial Foraging Optimization

2.1Bacterial Foraging Optimization

Original BFO tackles problem by using three operators, including chemotaxis, reproduction, and elimination/dispersal. For more detailed information, please refer to [9].

2.2 Multi-objective Comprehensive Learning Bacterial Foraging Optimization

Although BFO has been applied to many single objective optimization problems successfully, it cannot directly tackle multi-objective optimization problems (MOPs) with non-dominated solutions set instead of an absolute global best solution. Inspired by the comprehensive learning strategy used in MOCLPSO [10], it also was incorporated into MOBFO [11]and thus MOCLBFO [7] is put forward.

3 MOCLBFO for Portfolio Optimization Problem

3.1Portfolio Optimization Model

Markowiz [1] proposed mean-variance model with many assumptions. These strict assumptions made this PO model can’t get a good application in practical PO problems. In order to solve these problems, Li L.et.al [8] proposed an improved model, which has considered the transaction cost and no short selling and other factors. The expression of the new PO model is presented in Eq. (1-4) and the related parameters and definitions are shown in Table 1.

Table 1. Parameters and definitions of PO model

1.1 MOCLBFO for Portfolio Optimization Model

3.1.1 Encoding

When MOCLBFO is used to seek  solutions to the PO model, each particle is regarded as a potential feasible solution. Three kinds of information are carried by each bacterium, including the proportion of investment, the value of corresponding profit and risk [3].

Eq. (5) shows the coding of the bacteria. As is shown in Eq. (6), we sum up proportions of all asset firstly and then divide every proportion of asset by the so that the sum of all asset proportions is equal to 1. And the penalty function method is used to guarantee that each proportion is positive number.

3.1.2Four Key Mechanisms of MOCLBFO

Before describing computational steps of solving PO model, brief introduction about four key mechanisms of MOCLBFO [7] is given as follows.

3.1.2.1Health Evaluation

Every bacterium’s capacity to look for nutrients is different. The health index is calculated according to Eq. (7). The greater health index means that the bacterium is healthier, so greater probability would be given for this individual to reproduce. And the unhealthy bacteria would be given less probability to generate new offspring.

3.1.2.2Non-dominance Choice

In MOCLBFO, the pareto solutions which are obtained in the optimization process would be stored to external archive, but the size of the external archive is limited. Therefore, the maintenance and management of the external archive is very important. The process of non-dominance choice is the same as the mechanism in [7].

3.1.2.3Comprehensive Learning Mechanism

In this algorithm, every individual has capacity to learn from other bacterium or the bacterial group by dimensions and the pareto solutions are stored in the external archive. As Eq. (8) shows, the computational formula of the moving direction of bacterium at dimension consists of four parts, including the original direction, random direction, the best previous position, and the position of random bacterium in the external archive.

（8）

（9）

（10）

（11）

Among the equations above,, and are both known numbers. denotes the size of the bacteria and mean the random individuals in the group. denotes the probability of learning.

3.1.2.4 Constrained Boundary Control

Boundary control affects the validity of the border solutions and the diversity of bacteria species. Bacteria change direction randomly in the process of chemotaxis, which may let them exceed the prescribed area. If bacteria are allowed to leave the given area, the border solutions may not been obtained. Otherwise, the border solutions may be infeasible. The boundary control rules are shown in Eq. (12), and are the lower and upper boundaries. Besides, is a fixed value given beforehand.

(12)

3.1.3Computational Steps of MOCLBFO Algorithm for PO model

Based on the mechanisms mentioned above, an algorithm which is designed for solving PO model is proposed and the experiment is performed in MATLAB environment. MOCLPSO is chosen as the competing algorithm. The two algorithms have the same experimental settings, e.g., the population sizes are both set to 200.The pseudo-code of MOCLBFO for PO model is shown in Table 2.

3.2 Experimental Data

The data of 10 assets which are collected from the financial.sina.com.cn are chosen as the example. The detailed data are from the beginning of 2013 to the third quarter of 2016. After collecting the related data, we use MATLAB to calculate the mean value and the covariance matrix of return on net assets (RONA) of the related companies. The detailed data are presented as follows:

4 Experimental Results and Analyses

Table 3 shows the data obtained by MOCLBFO and MOCLPSO when the investment profit is the largest, including the investment proportion of the ten assets and the corresponding value of profit and risk. Similarly, Table 4 presents the corresponding results when the risk is the lowest. The best value is in italics. Fig.1 presents the pareto optimal front. Two conclusions can be drawn.

• MOCLBFO and MOCLPSO can both obtain Pareto set that make objective function satisfied. But much better performance of MOCLBFO can be observed. For example, MOCLBFO’s Pareto set has better diversity.

• The convergence curve with increasing trend indicates that the greater the investment profit, the greater the investment risk. It gives investors a revelation that they should allocate the proportion of assets based on their ability to bear the risk, rather than blindly pursuit high returns.

Fig. 1. Convergence curve of MOCLBFO and MOCLPSO

Table 3. The experimental results of the

5 Conclusions

In this paper, based on MOCLBFO, we propose a specific algorithm for solving a multi-objective PO model with two conflicting objectives and two constraints. In order to test the validity of MOCLBFO, MOCLPSO is chosen as the competing algorithm. Experimental tests on ten assets PO problem demonstrate that MOCLBFO is outstanding in addressing MOPs. In the future, attention should be placed on building a new PO model that considers more realistic constraints and studying more powerful MOBFO variants. In addition, we can also consider using MOCLBFO and another MOBFO variants to address other complicated MOPs.

Acknowledgment

This work is partially supported by The National Natural Science Foundation of China (Grants Nos. 71571120, 71271140, 61603310, 71471158, 71001072, 61472257), Natural Science Foundation of Guangdong Province (2016A030310074) and Shenzhen Science and Technology Plan (CXZZ20140418182638764), the Fundamental Research Funds for the Central Universities Nos. XDJK2014C082, XDJK2013B029, SWU114091.

【References】

1.Markowitz, H.: Portfolio selection. J. Finance. 7(1), 77–91 (1952)

2.Shi, N. Z., Lai, M., Zheng, S. R., Zhang, B. X.: Optimal algorithms and intuitive explanations for markowitz's portfolio selection model and sharpe's ratio with no short-selling. Science China Mathematics, 51(11), 2033(2008)

3.Niu, B., Bi, Y., Xie, T.: Structure-redesign-based bacterial foraging optimization for portfolio selection. In: Han, K., Gromiha, M., Huang, D.-S. (eds.) ICIC 2014. LNCS, vol.8590, pp. 424–430. Springer, Heidelberg (2014)

4.Davis, M. H. A., Norman, A. R.: Portfolio selection with transaction costs. Mathematics of Operations Research, 15(4), 676-713(1988)

5.Chen, W., Zhang, R.T, Yang, L.: Portfolio selection model based on the improved particle swarm optimization. Computer science, 36(1), 146-147 (2009)

6.Ayodele, A. A., Charles, K. A.: Improved constrained portfolio selection model using particle swarm optimization. Indian Journal of Science and Technology, 8(31) (2015)

7.Tan L.J., Wang H., Yang C*., Niu B*:. Natural Computing. Online (2017)

8.Li, L., Xue, B., Tan, L.J., Niu, B.: Improved particle swarm optimizers with application on constrained portfolio selection. In: Huang, D.-S., Zhao, Z., Bevilacqua, V., Figueroa, J.C. (eds.) ICIC 2010. LNCS, vol. 6215, pp. 579–586. Springer, Heidelberg (2010)

9.Passino, K.M.: Biomimicry of bacterial foraging for distributed optimization and control. IEEE Control Systems Magazine, 52–67 (2002)

10.Huang V.L., Suganthan P. N., Liang J.J.: Comprehensive learning particle swarm optimizer for solving multiobjective optimization problems. International Journal of Intelligent Systems, 21(2):209–226(2006)

11.Niu, B., Wang, H., Wang, J., Tan, L.J.: Multi-objective bacterial foraging optimization. Neurocomputing, 116(116), 336-345(2013）