Complexity

Volume 2018, Article ID 7032402, 8 pages

https://doi.org/10.1155/2018/7032402

## A Novel Decision-Making Approach to Fund Investments Based on Multigranulation Rough Set

School of Economics and Management, Jiangsu University of Science and Technology, Zhenjiang 212003, China

Correspondence should be addressed to Xima Yue; moc.621@euyamix

Received 11 July 2018; Accepted 2 September 2018; Published 14 October 2018

Guest Editor: Bernardo A. Furtado

Copyright © 2018 Xima Yue and Xiang Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Fund investment is a hot issue in today’s society. How to choose a project for investment is affected by many factors. In view of this problem, this paper starts from the granular computing point of view and combines the multigranulation rough set decision-making method to construct a fund investment decision information system; then, the fund investment decision information system is reduced under different thresholds, and the decision rules are extracted through reduction. And from the aspects of decision accuracy and rule accuracy, the rules are analyzed. Finally, decision rules are used to give the decision of the fund investment project. This study provides a new approach to fund management.

#### 1. Introduction

Fund investment management [1] is a hot issue in today’s society. How to choose a better one from some of the possible fund projects or to find some direction from the existing successful fund project before investment is a problem that every decision maker needs to think about. At present, many scholars have studied how to carry out project selection and malpractice from the pension fund industry and social insurance industry [2–6].

In order to enable investors to better invest in projects, many scholars have carried out many researches on investment methods in recent years. Lu [7] uses mathematical methods to analyze the major Reynolds index, SHARP index, Jansen index, and M2 index, which are mainly used in the performance evaluation of the fund, and deduces the relationship between each other by mathematical methods. It also expounds the conditions for the use of indicators in the performance evaluation of funds. Peng [8] systematically introduced the theories and methods of performance evaluation of open-end funds and summarized the research results of fund performance evaluation both at home and abroad. Then, according to the scientific, systematic, and feasible principles, we select 9 indicators that can reflect the performance level of the open-end fund and use the analytic hierarchy process to construct the system of the performance evaluation of the open-end fund in China. Then, we evaluated the index factors of sample funds in the sample period, respectively. Finally, combined with fuzzy comprehensive evaluation method and grey comprehensive evaluation method, we get the overall evaluation results of an open-end fund performance. From the quantitative and qualitative point of view, Xu [9] selected the VaR method combining the GARCH model and the fuzzy comprehensive evaluation method using the analytic hierarchy process to further construct the risk assessment system of China’s open fund.

Rough set theory [10, 11] is a mathematical method proposed by Professor Pawlak in 1982 to effectively analyze and deal with inaccurate, inconsistent, and incomplete information. After nearly thirty years of development, rough set theory has been widely used in the fields of pattern recognition, machine learning, decision analysis, knowledge acquisition, and data mining. The classical rough set is an equivalent class derived from a single undiscernible two-element relation on a domain. The following and upper approximation sets are used to rough the unknown concept. Granular computing is a new discipline with rapid development. It integrates many theoretical research achievements such as rough set, fuzzy set, and artificial intelligence. At present, rough set theory has become an important tool for granular computing. Literatures [12–14] from the angle of granular computing (parallel multiple grain structures), the knowledge particles (equivalent classes) derived from a single indiscernible two-element relation, are analyzed to approximate the inadequacies of the rough sets of the unknown concepts. Furthermore, the concept of multigranulation rough sets is proposed. An optimistic multigranulation rough set and a pessimistic multigranulation rough set model are given. The Pawlak rough set is extended from a single grain structure to a number of granular structures, and it is proved that the classic rough set model of Pawlak is a special case of the multigranularity rough set. In addition, many researchers have extended the multigranularity rough set. Xu et al. have done a lot of work on multigranulation rough sets [15–20]. Yang and others studied the multigranularity rough set [21–25] based on incomplete information system, test cost sensitivity, and hierarchical structure.

In order to solve the problem of project investment fund, this paper constructs the fund investment decision information system from the perspective of granular computing and combined with multigranular rough set decision method. Then reduce the information table, extract the rules from the simplified information table, and perform the rule analysis. Finally, the decision conclusions of the fund investment are given. The main contributions of this article are as follows: (1)This paper proposes a method for constructing a project fund investment decision information system, which is the premise that we use rough sets to solve project investment problems(2)In this paper, the generalized multigranularity rough set model is used to reduce the fund project information system, and then the rules are extracted on the simplified information system. Finally, the fund investment decision is given from these rules

The remaining structure of this paper is shown as follows: in the second part, the related knowledge of multigranularity rough sets is introduced. In the third part, the cleaning information system of the fund investment project is constructed. In the fourth part, the fund investment decision based on the multigranularity rough set is studied. Finally, the conclusion is given in the fifth part.

#### 2. Preliminaries

The section recalls necessary concepts and preliminaries required in the sequel of our work. Detailed description of the theory can be found in [12, 13, 26–29].

An information system with decisions is an ordered quadruple , where (i) is a nonempty finite set of objects(ii) is a nonempty finite attribute set(iii) denotes the set of condition attributes(iv) denotes the set of decision attributes and (v) is the value of on , the domain of (vi) is the value of on , the domain of

In an information system, the equivalence class of an object with respect to an attribute subset of is a granularity from the viewpoint of granular computing. A partition of the universe is a granular structure. Rough set proposed by Pawlak is a single granularity rough set model, and the granular structure in this model is induced by the indiscernibility relation of the attribute set. In general, the above cases cannot always be satisfied or required in practical problems. In the three cases referred in [12], there are limitations in single granularity rough set for addressing practical problems with multiple partitions, and multigranulation rough set can now be used to effectively solve these problems. Under those circumstances, we must describe a target concept through multiple binary relations on the universe according to user’s requirements or targets of problem solving. In the literatures [12, 13, 26, 27], to apply rough set theory to practical problems widely, multigranulation rough set model has been studied based on multiple equivalence relations.

Let be an information system, and . Then or is referred to as a granularity. The equivalence class of an object with respect to is defined as

The lower and upper approximation sets of with respect to single are defined as follows:

Considering further studies on multigranulation rough set, we now review the two basic forms of multigranulation rough set model.

*Definition 1 (see [12]). *Let be an information system, and , . The optimistic multigranulation lower and upper approximation sets of with respect to single are defined as follows:
where “” means the logical operator “or,” which represents that the alternative conditions are satisfied, and “” means the logical operator “and,” which represents that all of the conditions are satisfied.

The set is definable if and only if . Otherwise, is rough. and are referred to as optimistic lower and upper approximation sets, respectively.

From the above definition, the operators “” and “” can be exchanged between the optimistic lower approximation set and the optimistic upper approximation set. Corresponding to optimistic multigranulation rough set, pessimistic multigranulation rough set model can be defined in the following.

*Definition 2 (see [12]). *Let be an information system, and , . The optimistic multigranulation lower and upper approximation sets of with respect to single are defined as follows:
The set is definable when and only when . Otherwise, is rough. and are referred to as pessimistic lower and upper approximation sets, respectively.

The uncertainty of a concept in a multigranulation rough set model is also due to the existence of a boundary region. The greater the boundary of a concept is, the lower its accuracy is, and the coarser the concept is. Similar to the measures in the Pawlak rough set model, the accuracy and roughness measures in optimistic multigranulation rough set and pessimistic multigranulation rough set were defined in the same way [12]. As generalizations of the Pawlak rough set model, we only show the relations among optimistic multigranulation rough set, pessimistic multigranulation rough set, and single granularity rough set in the following.

Proposition 1 (see [12]). *Let be an information system, and , . The following properties hold:
*(1)*(2)**(3)**(4)**(5)**(6)**In addition, there are many related properties as well as proof, please refer to [12, 28].**In order to express generalized multigranulation rough sets, we first introduce a characteristic function, which is called support feature function.*

*Definition 3 (see [30]). *Let be an information system, and , , suppose
is called ’s support feature function for , which is used to describe the inclusion relation between equivalence class and concept , which indicates whether object accurately supports by .

The optimistic multigranular rough set and the pessimistic multigranular rough set are generalizations of two multigranular rough set models. We will propose a new multigranular rough set model with the parameter . We have introduced this parameter to implement the conceptual description of objects that support the concept at the most granular levels. Objects that may portray the concept are ignored below the corresponding level. The new model is expressed as follows.

*Definition 4 (see [30]). *Let be an information system, and , , is called ’s support feature function for . For any , the lower approximation and upper approximation of for are defined as follows:
The set is definable if ; otherwise, is a rough set. We denote this generalized multigranulation rough set model as GMGRS, and as the information level for .

*The multigranulation rough set is a generalization of the classical rough set. Since several attributes in the information system can have different effects on the decision-making effect, when these effects cannot be performed simultaneously, but separately and independently, we cannot use classical rough set theory to treat these attributes as a whole through an indistinguishable relationship for system reduction and rule extraction. Therefore, according to the general process of rough set decision, we can get the specific steps of multigranular rough set decision and provide a theoretical model for decision analysis of fund project investment in multigranular environment.*

*3. Construction of Decision Information System for Fund Investments*

*Fund has become an increasingly important source of financing for people. For a decision maker, one may need to adopt a better one from some possible fund projects or find some directions from existing successful fund projects before investing. How to do it? We will propose a novel decision-making fund investments based on multigranulation rough set. This section mainly focuses to build fund investment decision information system.*

*The flow chart of our multigranulation decision-making model for fund investment is shown in Figure 1.*