A soft set based approach for the decision-making problem with heterogeneous information

: This paper proposes the concept of a neighborhood soft set and its corresponding decision system, named neighborhood soft decision system to solve decision-making (DM) problems with heterogeneous information. Firstly, we present the deﬁnition of a neighborhood soft set by combining the concepts of a soft set and neighborhood space. In addition, some operations on neighborhood soft sets such as “restricted / relaxed AND” operations and the degree of dependency between two neighborhood soft sets are deﬁned. Furthermore, the neighborhood soft decision system and its parameter reduction, core attribute are also deﬁned. According to the core attribute, we can get decision rules and make the optimal decision. Finally, the algorithm of DM with heterogeneous information based on the neighborhood soft set is presented and applied in the medical diagnosis, and the comparison analysis with other DM methods is made.


Introduction
Various mathematical theories, such as fuzzy set theory [6,11], rough set theory [38,39], vague set theory [2,31], etc., have been proposed by researchers to deal with vagueness and uncertainty in practical problems in engineering, economics, social science, medical science and so on. However, Molodtsov [1] pointed out that the parameterization tools of the above aforementioned theories were inadequate due to inherent limitation. Then he instead proposed soft set theory, which is a new mathematical tool to deal with uncertain problems. It is free from the inherent limitations, and its parameterization tools are adequate to process uncertainties. Objects can be described based on soft recognition [9,19,20] and so on. By using some gathering rules in neighborhood space, objects can be classified into several groups according to their similarity and some neighborhood granules can be generated. More importantly, these gathering rules and neighborhood granules can be generated from a heterogeneous information environment. That means there is no limitation on the types of data in description on objects, or no need to transfer different types of data into the same in the process of DM. On the contrary, neighborhood space can process the heterogeneous information directly. Considering soft sets have the advantage of providing adequate parameterization tools and neighborhood space is capable of classifying objects with heterogeneous information, this paper proposes a new method to deal with DM problems which contain heterogeneous information by combining soft sets with neighborhood space.
The rest of this paper is organized as follows: Section 2 introduces the basic definitions of soft sets and neighborhood space. The concept of a neighborhood soft set which is a combination of the soft set and neighborhood space is presented in section 3. Besides, the operation rules on the neighborhood soft set are discussed in this section. In section 4, the definition of a neighborhood soft decision system is proposed and the method of DM under heterogeneous information environment based on the neighborhood soft decision system is proposed, followed by an illustrative example and a comparison analysis. Section 5 applies the new method in the medical diagnosis, and the last section discusses our main conclusions.

Soft sets
Definition 1. [1] Suppose that U is an initial universe set and A is a set of parameters. Let P(U) denote the set of all subsets of U, a pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P(U). (2.1) Clearly, a soft set is a mapping from parameters to P(U), and it is not a set, but a parameterized family of subsets of U. For e ∈ A, F(e) can be considered as the set of e-approximate elements of the soft set (F, A). In this case, to define a soft set means to point out cheap houses, beautiful houses, and so on. The soft set (F, A) describes the 'attractiveness of the houses' which Mr. X is going to buy. Thus, we can view the soft set (F, A) as a collection of approximations as below: in the green surroundings = {h 1 } .
The soft set of 'attractiveness of the houses' in Example 2.1 can also be tabulated as in Table 1.
x n } be an initial universe set, and A = {e 1 , e 2 , · · · , e m } be a set of parameters. ∀x i ∈ U and B ⊆ A, the neighborhood of x i in the subspace B denoted by δ B (x i ) is defined as: where δ is an arbitrary small nonnegative number and ∆ is a metric function which satisfies: is the value function of object x in the kth dimension.
From Definition 2, it is obvious that the family of neighborhood granules {δ B (x i )|x i ∈ U} forms an elemental granule system, which gathers similar objects from the universe set, rather than partitions off it into several mutual exclusive subsets.

The concept of a neighborhood soft set
Based on Definition 1 and Definition 2, we can combine the concepts of soft sets and neighborhood space to get the following definition of a neighborhood soft set.
Definition 3. Let U = {x 1 , x 2 , · · · , x n } be an initial universe set and A = {e 1 , e 2 , · · · , e m } be a set of parameters. A pair (F δ , A) is called a neighborhood soft set over U, where F δ is a mapping given by: where P δ (U) denotes the set of all neighborhoods of each object in U. For any x i ∈ U(i = 1, 2, · · · , n) and e k ∈ A(k = 1, 2, · · · , m), let δ e k (x i ) denotes the neighborhood of x i on e k and where j = 1, 2, · · · , n and δ is a threshold parameter which defines the range of the neighborhood δ e k (x i ). δ can be defined on any type of data but should be small enough. We can calculate the range according to the Eq (2.3), and for the purpose of simplicity, let P = 1, then where f (x i , e k ) is the value function of the object x i on the parameter e k . Obviously, a neighborhood soft set is also a special case of a soft set, because it is still a mapping from parameters to the universe. For e ∈ A, F δ (e) can be considered as the set of e-approximate elements of the neighborhood soft set (F δ , A). Unlike the other subtypes of soft sets, such as fuzzy soft sets, interval-valued fuzzy soft sets and so on, which express information by a uniform type of data (the discrete data 0 or 1, or the interval number between 0 and 1), neighborhood soft sets store information by neighborhood granules, which are determined by the threshold parameter δ and are the sets of approximate objects of x i on parameter e k .
According to the Definition 3, there is no restriction on the data types of the threshold parameter δ. It can be any form if it can weigh the distance between two objects. We can define it according to the actual situation. For example, for numerical data, δ can be an arbitrarily small nonnegative number, which denotes the maximum acceptable difference between two objects; for semantic data, δ can be a word/sentence, which can identify the difference of objects precisely. Therefore, neighborhood soft sets can process the heterogeneous information in DM problems. The following example can make this easier to understand. Example 2. Let U = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } be an initial universe set which represents a set of six houses, and A = {e 1 , e 2 , e 3 , e 4 } be a parameter set which describes the status of the houses. Specifically, e 1 denotes the area of house; e 2 denotes the appearance of house; e 3 describes the public transportation of house; and e 4 represents the price of house. The data are listed in Table 2: x 2 83 ordinary 7 11 x 3 75 beautiful 8 12.5 x 4 69 ordinary 3 9 x 5 91 beautiful 7 12 x 6 155 beautiful 0 15 From Table 2 we can see that the types of data which describe the six houses are different: e 1 and e 4 are continuous variables, e 2 is a semantic variable, e 3 is a discrete variable. For the four variables we specify the following threshold parameters δ respectively: For e 1 , let δ 1 = 10m 2 . According to the definition of neighborhood soft sets, it means if the difference between the area of two houses x i and x j is not bigger than 10 m 2 , then they can be regarded as similar area, and x j should be in the neighborhood of x i . Analogously, we can define that δ 2 = "the same", δ 3 = 2 bus/train routes, and δ 4 =1.5 thousand yuan/m 2 .
Then according to Definition 3, a neighborhood soft set (F δ , A) can be used to describe the six houses on the shortlist, the mapping from parameters to the universe in (F δ , A) is given as follows: where: In Example 2, the raw data which describes the houses can be processed directly by the neighborhood soft set defined by this paper. It is obvious that through setting the value of δ carefully, the new mapping method under the framework of neighborhood soft sets is capable of processing various types of data, and there is no need to transform the different types of data into the same. Therefore, neighborhood soft sets can provide a holistic approach to process heterogeneous information directly and precisely. It can classify the universe U into several categories through finding the neighborhood of each object. Figure 1 illustrates an example of classification based on two parameters in the neighborhood soft set (F δ , A): e 1 and e 3 , which is a simplified version of Example 2. Consider x 3 , its neighborhood on parameter e 1 with δ 1 can be defined by the space between parallel dashed lines R 1 and R 2 , so x 2 , x 4 , and of course including x 3 itself, are in the neighborhood of x 3 based on e 1 . That means the houses x 2 , x 3 and x 4 in U can be treated as similar to x 3 in terms of the area of house; similarly, x 2 , x 5 and also x 3 itself are in the neighborhood of x 3 based on e 3 with δ 3 . If both e 1 and e 3 are considered, then only x 2 and x 3 are in the neighborhood of x 3 . Based on the definition of neighborhood soft sets, the following properties of neighborhood soft sets can be derived: ii. Symmetric: can be given to prove it: Definition 4. Let U be an initial universe set and E be a set of parameters. Suppose that A, B ⊆ E, (F δ , A), and (G δ , B) are two neighborhood soft sets, we say that is a neighborhood soft subset of (F δ , A).

Example 3. Given two neighborhood soft sets
Here U is the set of houses on the list.
Definition 6. Let U be an initial universe set, E be a set of parameters, and A, B ⊆ E. (F δ , A) is called a null neighborhood soft set (with respect to the parameter set A), denoted by ∅ A δ , if F δ (e) = ∅ for all e ∈ A; (G δ , B) is called a whole neighborhood soft set (with respect to the parameter set B), denoted However, because of the reflexive in neighborhood soft sets as proved above, i.e. x i ∈ F δ (e) x i . So F δ (e) ∅, the null neighborhood soft set does not exist.

Restricted/relaxed AND on a neighborhood soft set and a subset
Definition 7. Let U = {x 1 , x 2 , · · · , x n } be an initial universe set, (F δ , A) be a neighborhood soft set defined on U and X be a subset of U. The operation of "(F δ , A) restricted AND X" denoted by (F δ , A)∧ δ X is given by: } be the set of houses which are preferred by most consumers and are the bestselling houses on the market. Then From the above definition, we have: Proof. For ∀e ∈ A, and x i ∈ U, according to the reflexive property: x n } be an initial universe set, (F δ , A) be a neighborhood soft set defined on U and X be a subset of U. The operation of "(F δ , A) relaxed AND X" denoted by (F δ , A)∧ δ X is given by: Example 5. Continue Example 4: Similarly, from the above definition, we have: Proof. Straightforward.

The degree of dependency between two neighborhood soft sets
To explore the ability of classification of neighborhood soft sets, we give the definition of the degree of dependency between two neighborhood soft sets.
Definition 9. Suppose that (F δ , A) and (G δ , B) are two neighborhood soft sets over U, where A∩B = ∅. We say (F δ , A) has a 'k degree of dependency' on (G δ , B) denoted by k (F δ , A), (G δ , B) and where | * | denotes the number of elements in a set. Apparently, k (F δ , A), (G δ , B) is the ratio of the number of elements in two sets, one is the result of (F δ , A) restricted AND G δ (ε j )/x i , the other is U, and: i. k(0 ≤ k ≤ 1), we say (F δ , A) is partially depended on (G δ , B), which measures the degree of approximation in classification between two neighborhood soft sets.
ii. If k = 1 we say (F δ , A) is completely depended on (G δ , B), which means the results of classification by the two neighborhood soft sets are exactly the same.
iii. If k = 0 we say (F δ , A) is not depended on (G δ , B), which means the results of classification by the two neighborhood soft sets are completely different.
Through the definition of the degree of dependency between two neighborhood soft sets, we can compare the similarity of classification results between two neighborhood soft sets.
Example 6. Let (F δ , A) and (G δ , B) be two neighborhood soft sets, and

The concept of a neighborhood soft decision system
To store the heterogeneous information in DM problems and develop a DM method under a heterogeneous information environment based on neighborhood soft sets, it is necessary to introduce neighborhood soft decision systems.
Definition 10. Suppose that (F δ , A) and (G δ , B) are two neighborhood soft sets over a common universe U, where A ∩ B = ∅. Then the information system U, (F δ , A), (G δ , B) is called a neighborhood soft decision system over the common universe U, where A is a condition attributes set, B is a decision attributes set, and (F δ , A) is a condition neighborhood soft set, (G δ , B) is a decision neighborhood soft set.
In a neighborhood soft decision system U, (F δ , A), (G δ , B) , the condition attributes set A is used for describing characteristics of each alternative object. The decision attributes set B is used for describing the rating of each alternative object.
Then, the degree of dependency of (U, (F δ , A), (G δ , B)) is defined as the degree of dependency between (F δ , A) and (G δ , B), and is denoted by k U, (F δ , A), (G δ , B) (0 ≤ k ≤ 1). It provides a measure of the similarity of classification results between the condition neighborhood soft set (F δ , A) and the decision neighborhood soft set (G δ , B).
Example 7. Reconsider Example 2. Let A = {e 1 , e 2 , e 3 , e 4 } be a condition attribute set, (F δ , A) is the corresponding condition neighborhood soft set. Table 3 listed the rating of sold houses. According to Definition 3, let δ = 4%, which means the two houses are in the same class if their difference of rating is no more than 4%, we get another neighborhood soft set (G δ , B), where B = {ε 1 } is the decision attribute set. The two neighborhood soft sets (F δ , A) and (G δ , B) are given as follows: Apparently according to Definition 10, U, (F δ , A), (G δ , B) can be called a neighborhood soft decision system, and its degree of dependency is:

The reduction of a neighborhood soft decision system
To explore which variables in the condition neighborhood soft set are decisive for DM, we give the definition of the reduction of a neighborhood soft decision system based on the degree of dependency of a neighborhood soft decision system. Definition 11. Let (U, (F δ , A), (G δ , B)) be a neighborhood soft decision system, C ⊆ A, we say attribute subset C is a reduction of (U, (F δ , A), (G δ , B)) if: i. k (U, (F δ , C), (G δ , B)) = k (U, (F δ , A), (G δ , B)); ii. ∀e ∈ C, k (U, (F δ , C), (G δ , B)) >k (U, (F δ , (C − e)), (G δ , B)).
Obviously, according to Definition 4, (F δ , C) is a neighborhood soft subset of (F δ , A) since C ⊆ A. Through Definition 11, we can compare the degree of dependency between each neighborhood soft subset of (F δ , A) and (G δ , B), and identify the subset C which not only contains the minimum number of variables (the key attributes), but also produces the same degree of dependency between (F δ , C) and (G δ , B) as that between (F δ , A) and (G δ , B).
Example 8. Consider the neighborhood soft decision system in Example 7. Our aim is to figure out which attributes in the condition attribute set A are the major factors in deciding which house is the most favorite house for Mr. X. The burden to make a good choice can be reduced by getting rid of unnecessary information or attributes. Therefore, the reduction of (U, (F δ , A), Moreover, subset C = {e 1 } can not be reduced anymore, so it is a reduction of U, (F δ , A), (G δ , B) . That is to say, the area of house is the major factor in determining which house is the best for Mr. X.
It should be noted that C = {e 1 } is the sole reduction of U, (F δ , A), (G δ , B) , because the k between the other minimal neighborhood soft subsets of (F δ , A) and (G δ , B) is not equal to k (F δ , A), (G δ , B) .
Based on the definition of the reduction of a neighborhood soft decision system, we can define the core attribute of a neighborhood soft decision system and the core attribute set of a neighborhood soft decision system as follows: Definition 12. The attribute e(e ∈ A) is a core of a neighborhood soft decision system if it belongs to every reduction of a neighborhood soft decision system. Definition 13. The attribute set C(C ⊆ A) is a core attribute set of a neighborhood soft decision system, if all of the elements in C are the core attributes. Example 9. In Example 8, the attribute e 1 belongs to the unique (every) reduction of k (F δ , A), (G δ , B) , so e 1 is a core of it.
In addition, in the attribute set C = {e 1 }, all of its elements (e 1 ) is a core of k (F δ , A), (G δ , B) , so C is a core attribute set of it.

Classification rules of a neighborhood soft decision system
To make a decision based on the neighborhood soft decision system, we need to classify objects in the first place. The follows are classification rules based on the neighborhood soft decision system: In a neighborhood soft decision system, if F δ (e)/x i ∩ F δ (e)/x j = ∅, then x i and x j are in different groups, otherwise they are in the same group.
For the condition attribute e 1 , we have: Therefore, for the condition attribute e 1 , the two objects x 1 and x 2 are not the same. From the raw data on e 1 , we can regard x 1 as a medium area house, and x 2 as a small area house. For the decision attribute ε 1 , we have: Therefore, for the decision attribute ε 1 , the two objects x 1 and x 2 are the same. From the raw data on ε 1 , we can regard x 1 and x 2 as two good-selling houses.

Decision rules of a neighborhood soft decision system
Based on the above classification, we can get the decision rules of the neighborhood soft decision system. Through classification rules, the alternatives in a neighborhood soft decision system can be classified into several categories, each category can be seen as a decision rule, that means we can make a decision according to the relationship between the condition attributes and its decision attributes of each category. Therefore, a neighborhood soft decision system is a set of rules actually.
Example 11. In Example 10, there are two categories for condition attribute e 1 in (U, (F δ , A), (G δ , B)), so we can get two decision rules: (1) If the area of house is like x 1 (medium area house), then its rating is like x 1 and x 2 (good-selling house); (2) If the area of house is like x 2 (small area house), then its rating is like x 1 and x 2 (good-selling house).
That is to say, if a house is a medium area house or a small area house, then it is a good-selling house. However, we cannot decide whether a big area house is a good-selling house or a bad-selling house based on the above decision rules, and the final decision rules may be different when all objects are considered.

The algorithm of DM based on neighborhood soft sets
This section attempts to demonstrate how to apply the newly developed neighborhood soft sets to DM problems with heterogeneous information.
Based on the above definitions about the neighborhood soft set and the neighborhood soft decision system, the algorithm of DM under heterogeneous information environment is given as follows. Figure 2 shows the flow of the new algorithm to get the classification rules and decision rules, and then get the optimal decision.
In step 2, according to the Eq (3.5), the degree of dependency of U, (F δ , A), (G δ , B) is given by In step 3, the degree of dependency between the neighborhood soft subset (F δ , e i ) and (G δ , B) in Example 7 is given by: e 4 ), (G δ , B) = 1/3.
Then according to Definition 11, we can conclude that attribute e 1 is the reduction of U, (F δ , A), (G δ , B) , because k 1 = k (F δ , e 1 ), (G δ , B) = k (F δ , A), (G δ , B) = 2/3. That means the rating of sold houses is mainly determined by e 1 , i.e. the area of a house.
In step 4, for the reduction attribute e 1 : We can classify objects according to the reduction attribute e 1 , as demonstrated in Table 4: Table 4. Classification according to e 1 .
Then we get the classification result: {{x 1 }, {x 2 , x 3 , x 4 , x 5 }, {x 6 }}, which means the objects were categorized into three classes by condition attribute e 1 : medium area house x 1 , small area houses x 2 , x 3 , x 4 , x 5 , and large area house x 6 .
For the decision attribute ε 1 : We can classify objects according to the decision attribute ε 1 , as shown in Table 5: Table 5. Classification according to ε 1 .
And the final decision rules can be gotten as follows: 1) If the area of a house is medium, then it is a best-selling house; 2) If the area of a house is small, then it may be a best-selling house or may not; 3) If the area of a house is large, then it is a best-selling house. In step 5, according to the above rules, the optimal choice(s) for Mr. X is to buy a medium area house or a large area house, depends on his budgets, because both of them are good-selling houses.

Comparison analysis
DM methods based on soft sets have been analyzed by many researchers [4,5,17,36]. The common feature of these methods, when dealing with heterogeneous information, is to homogenize the variables or attributes. That means different types of data have to be transformed into the same one before applying other DM methods based on soft sets. Take the research of Maji [17] for example, one of the classical works in this field, Maji [17] presented the rough mathematics soft sets (RMSS) method of decision-making by combining soft sets with rough mathematics of Pawlak [38]. This section compares RMSS with the neighborhood soft sets (NSS) we proposed in this paper to the problem of choosing the best house in Example 2.

Results from RMSS
The first step of RMSS is to construct a soft set (F, A) according to the raw data set. However, the raw data in Example 2 contains various types of variables which can not be described by soft set theory. In order to implement RMSS, all of the variables should be transformed into binary variables with values of "0" and "1", where "1" indicates "cheap", "beautiful", "big", and "good location" respectively, and "0" indicates otherwise as in table 6. Then the soft set (F, A) is given as follows according to the mapping F from A to U, which consists of the sets of "cheap houses", "beautiful houses", "big houses", and "good location houses": In (F, A), F(e 1 ) = {cheap houses} = {x 1 , x 6 }, for example, is a subset in U, in which the houses x 1 and x 6 are cheap in price, while the houses x 2 , x 3 , x 4 and x 5 are not cheap in price. F(e 2 ), F(e 3 ) and F(e 4 ) are defined in the same manner.
The choice value c i of each house is given by: where i = 1, · · · , n, j = 1, · · · , m, h i j is the value of the house x i on the attribute e j . Table 6 is the tabulation of (F, A) with the choice values presented in the last column. The optimal choices are the houses with the largest choice value, i.e. x 1 , x 3 , x 5 and x 6 . Table 6. Tabular representation of (F, A).

Comparison
From the results of RMSS and NSS, the differences between them can be summarized as follows: Firstly, decision-making with the NSS method is much simpler than with the RMSS method. The former has the capacity to reduce redundant parameters, so decisions can be made based on less but essential parameters and the burden of decision-making is lessened. In RMSS, however, the final decision had to be made based on all attributes. In the house buying example, four houses were determined as the best houses in both methods. With limited budgets, consumers may have to choose one from the four best alternatives. The NSS method simplified the core condition attributes to only one: The area of the houses. Therefore, the final decision can be derived relying on this core attribute and her/his budget. But in the RMSS method, consumers still need to go through all the attributes of the best alternatives to determine her/his final choice. Therefore, the difficulties of making decision with RMSS method were not fundamentally reduced. Secondly, in RMSS method, serious loss of information was obvious. For example, the houses were classified into big and non-big roughly by the area of them. However, the difference of area between houses within one class could be significant for decision-making. Therefore, the RMSS method based on the binary variables may drop a lot of valuable information during transferring. In the NSS method, heterogeneous information in decisionmaking problems can be integrated straightway, and information losing or distortion can be prevented. Thirdly, the RMSS method did not make full use of the decision values in the process of decisionmaking, and there was no connection between the conditional attributes and the decision attributes. A single decision rule was generated by ranking the objectives according to their choice values, the objectives with the maximum choice value were selected as the optimum. On the contrary, the NSS method can generate decision rules by making a connection between the conditional parameters and decision parameters, and the decision can be made based on multiple decision rules, which is more suitable for the actual decision-making environment because of its variety and convenience.

Application in the medical diagnosis
In this section, we use the newly developed method to facilitate the diagnosis of heart disease. We got a dataset of 297 patients from UCI machine learning repository, which contains 14 parameters for their heart disease diagnosis. Among all of the parameters, the classes of heart disease was regarded as the decision attribute, and the other 13 variables were regarded as the condition attributes for describing the patients. This dataset was chosen because it contains various data types, including binary variables, categorical variables, discrete variables and continuous numerical variables. So the medical diagnosis of heart disease is a practical decision-making problem under a heterogeneous information environment, which can be solved by the NSS method proposed by this paper.
We randomly selected 267 observations for training the NSS algorithm, and used the rest 30 observations for testing the predictive accuracy of the trained decision rules. We also applied the RMSS method in the testing dataset to predict the patients' type of heart disease. The prediction errors of the two methods are shown in Table 7, and it is obvious that our method outperforms RMSS. Moreover, in the process of diagnosis, even though there were more steps in our algorithm compared with Maji's, decision-making based on our algorithm relies on fewer parameters (derived from parameter reduction) and more precise decision rules (obtained from the connection between the conditional attributes and the decision attributes), which can make the decision-making simpler and more efficient. But in Maji's method, there is no parameter reduction involved, and no decision rules specifically defined. As a result, decision-making and prediction using Maji's method is not efficient and less accurate.

Conclusions
Based on the research of Molodtsov [1], this paper proposed the concepts of neighborhood soft sets and the neighborhood soft decision system. After that, a new decision-making method under heterogeneous information environment was presented. This method not only can provide adequate parameterization tools inherited from the properties of soft sets, but also can integrate heterogeneous information directly with the help of neighborhood space. More accurate decision can be derived based on the new method, because information losing or distortion caused by transformation of information can be avoided. An example of choosing the best houses was used to demonstrate the operation of the newly developed decision-making method. With the same example, we also compared our method with the decision-making method of Maji [17]. The results showed not only the advantage of our method to process heterogeneous information, but also its capacity to develop concise and effective decision rules. Moreover, the decision-making method proposed by this paper can be applied to a wide range of areas such as feature selection, evaluation and forecasting problems with heterogeneous information.