Abstract
Combining rough sets and fuzzy soft sets, we propose an algorithm to obtain the optimal decision program. In this algorithm, firstly, according to fuzzy soft sets, we build up information systems; secondly, we compute the significance of each parameter according to rough set theory; thirdly, combining subjective bias, we give an algorithm to obtain the comprehensive weight of each parameter; at last, we put forward a method to choose the optimal program. Example shows that the optimal algorithm is effective and rational.
1. Introduction
The real world is full of uncertainty, imprecision, and vagueness. Actually most of the concepts we meet in everyday life are vague than precise. So many authors have become interested in modeling vagueness recently. Traditional tools are not always successful to solve these problems. While a wide range of existing theories such as probability theory, fuzzy set theory, rough set theory [1, 2], vague set theory [3], and interval mathematics [4] are well known and often useful mathematical approaches to model vagueness, each of these theories has its inherent difficulties. In 1999, Molodtsov initiated soft set theory as a new mathematical tool for dealing with uncertainties, that is, free from the difficulties affecting existing methods [5]. This theory has proven useful in many different fields such as decision making [6, 7], data analysis [8], and forecasting.
The soft set model can be combined with other mathematical models. Maji et al. presented the concept of fuzzy soft set [9, 10] which is based on a combination of the fuzzy set and soft set models. By combining the interval-valued fuzzy set and soft set, Yang et al. introduced the concept of the interval-valued fuzzy soft set [11]. In [12], the authors applied the notion of soft sets to the theory of BCK/BCI-algebras. Feng et al. established an interesting connection between rough sets and soft sets [13].
Based on [9, 10], and combining rough set theory, in this paper, we propose an algorithm to obtain the optimal decision program.
2. Rough Sets and Fuzzy Soft Sets
In this section, we recall the basic notions of rough sets, soft sets, and fuzzy soft sets.
An information system is a pair , where is a nonempty, finite set called the universe and is a nonempty, finite set of primitive attributes. Every primitive attribute is a total function , where is the set of values of , called the domain of .
With every subset of attributes , we associate a binary relation , called an indiscernibility relation, and defined thus , where denotes the value of the object with respect to attribute .
Obviously is an equivalence relation and . If , then and are indiscernible by . The partition generated by is denoted by , which is further abbreviated as .
Let be an information system and let ; then [2],(1)knowledge depends on knowledge if and only if ind ind, denoted as ;(2)knowledge and are equivalent, denoted as , if and only if and ;(3)knowledge and are independent, denoted as , if and only if neither nor holds.
Obviously , if and only if ind ind.
Let be an information system and let ; we say that is dispensable in if ind ind; otherwise is indispensable in information system . The information system is independent if each is indispensable; otherwise the information system is dependent. is a reduction of if is independent and ind . The set of all indispensable relations in will be called the core of and will be denoted as core.
The following conditions are equivalent [1]:(1);(2)ind ind (;(3), where , ;(4) for all .
The above properties demonstrate that if depends on then knowledge is superfluous within the information system, in the sense that the knowledge and provide the same characterization of objects.
Let be an initial universe of objects and (simply denoted by ) the set of parameters in relation to objects in . Parameters are often attributes, characteristics, or properties of objects. Let denote the power set of and . Following [8, 10], the concept of soft sets is defined as follows.
Definition 1 (see [5]). A pair is called a soft set over , where is a mapping given by .
By Definition 1, a soft set over the universe can be regarded as a parameterized family of subsets of the universe , which gives an approximate (soft) description of the objects in . As pointed in [5], for any parameter , the subset may be considered as the set of -approximate elements in the soft set . It is worth noting that may be arbitrary: some of them may be empty, and some may have nonempty intersection. For illustration, Molodtsov considered several examples in [5]. Similar examples were also discussed in [7, 8].
Maji et al. [9] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced in [9] the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of (crisp) soft sets.
Definition 2 (see [10]). Let be the set of all fuzzy subsets in a universe . Let be a set of parameters and . A pair is called a fuzzy soft set over , where is a mapping given by .
In the above definition, fuzzy subsets in the universe are used as substitutes for the crisp subsets of . Hence it is easy to see that every soft set may be considered as a fuzzy soft set. Generally speaking, is a fuzzy subset in and it is called the fuzzy approximate value set of the parameter . Following the standard notations, can be written as .
It is well known that the notion of fuzzy sets provides a convenient tool for representing vague concepts by allowing partial memberships. In the definition of a fuzzy soft set, fuzzy subsets are used as substitutes for the crisp subsets. Hence every soft set may be considered as a fuzzy soft set. In addition, by analogy with soft sets, one easily sees that every fuzzy soft set can be viewed as an information system and be represented by a data table with entries belonging to the unit interval . For illustration, we consider the following example.
Example 3 (see [10]). Suppose that there are six houses in the universe , , , , , and the set of parameters is given by , where stand for “beautiful,” “modern,” “cheap,” “in good repair,” “wooden," and “in green surroundings,” respectively. Let be consisting of the parameters that Mr. X is interested in buying a house. This means that out of the available houses in Mr. X wants to buy the house which is qualified with the attributes in to the utmost extent. Now all the available information on houses under consideration can be formulated as a fuzzy soft set describing “attractiveness of houses” that Mr. X is going to buy. Table 1 gives the tabular representation of the fuzzy soft set . We can view the fuzzy soft set as the collection of the following fuzzy approximations:
3. The Comprehensive Weight of Each Parameter under Threshold
3.1. Ascertain the -Level
Give a fuzzy soft set over , we use the following algorithm to ascertain the -level.
Step 1. Turn fuzzy soft set into an information system, such as Table 1.
Step 2. Build up fuzzy similarity matrix . We use formula to compute the similarity degree of objects and . Let ; obviously, it is a fuzzy similarity matrix.
Step 3. Turn into a fuzzy equivalence matrix ; that is, we compute the transitive closure of .
Step 4. Choose the optimal classification threshold to clustering.
In an equivalence matrix , we sorted the elements in from big to small; that is, ; let , where and are the number of objects in the th and clustering, respectively. If , then we choose as the best threshold [14].
3.2. The Objective Weight of Each Parameter under Threshold
In the following, combining rough set theory and fuzzy clustering technic, we put forward an algorithm to get objective weight of each parameter from Table 1.
Input a fuzzy soft set over ; are the parameter sets; the processes of parameter weight algorithm are as such.
Step 1. Choose the optimal threshold according to Section 3.1.
Step 2. Classify the objects under the optimal threshold , and we regard them as the classes of .
Step 3. Delete each from ; in the same way, we can obtain the classes under ; we regard them as the classes of if ; then is indispensable in , and the significance of is
Step 4. Normalize the significance of each parameter; we get , where .
3.3. The Comprehensive Weight of Each Parameter
In the application of actual problems, different customers have different requests; that is, different customers have different subjective weights of parameters. Suppose there are parameters, and , are the subjective and objective weights of the th parameter, respectively. We denote ; , where , , and .
Suppose the comprehensive weight of the th parameter is ; we denote , where , . In order to take into account the bias of not only subjective but also objective information, we can build up the following decision model [15]: where is the bias coefficient; it reflects the bias degree of customers to subjective and objective.
We do a Lagrange function based on the above formula:
Let
Solve the above equations; we can obtain .
4. Algorithm for Selection of the Optimal Program
At present there are many methods to handle decision making problems in fuzzy soft sets. In [10] the decision depends on the score , where signifies the number of parameters of relatively larger membership value of object . In [16] the decision depends on choice value , where signifies the sum of the membership values of all parameters of object . The decision that results is not always the same according to the two methods. In both of these algorithms, the authors did not think over the significance of each parameter, whereas in the actual problems, different parameters have different significance, and the significance of parameters will affect the result of decision. In this paper, combining rough sets and fuzzy soft sets, we put up with a weighted comprehensive algorithm to choose the optimal program.(1)Input the fuzzy soft sets over .(2)Compute the significance of each parameter .(3)Compute the score of , , where .(4)The decision is if .(5)If has more than one value, then any one of may be chosen.
5. Example
In the following, we use an example to account for the effectiveness of this optimal decision algorithm. If there are eight invest programs, which denote program 1, program 2, program 3, program 4, program 5, program 6, program 7, program 8, each program is described by ten factor indexes, denoted by payback period of investment), estimate of sales), (overall labor productivity), (expected output tax rate), (expected output of per kilowatt), (expected transportation amount per ton kilometer), (expected annual foreign exchange volume), (level of employment), (technical level), and (benefit of bioenvironment)} [14]; according to the experts, we have
Step 1. Turn fuzzy soft set into an information system (Table 2).
According to the above fuzzy soft set, we can get the following information system.
Step 2. Compute the similarity degree of objects and according to ; thus we obtain the similarity degree matrix .
Through computing, we obtain as follows:
Step 3. Compute the transitive closure of according to [14].
Through computing, we obtain an equivalence relation as follows:
Step 4. Ascertain the optimal threshold.
According to the fuzzy equivalence matrix , we choose the optimal threshold; since there is no meaning each object is in a class or all the objects are in a class; we do not think over the rate of change at this time; through computing, we have the following rate of change:
Since , so we choose as the threshold of classification. And the classes are as follows: , , , , , , , and .
Step 5. Compute the objective weight of each parameter under .
Delete from , under the threshold = ; in the same way, we can get , , , , , , , and ; the significance of parameter is
In the same way, we can get
Normalize the significance of each parameter; we can get the objective weight of each parameter; through computing, we have , , , , , , , , , and .
Step 6. Compute the comprehensive weight of each parameter.
Suppose that the subjective weight of each parameter is equal; that is, , , , , , , , , , and , and the bias coefficient of some customers is ; according to the formula , we can get the comprehensive weight of each parameter; through computing, we have , , , , , , , , , and .
Step 7. Choose the optimal program.
According to formula , through computing, we have , , , , , , , and . Since , we choose program 1 as the optimal program.
Since different decision makers have different preferences, that is, different decision makers have different subjective weights, so, they will have different choices according to the above algorithm.
6. Conclusion
Combining rough sets with fuzzy soft set theory, in this paper, we give an algorithm to choose the optimal program. During this algorithm, we think over not only the objective weight but also the subjective bias; this optimal program selection algorithm is effective and rational.
Conflict of Interests
The authors declared that they have no conflict of interests to this work.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11371130 and no. 11401052) and the Science and Technology Program of Hunan of China (no. 2013FJ4037).