Hindawi Publishing Corporation ξ e Scientiο¬c World Journal Volume 2014, Article ID 514369, 6 pages http://dx.doi.org/10.1155/2014/514369
Research Article An Algorithm to Select the Optimal Program Based on Rough Sets and Fuzzy Soft Sets Liu Wenjun1,2 and Li Qingguo1 1 2
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China School of Mathematics and Computer Science, Changsha University of Science and Technology, Changsha, Hunan 410076, China
Correspondence should be addressed to Liu Wenjun;
[email protected] Received 30 June 2014; Accepted 28 July 2014; Published 28 August 2014 Academic Editor: Yunqiang Yin Copyright Β© 2014 L. Wenjun and L. Qingguo. 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. 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 intervalvalued 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 π = (π, π΄), where π is a nonempty, finite set called the universe and π΄ is a nonempty, finite set of primitive attributes. Every primitive attribute π β π΄ is a total function π : π β ππ , where ππ is the set of values of π, called the domain of π. With every subset of attributes π΅ β π΄, we associate a binary relation ind(π΅), called an indiscernibility relation, and defined thus ind(π΅) = {(π₯, π¦) β π2 : for every π β π΅, π(π₯) = π(π¦)}, where π(π₯) denotes the value of the object π₯ with respect to attribute π. Obviously ind(π΅) is an equivalence relation and ind(π΅) = βπβπ΅ ind(π). If (π₯, π¦) β ind(π΅), then π₯ and π¦ are indiscernible by π΅. The partition generated by ind(π΅) is denoted by π/ind(π΅), which is further abbreviated as π/π΅.
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π (program) π’1 π’2 .. . π’π
π1 πΉ(π1 )(π’1 ) πΉ(π1 )(π’2 ) .. . πΉ(π1 )(π’π )
Let π = (π, π΄) be an information system and let π΅, πΆ β π΄; then [2], (1) knowledge π΅ depends on knowledge πΆ if and only if ind(πΆ) β ind(π΅), denoted as πΆ β π΅; (2) knowledge π΅ and πΆ are equivalent, denoted as π΅ β
πΆ, if and only if πΆ β π΅ and π΅ β πΆ; (3) knowledge π΅ and πΆ are independent, denoted as π΅ β πΆ, if and only if neither πΆ β π΅ nor π΅ β πΆ holds. Obviously πΆ β
π΅, if and only if ind(πΆ) = ind(π΅). Let π = (π, π΄) be an information system and let π
β π΄; we say that π
is dispensable in π if ind(π΄) = ind(π΄ β {π
}); otherwise π
is indispensable in information system π. The information system π = (π, π΄) is independent if each π
β π΄ is indispensable; otherwise the information system π = (π, π΄) is dependent. π΅ β π΄ is a reduction of π΄ if π΅ is independent and ind(π΅) = ind(π΄). The set of all indispensable relations in π΄ will be called the core of π΄ and will be denoted as core(π΄). The following conditions are equivalent [1]: (1) πΆ β π΅; (2) ind (πΆ βͺ π΅) = ind (πΆ); (3) PosπΆ(π΅) = π, where PosπΆ(π΅) = βπβπ/π΅ πΆπ, πΆπ = βͺ{π β π/πΆ | π β π}; (4) πΆπ = π for all π β π/π΅. The above properties demonstrate that if π΅ depends on πΆ then knowledge π΅ is superfluous within the information system, in the sense that the knowledge πΆ βͺ π΅ and πΆ provide the same characterization of objects. Let π be an initial universe of objects and πΈπ (simply denoted by πΈ) the set of parameters in relation to objects in π. Parameters are often attributes, characteristics, or properties of objects. Let π(π) denote the power set of π and π΄ β πΈ. Following [8, 10], the concept of soft sets is defined as follows. Definition 1 (see [5]). A pair (πΉ, π΄) is called a soft set over π, where πΉ is a mapping given by πΉ : π΄ β π(π). By Definition 1, a soft set (πΉ, π΄) over the universe π can be regarded as a parameterized family of subsets of the universe π, which gives an approximate (soft) description of the objects in π. As pointed in [5], for any parameter π β π΄, the subset πΉ(π) β π may be considered as the set of π-approximate elements in the soft set (πΉ, π΄). It is worth noting that πΉ(π) 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].
π2 πΉ(π2 )(π’1 ) πΉ(π2 )(π’2 ) .. . πΉ(π2 )(π’π )
β
β
β
β
β
β
β
β
β
.. . β
β
β
ππ πΉ(ππ )(π’1 ) πΉ(ππ )(π’2 ) .. . πΉ(ππ )(π’π )
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 F(π) be the set of all fuzzy subsets in a universe π. Let πΈ be a set of parameters and π΄ β πΈ. A pair (πΉ, π΄) is called a fuzzy soft set over π, where πΉ is a mapping given by πΉ : π΄ β F(π). In the above definition, fuzzy subsets in the universe π are used as substitutes for the crisp subsets of π. Hence it is easy to see that every soft set may be considered as a fuzzy soft set. Generally speaking, πΉ(π) is a fuzzy subset in π and it is called the fuzzy approximate value set of the parameter π. Following the standard notations, πΉ(π) can be written as πΉ(π) = {(π₯, πΉ(π)(π₯)) : π₯ β π}. 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 [0, 1]. For illustration, we consider the following example. Example 3 (see [10]). Suppose that there are six houses in the universe π = {β1 , β2 , β3 , β4 , β5 , β6 } and the set of parameters is given by πΈ = {π1 , π2 , π3 , π4 , π5 , π6 }, where ππ (π = 1, 2, . . . , 6) stand for βbeautiful,β βmodern,β βcheap,β βin good repair,β βwooden,β and βin green surroundings,β respectively. Let π΄ = {π1 , π2 , π3 } β πΈ be consisting of the parameters that Mr. X is interested in buying a house. This means that out of the available houses in π Mr. X wants to buy the house which is qualified with the attributes in π΄ to the utmost extent. Now all the available information on houses under consideration can be formulated as a fuzzy soft set πΊ = (πΉ, π΄) describing βattractiveness of housesβ that Mr. X is going to buy. Table 1 gives the tabular representation of the fuzzy soft set πΊ = (πΉ, π΄). We can view the fuzzy soft set πΊ = (πΉ, π΄) as the collection of the following fuzzy approximations: πΉ (π1 ) = beautiful houses = {(β1 , 0.4) , (β2 , 0.6) , (β3 , 0.5) , (β4 , 0.9) , (β5 , 0.3) , (β6 , 0.6)} , πΉ (π2 ) = modern houses = {(β1 , 1.0) , (β2 , 0.5) , (β3 , 0.5) , (β4 , 0.5) , (β5 , 0.7) , (β6 , 0.4)} ,
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πΉ (π3 ) = cheap houses = {(β1 , 0.5) , (β2 , 0.6) , (β3 , 0.8) , (β4 , 0.2) , (β5 , 0.9) , (β6 , 0.5)} . (1)
3. The Comprehensive Weight of Each Parameter under π Threshold 3.1. Ascertain the π-Level. Give a fuzzy soft set (πΉ, π΄) over π, we use the following algorithm to ascertain the π-level.
parameters, and πΌπ , π½π (π = 1, 2, . . . , π) are the subjective and objective weights of the πth parameter, respectively. We denote πΌ = (πΌ1 , πΌ2 , . . . , πΌπ ); π½ = (π½1 , π½2 , . . . , π½π ), where π βπ π=1 πΌπ = 1, βπ=1 π½π = 1, and πΌπ , π½π β₯ 0 (π = 1, 2, . . . , π). Suppose the comprehensive weight of the πth parameter is π€π ; we denote π = (π€1 , π€2 , . . . , π€π ), where βπ π=1 π€π = 1, π€π β₯ 0 (π = 1, 2, . . . , π). 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]: π
Step 1. Turn fuzzy soft set (πΉ, π΄) into an information system, such as Table 1.
min
Step 2. Build up fuzzy similarity matrix π
. We use formula πππ = (1/π) βπ π=1 (1 β |πΉ(ππ )(π’π ) β πΉ(ππ )(π’π )|) (π, π = 1, 2, . . . , π) to compute the similarity degree of objects π’π and π’π . Let π
= (πππ )πΓπ ; obviously, it is a fuzzy similarity matrix.
s.t
Step 3. Turn π
into a fuzzy equivalence matrix π
β ; that is, we compute the transitive closure of π
. Step 4. Choose the optimal classification threshold to clustering. In an equivalence matrix π
β , we sorted the elements π π in β π
from big to small; that is, 1 > π 1 > π 2 > β
β
β
> π π > 0; let πΆπ = (π πβ1 β π π )/(ππ β ππβ1 ), where ππ and ππβ1 are the number of objects in the πth and π β 1th clustering, respectively. If πΆπ = maxπ (πΆπ ), then we choose π π as the best threshold [14]. 3.2. The Objective Weight of Each Parameter under π 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 (πΉ, π΄) over π, π = {π’1 , π’2 , . . . , π’π }; π΄ = {π1 , π2 , . . . , ππ } are the parameter sets; the processes of parameter weight algorithm are as such. Step 1. Choose the optimal threshold π according to Section 3.1. Step 2. Classify the objects under the optimal threshold π, and we regard them as the classes of π/π΄. Step 3. Delete each ππ β π΄ (π = 1, 2, . . . , π) from π΄; in the same way, we can obtain the classes under π; we regard them as the classes of π/(π΄ β {ππ }) if π/(π΄ β {ππ }) =ΜΈ π/π΄; then ππ is indispensable in π΄, and the significance of ππ is σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨Posπ΄ (π΄ β {ππ }) β© Pos(π΄β{ππ }) (π΄)σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨. πππ = 1 β |π|
(2)
Step 4. Normalize the significance of each parameter; we get πΌ = (πΌ1 , πΌ2 , . . . , πΌπ ), where πΌπ = ππ (ππ )/(βπ π=1 ππ (ππ )). 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 π
2
2
πΉ (π) = β [π’(π€π β πΌπ ) + (1 β π’) (π€π β π½π ) ] π=1
π
(3)
β π€π = 1,
π=1
π€π β₯ 0,
π = 1, 2, . . . , π,
where 0 < π’ < 1 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: π
2
2
πΏ (π, π) = β [π’(π€π β πΌπ ) + (1 β π’) (π€π β π½π ) ] π=1
π
(4)
+ π ( β π€π β 1) . π=1
Let ππΏ = 2π’ (π€π β πΌπ ) + 2 (1 β π’) (π€π β π½π ) + π = 0 ππ€π π ππΏ = βπ€π β 1 = 0. ππ π=1
(5)
Solve the above equations; we can obtain π€π = π’πΌπ + (1 β π’)π½π , π = 0.
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 π π , where π π signifies the number of parameters of relatively larger membership value of object π’π . In [16] the decision depends on choice value ππ , where ππ signifies the sum of the membership values of all parameters of object π’π . 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
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fuzzy soft sets, we put up with a weighted comprehensive algorithm to choose the optimal program. (1) Input the fuzzy soft sets (πΉ, π΄) over π. (2) Compute the significance of each parameter ππ β π΄. (3) Compute the score π π of π’π , βπ, where π π = βπ π=1 πΉππ (π’π ) β
π(ππ ). (4) The decision is π’π if π π = maxπβ{1,2,...,π} π π . (5) If π has more than one value, then any one of π’π 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 π = {program 1, program 2, program 3, program 4, program 5, program 6, program 7, program 8} = {π’1 , π’2 , π’3 , π’4 , π’5 , π’6 , π’7 , π’8 }, each program is described by ten factor indexes, denoted by π΄ = {π1 (payback period of investment), π2 (estimate of sales), π3 (overall labor productivity), π4 (expected output tax rate), π5 (expected output of per kilowatt), π6 (expected transportation amount per ton kilometer), π7 (expected annual foreign exchange volume), π8 (level of employment), π9 (technical level), and π10 (benefit of bioenvironment)} [14]; according to the experts, we have πΉ (π1 ) = {(π’1 , 0.5) , (π’2 , 0.3) , (π’3 , 0.6) , (π’4 , 0.5) , (π’5 , 0.4) , (π’6 , 0.2) , (π’7 , 0.4) , (π’8 , 0.5)} ; πΉ (π2 ) = {(π’1 , 0.8) , (π’2 , 0.5) , (π’3 , 0.4) , (π’4 , 0.4) , (π’5 , 0.6) , (π’6 , 0.3) , (π’7 , 0.7) , (π’8 , 0.6)} ; πΉ (π3 ) = {(π’1 , 0.4) , (π’2 , 0.7) , (π’3 , 0.5) , (π’4 , 0.3) , (π’5 , 0.4) , (π’6 , 0.3) , (π’7 , 0.3) , (π’8 , 0.7)} ; πΉ (π4 ) = {(π’1 , 0.8) , (π’2 , 0.6) , (π’3 , 0.4) , (π’4 , 0.8) , (π’5 , 0.3) , (π’6 , 0.4) , (π’7 , 0.5) , (π’8 , 0.3)} ; πΉ (π5 ) = {(π’1 , 0.7) , (π’2 , 0.5) , (π’3 , 0.6) , (π’4 , 0.4) , (π’5 , 0.4) , (π’6 , 0.3) , (π’7 , 0.6) , (π’8 , 0.3)} ; πΉ (π6 ) = {(π’1 , 0.6) , (π’2 , 0.8) , (π’3 , 0.7) , (π’4 , 0.2) , (π’5 , 0.4) , (π’6 , 0.5) , (π’7 , 0.4) , (π’8 , 0.5)} ; πΉ (π7 ) = {(π’1 , 0.3) , (π’2 , 0.5) , (π’3 , 0.4) , (π’4 , 0.4) , (π’5 , 0.5) , (π’6 , 0.5) , (π’7 , 0.3) , (π’8 , 0.6)} ; πΉ (π8 ) = {(π’1 , 0.6) , (π’2 , 0.4) , (π’3 , 0.3) , (π’4 , 0.5) , (π’5 , 0.4) , (π’6 , 0.6) , (π’7 , 0.3) , (π’8 , 0.5)} ; πΉ (π9 ) = {(π’1 , 0.5) , (π’2 , 0.6) , (π’3 , 0.7) , (π’4 , 0.3) , (π’5 , 0.3) , (π’6 , 0.6) , (π’7 , 0.4) , (π’8 , 0.6)} ;
πΉ (π10 ) = {(π’1 , 0.3) , (π’2 , 0.6) , (π’3 , 0.5) , (π’4 , 0.5) , (π’5 , 0.4) , (π’6 , 0.7) , (π’7 , 0.6) , (π’8 , 0.4)} . (6) Step 1. Turn fuzzy soft set (πΉ, π΄) 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 π’π and π’π according to πππ = (1/π) βπ π=1 (1 β |πΉ(ππ )(π’π ) β πΉ(ππ )(π’π )|); thus we obtain the similarity degree matrix π
= (πππ )πΓπ . Through computing, we obtain π
as follows: 1 0.57 0.61 ( (0.61 π
=( (0.61 0.56 0.61 (0.52
0.57 1 0.66 0.6 0.65 0.6 0.64 0.65
0.61 0.66 1 0.62 0.64 0.6 0.64 0.65
0.61 0.6 0.62 1 0.69 0.59 0.69 0.69
0.61 0.65 0.64 0.69 1 0.6 0.69 0.72
0.56 0.6 0.6 0.59 0.6 1 0.59 0.6
0.61 0.64 0.64 0.69 0.69 0.59 1 0.69
0.52 0.65 0.65 ) 0.69) . 0.72) ) 0.6 0.69 1 ) (7)
Step 3. Compute the transitive closure of π
according to [14]. Through computing, we obtain an equivalence relation π
β as follows: 1 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 1 0.66 0.65 0.65 0.6 65 0.65 0.61 0.66 1 0.65 0.65 0.6 0.65 0.65 ( ) (0.61 0.65 0.65 1 0.69 0.6 0.69 0.69) β π
=( ). (0.61 0.65 0.65 0.69 1 0.6 0.69 0.72) 0.6 0.6 0.6 0.6 0.6 1 0.6 0.6 0.61 0.65 0.65 0.69 0.69 0.6 1 0.69 (0.61 0.65 0.65 0.69 0.72 0.6 0.69 1 ) (8) Step 4. Ascertain the optimal threshold. According to the fuzzy equivalence matrix π
β , 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: πΆ1 =
1 β 0.72 = 0.14; 2β0
πΆ2 =
0.72 β 0.69 = 0.015; 4β2
πΆ3 =
0.69 β 0.66 = 0.015; 6β4
(9)
0.66 β 0.61 = 0.005. 7β6 Since πΆ1 = max{πΆ1 , πΆ2 , πΆ3 , πΆ4 }, so we choose π = 0.72 as the threshold of classification. And the classes are as follows: π/π΄ = {{π’1 }, {π’2 }, {π’3 }, {π’4 }, {π’5 , π’8 }, {π’6 }, and {π’7 }}. πΆ4 =
Step 5. Compute the objective weight of each parameter under π = 0.72.
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5 Table 2: Information system according to fuzzy soft set (πΉ, π΄).
π π’1 π’2 π’3 π’4 π’5 π’6 π’7 π’8
π1 0.5 0.3 0.6 0.5 0.4 0.2 0.4 0.5
π2 0.8 0.5 0.4 0.4 0.6 0.3 0.7 0.6
π3 0.4 0.7 0.5 0.3 0.4 0.3 0.3 0.7
π4 0.8 0.6 0.4 0.8 0.3 0.4 0.5 0.3
π5 0.7 0.5 0.6 0.4 0.4 0.3 0.6 0.3
Delete π1 from π΄, under the threshold π = 0.72; in the same way, we can get π/(π΄ β {π1 }) = {{π’1 }, {π’2 }, {π’3 }, {π’4 }, {π’5 }, {π’6 }, {π’7 }, and {π’8 }} =ΜΈ π/π΄; the significance of parameter π1 is σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨Posπ΄β{π1 } (π΄) β© Posπ΄ (π΄ β {π1 })σ΅¨σ΅¨σ΅¨ 1 σ΅¨ σ΅¨= . (10) ππ1 = 1 β 4 |π| In the same way, we can get 1 ππ2 = ; 4
ππ3 = 0;
3 ππ7 = ; 8
ππ8 = 0;
3 ππ4 = ; 8
1 ππ5 = ; 4
ππ9 = 0;
1 ππ10 = . 4
1 ππ6 = ; 4
π6 0.6 0.8 0.7 0.2 0.4 0.5 0.4 0.5
π7 0.3 0.5 0.4 0.4 0.5 0.5 0.3 0.6
π8 0.6 0.4 0.3 0.5 0.4 0.6 0.3 0.5
π9 0.5 0.6 0.7 0.3 0.3 0.6 0.4 0.6
π10 0.3 0.6 0.5 0.5 0.4 0.7 0.6 0.4
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 (11)
Normalize the significance of each parameter; we can get the objective weight of each parameter; through computing, we have π½ = (0.125, 0.125, 0, 0.1875, 0.125, 0.125, 0.1875, 0, 0, and 0.125). Step 6. Compute the comprehensive weight of each parameter. Suppose that the subjective weight of each parameter is equal; that is, πΌ = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, and 0.1), and the bias coefficient of some customers is π’ = 0.4; according to the formula π€π = π’πΌπ + (1 β π’)π½π , we can get the comprehensive weight of each parameter; through computing, we have π = (0.115, 0.115, 0.04, 0.1525, 0.115, 0.115, 0.1525, 0.04, 0.04, and 0.115). Step 7. Choose the optimal program. According to formula π π = β10 π=1 π€π β
πΉ(ππ )(π’π ), through computing, we have π 1 = 0.5613, π 2 = 0.5463, π 3 = 0.504, π 4 = 0.457, π 5 = 0.419, π 6 = 0.4273, π 7 = 0.4725, and π 8 = 0.4738. Since π 1 = max10 π=1 {π π }, 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
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).
References [1] Z. Pawlak, βRough sets,β International Journal of Computer and Information Sciences, vol. 11, no. 5, pp. 341β356, 1982. [2] Z. Pawlak, Rough Sets. Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Boston, Mass, USA, 1991. [3] W. Gau and D. J. Buehrer, βVague sets,β IEEE Transactions on Systems, Man and Cybernetics, vol. 23, no. 2, pp. 610β614, 1993. [4] M. B. GorzaΕczany, βA method of inference in approximate reasoning based on interval-valued fuzzy sets,β Fuzzy Sets and Systems, vol. 21, no. 1, pp. 1β17, 1987. [5] D. A. Molodtsov, βSoft set theory-first results,β Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19β31, 1999. [6] B. Cagnabm and S. Enginoglu, βSoft matrix theory and its decision making,β Computers & Mathematics with Applications, vol. 59, no. 10, pp. 3308β3314, 2010. [7] N. Cagman and S. Enginoglu, βSoft Set theory and uni-int decision making,β European Journal of Operational Research, vol. 207, no. 2, pp. 848β855, 2010. [8] Y. Zou and Z. Xiao, βData analysis approaches of soft sets under incomplete information,β Knowledge-Based Systems, vol. 21, no. 8, pp. 941β945, 2008. [9] P. K. Maji, R. Biswas, and A. R. Roy, βFuzzy soft sets,β Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589β602, 2001. [10] A. R. Roy and P. K. Maji, βA fuzzy soft set theoretic approach to decision making problems,β Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412β418, 2007. [11] X. B. Yang, T. Y. Lin, J. Y. Yang, Y. Li, and D. Y. Yu, βCombination of interval-valued fuzzy set and soft set,β Computers & Mathematics with Applications, vol. 58, no. 3, pp. 521β527, 2009.
6 [12] Y. B. Jun, βSoft BCK/BCI-algebras,β Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1408β1413, 2008. [13] F. Feng, X. Y. Liu, and V. L. Leoreanu-Fotea, βSoft sets and soft rough sets,β Information Sciences, vol. 181, no. 6, pp. 1125β1137, 2011. [14] S. L. Chen, J. G. Li, and X. G. Wang, Fuzzy Set Theory and Application, Science Press, Beijing, China, 2006. [15] P. Xiong, H. B. Cheng, and X. P. Wu, βMethod of weighting ordering based on rough sets and its application,β Journal of Naval University of Engineering, vol. 1, no. 2003, pp. 53β56, 2003. [16] Z. Kong, L. Gao, and L. Wang, βComment on a fuzzy soft set theoretic approach to decision making problems,β Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 540β542, 2009.
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