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Applied Mathematical Modelling 37 (2013) 4474–4485

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Bijective soft set decision system based parameters reduction under fuzzy environments Ke Gong a,⇑, Panpan Wang a, Zhi Xiao b a b

School of Management, Chongqing Jiaotong University, Chongqing 400074, PR China School of Economics and Business Administration, Chongqing University, Chongqing 400030, PR China

a r t i c l e

i n f o

Article history: Received 16 January 2012 Received in revised form 29 August 2012 Accepted 25 September 2012 Available online 8 October 2012 Keywords: Bijective fuzzy soft set Parameters reduction Weight Bijective soft decision system

a b s t r a c t Gong et al. (2010) and Xiao et al. (2010) have proposed the notion of bijective soft set and exclusive disjunctive soft set, respectively, which is a subtype of soft set. On the basis of their work, this paper extends these notions to fuzzy environments, and formulates the concept of bijective fuzzy soft set, which can deal with more uncertain problems. Moreover, this paper proposes two parameters reduction algorithms: one (Algorithm 1) is based on bijective fuzzy soft system, and the other (Algorithm 2) takes weight of an element into consideration. Since the threshold plays an important role in these algorithms, we proposed an algorithm (Algorithm 3) to decide the optimal value of threshold specially. Afterwards, an example analysis of the two parameters reduction algorithms is given and the result shows that the two algorithms lead to the same parameters reduction of a bijective fuzzy soft system. Since Algorithm 2 considers the detail weights of elements, thus it can be used in more uncertain problems, such as time series analysis problems, than Algorithm 1. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Dealing with uncertainties is a major problem in many areas such as economics, engineering, environmental science, medical science and social sciences. Classical methods, such as probability theory, fuzzy set theory, rough set theory and interval mathematics play important roles in modeling uncertainty. However, with the increasing quantity and type of uncertainties, classical methods show their own difficulties as pointed out by Molodtsov [1]. To avoid these difficulties, Molodtsov [1] proposed soft set theory for modeling uncertainty, which has shown wide application prospects in smoothness of functions, game theory, operations research, etc. Recently, soft sets theory has been developed rapidly and focused by some scholars in theory and practice. Based on the definitions initiated by Molodtsov [1], Maji et al. [2] proposed some operations on soft set theory and verified De Morgan’s law. Aktasß and Çag˘man [3] gave a definition of soft groups, and derive their basic properties using Molodtsov’s definition of the soft sets. Jun [4] applied the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras and introduced the concept of soft BCK/BCI-algebras and subalgebras. Feng et al. [5] then introduced the notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring, homomorphisms and investigated several related properties of them. Ali and Feng et al. [6] corrected some mistakes of former studies and proposed some new operations on soft sets. Then Jun [7] gave an extending study on soft p-ideals of BCI-algebras. Çag˘man [8] defined soft matrices and their operations which were more functional to make theoretical studies in the soft set theory. In [9,10], notions of soft topology on a soft set and several properties of soft topology spaces were defined. ⇑ Corresponding author. E-mail address: [email protected] (K. Gong). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.067

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However, in real life many problems are imprecise in nature. Classical soft set theory is not capable of successfully dealing with such problems. Maji et al. [11] defined the fuzzy soft sets. Afterwards, many researchers have worked on this concept. Fuzzy soft set theory has been used to deal with imprecision, and its combinations with the previous branches of soft set may be found in [12–15]. In [16], a direct proof was given that Pawlak’s and Iwinski’s rough sets can be considered as soft sets. Ref. [17,18] initiated concepts of soft rough sets, soft-rough fuzzy sets, rough-soft sets, soft-rough sets, soft-rough fuzzy sets. Then the concepts of interval-valued fuzzy soft sets [19], intuitionistic fuzzy soft sets [20], vague soft sets [21] and generalized fuzzy soft sets were [22] introduced as some further extensions of fuzzy soft sets. Gong and Xiao [23] proposed the concept of bijective soft sets and defined some of its operations, and extended it into exclusive disjunctive soft sets [24] subsequently. Xiao et al. [25] proposed the concept of D-S generalized fuzzy soft sets to deal with problems of subjective evaluation and uncertain knowledge, and integrated the fuzzy cognitive map (FCM) and fuzzy soft set model for solving the supplier selection problem [26] afterwards. In parameters reduction aspect, Maji and Roy [27] introduced the concept of parameters reduction of soft sets and gave an application of it in decision making. Chen and Tsang [28] proved the results of soft set reductions offered in are incorrect and proposed a reasonable definition of parameters reduction of soft sets. In [29], the problems of suboptimal choice and added parameter set were discussed in the reduction of soft sets and a new definition of normal parameters reduction was introduced. Zou et al. [30] proposed the parameters reduction method of invariant optimal choice. Xiao et al. [31] introduced the decision parameters, and proposed two bijective soft system based parameters reduction algorithms. The purpose of this paper is to further extend the concept of soft sets and to find relevant parameters reduction algorithms, we introduce the bijective fuzzy soft sets, which is a new type of bijective soft set introduced by Gong et al. [23]. This paper proposes the restricted AND and the relaxed AND operation on a bijective fuzzy soft set and boundary region, the dependency between two bijective fuzzy soft sets, and the bijective fuzzy soft system dependency. Then the above results are used for formulating the parameters reduction algorithms of bijective soft system in fuzzy environments (Algorithm 1). Additionally, another algorithm (Algorithm 2), which takes the weights of elements into account, is inferred. To be objective, we give an extra algorithm (Algorithm 3) to choose an optimal cut level k, since k plays an important role in the above algorithms. Based on Algorithm 3, an example analysis of the two parameters reduction algorithms is given. The result shows that the two algorithms lead to the same parameters reduction of a bijective fuzzy soft system. Since Algorithm 2 considers the detail weights of elements, thus it can be used in more uncertain problems than Algorithm 1. The rest of the paper is organized as follows. Section 2 introduces the basic principles of fuzzy soft sets and bijective soft sets. Section 3 gives the concepts of bijective fuzzy soft set with its operations and two algorithms. Section 4 gives an example analysis of the two algorithms. Finally, Section 5 presents some conclusions from the research. 2. Preliminary 2.1. Soft set Let U be a common universe and let E be a set of parameters. Definition 2.1 (See [1]). SOFT SET. A pair (F, E) is called a soft set (over U) if and only if F is a mapping of E into the set of all subsets of the set U, where F is a mapping given by F : E ? P(U). In other words, the soft set is a parameterized family of subsets of the set U. Every set F(e)(e 2 E), from this family may be considered the set of e-elements of the soft sets (F, E), or as the set of e-approximate elements of the soft set. To illustrate this idea, let us consider the following example. Example 1. Let universe U = {h1, h2, h3, h4} be a set of houses, a set of parameters E = {e1, e2, e3, e4} be a set of status of houses which stand for the parameters ‘‘beautiful’’, ‘‘cheap’’, ‘‘in green surroundings’’, and ‘‘in good location’’ respectively. Consider the mapping F be a mapping of E into the set of all subsets of the set U. Now consider a soft set (F, E) that describes the ‘‘attractiveness of houses for purchase’’. According to the data collected, the soft set (F, E) is given by

ðF; EÞ ¼ fðe1 ; fh1 ; h3 ; h4 gÞ; ðe2 ; fh1 ; h2 gÞ; ðe3 ; fh1 ; h3 gÞ; ðe4 ; fh2 ; h3 ; h4 gÞg; where F(e1) = {h1, h3, h4}, F(e2) = {h1, h2}F(e3) = {h1, h3} and F(e4) = {h2, h3, h4}.In order to store a soft set in computer, a twodimensional table is used to represent the soft set (F, E). Table 1 is the tabular form of the soft set (F, E). If hi 2 F(ej), then hij = 1, otherwise hij = 0, where hij are the entries (see Table 1). Table 1 The tabular representation of (F, E). U

e1

e2

e3

e4

h1 h2 h3 h4

1 0 1 1

1 1 0 0

1 0 1 0

0 1 1 1

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Definition 2.2 (See [2]). NULL SOFT SET. A soft set (F, A) over U is said to be a NULL soft set denoted by U, if e 2 A, F(e) = £ . 2.2. Fuzzy soft set Let U be a common universe and let E be a set of parameters. ~ ~ AÞ is called a fuzzy soft set Definition 2.3 (See [32]). Suppose A  E. Let }ðUÞ be the set of all fuzzy subsets in U. A pair ð}; ~ over U, where e F is a mapping given by e F : A ! }ðUÞ. ~ AÞ; ð~f; BÞ, then ‘‘ð}; ~ AÞ AND ð~f; BÞ’’ denoted by is defined by Definition 2.4 (See [32] ). If and are two fuzzy soft sets ð}; ~ ~ ~ ~ ~ AÞ ^ ðf; BÞ ¼ ð1; A  BÞ, where 1ða; bÞ ¼ }ð ~ aÞ \ fðbÞ; 8ða; bÞ 2 A  B. ð}; Example 2. Suppose that there are six houses under consideration, namely the universe U = {x1, x2, x3, x4, x5, x6}, and the parameter set A = {e1, e2, e3, e4, e5, e6}, where ei stand for ‘‘beautiful’’, ‘‘modern’’, ‘‘large’’, ‘‘cheap’’, ‘‘in good repair’’ and ‘‘in green surroundings’’ respectively. Suppose Mr. X wants to buy a house which satisfies the criteria in A to the utmost extent. All the ~ AÞ. The table representation of ð}; ~ AÞ is available information on these houses can be characterized by a fuzzy soft set ð}; shown in Table 2. In Table 2, we can see that the precise evaluation for an alternative to satisfy a criterion is unknown while the lower and upper approximations of such an evaluation are given. For example, we cannot present the precise degree of how beautiful house x1 is, however, house x1 is at least beautiful on the degree of 0.3 and it is at most beautiful on the degree of 0.5. ~; A  BÞ, ~ AÞ; ð~f; BÞ is defined by ð}; ~ AÞ ^ ð~f; BÞ ¼ ð1 Definition 2.5 (See [19]). The ‘‘AND’’ operation on the two fuzzy soft sets ð}; ~ ~ ~ aÞ \ fðbÞ; 8ða; bÞ 2 A  B. where 1ða; bÞ ¼ }ð ~ ; A  BÞ, ~ AÞ; ð~f; BÞ is defined by ð}; ~ AÞ _ ð~f; BÞ ¼ ðu Definition 2.6 (See [19]). The ‘‘OR’’ operation on the two fuzzy soft sets ð}; ~ ða; bÞ ¼ }ð ~ aÞ [ ~fðbÞ; 8ða; bÞ 2 A  B. where u ~ AÞ and ð~f; BÞ are two fuzzy soft sets, we say that ð}; ~ AÞ is a fuzzy soft subset of ð~f; BÞ Definition 2.7 (See [19]). Suppose that ð}; if and only if (1) A  B; ~ eÞ is a fuzzy subset of ~fðeÞ; (2) 8e 2 A; }ð ~ ~f; BÞ. ~ AÞð Which can be denoted by ð}; 2.3. Bijective soft set

Definition 2.8 (See [23]). Let (F, B) be a soft set over a common universe U, where F is a mapping F : B ? P(U) and B is nonempty parameter set. We say that (F, B) is a bijective soft set, if (F, B) such that S (1) e2BF(e) = U. (2) For any two parameters ei, ej 2 B, ei – ej, F(ei) \ F(ej) = £ . In other words, Suppose Y # P(U) and Y = {F(e1),F(e2), . . . , F(en)}, e1, e2, . . . , en 2 B. From Definition 2.8, the mapping F : B ? P(U) can be transformed to the mapping F : B ? Y, which is a bijective function. i.e. for every y 2 Y, there is exactly one parameter e in B such that F(e) = y and no unmapped element remains in both B and Y.

Table 2 ~ AÞ. Fuzzy soft set ð};

x1 x2 x3 x4 x5 x6

e1

e2

e3

e4

e5

e6

0.30 0.50 0.60 0.25 0.85 0.90

0.65 0.20 0.55 0.30 0.60 0.90

0.65 0.90 0.30 0.05 0.20 0.55

0.65 0.20 0.30 0.30 0.20 0.75

0.40 0.80 0.60 0.25 0.85 0.40

0.65 0.90 0.70 0.20 0.40 0.55

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Example 3. Suppose that U = {x1, x2, x3, x4, x5, x6, x7} is a common universe, (F, E) is a soft set over U, E = {e1, e2, e3, e4}. The mapping of (F, E) is given below:

Fðe1 Þ ¼ fx1 ; x2 ; x3 g Fðe2 Þ ¼ fx4 ; x5 ; x6 g Fðe3 Þ ¼ fx7 g Fðe4 Þ ¼ fx4 ; x5 ; x6 ; x7 g From Definition 2.6, (F, {e1, e2, e3}) and (F, {e1, e4}) are bijective soft sets. While (F, {e1, e2}) and (F, {e1, e3}) are not bijective soft sets. Definition 2.9 (See [23]). (restricted AND operation on a bijective soft set and a subset of universe) Let U = {x1, x2, . . . , xn} be a common universe, X be a subset of U, and (F, E) be a bijective soft set over U. The operation of ‘‘(F, E) restricted AND X’’denoted S by ðF; EÞ ^ X is defined by e2E{F(e) : F(e) # X}. 

Definition 2.10 (See [23]). (relaxed AND operation on a soft set and a subset of universe) Let set U = {x1, x2, . . . , xn} be a common universe, X be a subset of U, and (F, E) be a bijective soft set over U. The operation of ‘‘(F, E) relaxed AND X’’ denoted by S ~ is defined by e2E {F(e) : F(e) \ X – £ }. ðF; EÞ^X, 3. Bijective fuzzy soft set To formulate the concept of bijective fuzzy soft set, we will give an example below firstly. And it will be used to illustrate some notions of this section. Example 4. Let U = {x1, x2, x3, x4, x5, x6} be a common universe, is a set of six stores. Suppose that the six stores are ~ EÞ over a common universe U. E denotes the parameter set, E = E1 [ E2 [ E3 [ E4. E1 characterized by a fuzzy soft set ð}; describes the empowerment of sales personnel. E2 describes the perceived quality of merchandise. E3 describes the high traffic location. And E4 describes store profit or loss. The sets of these parameters are E1 = {high, med., low}, E2 = ~i ; Ei Þ is soft subset of ð}; ~ EÞ, where i = 1,2,3,4. The {good, avg.}, E3 = {no, yes} and E4 = {profit, loss}, respectively. And ð} mapping of each fuzzy soft set over U is shown in Tables 3–6. 3.1. Concept of bijecive fuzzy soft set ~ EÞ be a fuzzy soft set over universe U, x be an element of U and e be a parameter of E. nxk ðeÞ denotes the Definition 3.1. Let ð}; ~ EÞ, defined by characteristic function of the k-level soft set of the fuzzy soft set ð};

(

nxk ðeÞ ¼

1 if 0

l}ðeÞ ~ ðxÞ P k;

else;

where

l}ðeÞ ~ ðxÞ; k 2 ½0; 1:

ð3:1Þ

Table 3 ~1 ; E1 Þ. The fuzzy soft set ð}

x1 x2 x3 x4 x5 x6

High

Med

Low

0.75 0.55 0.60 0.30 0.70 0.80

0.30 0.80 0.85 0.55 0.80 0.55

0.30 0.70 0.60 0.80 0.55 0.25

Table 4 ~2 ; E2 Þ. The fuzzy soft set ð}

x1 x2 x3 x4 x5 x6

Good

Avg.

0.80 0.80 0.80 0.30 0.70 0.55

0.30 0.55 0.60 0.85 0.85 0.80

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x1 x2 x3 x4 x5 x6

No

Yes

0.75 0.75 0.80 0.80 0.70 0.60

0.30 0.50 0.60 0.30 0.85 0.80

Table 6 ~4 ; E4 Þ. The fuzzy soft set ð}

x1 x2 x3 x4 x5 x6

Profit

Loss

0.90 0.50 0.85 0.20 0.60 0.80

0.30 0.80 0.60 0.85 0.85 0.60

~ EÞ as follows: Example 5. Consider Example 4, we obtain the 0.75-level soft set (F, E) of the fuzzy soft set ð};

F 1 ðhighÞ ¼ f1 n x1 ; 0 n x2 ; 0 n x3 ; 0 n x4 ; 0 n x5 ; 1 n x6 g;

F 1 ðmed:Þ ¼ f0 n x1 ; 1 n x2 ; 1 n x3 ; 0 n x4 ; 1 n x5 ; 0 n x6 g;

F 1 ðlowÞ ¼ f0 n x1 ; 0 n x2 ; 0 n x3 ; 1 n x4 ; 0 n x5 ; 0 n x6 g; F 2 ðgoodÞ ¼ f1 n x1 ; 1 n x2 ; 1 n x3 ; 0 n x4 ; 0 n x5 ; 0 n x6 g;

F 2 ðavg:Þ ¼ f0 n x1 ; 0 n x2 ; 0 n x3 ; 1 n x4 ; 1 n x5 ; 1 n x6 g

F 3 ðnoÞ ¼ f1 n x1 ; 1 n x2 ; 1 n x3 ; 1 n x4 ; 0 n x5 ; 0 n x6 g;

F 3 ðyesÞ ¼ f0 n x1 ; 0 n x2 ; 0 n x3 ; 0 n x4 ; 1 n x5 ; 1 n x6 g;

F 4 ðprofitÞ ¼ f1 n x1 ; 0 n x2 ; 1 n x3 ; 0 n x4 ; 0 n x5 ; 1 n x6 g; F 4 ðlossÞ ¼ f0 n x1 ; 1 n x2 ; 0 n x3 ; 1 n x4 ; ;1 n x5 ; 0 n x6 g: where k equals to 0.75. ~ BÞ over a common universe U, B is a nonempty parameter Definition 3.2. Let (F, B) be the k-level soft set of a fuzzy soft set ð}; ~ BÞ is a k-level bijective fuzzy soft set, if and only if (F, B) is a bijective soft set. set. We say that ð}; ~i ; Ei Þ; i ¼ 1; 2; 3; 4; are bijective Example 6. Let us reconsider the fuzzy soft sets given in Example 5. From Definition 3.2, ð} fuzzy soft sets respectively. ~; CÞ ¼ ð}; ~ AÞ ^ ~fðBÞ ~ AÞ and ð~f; BÞ are two k-level bijective fuzzy soft sets over common universe U. ð1 Theorem 3.1. Suppose that ð}; is a k-level bijective fuzzy soft set. Proof. Straightforward. h ~; CÞ ¼ ð}; ~ AÞ [~ ~fðBÞ ~ AÞ is a k-level bijective fuzzy soft set over U andð~f; BÞ is a Null soft set over U. ð1 Theorem 3.2. Suppose that ð}; is a k-level bijective fuzzy soft set. Proof. Straightforward. h ~ EÞ is a k-level bijective fuzzy soft set over ~fðUÞ, and (F, E) is a k-level soft set of ð}; ~ EÞ, a Definition 3.3. Suppose that ð}; ~ EÞ and (F, E) is called the combined k-level bijective fuzzy soft set of ð}; ~ EÞ, denoted by (}, E) and defined by product of ð};

ð}; EÞ ¼ f e F k ðeÞj e F k ðeÞ ¼ l}ðeÞ ~ ðxÞ  nxk ðeÞ; e 2 E; x 2 Ug: Example 7. By Definition 3.3, we get a 0.75-level combined bijective fuzzy soft set (}1, E1) as following:

ð}1 ; E1 Þ ¼ f e F k ðeÞj e F k ðeÞ ¼ l}ðeÞ ~ ðxÞ  nxk ðeÞ; e 2 E1 ; x 2 Ug;

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Then,

8 9 < highf0:75  1 n x1 ; 0 n x2 ; 0 n x3 ; 0 n x4 ; 0 n x5 ; 0:80  1 n x6 g; = ð}1 ; E1 Þ ¼ med:f0 n x1 ; 0:80  1 n x2 ; 0:85  1 n x3 ; 0 n x4 ; 0:80  1 n x5 ; 0 n x6 g; : ; lowf0 n x1 ; 0 n x2 ; 0 n x3 ; 0:80  1 n x4 ; 0 n x5 ; 0 n x6 g ¼ fhighf0:75 n x1 ; 0:80 n x6 g; med:f0:80 n x2 ; 0:85 n x3 ; 0:80 n x5 g; lowf0:80 n x4 gg:

~1 ; E1 Þ can be shown as follows: Thus, the combined tabular representation of ð} ~ EÞ is a k-level bijective fuzzy soft set over ~fðUÞ. (}, E) is a combined k-level bijective fuzzy Definition 3.4. Suppose that ð}; ~ EÞ. Then the total membership degree of (}, E) can be defined as: soft set of ð};

Te E ðxi Þk ¼

X ðxij Þk ; for all e 2 E1 : xi 2U

~ EÞ. Here, xij are the entries in the combined tabular representation of ð}; e E ðxi Þ ¼ P From Table 7, we can get T k xi 2U ðxij Þk ¼ 0:75 þ 0:80  4 þ 0:85 ¼ 4:80, for all e 2 E1. xij are the entries in the 1 ~1 ; E1 Þ. combined tabular representation of ð} Definition 3.5 (restricted AND operation on a bijective fuzzy soft set and a subset of universe ). Let U = {x1, x2, . . . , xn} be a e be a fuzzy subset of U, and ð}; ~ EÞ be a k-level bijective fuzzy common universe, and ~fðUÞ be the set of all fuzzy subsets in U. X e e is defined by S f e e ~ ~e ~ EÞ restricted AND X ’’ denoted by ð}; ~ EÞ ^ X soft set over fðUÞ. The operation of ‘‘ð}; e2E F k ðeÞ : F k ðeÞ X g.  Here, e F k ðeÞ ¼ l}ðeÞ ðxÞ  n ðeÞ; x 2 U; e 2 E. xk ~ ~ l}ðeÞ 0 6 l}ðeÞ ~ ðxÞ are membership of x to }ðeÞ; ~ ðxÞ 6 1; 0 6 k 6 1. ~ AÞ be a k-level bijective fuzzy soft set over ~fðUÞ, and ~fðUÞ ¼ f0:90 n x1 ; 0:85 n x2 ; 0:90 n x3 ; 0:90 n x4 g. Example 8. Let ð}; Suppose the following:

ð}; AÞ ¼ fe1 f0:90 n x1 g; e2 f0:85 n x2 ; 0:90 n x3 g; e3 f0:90 n x4 gg e ¼ f0:90 n x2 ; 90 n x3 g X From Definition 3.5, we can write

e ¼ f0:85 n x2 ; 0:90 n x3 g: ~ AÞ ^ X ð}; 

Definition 3.6 (relaxed AND operation on a soft set and a subset of universe). Let set U = {x1, x2, . . . , xn} be a common universe, e be a fuzzy subset of U, and ð}; ~ EÞ be a k-level bijective fuzzy soft set over ~fðUÞ. and ~fðUÞ be the set of all fuzzy subsets in U. X S e e e – £g ~ ~ ~ The operation of ‘‘ð}; EÞ relaxed AND X’’ denoted by ð}; EÞ^ X , is defined by e2E f e F k fegj e F k ðeÞ \ X Here, e F k ðeÞ ¼ l}ðeÞ ~ ðxÞ  nxk ðeÞ; x 2 U; e 2 E, ~ l}ðeÞ ðxÞ are the degree of the membership of x to }ðeÞ; 0 6 l}ðeÞ ~ ~ ðxÞ 6 1; 0 6 k 6 1. ~ AÞ be a k-level bijective fuzzy soft set over ~fðUÞ, and ~fðUÞ ¼ f0:90 n x1 ; 0:85 n x2 ; 0:90 n x3 ; 0:85 n x4 g. Example 9. Let ð}; Suppose the following:

ð}; AÞ ¼ fe1 f0:90 n x1 g; e2 f0:85 n x2 ; 0:90 n x3 ; 0:85 n x4 gg e ¼ f0:90 n x2 ; 90 n x3 g X From Definition 3.6, we can write

e ¼ f0:85 n x2 ; 0:90 n x3 ; 0:85 n x4 g ~ AÞ^~X ð};

Table 7 ~1 ; E1 Þ. The combined tabular representation of bijective fuzzy soft set ð}

x1 x2 x3 x4 x5 x6

High

Med

Low

0.75 0 0 0 0 0.80

0 0.80 0.85 0 0.80 0

0 0 0 0.80 0 0

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~ AÞ; ð~f; CÞ are two k-level bijective fuzzy soft Definition 3.7 (Dependency between two bijective fuzzy soft sets). Suppose that ð}; S ~ AÞ is said to depend on ð~f; CÞ to a degree ~ AÞ ^ ~fðeÞ. ð}; sets over a common universe U, where A \ C = £. ð1; PÞ ¼ e2C ð};  ~ AÞ)k ð~f; CÞ, if kk(0 6 kk 6 1), denoted ð}; 

~ AÞ; ð~f; CÞÞ ¼ kk ¼ ck ðð};

Te P ðxi Þ Te A ðxi Þ

! ð3:2Þ k

e A ðxi Þ ¼ P e P ðxi Þ ¼ P ~ AÞ, and T where T k xi 2U ðxij Þk ; xij are the entries in the table of ð}; k xi 2U ðxij Þk ; xij are the entries in the combined tabular representation of (1, P) (see Definition 3.4). The concept of dependency is to describe a degree of bijective fuzzy soft set in classifying the other one. ~ AÞ is full depended on ð~f; CÞ. If kk = 1 we say ð}; ~ AÞ is not depended on ð~f; CÞ. If kk = 0 we say ð}; To illustrate this concept, we will give an example below. Example 10. Let us reconsider the k-level bijective fuzzy soft sets given in Example 5.

}4 ðprofitÞ ¼ f0:90 n x1 ; 0:85 n x3 ; 0:80 n x6 g; }4 ðlossÞ ¼ f0:80 n x2 ; 0:85 n x4 ; 0:85 n x5 g ð}1 ; E1 Þ ¼ ff0:75 n x1 ; 0:80 n x6 g; f0:80 n x2 ; 0:85 n x3 ; 0:80 n x5 g; f0:80 n x4 gg ~ 1 ; E1 Þ ^ } ~4 ðprofitÞ ¼ f0:75 n x1 ; 0:80 n x6 g ð} 

~ 1 ; E1 Þ ^ } ~4 ðlossÞ ¼ f0:80 n x4 g: ð} 

From Definition 3.7, we can write

~1 ; E1 Þ; ð} ~4 ; E4 ÞÞ ¼ kk ¼ ck ðð}

0:75 þ 0:80 þ 0:80 ¼ 0:4895: 0:75 þ 0:80  4 þ 0:85

Definition 3.8 (Bijective fuzzy soft decision system). Let set U = {x1, x2, . . . , xn} be a common universe, and ~fðUÞ be the set of all ~i ; Ei Þ ði ¼ 1; 2; 3; . . . ; nÞ are n k-level bijective fuzzy soft sets over ~fðUÞ, where any Ei \ Ej = fuzzy subsets in U.Suppose that ð} £ (i = 1,2,3, . . . , n;j = 1,2,3, . . . , n;i – j). ð~f; BÞ is a k-level bijective fuzzy soft set over a common universe U, B \ Ei = £ ~ EÞ ¼ [~ni¼1 ð} ~i ; Ei Þ. The triple ðð}; ~ EÞ; ð~f; BÞ; UÞk is called k-level (i = 1,2,3, . . . , n), and we call it the decision soft set. Suppose ð}; bijective fuzzy soft decision system over a common universe U. ~i ; Ei Þ; ð} ~4 ; E4 Þ; UÞ0:75 . This bijective fuzzy In Example 5, we can consider a k-level bijective fuzzy soft decision system ð[~3i¼1 ð} soft decision system set describes the profit ability of stores and other information. ~ EÞ; ð~f; BÞ; UÞk be a k-level bijective fuzzy soft decision Definition 3.9 (Bijective fuzzy soft decision system dependency). Let ðð}; ~ EÞ ¼ [~ni¼1 ð} ~i ; Ei Þ and ð} ~i ; Ei Þ is k-level bijective fuzzy soft set. ð}; ~ EÞ is called condition soft set. The k-level system, where ð}; ~1 ; E1 Þ ^ ð} ~2 ; E2 Þ ^    ^ ð} ~n ; En Þ and ð~f; BÞ is called k-level bijective fuzzy soft bijective fuzzy soft dependency between ð} ~ EÞ; ð~f; BÞ; UÞk , denoted by jk and defined by decision system dependency of ðð};



jk ¼ ck ^ni¼1 ð}~i ; Ei Þ; ð~f; BÞ



ð3:3Þ

  ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 is a 0.75-level bijective fuzzy soft decision system on how to choose Example 11. Suppose that [~3i¼1 ð} ~1 ; E1 Þ ^ ð} ~2 ; E2 Þ ^ ð} ~3 ; E3 Þ is given in Table 8. profitable stores. The combined tabular form of ð} ~; CÞ, ~1 ; E1 Þ ^ ð} ~2 ; E2 Þ ^ ð} ~3 ; E3 Þ ¼ ð1 Suppose that ð}

Table 8 ~ 1 ; E1 Þ ^ ð } ~ 2 ; E2 Þ ^ ð } ~3 ; E3 Þ. The combined tabular form of ð}

x1 x2 x3 x4 x5 x6

e1

e2

e3

e4

e5

0.75 0 0 0 0 0

0 0.75 0.80 0 0 0

0 0 0 0.80 0 0

0 0 0 0 0.80 0

0 0 0 0 0 0.80

Where e1= high and good and no, e2= med and good and no, e3 = low and avg. and no, e4= med and avg. and yes, e5= high and avg. and yes.

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}4 ðprofitÞ ¼ f0:90 n x1 ; 0:85 n x3 ; 0:80 n x6 g; }4 ðlossÞ ¼ f0:80 n x2 ; 0:85 n x4 ; 0:85 n x5 g ð1; CÞ ¼ ff0:75 n x1 g; f0:75 n x2 ; 0:80 n x3 g; f0:80 n x4 g; f0:80 n x5 g; f0:80 n x6 gg ~; CÞ ^ } ~4 ðprofitÞ ¼ f0:75 n x1 ; 0:80 n x6 g; ð1 

~; CÞ ^ } ~4 ðlossÞ ¼ f0:80 n x4 ; 0:80 n x5 g: ð1 

From Definition 3.9, we can write

j0:75 ¼ c0:75 ðð1~; CÞ; ð}~4 ; E4 ÞÞ ¼

0:75 þ 0:80 þ 0:80 þ 0:80 ¼ 0:6702: 0:75  2 þ 0:80  4

~ EÞ; ~f; B; UÞk be a k-level bijective fuzzy soft decision system, where ð}; ~ EÞ ¼ [~ni¼1 ð} ~i ; Ei Þ and ð} ~i ; Ei Þ is k-level Theorem 3.3. Let ðð}; ~ bijective fuzzy soft set. jk is the bijective fuzzy soft decision system dependency ofðð }; EÞ; ð~f; BÞ; UÞk . The dependency between   ~ ~i ; Ei Þ, where m 6 n, and ð~f; BÞ isck ^m ^m i¼1 ð} i¼1 ðF i ; Ei Þ; ðf; BÞ . And





ck ^mi¼1 ð}~i ; Ei Þ; ð~f; BÞ 6 jk

ð3:4Þ

In other words, the condition soft set of bijective fuzzy soft decision system can explain the most detailed classification of decision fuzzy soft sets. And removing some bijective fuzzy soft sets of it may lose some information of the decision fuzzy soft set. For example, a profitable store may be a store with high empowerment of sales personnel and good perceived quality of merchandise. But if we only know the perceived quality of merchandise of the store, we cannot judge its profit ability exactly for the absence information of other factors that affect the profit ability. Thus, more information (bijective fuzzy soft set) can result in bigger dependency on decision fuzzy soft set. Proof. Suppose that

~; CÞ ¼ ^ni¼1 ð} ~ ; KÞ ¼ ^m ~i ; Ei Þ; ðu ~i ; Ei Þ; ð e ð1 F 1 ; P1 Þ ¼ i¼1 ð}

[ [ ~; CÞ ^ ~fðeÞ; ð e ~ ; KÞ ^ ~fðeÞ: F 2 ; P 2 Þ ¼ ðu ð1 e2B



e2B



From Definitions 3.7 and 3.5,

jk ¼





~i ; Ei Þ; ð~f; BÞ ^ni¼1 ð}

k

¼

! Te P1 ðxi Þ ; Te C ðxi Þ

  ck ^mi¼1 ð}~i ; Ei Þ; ð~f; BÞ ¼

k

! Te P2 ðxi Þ ; Te K ðxi Þ k

P e C ðxi Þ ¼ P e ~ Where T k xi 2U ðxij Þk ; xij are the entries in the combined tabular representation of ð1; CÞ; T K ðxi Þk ¼ xi 2U ðxij Þk ; xij are the ~ KÞ (see Definition 3.4). entries in the combined tabular representation of ð/; S S S ~; CÞ ^ ~fðeÞ ¼ e2B a2C f e Obviously, e2B ð1 F k ðaÞ : e F k ðaÞ~~fðeÞg,

[

e2B

~ ; KÞ ^ ~fðeÞ ¼ ðu 

 [ [

fe F k ðaÞ : e F k ðaÞ ^ ~fðeÞ – £g:

e2Ba2K

From Definition 2.5,

1~ðe1 ; e2 ; . . . ; en Þ ¼ }~1 ðe1 Þ \ }~2 ðe2 Þ \    \ }~m ðem Þ \    \ }~n ðen Þ; 8ðe1 ; e2 ; . . . ; en Þ 2 E1  E2      En ~ ðe1 ; e2 ; . . . ; em Þ ¼ } ~1 ðe1 Þ \ } ~2 ðe2 Þ \    \ } ~m ðem Þ; 8ðe1 ; e2 ; . . . ; em Þ 2 E1  E2      Em u *n>m ~ðe1 ; e2 ; . . . ; en Þ~u ~ ðe1 ; e2 ; . . . ; em Þ )1 S Since ð~f; BÞ is a k-level bijective fuzzy soft set, then the k-level fuzzy soft set of ð~f; BÞ satisfies e2B ~fðeÞk ¼ ~fðUÞ. S S S S Therefore, e2B a2C f e F k ðaÞ : e F k ðaÞ~~fðeÞg P e2B a2K f e F k ðaÞ : e F k ðaÞ ^ ~fðeÞ – q£g,

Te P1 ðxi Þk P Te P2 ðxi Þk ; e C ðxi Þ 6 T e K ðxi Þ ; And T k k

  ~i ; Ei Þ; ð~f; BÞ : Thus; jk P ck ^m i¼1 ð}



~ EÞ; ð~f; BÞ; UÞk be a k-level bijective fuzzy soft decision system, where ð}; ~ EÞ ¼ [~ni¼1 ð} ~i ; Ei Þ and ð} ~i ; Ei Þ is Definition 3.10. Let ðð}; ~ ~ ~ k-level fuzzy bijective soft set, [m ð } ; E Þ ð }; EÞ . j is the bijective fuzzy soft decision system dependency of i i k i¼1   ~ EÞ; ð~f; BÞ; UÞ . If c ~i ; Ei Þ; ð~f; BÞ ¼ jk we say [m ð} ~i ; Ei Þ is a reduct of a k-level bijective fuzzy soft decision system ðð}; ^m ð} k

½t 1 ;t 2 

i¼1

i¼1

~ EÞ; ð~f; BÞ; UÞk . ðð}; Example 12. Let us reconsider the k-level bijective fuzzy soft sets given in Example 5.   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 is a 0.75-level bijective fuzzy soft decision system on how to choose profitable Suppose that [~3i¼1 ð} ~1 ; E1 Þ ^ ð} ~2 ; E2 Þ; ð} ~4 ; E4 ÞÞ ¼ j0:75 ¼ 0:6702. stores. c0:75 ðð}

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~} ~1 ; E1 Þ[ð ~2 ; E2 Þ is a reduct of ð[~3i¼1 ð} ~i ; Ei Þ; ð} ~4 ; E4 Þ; UÞ0:75 . Thus ð} 3.2. Bijective soft set decision system based parameters reduction under fuzzy environments Algorithm 1   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 ; Step 1. Construct k-level bijective fuzzy soft decision system [~3i¼1 ð} ~j ; Ej Þ and ð} ~4 ; E4 Þ, where 0 < j 6 3, by Definition 3.7; Step 2. Calculate each dependency between ^ð}   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 by Definition 3.9; Step 3. Calculate bijective fuzzy soft decision system dependency of [~3i¼1 ð} Step 4. Find reduct bijective fuzzy soft sets with respect to bijective fuzzy soft decision system  3  ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 by Definition 3.10; [~i¼1 ð}   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 . Step 5. Obtain decision rules by reducted bijective fuzzy soft decision system [~3i¼1 ð}

3.3. Bijective soft set decision system based weighted parameters reduction under fuzzy environments ~ AÞ; ð~f; CÞ are two k-level bijective Definition 3.11 (Weighted dependency between two bijective fuzzy soft sets). Suppose that ð}; fuzzy soft sets over a common universe U, where A \ C = £. W = [w1, w2, . . . , wn] is a weighted vector corresponds to elements P S ~ AÞ is said to depend on ð~f; CÞ to a degree kkw(0 6 kkw 6 1), denoted ~ AÞ ^ ~fðeÞ. ð}; in U, wi 2 ½0; 1; wi 2W wi ¼ 1. ð1; PÞ ¼ e2C ð};  ~ ~ AÞ)kw ðf; CÞ, if ð}; 

P e F k ðeÞ  w½x ~ ~ AÞ; ðf; CÞÞ ¼ Px2U;e2P kk w ¼ ck wðð}; ; e x2U;e2A F k ðeÞ  w½x

ð3:5Þ

where e F k ðeÞ ¼ l}ðeÞ ~ ðxÞ  nxk ðeÞ; e 2 P; x 2 U; nxk ðeÞ denotes the characteristic function of a k-level bijective fuzzy soft set, and l}ðeÞ ~ ðxÞ denotes the membership of x to e; l}ðeÞ ~ ðxÞ 2 ½0; 1. Example 13. Let’s reconsider the 0.75-level bijective fuzzy soft sets given in Example 5. Suppose that W = [0.16,0.13,0.19,0.23,0.10,0.19].

}4 ðprofitÞ ¼ f0:90 n x1 ; 0:85 n x3 ; 0:80 n x6 g; }4 ðlossÞ ¼ f0:80 n x2 ; 0:85 n x4 ; 0:85 n x5 g ð}1 ; E1 Þ ¼ ff0:75 n x1 ; 0:80 n x6 g; f0:80 n x2 ; 0:85 n x3 ; 0:80 n x5 g; f0:80 n x4 gg ~ 1 ; E1 Þ ^ } ~4 ðprofitÞ ¼ f0:75 n x1 ; 0:80 n x6 g ð} 

~ 1 ; E1 Þ ^ } ~4 ðlossÞ ¼ f0:80 n x4 g: ð} 

From Definition 3.11, we can write

~1 ; E1 Þ; ð} ~4 ; E4 ÞÞ ¼ kk w ¼ ck wðð}

0:75  0:16 þ 0:80  0:23 þ 0:80  0:19 ¼ 0:5689: 0:75  0:16 þ 0:80  ð0:13 þ 0:23 þ 0:10 þ 0:19Þ þ 0:85  0:19

Similarly, we can define the weighted bijective fuzzy soft decision system dependency as follows: ~ EÞ; ð~f; BÞ; UÞk be a k-level bijective fuzzy Definition 3.12 (Weighted bijective fuzzy soft decision system dependency). Let ðð}; ~ EÞ ¼ [~ni¼1 ð} ~i ; Ei Þ and ð} ~i ; Ei Þ is k-level bijective fuzzy soft set. W = [w1, w2, . . . , wn] is a weighted soft decision system, where ð}; P ~ EÞ is called condition soft set. The k-level bijective fuzzy soft vector corresponds to elements in U; wi 2 ½0; 1; wi 2W wi ¼ 1. ð}; ~1 ; E1 Þ ^ ð} ~2 ; E2 Þ ^    ^ ð} ~n ; En Þ and ð~f; BÞ is called weighted k-level bijective fuzzy soft decision system dependency between ð} ~ EÞ; ð~f; BÞ; UÞk , denoted by jkw and defined by dependency of ðð};





jk w ¼ ck w ^ni¼1 ð}~i ; Ei Þ; ð~f; BÞ : Algorithm 2   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 ; Step 1. Construct k-level bijective fuzzy soft decision system [~3i¼1 ð} ~j ; Ej Þ and ð} ~4 ; E4 Þ, where 0 < j 6 3, by Definition 3.11; Step 2. Calculate each weighted dependency between ^ð}   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 by DefiniStep 3. Calculate weighted bijective fuzzy soft decision system dependency of [~3i¼1 ð} tion 3.12; Step 4. Find reduct bijective fuzzy soft sets with respect to bijective fuzzy soft decision system  3  ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 by Definition 3.10; [~i¼1 ð}   ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 . Step 5. Obtain decision rules by reducted bijective fuzzy soft decision system [~3i¼1 ð}

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4. An example analysis of the two algorithms and discussion 4.1. How to choose k Obviously, k plays an important role in the above algorithms. k helps to cut the fuzzy soft set into a bijective soft set. The higher value k holds, more credible and useful decision rules and parameters reduction become. On the other hand, if k is given by decision maker then the result is very subjective. Thus, we propose the algorithm below (Algorithm 3) to get a larger k to cut. Firstly, we give the judgment condition of bijective soft set below which can be used in the next algorithm. Proposition 1 (Judgment condition of bijective soft set). Let (F, E) be a bijective soft set over a common universe U, aij be the entities in the tabular form of (F, E). U = {x1, . . . , xm}, E = {e1, . . . , en}, i = 1, . . . , m, j = 1, . . . , n. Then aij satisfies: Pm Pn (iii) i¼1 j¼1 aij ¼ m; P P (iv) For any row in the table, nj¼1 a1j ¼    ¼ nj¼1 amj ¼ 1. Here, i = 1, . . . , m, j = 1, . . . , n. Algorithm 3. ~ EÞ is a fuzzy soft set and Suppose that ð}; S ~ ðxÞ. x2U;e2E l}ðeÞ

l}ðeÞ ~ ðxÞ is the membership function of each element and parameter.

Step 1 set M ¼

Remove entities that are lower than 0.5 in M. Step 2 While M is not empty cut the fuzzy soft set with maxM and obtain a soft set (F, E). If (F, E) satisfies the judgment condition then return koptimal = maxM Else M = M  maxM End while Step 3 If M is empty and koptimal is not obtained then output ‘the fuzzy soft set cannot be cut to a fuzzy bijective soft set’ Else output koptimal

~1 ; E1 Þ given in Example 4. Example 14. Consider the fuzzy soft set ð} From Algorithm 3, Step 1 a41, a12, a13, a63 are removed, M = [aij]63  {a41, a12, a13, a63}; Step 2 we can get

max M ¼ 0:85; max M ¼ 0:80; cut the fuzzy soft set with 0.75and obtain a bijective soft set (F, E): P P P ½aij 63 ¼ 6; For any row in the table, 3j¼1 ½a1j 13 ¼    ¼ 3j¼1 ½a6j 13 ¼ 1

koptimal ¼ max M ¼ 0:75 Step 3 output koptimal = 0.75. ~3 ; E3 Þ; ð} ~4 ; E4 Þ, the optimal cut levels are 0.75,0.75,0.80 respectively. ~2 ; E2 Þ; ð} Similarly, for fuzzy soft sets ð} 4.2. An example analysis of the two algorithms ~1 ; E1 Þ; ð} ~2 ; E2 Þ; ð} ~3 ; E3 Þ; ð} ~4 ; E4 Þ, we can use Algorithm 3 Consider the fuzzy soft set given in Example 4. For fuzzy soft sets ð} ~4 ; E4 Þ, 0.75-level to obtain the optimal cut levels, which are 0.75, 0.75,0.75,0.80 respectively. Meanwhile, for fuzzy soft set ð} soft set of it is still a bijective soft set. Therefore, we make 0.75 as the optimal cut level of the fuzzy soft decision system   S3 ~i ; Ei Þ; ð} ~4 ; E4 Þ; U . i¼1 ð} ~ EÞ ¼ [~ni¼1 ð} ~i ; Ei Þ. The tabular representation of the 0.75-level bijective fuzzy soft set ð}; ~ EÞ can be shown in Suppose that ð}; Table 9.

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Table 9 ~ EÞ. The tabular representation of the 0.75-level bijective fuzzy soft set ð};

x1 x2 x3 x4 x5 x6

High

Med

Low

Good

Avg.

No

Yes

Profit

Loss

0.75 0 0 0 0 0.80

0 0.80 0.85 0 0.80 0

0 0 0 0.80 0 0

0.80 0.80 0.80 0 0 0

0 0 0 0.85 0.85 0.80

0.75 0.75 0.80 0.80 0 0

0 0 0 0 0.85 0.80

0.90 0 0.85 0 0 0.80

0 0.80 0 0.85 0.85 0

  ~i ; Ei Þ; ð} ~4 ; E4 Þ; U 0:75 . In Algorithm 1, we construct 0.75-level bijective fuzzy soft decision system [e3i¼1 ð} In the step 2, we can calculate each dependency between condition bijective soft sets and the decision bijective soft set as following.

c0:75 ððF 1 ; E1 Þ; ðF 4 ; D4 ÞÞ ¼ 0:4895; c0:75 ððF 2 ; E2 Þ; ðF 4 ; D4 ÞÞ ¼ 0; c0:75 ððF 3 ; E3 Þ; ðF 4 ; D4 ÞÞ ¼ 0; c0:75 ððF 3 ; E3 Þ ^ ðF 1 ; E1 Þ; ðF 4 ; D4 ÞÞ ¼ 0:6702; c0:75 ððF 2 ; E2 Þ ^ ðF 1 ; E1 Þ; ðF 4 ; D4 ÞÞ ¼ 0; 6632; c0:75 ððF 2 ; E2 Þ ^ ðF 3 ; E3 Þ; ðF 4 ; D4 ÞÞ ¼ 0:1702: In the step 3, we can obtain the bijective soft decision system dependency j = 0.6702.   In the step 4, since j0.75 = c0.75((F3, E3) ^ (F1, E1), (F4, D4)), we can obtain the reduct of [~3i¼1 ðF i ; Ei Þ; ðF 4 ; E4 Þ; U 0:75 is ~ 1 ; E1 Þ. ðF 3 ; E3 Þ[ðF If the representation ability of each element differs, then we should choose weighted bijective fuzzy soft decision system to parameters reduction process. Suppose that W = [0.16,0.13,0.19,0.23,0.10,0.19]. In the step 2, we can calculate each dependency between condition bijective soft sets and the decision bijective soft set as following.

c0:75 wððF 1 ; E1 Þ; ðF 4 ; D4 ÞÞ ¼ 0:5689; c0:75 wððF 2 ; E2 Þ; ðF 4 ; D4 ÞÞ ¼ 0; c0:75 wððF 3 ; E3 Þ; ðF 4 ; D4 ÞÞ ¼ 0; c0:75 wððF 3 ; E3 Þ ^ ðF 1 ; E1 Þ; ðF 4 ; D4 ÞÞ ¼ 0:6824; c0:75 wððF 2 ; E2 Þ ^ ðF 1 ; E1 Þ; ðF 4 ; D4 ÞÞ ¼ 0:6768; c0:75 wððF 2 ; E2 Þ ^ ðF 3 ; E3 Þ; ðF 4 ; D4 ÞÞ ¼ 0:2342: In the step 3, we can obtain the bijective soft decision system dependency j = 0.6824, a little higher than the result of algorithm 1.   In the step 4, since j0.75w = c0.75w((F3, E3) ^ (F1, E1), (F4, D4)), we can obtain the reduct of [~3i¼1 ðF i ; Ei Þ; ðF 4 ; E4 Þ; U 0:75 is ~ 1 ; E1 Þ, which is the same as Algorithm 1. ðF 3 ; E3 Þ[ðF If different time series correspond to different weights, then Algorithm 2 becomes more applicable in weighted situations or time series analysis than Algorithm 1.

5. Conclusion This paper proposed the concept of bijective fuzzy soft set and defined some operations on it, such as, the restricted AND, the relaxed AND operation on a bijective fuzzy soft set, dependency between two bijective fuzzy soft sets, and reduction of bijective fuzzy soft set with respect to bijective fuzzy soft decision system. Based on these definitions, this paper proposed two parameters reduction algorithms of a bijective fuzzy soft system: one (Algorithm 1) is suitable for bijective fuzzy soft system, and the other (Algorithm 2) takes weight of an element into consideration. As a critical factor in these algorithms, k helps to cut the fuzzy soft set into a bijective soft set and influences the corresponding decision rules and parameters reduction we get. Since the result given by decision makers may be very subjective, we proposed an algorithm (Algorithm 3) to decide the optimal value of k specially. On the basis of that, an example analysis of the two parameters reduction algorithms is given. It reveals that the two algorithms lead to the same parameters reduction of a bijective fuzzy soft system. Since Algorithm 2 considers the detail weights of elements, thus it can be used in more uncertain problems, such as time series analysis problems, than Algorithm 1.

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Acknowledgements Our work is supported by National Science Foundation Project of CQ CSTC (cstcjjA30007), Science and Technology Research Project of Chongqing Municipal Education Commission (KJ110414), and Humanity and Social Science Youth foundation of Ministry of Education of China (12YJC630053). The authors express sincere gratefulness to the anonymous reviewers for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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