European Journal of Operational Research 207 (2010) 848–855

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Soft set theory and uni–int decision making

Naim Çagman *, Serdar EnginogluDepartment of Mathematics, Faculty of Arts and Sciences, Gaziosmanpas�a University, 60250 Tokat, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 December 2008Accepted 3 May 2010Available online 12 May 2010

Keywords:Soft setsSoft operationsSoft productsuni–int Decision functionuni–int Decision making

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.05.004

* Corresponding author.E-mail addresses: ncagman@gop.edu.tr (N. Çagma

We firstly redefine the operations of Molodtsov’s soft sets to make them more functional for improvingseveral new results. We also define products of soft sets and uni–int decision function. By using these newdefinitions we then construct an uni–int decision making method which selects a set of optimum ele-ments from the alternatives. We finally present an example which shows that the method can be success-fully applied to many problems that contain uncertainties.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Problems in many fields involve data that contain uncertainties. Uncertainties may be dealt with using a wide range of existing theoriessuch as theory of probability, fuzzy set theory [29], intuitionistic fuzzy sets [3], vague sets [7], theory of interval mathematics [8], rough settheory [20], etc. All of these theories have their own difficulties which are pointed out in [18]. To overcome these difficulties, Molodtsov[18] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties.

In [16,18], Molodtsov pointed out several directions for the applications of soft sets, such as smoothness of functions, game theory, oper-ations research, Riemann-integration, Perron integration, probability, theory of measurement and so on. At present, works on soft set the-ory and its applications are progressing rapidly. The help of rough mathematics of Pawlak [20], Maji et al. [14] defined a parameterreduction on soft sets, and presented an application of soft sets in a decision making problem. Chen et al. [5] and Kong et al. [11] presenteda new definition of the parameter reduction. Xiao et al. [27], and Pei and Miao [22] discussed the relationship between soft sets and infor-mation systems. They showed that soft sets are a class of special information systems.

Maji et al. [13] published a detailed theoretical study on soft sets. By using this study, the algebraic structure of soft set theory has beenstudied increasingly in recent years. Aktas� and Çagman [2] gave a definition of soft groups. They also compared soft sets to the related con-cepts of fuzzy sets and rough sets. Jun [9] introduced the notion of soft BCK/BCI-algebras and soft subalgebras. Jun and Park [10] dealt withthe algebraic structure of BCK/BCI-algebras by applying soft set theory. Park et al. [21] introduced the notion of soft WS-algebras and thenderived their basic properties. Feng et al. [6] initiated the study of soft semirings by using the soft set theory and investigated several re-lated properties. Sun et al. [24] introduced a basic version of soft module theory, which extends the notion of module by including somealgebraic structures in soft sets.

A soft set is a parameterized family of subsets of the universe. In the soft set theory, the parameters are fuzzy concepts in real worldfrom the viewpoint of fuzzy set theory. Some researchers have worked on fuzzy soft sets. Majumdar and Samanta [15] introduced severalsimilarity measures of fuzzy soft sets. Roy and Maji [23] presented some results on an application of fuzzy soft sets in decision makingproblem. Yang et al. [28] defined the reduction of fuzzy soft sets and then analyzed a decision making problem by fuzzy soft sets. Basedon the theory of soft sets, the analysis was developed in [17], and the notions of soft number, soft derivative, soft integral, etc. are formu-lated. This technique is applied to soft optimization problems by Kovkov et al. [12]. Xiao et al. [25] introduced the soft set theory for fore-casting the export and import volume in international trade. They proposed a combined forecasting approach based on the fuzzy soft sets.Xiao et al. [26] introduced soft set theory into the research of business competitive capacity evaluation.

ll rights reserved.

n), serdarenginoglu@gop.edu.tr (S. Enginoglu).

N. Çagman, S. Enginoglu / European Journal of Operational Research 207 (2010) 848–855 849

Furthermore, Zou and Xiao [30] presented data analysis approaches of soft sets under incomplete information. These approaches pre-sented in [30] are preferable for reflecting actual states of incomplete data in soft sets. Mushrif et al. [19] proposed a new algorithm forclassification of the natural textures. Ali et al. [1] introduced the analysis of several operations on soft sets.

Up to the present, the applications of the soft set theory generally solve problems with the help of the rough sets or fuzzy soft sets. Inthis paper, we first redefine the operations of soft sets which are more functional to make theoretical studies of soft set theory in moredetail and improve several results. We then introduce soft decision making method which selects a set of optimum elements from thealternatives without using the rough sets and fuzzy soft sets. We finally give an example which shows that the method can be successfullyapplied to many fields.

The presentation of the rest of the paper is organized as follows. In the next section, the operations of soft sets are redefined. In Section 3,the products of soft sets are defined and their basic properties are studied. In Section 4, uni–int decision function is defined to constructuni–int decision making method, and the method is applied to a problem. In the final section, some concluding comments are presented.

2. Soft set theory

In this section, we give new definitions and various results on soft set theory. Throughout this work, U refers to an initial universe, E is aset of parameters, P(U) is the power set of U, and A # E.

Definition 1. A soft set FA on the universe U is defined by the set of ordered pairs

FA ¼ fðx; fAðxÞÞ : x 2 E; fAðxÞ 2 PðUÞg;

where fA : E ? P(U) such that fA(x) = ; if x R A.Here, fA is called approximate function of the soft set FA. The subscript A in the notation fA indicates that fA is the approximate function of

FA. The value of fA(x) is a set called x-element of the soft set for all x 2 E. It is worth noting that the sets fA(x) may be arbitrary. Some of themmay be empty, some may have nonempty intersection.

From now on, we will use S(U) instead of the set of all soft sets over U.To illustrate this idea, let us consider the following example.

Example 1. A soft set FA describes the capability of the candidates who are wanted to fill a position for a company. Suppose that there aresix candidates in the universe U = {ul,u2,u3,u4,u5,u6} under consideration, and E = {x1,x2,x3,x4,x5} is the set of parameters. The xi

(i = 1,2,3,4,5) stand for the parameters ‘‘experience”, ‘‘computer knowledge”, ‘‘young age”, ‘‘higher education”, and ‘‘good health”,respectively. In this case, to define a soft set means to point out experienced workers, computer knowledge workers, and so on.

Assume that A = {x1,x3,x4} # E and fA(x1) = {u2,u4}, fA(x3) = U, and fA(x4) = {u1,u3,u5}. Then we can view the soft set FA as consisting of thefollowing collection of approximations:

FA ¼ fðx1; fu2;u4gÞ; ðx3;UÞ; ðx4; fu1; u3; u5gÞg:

Definition 2. Let FA 2 S(U). If fA(x) = ; for all x 2 E, then FA is called a empty soft set, denoted by FU.fA(x) = ;means that there is no element in U related to the parameter x 2 E. Therefore, we do not display such elements in the soft sets, as

it is meaningless to consider such parameters.

Definition 3. Let FA 2 S(U). If fA(x) = U for all x 2 A, then FA is called A-universal soft set, denoted by FeA . If A = E, then the A-universal soft setis called universal soft set denoted by FeE .

Example 2. Let U = {u1,u2,u3,u4,u5} be a universal set and E = {x1,x2, x3,x4} be a set of parameters.If A = {x2,x3,x4} and fA(x2) = {u2,u4}, fA(x3) = ;, fA(x4) = U, then the soft set FA is written by FA = {(x2, {u2,u4}), (x4,U)}.If B = {x1,x3} and fB(x1) = ;, fB(x3) = ;, then the soft set FB is an empty soft set. That is, FB = FU.If C = {x1,x2} and fC(x1) = U, fC(x2) = U, then the soft set FC is a C-universal soft set. That is, FC ¼ FeC .If D = E, and fD(xi) = U for all xi 2 E, i = 1,2,3,4, then the soft set FD is the universal soft set. That is, FD ¼ FeE .

Definition 4. Let FA, FB 2 S(U). Then, FA is a soft subset of FB, denoted by FAf# FB, if fA(x) # fB(x) for all x 2 E.

Remark 1. FAf# FB does not imply that every element of FA is an element of FB. Therefore the definition of classical subset is not valid for thesoft subset. For example, let U = {u1,u2,u3,u4} be a universal set of objects and E = {x1,x2,x3} be the set of all parameters. If A = {x1} andB = {x1,x3}, and FA = {(x1, {u2,u4})}, FB = {(x1, {u2,u3,u4}), (x3, {u1,u5})}, then for all e 2 FA, fA(x) # fB(x) is valid. Hence FAf# FB. It is clear that,(x1, fA(x1)) 2 FA but (x1, fA(x1)) R FB.

Proposition 1. If FA, FB 2 S(U), then

i. FAf# FeE .ii. FUf# FA.

iii. FAf# FA.iv. FAf# FB and FBf# FC ) FAf# FC .

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Proof. They can be proved by using the approximate functions of the soft sets. For all x 2 E,

i. fAðxÞ# feEðxÞ, since fA(x) # U.ii. fU(x) # fA(x), since ; # fA(x).

iii. fA(x) # fA(x), since fA(x) = fA(x).iv. fA(x) # fB(x) and fB(x) # fC(x)) fA(x) # fC(x). h

Definition 5. Let FA, FB 2 S(U). Then, FA and FB are soft equal, denoted by FA = FB, if and only if fA(x) = fB(x) for all x 2 E.

Proposition 2. If FA, FB, FC 2 S(U), then

i. FA = FB and FB = FC, FA = FC.ii. FAf# FB and FBf# FA () FA ¼ FC .

Proof. The proof follows from the approximate functions. For all x 2 E,

i. fA(x) = fB(x) and fB(x) = fC(x),fA(x) = fC(x).ii. fA(x) # fB(x) and fB(x) # fA(x), fA(x) = fB(x). h

Definition 6. Let FA 2 S(U). Then, complement of FA, denoted by F~cA, is a soft set defined by the approximate function

fA~c ðxÞ ¼ f cA ðxÞ; for all x 2 E;

where f cA ðxÞ is complement of the set fA(x), that is, f c

A ðxÞ ¼ U n fAðxÞ for all x 2 E.It must be noted that to keep clear of confusion, we use two different symbols ~c and c which represent the complement of soft sets and

classical sets, respectively. But, in the subscripts, the symbol ~c indicates that fA~c is the approximate function of F~cA, is not a set operation.

Proposition 3. If FA 2 U, then

i. ðF~cAÞ

~c ¼ FA.ii. F~c

U ¼ FeE .

Proof. By using the approximate functions of the soft sets we can easily get the proofs. For all x 2 E,

i. f cAðxÞ� �c ¼ fAðxÞ.

ii. f cUðxÞ ¼ U n fUðxÞ ¼ U n ; ¼ U ¼ feEðxÞ. h

Definition 7. Let FA, FB 2 S(U). Then, union of FA and FB, denoted by FA ~[FB, is a soft set defined by the approximate function

fA ~[BðxÞ ¼ fAðxÞ [ fBðxÞ; for all x 2 E:

Proposition 4. If FA, FB, FC 2 S(U), then

i. FA ~[FA ¼ FA FA ~[FU ¼ FA.ii. FA ~[FeE ¼ FeE .

iii. FA ~[F~cA ¼ FeE .

iv. FA ~[FB ¼ FB ~[FA.

A B C A B C v. ðF ~[F Þ ~[F ¼ F ~[ðF ~[F Þ.

vi. ðFA ~[FBÞ ~[FC ¼ FA ~[ðFB ~[FCÞ.

Definition 8. Let FA, FB 2 S(U). Then, intersection of FA and FB, denoted by FA ~\FB, is a soft set defined by the approximate function

fA ~\BðxÞ ¼ fAðxÞ \ fBðxÞ; for all x 2 E:

Proposition 5. If FA, FB, FC 2 S(U), then

i. FA ~\FA ¼ FA.ii. FA ~\FU ¼ FU.

iii. FA ~\FeE ¼ FA.iv. FA ~\F~c

A ¼ FU.

v. FA\FB ¼ FB\FA.

vi. ðF ~\F Þ ~\F ¼ F ~\ðF ~\F Þ.

~ ~

A B C A B C

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Proposition 6. Let If FA, FB 2 S(U). Then De Morgan’s laws are valid

i. ðFA ~[FBÞ~c ¼ F~c

A~\F~c

B.ii. ðFA ~\FBÞ~c ¼ F~c

A~[F~c

B.

Proof. The proofs can be made by using the respective approximate function. For all x 2 E,

i. fðA ~[BÞ~c ðxÞ ¼ f cA ~[BðxÞ ¼ ðfAðxÞ [ fBðxÞÞc ¼ ðfAðxÞÞc \ ðfBðxÞÞc ¼ fA~c ~\B~c ðxÞ.

The proof of ii. can be made similarly. h

Proposition 7. If FA, FB, FC 2 S(U), then

i. FA ~[ðFB ~\FCÞ ¼ ðFA ~[FBÞ ~\ðFA ~[FCÞ.ii. FA ~\ðFB ~[FCÞ ¼ ðFA ~\FBÞ ~[ðFA ~\FCÞ.

Proof. The proofs can be made by using the respective approximate functions. For all x 2 E,

i: f A ~[ðB ~\CÞðxÞ ¼ fAðxÞ [ fB ~\CðxÞ¼ fAðxÞ [ ðfBðxÞ \ fCðxÞÞ¼ ðfAðxÞ [ fBðxÞÞ \ ðfAðxÞ [ fCðxÞÞ¼ fA ~[BðxÞ \ fA ~[CðxÞ¼ fðA ~[BÞ ~\ðA ~[CÞðxÞ:

The proof of ii. can be made by similar way. h

Definition 9. Let FA, FB 2 S(U). Then, difference of FA and FB, denoted by FA ~\FB, is a soft set defined by the approximate function

fA ~nBðxÞ ¼ fAðxÞ n fBðxÞ; for all x 2 E:

It must be noted that the binary soft sets operations, ~\, ~[ and ~n are used in the subscripts of approximate functions, are not classical setoperations. They indicate that fA ~\B, fA ~[B and fA ~nB are the approximate function of FA ~\FB, FA ~[FB and FA

~nFB, respectively.

Proposition 8. If FA, FB, FC 2 S(U), then

i. FA~nFB ¼ FA ~\F~c

B.ii. FA

~nFB ¼ FU () FAf# FB.iii. A \ B ¼ ; ) FA

~nFB ¼ FA and FB~nFA ¼ FB.

Proof. They can be proved by using the approximate functions of the soft sets. For all x 2 E,

i. fA ~nBðxÞ ¼ fAðxÞ n fBðxÞ ¼ fAðxÞ \ fBðxÞc .ii. fA(x) n fB(x) = fU(x) = ; , fA(x) # fB(x).

iii. A \ B = ; ) fA(x) n fB(x) = fA(x) and fB(x) n fA(x) = fB(x). h

3. Products of soft sets

Until now, we defined the binary operations of soft sets which depend on an approximate function of one variable. Now, we define prod-ucts of the soft sets which are binary operations of soft sets depending on an approximate function of two variables. We have four kinds ofproducts in the soft set theory. They are And-product, Or-product, And–Not-product and Or–Not-product that are denoted by ^-product, _-product, Z-product and Y-product, respectively.

Definition 10. If FA, FB 2 S(U), then ^-product of two soft sets FA and FB, denoted by FA ^ FB, is a soft set defined by the approximate function

fA^B : E� E! PðUÞ; f A^Bðx; yÞ ¼ fAðxÞ \ fBðyÞ:

Definition 11. If FA, FB 2 S(U), then _-product of two soft sets FA and FB, denoted by FA _ FB, is a soft set defined by the approximate function

fA_B : E� E! PðUÞ; f A_Bðx; yÞ ¼ fAðxÞ [ fBðyÞ:

Definition 12. If FA, FB 2 S(U), then Z-product of two soft sets, denoted by FAZFB, is a soft set defined by the approximate function

fAZB : E� E! PðUÞ; f AZBðx; yÞ ¼ fAðxÞ \ f cB ðyÞ:

It is clear to see that FAZFB ¼ FA ^ F~cB.

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Definition 13. If FA, FB 2 S(U), then Y-product of two soft sets, denoted by FA Y FB, is a soft set defined by the approximate function

fAYB : E� E! PðUÞ; f AYBðx; yÞ ¼ fAðxÞ [ f cB ðyÞ:

It is clear to see that FAYFB ¼ FA _ F~cB.

It must be noted the binary product operations, _, ^, Z and Y are used in the subscripts of approximate functions, are not classical setoperations. They indicate that fA_B, fA^B, fAZB and fAYB are the approximate function of FA _ FB, FA ^ FB, FAZFB and FA Y FB, respectively.

Let us illustrate above definitions in the following example.

Example 3. Let U = {u1,u2,u3,u4,u5} be a universal set and E = {x1,x2,x3,x4} be the set of all parameters. Assume that A = {x2,x3,x4} andB = {x1,x3,x4} are two subsets of E. Then we can write the following soft sets,

FA ¼ fðx2; fu2;u3;u4;u5gÞ; ðx3; fu1; u2; u3gÞ; ðx4; fu1; u2;u5gÞg;FB ¼ fðx1; fu1;u2gÞ; ðx3; fu3;u4; u5gÞ; ðx4;UÞg:

Now, we can find FA ^ FB as follows:

fððx2; x1Þ; fu2gÞ; ððx2; x3Þ; fu3;u4;u5gÞ; ððx2; x4Þ; fu2;u3;u4;u5gÞ;ððx3; x1Þ; fu1;u2gÞ; ððx3; x3Þ; fu3gÞ; ððx3; x4Þ; fu1; u2; u3gÞ;ððx4; x1Þ; fu1;u2gÞ; ððx4; x3Þ; fu5gÞ; ððx4; x4Þ; fu1; u2; u5gÞg:

Similarly, we can also find the others products FA _ FB, FAZFB and FA Y FB.It must be noted that ^, _, Z and Y are not commutative.

Proposition 9. If FA, FB, FC 2 S(U), then

i. FA _ (FB _ FC) = (FA _ FB) _ FC.ii. FA ^ (FB ^ FC) = (FA ^ FB) ^ FC.

It must be noted that Z and Y are not associative.

Proposition 10. If FA, FB 2 S(U), then the following De Morgans types of results are true.

i. ðFA _ FBÞ~c ¼ F~c

A ^ F~cB.

ii. ðFA ^ FBÞ~c ¼ F~c

A _ F~cB.

iii. ðFAYFBÞ~c ¼ F~c

AZF~cB.

iv. ðFAZFBÞ~c ¼ F~c

AYF~cB.

Proof. The proofs can be made by using the respective approximate functions. For all x, y 2 E,

i: f ðA_BÞ~c ðx; yÞ ¼ f cðA_BÞðx; yÞ

¼ ðfAðxÞ [ fBðyÞÞc

¼ f cA ðxÞ \ f c

B ðyÞ¼ fA~c^B~c ðx; yÞ:

ii: f ðAYBÞ~c ðx; yÞ ¼ f cðAYBÞðx; yÞ

¼ fAðxÞ [ f cB ðyÞ

� �c

¼ f cA ðxÞ \ fBðyÞ

¼ f cA ðxÞ \ f c

B ðyÞ� �c

¼ fA~cZB~c ðx; yÞ:

The proofs of ii. and iv. can be made similarly. h

4. uni–int Decision making

In this section, we define uni–int operators and uni–int decision function for the ^-product to construct a uni–int decision making meth-od. The method reduces a set to its subset according to the parameters of decision makers. Therefore, decision makers works on small num-ber of alternatives instead of large numbers.

Throughout this section, assume that ^(U) is a set of all ^-products of the soft sets over U.

Definition 14. Let FA ^ FB 2 ^(U). Then uni–int operators for the ^-products, denoted by unixinty and uniyintx, are defined, respectively,as

unixinty : ^ðUÞ ! PðUÞ; unixintyðFA ^ FBÞ ¼ [x2Að\y2BðfA^Bðx; yÞÞÞ;uniyintx : ^ðUÞ ! PðUÞ; uniyintxðFA ^ FBÞ ¼ [y2Bð\x2AðfA^Bðx; yÞÞÞ:

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Each of them transforms the ^-product FA ^ FB into a subset of the universe U. The operator unixinty is called altered operator of uniyintx,denoted by alt-unixinty or vice versa. That is, alt-unixinty = uniyintx and alt-alt-unixinty = unixinty.

The values of unixinty(FA ^ FB) and uniyintx(FA ^ FB) are called uni–int set and alt-uni–int set of FA ^ FB, respectively.

It is clear to see that unixinty(FA ^ FB) – uniyintx(FA ^ FB), but following proposition is true.

Proposition 11. Let FA ^ FB 2 ^(U). Then,

unixintyðFA ^ FBÞ ¼ uniyintxðFB ^ FAÞ:

Proof. The proof is easy from Definition 14. h

Definition 15. Let FA ^ FB 2 ^(U). Then uni–int decision function for the ^-products, denoted by uni–int, is defined by,

uni—int : ^ðUÞ ! PðUÞ;uni—intðFA ^ FBÞ ¼ unixintyðFA ^ FBÞ [ uniyintxðFA ^ FBÞ

that reduces the size of the universe U. Hence, the values uni–int(FA ^ FB) is a subset of U called uni–int decision set of FA ^ FB.We know that ^-product is not commutative, but following proposition is true.

Proposition 12. Let FA ^ FB 2 ^(U). Then

uni—intðFA ^ FBÞ ¼ uni—intðFB ^ FAÞ:

Proof. The proof is trivial from Proposition 11. h

Assume that a set of alternatives and a set of parameters are given. Then we select a set of optimum alternatives by the uni–int decisionmaking method which is organized as in the following algorithm: According to the problem,

Step 1: Choose feasible subsets of the set of parameters,Step 2: Construct the soft sets for each set of parameters,Step 3: Find the ^-product of the soft sets,Step 4: Compute the uni–int decision set of the product.

Note that obtained uni–int decision set is not small enough to work on it, subset of the decision set can be reached by the method.Now, we can represent an application of soft set theory in the uni–int decision making problem for the ^-products.

Example 4. Assume that a company wants to fill a position. There are 48 candidates who fill in a form in order to apply formally for theposition. There are two decision makers; one of them is from the department of human resources and the other one is from the board ofdirectors. They want to interview the candidates, but it is very difficult to make it all of them. Therefore, by using the uni–int decisionmaking method, the number of candidates are reduced to a suitable one.

Assume that the set of candidates U = {u1,u2, . . . ,u48} which may be characterized by a set of parameters E = {x1,x2 , . . . ,x7}. Fori = 1,2, . . . ,7, the parameters xi stand for ‘‘experience”, ‘‘computer knowledge”, ‘‘training”, ‘‘young age”, ‘‘higher education”, ‘‘marriagestatus” and ‘‘good health”, respectively.

Now, we can apply the uni–int–^ as follows:

Step 1: The decision makers considers set of parameters, A = {x ,x ,x ,x } and B = {x ,x ,x }, respectively, to evaluate the candidates.

1 2 4 7 1 2 5

Step 2: The decision makers seriously investigate CV of the candidates. After a serious discussion each candidate is evaluated from point ofview of the goals and the constraint according to a chosen subset A, B # E. Then the decision makers constructs the following two soft setsover U according to their parameters, respectively,

FA ¼ fðx1; fu4;u7;u13;u21;u28; u31; u32;u36;u39; u41;u43;u44;u48gÞ;ðx2; fu1;u3;u13;u18; u19;u21;u22;u24; u28;u32;u36;u42;u44;u46gÞ;ðx4; fu2;u3;u13;u15; u18;u23;u25;u28; u30;u33;u36;u38;u42;u43gÞ;ðx7; fu1;u5;u12;u13; u17;u20;u24;u28; u29;u34;u36;u41;u45;u47gÞg

FB ¼ fðx1; fu3;u4;u5;u8; u14;u21;u22;u26; u27;u34;u35;u37;u40;u42;u46gÞ;ðx2; fu1;u4;u7;u10;u11; u13;u15;u21;u29; u30; u32;u36;u42;u43;u45gÞ;ðx5; fu2;u4;u8;u9;u12;u13; u14;u16;u17;u21; u23;u28;u36;u42;u44gÞg:

Step 3: Now, we can find the ^-product FA ^ FB of the soft sets FA and FB as follows:

fððx1; x1Þ; fu4;u21gÞ; ððx1; x2Þ; fu4;u7;u13;u21;u32; u36;u43gÞ;ððx1; x5Þ; fu4;u13; u21;u28;u36;u44gÞ; ððx2; x1Þ; fu3;u21;u22;u42;u46gÞ;ððx2; x2Þ; fu1;u13; u21;u32;u36;u42gÞ; ððx2; x5Þ; fu13; u21;u28;u36;u42;u44gÞ;ððx4; x1Þfu3;u42gÞ; ððx4; x2Þ; fu13;u15;u30; u36;u42;u43gÞ;

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ððx4; x5Þ; fu2;u13;u23; u28; u36;u42gÞ; ððx7; x1Þ; fu5;u34gÞ;ððx7; x2Þ; fu1;u13;u29; u36; u45gÞ; ððx7; x5Þ; fu12;u13;u17;u28;u36gÞg

Step 4: Finally, we can find a decision set uni–int(FA ^ FB) as follows:

unixintyðFA ^ FBÞ ¼ [x2Að\y2BðfA^Bðx; yÞÞÞ ¼[

\ffu4;u21g; fu4;u7;u13;u21;u32;u36;u43g;fu4;u13;u21;u28;u36;u44gg;

\ffu3;u21;u22;u42;u46g; fu1;u13;u21;u32;u36;u42g;fu13;u15;u30;u36;u42;u44gg;

\ffu3;u42g; fu13;u15;u30;u36;u42;u43g;fu2;u13;u23;u28;u36;u42gg;

\ffu5;u34g; fu1;u13;u29;u36;u45g;fu12;u13;u17;u28;u36gg

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

¼ [ffu4;u21g; fu42g; fu42g; ;g ¼ fu4;u21;u42g

and

uniyintxðFA ^ FBÞ ¼ [y2Bð\x2AðfA^Bðx; yÞÞÞ ¼[

\ffu4;u21g; fu3;u21;u22;u42;u46g; fu3;u42g; fu5;u34gg;\ffu4;u7;u13;u21;u32;u36;u43g; fu1;u13;u21;u32;u36;u42g;

fu13;u15;u30;u36;u45g; fu1;u13;u29;u36;u42;u44gg;\ffu4;u13;u21;u28;u36;u44g; fu13;u15;u30;u36;u42;u43g;

fu2;u13;u23;u28;u36;u42gg; fu12;u13;u17;u28;u36gg

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼ [f;; fu13;u36g; fu13; u36gg ¼ fu13;u36g:

Hence, the decision makers to make interviewing invite the candidates which are elements of the following uni–int decision set.

uni� intðFA ^ FBÞ ¼ unixintyðFA ^ FBÞ [ uniyintxðFA ^ FBÞ ¼ fu4;u21;u42g [ fu13;u36g ¼ fu4;u13;u21;u36;u42g:

5. Conclusion

Main aim of this paper is redefine the operations of soft sets and construct a uni–int decision making method. The method is constructedby using the union-intersection decision function for the ^-products. It should be easy that, a group of a similar type uni–int decision mak-ing methods can be constructed for the _-product, Z-product and Y-product of the soft sets. In addition, by using intersection–union (int–uni), union-union (uni–uni) and intersection-intersection (int–int) decision functions other groups of similar type decision making methodscan also be constructed by the similar way with some small modifications. In [4], Çagman and Enginoglu already introduced a matrix rep-resentation of this work that gives several advantages to compute applications of the soft set theory. We hope that this works would bebeneficial to extend the proposed method to subsequent studies.

Acknowledgment

The authors are grateful for financial support from the Research Fund of Gaziosmanpasa University under Grant No.: 2007/11.

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