﻿ Fuzzy Soft Expert Set and Its Application

Applied Mathematics
Vol.5 No.9(2014), Article ID:46144,20 pages DOI:10.4236/am.2014.59127

Fuzzy Soft Expert Set and Its Application

Shawkat Alkhazaleh1, Abdul Razak Salleh2*

1Department of Mathematics, Faculty of Science and Art, Shaqra University, Shaqraa, KSA

2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Email: shmk79@gmail.com, *aras@ukm.my

Received 27 February 2014; revised 27 March 2014; accepted 3 April 2014

ABSTRACT

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh and Salleh (2011) define the concept of soft expert sets where the user can know the opinion of all experts in one model and give an application of this concept in decision making problem. So in this paper, we generalize the concept of a soft expert set to fuzzy soft expert set, which will be more effective and useful. We also define its basic operations, namely complement, union, intersection, AND and OR. We give an application of this concept in decision making problem. Finally, we study a mapping on fuzzy soft expert classes and its properties.

Keywords:Soft Set, Soft Expert Set, Fuzzy Soft Expert Set

1. Introduction

Many scientists wish to find appropriate solutions to some mathematical problems that cannot be solved by traditional methods. These problems lie in the fact that traditional methods cannot solve the problems of uncertainty in economy, engineering, medicine and the problems of decision-making and others. One of these solutions is fuzzy sets—the title of Zadeh’s first article about his new mathematical theory, which was published in a scientific journal in 1965. Since Zadeh published his new classic paper almost fifty years ago, fuzzy set theory has received more and more attention from researchers in wide range of scientific areas, especially in the past few years. The difference between a binary set and a fuzzy set is that in a “normal” set every element is either a member or a non-member of the set. Here, we see that it either has to be A or not A. In a fuzzy set, an element can be a member of a set to some degree and at the same time a non-member to some degree of the same set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the closed unit interval [0,1]. Fuzzy sets generalise classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Therefore, a fuzzy set A in a universe of discourse X is a function, usually this function is referred to as the membership function and denoted by. Some mathematicians use the notation to denote the membership function instead of. A fuzzy set A is written symbolically in various

Molodtsov [1] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. After Molodtsov’s work, some operations and application of soft sets were studied by Chen et al. [2] , Maji et al. [3] and Maji et al. [4] . Also Maji et al. [5] have introduced the concept of fuzzy soft set, a more general concept, which is a combination of fuzzy set and soft set and studied its properties and also Roy and Maji [6] used this theory to solve some decision making problems. Alkhazaleh et al. [7] introduced the concept of soft multisets as a generalization of soft set. They also defined the concepts of fuzzy parameterized interval-valued fuzzy soft set [8] and possibility fuzzy soft set [9] and gave their applications in decision making and medical diagnosis. Alkhazaleh and Salleh [10] introduced the concept of a soft expert set, where the user can know the opinion of all experts in one model without any operations. Even after any operation the user can know the opinion of all experts. So in this paper, we introduce the concept of a fuzzy soft expert set, which will be more effective and useful which is a combination of fuzzy set and soft expert set. We also define its basic operations, namely complement, union, intersection, AND and OR and study their properties. We give an application of this concept in decision making problem. Finally, we study a mapping on fuzzy soft expert classes and its properties.

2. Preliminaries

In this section, we recall some basic notions related to this work. Molodtsov defined soft set in the following way. Let U be a universe and E be a set of parameters. Let denote the power set of U and.

Definition 1 [1] A pair is called a soft set over U where F is a mapping

In other words, a soft set over U is a parameterized family of subsets of the universe U For may be considered as the set of -approximate elements of the soft set.

Definition 2 [5] Let U be an initial universal set and let be a set of parameters. Let denote the power set of all fuzzy subsets of. Let A pair is called a fuzzy soft set over U where F is a mapping given by

Let U be a universe, E a set of parameters, X a set of experts (agents), and O a set of opinions. Let Z = E × X × O and.

Definition 3 [10] A pair is called a soft expert set over where is a mapping given by

where denotes the power set of

Definition 4 [10] For two soft expert sets and over U, is called a soft expert subset of if 1.

2.

This relationship is denoted by. In this case is called a soft expert superset of.

Definition 5 [10] Two soft expert sets and over U are said to be equal if is a soft expert subset of and is a soft expert subset of

Definition 6 [10] Let be a set of parameters and X a set of experts. The NOT set of, denoted by, is defined by

Definition 7 [10] An agree-soft expert set over U is a soft expert subset of defined as follows:

Definition 8 [10] A disagree-soft expert set over is a soft expert subset of defined as follows:

Definition 9 [10] The complement of a soft expert set is denoted by and is defined by where is a mapping given by

Definition 10 [10] The union of two soft expert sets and over U, denoted by , is the soft expert set where and

Definition 11 [10] The intersection of two soft expert sets and over U, denoted by , is the soft expert set where and

Definition 12 [10] If and are two soft expert sets over U then “AND” denoted by, is defined by

where.

Definition 13 [10] If and are two soft expert sets over then “OR” denoted by, is defined by

where.

3. Fuzzy Soft Expert Set

In this section, we introduce the definition of a fuzzy soft expert set and give basic properties of this concept.

Let U be a universe, a set of parameters, a set of experts (agents), and a set of opinions. Let and.

Definition 14 A pair is called a fuzzy soft expert set over where is a mapping given by

where denotes the set of all fuzzy subsets of

Example 1 Suppose that a company produces new types of products and wants to take the opinion of some experts about these products. Let be a set of products, is a set of decision parameters where denotes the parameters “easy to use”, “quality” and “cheap”. Let be a set of experts. Suppose that

Then we can view the fuzzy soft expert set (F,Z) as consisting of the following collection of approximations:

Definition 15 For two fuzzy soft expert sets and over, is called a fuzzy soft expert subset of if 1.

2. is fuzzy subset of

This relationship is denoted by. In this case is called a fuzzy soft expert superset of.

Definition 16 Two fuzzy soft expert sets and over U are said to be equal if is a fuzzy soft expert subset of and is a fuzzy soft expert subset of

Example 2 Consider Example 1. Suppose that the company takes the opinion of the experts once again after a month of using the products. Let

and

Clearly. Let and be defined as follows:

Therefore.

Definition 17 An agree-fuzzy soft expert set over U is a fuzzy soft expert subset of defined as follows:

Definition 18 A disagree-fuzzy soft expert set over U is a fuzzy soft expert subset of defined as follows:

Example 3 Consider Example 1. Then the agree-fuzzy soft expert set over is

and the disagree-fuzzy soft expert set over is

Definition 19 The complement of a fuzzy soft expert set is denoted by and is defined by where is a mapping given by

where c is a fuzzy complement.

Example 4 Consider Example 1. By using the basic fuzzy complement, we have

Proposition 1 If is a fuzzy soft expert set over then 1.

Proof From Definition 19 we have where. Now,

where

4. Union and Intersection

In this section, we introduce the definitions of union and intersection of fuzzy soft expert sets, derive their properties, and give some examples.

Definition 20 The union of two fuzzy soft expert sets and over U, denoted by , is the fuzzy soft expert set where and

where s is an s-norm.

Example 5 Consider Example 1. Let

and

Suppose and are two fuzzy soft expert sets over such that

By using basic fuzzy union (maximum) we have where

Proposition 2 If, and are three fuzzy soft expert sets over, then 1.

2.

Proof a. We want to prove that

By using definition 20 we have

We consider the case when as the other cases are trivial, then we have

.

We also consider her the case when as the other cases are trivial, then we have

.

b. The proof is straightforward.

Definition 21 The intersection of two fuzzy soft expert sets and over, denoted by, is the fuzzy soft expert set where and

where t is a t-norm.

Example 6 Consider Example 5. By using basic fuzzy intersection (minimum) we have where

Proposition 3 If, and are three fuzzy soft expert sets over, then 1.

2.

Proof a. We want to prove that

By using definition 21 we have

We consider the case when as the other cases are trivial, then we have

.

We also consider her the case when as the other cases are trivial, then we have

.

b. The proof is straightforward.

Proposition 4 If, and are three fuzzy soft expert sets over, then 1.

2.

Proof a. We want to prove that

By using definitions 20 and 21 we have

We consider the case when as the other cases are trivial, then we have

.

We also consider her the case when as the other cases are trivial, then we have

.

b. We want to prove that

By using definitions 20 and 21 we have

We consider the case when as the other cases are trivial, then we have

.

We also consider her the case when as the other cases are trivial, then we have

.

5. AND and OR Operations

In this section, we introduce the definitions of AND and OR operations for fuzzy soft expert sets, derive their properties, and give some examples.

Definition 22 If and are two fuzzy soft expert sets over then “AND” denoted by is defined by

such that, where t is a t-norm.

Example 7 Consider Example 1. Let

and

.

Suppose and are two fuzzy soft expert sets over such that

By using basic fuzzy intersection (minimum) we have where

Definition 23 If and are two fuzzy soft expert sets over U then “OR” denoted by is defined by

such that, where s is an s-norm.

Example 8 Consider Example 7. By using basic fuzzy union (maximum) we have where

Proposition 5 If and are two fuzzy soft expert sets over then [a.]

1.

2.

Proof a. Suppose that

Therefore, Now,

where

Now, take

Therefore,

Then and are the same. Hence, proved.

b. Suppose that

Therefore, Now,

where

Now, take

Therefore,

Then and are the same. Hence, proved.

Proposition 6 If, and are three fuzzy soft expert sets over then 1.

2.

3.

4.

Proof We give the proofs of a and b. [a.]

1. Suppose that,.

Therefore

,.

,.

.

2. Suppose that,.

Therefore

,.

,.

.

Remark The commutativity do not hold in AND and OR operations since.

6. An Application of Fuzzy Soft Expert Set in Decision Making

In this section, we present an application of fuzzy soft expert set theory in a decision making problem. Assume that a company wants to fill a position. There are four candidates who form the universe the hiring committee considers a set of parameters, the parameters stand for “experience”, “computer knowledge” and “good speaking” respectively. Let be a set of experts (Committee members). After a serious discussion the committee constructs the following fuzzy soft expert set

In Table 1 and Table 2 we present the agree-fuzzy soft expert set and disagree-fuzzy soft expert set respectively.

The following algorithm may be followed by the company to fill the position.

1. Input the fuzzy soft expert set.

2. Find an agree-fuzzy soft expert set and a disagree-fuzzy soft expert set.

3. Find for agree-fuzzy soft expert set.

4. Find for disagree-fuzzy soft expert set.

5. Find

6. Find m, for which Then is the optimal choice object. If has more than one value, then any one of them could be chosen by the company using its option.

Table 1. Agree-fuzzy soft expert set.

Table 2. Disagree-fuzzy soft expert set.

Now we use this algorithm to find the best choice for the company to fill the position. From Table 1 and Table 2, we have the following in Table 3:

Then so the committee will choose candidate 4 for the job.

7. Mapping on Fuzzy Soft Expert Classes

In this section, we introduce the notion of mapping on fuzzy soft expert classes. fuzzy soft expert classes are collections of fuzzy soft expert sets. We also define and study the properties of fuzzy soft expert images and fuzzy soft expert inverse images of fuzzy soft expert sets, and support them with example and theorems.

Definition 24 Let be a universe, a set of parameters, a set of experts (agents), and a set of opinions. Let. Then the collection of all fuzzy soft expert sets over U with a parameters from is called a fuzzy soft expert class and is denoted as.

Definition 25 Let and be fuzzy soft expert classes. Let and be mappings. Then a mapping is defined as follows:

For a fuzzy soft expert set in, is a fuzzy soft expert in obtained as follows:

For, and. is called a fuzzy soft expert image of the fuzzy soft expert set.

Definition 26 Let and be fuzzy soft expert classes. Let and be mappings. Then a mapping is defined as follows:

For a fuzzy soft expert set in, is a fuzzy soft expert set in obtained as follows:

For and. is called a fuzzy soft expert inverse image of the fuzzy soft expert set.

Example 9 Let, and let

Table 3. sj = cj − kj.

and

Suppose that and are fuzzy soft expert classes. Define and as follows:

, ,

.

Let and be two fuzzy soft expert sets over U and Y respectively such that

Then we define a mapping as follows:

For a soft expert set in, is a soft expert set in where

and is obtained as follows:

.

.

.

Then

.

.

.

Then

.

.

.

Then

.

.

.

Then

.

.

.

Then

Hence

.

Next for the soft expert inverse images, the mapping is defined as follows:

For a soft expert set in, is a soft expert set in where

and is obtained as follows:

.

.

.

Then

.

.

.

Then

By similar calculations, consequently, we get

Definition 27 Let be a mapping and and fuzzy soft expert sets in

. Then for, , the fuzzy soft expert union and intersection of fuzzy soft expert images

and are defined as follows:

Definition 28 Let be a mapping and and fuzzy soft expert sets in. Then for, , the fuzzy soft expert union and intersection of fuzzy soft expert inverse images and are defined as follows:

Theorem 1 Let be a mapping. Then for fuzzy soft expert sets and in the fuzzy soft expert class, [a.]

1..

2..

3..

4..

5. If, thenProof For (a), (b) and (e) the proof is trivial, so we just give the proof of (c) and (d).

c. For and, we want to prove that

For left hand side, consider. Then

(1.1)

such that where an interval-valued fuzzy union.

Considering only the non-trivial case, then Equation 0.1 becomes:

(1.2)

For right hand side and by using Definition 27, we have

(1.3)

From Equations (1.1) and (1.3), we get (c).

d. For and, and using Definition 27, we have

.

This gives (d).

Theorem 2 Let be mapping. Then for fuzzy soft experts, in the fuzzy soft expert class, we have:

1..

2..

3..

4..

5. If, then.

Proof We use the same method as in the previous proof.

8. Conclusion

In this paper, we have introduced the concept of fuzzy soft expert set which is more effective and useful and studied some of its properties. Also the basic operations on fuzzy soft expert set namely complement, union, intersection, AND and OR have been defined. An application of this theory has been given to solve a decisionmaking problem. We also studied a mapping on fuzzy soft expert classes and its properties.

Acknowledgements

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant AP-2013-009.

References

1. Molodtsov, D. (1999) Soft Set Theory—First Results. Computers & Mathematics with Applications, 37, 19-31. http://dx.doi.org/10.1016/S0898-1221(99)00056-5
2. Chen, D., Tsang, E.C.C., Yeung, D.S. and Wang, X. (2005) The Parameterization Reduction of Soft Sets and Its Application. Computers & Mathematics with Applications, 49, 757-763. http://dx.doi.org/10.1016/j.camwa.2004.10.036
3. Maji, P.K., Roy, A.R. and Biswas, R. (2003) Soft Set Theory. Computers & Mathematics with Applications, 54, 555- 562. http://dx.doi.org/10.1016/S0898-1221(03)00016-6
4. Maji, P.K., Roy, A.R. and Biswas, R. (2002) An Application of Soft Sets in a Decision Making Problem. Computers & Mathematics with Applications, 44, 1077-1083. http://dx.doi.org/10.1016/S0898-1221(02)00216-X
5. Maji, P.K., Roy, A.R. and Biswas, R. (2001) Fuzzy Soft Sets. Journal of Fuzzy Mathematics, 9, 589-602.
6. Roy, R. and Maji, P.K. (2007) A Fuzzy Soft Set Theoretic Approach to Decision Making Problems. Journal of Computational and Applied Mathematics, 203, 412-418. http://dx.doi.org/10.1016/j.cam.2006.04.008
7. Alkhazaleh, S., Salleh, A.R. and Hassan, N. (2011) Soft Multisets Theory. Applied Mathematical Sciences, 5, 3561- 3573.
8. Alkhazaleh, S., Salleh, A.R. and Hassan, N. (2011) Fuzzy Parameterized Interval-Valued Fuzzy Soft Set. Applied Mathematical Sciences, 5, 3335-3346.
9. Alkhazaleh, S., Salleh, A.R. and Hassan, N. (2011) Possibility Fuzzy Soft Set. Advances in Decision Sciences, 2011, 18 p.

NOTES

*Corresponding author.