Journal of Applied Mathematics

Volume 2012 (2012), Article ID 328195, 22 pages

doi:10.1155/2012/328195

Research Article

Generalized Fuzzy Soft Expert Set

Ayman A. Hazaymeh, Ismail B. Abdullah, Zaid T. Balkhi, and Rose I. Ibrahim

Faculty of Science and Technology, Universiti Sains Islam Malaysia, Bandar Baru Nilai, Negeri Sembilan, 71800 Nilai, Malaysia

Received 25 March 2012; Revised 14 May 2012; Accepted 14 May 2012

Academic Editor: Mina Abd-El-Malek

Copyright © 2012 Ayman A. Hazaymeh et al. 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.

Abstract.

In 2011 Alkhazaleh and Salleh defined the concept of soft expert sets where the user can know the opinion of all the experts in one model and give an application of this concept in decision-making problems. Also, they introduced the concept of the fuzzy soft expert set as a combination between the soft expert set and the fuzzy set. In 2010 Majumdar and Samanta presented the concept of a generalized fuzzy soft sets. The purpose of this paper is to combine the work of Alkhazaleh and Salleh (2011) and Majumdar and Samanta (2010), from which we can obtain a new concept: generalized fuzzy soft expert sets (GFSESs). We also introduce its operations, namely, complement, union intersection, “AND” and “OR”, and study their properties. The generalized fuzzy soft expert sets are used to analyze a decision-making problem. Also in our model the user can know the opinion of all experts in one model. In this work we also introduce the concept of a generalized fuzzy soft expert sets with multiopinions (four opinions), which will be more effective and useful. Finally, we give an application of this concept in decision-making problem.

1. Introduction

As much of the research completed in economics, engineering, environmental science, sociology, medical science, and other related fields involve uncertain data, it is not always possible to use classical methods when analyzing the information. This can be due to the fact that the information may come in different formats; to solve problems we use data in its many different forms. So we need new mathematical way free from difficulties of dealing with uncertain problems; this method must be efficient tools for dealing with diverse types of uncertainties and imprecision embedded in a system. 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] and Maji et al. [3, 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. In 2010 Majumdar and Samanta [7] introduced a concept of generalized fuzzy soft sets and their operations and application of generalised fuzzy soft sets in decision-making problem and medical diagnosis problem. Alkhazaleh and Salleh [8] introduced the concept of a soft expert set and fuzzy 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 generalised fuzzy soft expert set, which will be more effective and useful which is a combination of a fuzzy soft expert set and generalised fuzzy soft set. We also define its basic operations, namely complement, union, intersection, AND, and OR, and study their properties. Finally, we give an application of this concept in decision-making problem.

2. Preliminaries

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

Definition 2.1 (see [1]). A pair is called a soft set over , where is a mapping

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

Definition 2.2 (see [5]). Let 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 where is a mapping given by

Definition 2.3 (see [7]). Let be the universal set of elements and the universal set of parameters. The pair will be called a soft universe. Let and a fuzzy subset of ; that is, , where is the collection of all fuzzy subsets of . Let be a function defined as follows:

Then is called a generalized fuzzy soft set (GFSS in short) over the soft set . Here for each parameter indicates not only the degree of belongingness of the elements of in but also the degree of possibility of such belongingness which is represented by . So we can write as follows:

where and are the degree of belongingness and is the degree of possibility of such belongingness.

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

Definition 2.4 (see [8]). A pair is called a soft expert set over , where is a mapping given by

where denotes the power set of .

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

Definition 2.5 (see [9]). A pair is called a fuzzy soft expert set over , where is a mapping given by

where denotes all the fuzzy subsets of .

Definition 2.6 (see [9]). For two fuzzy soft expert sets and over , is called a fuzzy soft expert subset of if

(1); (2)for all is fuzzy subset of .

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

Definition 2.7 (see [9]). Two fuzzy soft expert sets and over are said to be equal if is a fuzzy soft expert subset of and is a fuzzy soft expert subset of .

Definition 2.8 (see [9]). An agree-fuzzy soft expert set over is a fuzzy soft expert subset of defined as follows:

Definition 2.9 (see [9]). A disagree-fuzzy soft expert set over is a fuzzy soft expert subset of defined as follows:

Definition 2.10 (see [9]). 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.

Definition 2.11 (see [9]). The union of two fuzzy soft expert sets and over , denoted by , is the fuzzy soft expert set where , and for all,

where is an s-norm.

Definition 2.12 (see [9]). The intersection of two fuzzy soft expert sets and over , denoted by , is the fuzzy soft expert set where , and for all ,

where is a t-norm.

Definition 2.13 (see [9]). If and are two fuzzy soft expert sets over , then “ AND ” denoted by is defined by

such that , for all , where is a t-norm.

Definition 2.14 (see [9]). If and are two fuzzy soft expert sets over , then “ OR ” denoted by is defined by

such that , for all , where is an s-norm.

3. Generalised Fuzzy Soft Expert Set

In this section we define the concept of the generalized fuzzy soft expert set and study some of its properties. Let be a universe set, a set of parameters, a set of experts (agents), and a set of opinions. Let and . be a fuzzy set of ; that is, .

Definition 3.1. A pair is called an generalized fuzzy soft expert set (GFSES in short) over , where is a mapping given by

where denotes the collection of all fuzzy subsets of . Here for each parameter , indicates not only the degree of belongingness of the elements of in but also the degree of possibility of such belongingness which is represented by .

Example 3.2. Let be a set of universe, and let a set of parameters. Let and . Let be a fuzzy set of ; that is, .

Define a function

as follows:

Then we can find the a generalized fuzzy soft expert sets as consisting of the following collection of approximations:

Definition 3.3. For two GFSESs and over , is called a generalized fuzzy soft expert set subset of if

(1);(2)for all is generalized fuzzy subset of .

Example 3.4. Consider Example 3.2. Let where is the rest of the statement and so forth

Since is a fuzzy subset of , clearly . Let and be defined as follows:

Therefore .

Definition 3.5. Two GFSES and over are said to be equal if is a GFSES subset of and is a GFSES subset of .

Definition 3.6. An agree-GFSES over is a GFSE subset of defined as follows:

Definition 3.7. A disagree-GFSES over is a GFSE subset of defined as follows:

Example 3.8. Consider Example 3.2. Then the agree-generalized fuzzy soft expert set over is

and the disagree-generalized fuzzy soft expert set over is

Definition 3.9. The complement of a generalized fuzzy soft expert set is denoted by and is defined by where is a mapping given by

where c is a generalized fuzzy complement and .

Example 3.10. Consider Example 3.2. By using the basic fuzzy complement, we have

Proposition 3.11. If is a generalized a fuzzy soft expert set over , then

(1), (2), (3).

Proof. The proof is straightforward.

4. Union and Intersection

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

Definition 4.1. The union of two GFSESs and over , denoted by , is the GFSESs such that and, for all ,

where is a generalized fuzzy soft expert sets.

Example 4.2. Consider Example 3.2. Let

Suppose and are two GFSESs over such that

Then where

Proposition 4.3. If , and are three GFSESs over , then

(1), (2).

Proof. The proof is straightforward.

Definition 4.4. The intersection of two GFSESs and over , denoted by , is the GFSES such that and, for all ,

where is an interval-valued fuzzy intersection.

Example 4.5. Consider Example 4.2; we have where

Proposition 4.6. If , , and are three GFSESs over , then

(1), (2).

Proof. The proof is straightforward.

Proposition 4.7. If , , and are three GFSESs over , then

(1), (2).

Proof. The proof is straightforward.

5. AND and OR Operations

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

Definition 5.1. If and are two GFSES over , then “ AND ” denoted by is defined by

such that , for all , where is GFSES.

Example 5.2. Consider Example 3.2. Let

Suppose and are two fuzzy soft expert sets over such that

Then

Definition 5.3. If and are two GFSES over , then “ OR ” denoted by is defined by

such that , for all , where is an generalized fuzzy union.

Example 5.4. Consider Example 5.2 we have

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

In this section, we present an application of the generalized fuzzy soft expert set theory in a decision making problem. Suppose that one of the broadcasting channels wants to invite experts to evaluate their show through the discussion of a controversial issue and obtain their opinion of the situation. The producers of the show used the following criteria to determine how to evaluate their findings. Their four alternatives are as follows: , suppose there are five parameters ; choose the experts for the programs. For the parameters stand for “this criteria to discriminate,” “this criteria is independent of the other criteria,” “this criteria measures one thing,” “the universal criteria”, and “the criteria that is important to some of the stakeholders.” Let be a set of committee members. From those findings we can find the most suitable choice for the decision. After a serious discussion, the committee constructs the following generalised fuzzy soft expert set:

In Tables 1 and 2 we present the agree-generalized fuzzy soft expert set and disagree-generalized fuzzy soft expert set. Now to determine the best choices, we first mark the highest numerical grade (underline) in each row in agree-generalized fuzzy soft expert set and disagree-generalized fuzzy soft expert set excluding the last column which is the grade of such belongingness of a expert against of parameters. Then we calculate the score of each of such expert in agree-generalized fuzzy soft expert set and disagree-generalized fuzzy soft expert set by taking the sum of the products of these numerical grades with the corresponding values of . Then we calculate the final score by subtracting the scores of expert in the agree-generalized fuzzy soft expert set from the scores of expert in disagree-generalized fuzzy soft expert set. The expert with the highest score is the desired expert.

Table 1: Agree-generalized fuzzy soft expert set.

0.4
0.1
0.0
0.2
0.3

0.3
0.1
0.4
0.7
0.4

0.5
0.6
0.8
0.9
0.1

0.3
0.6
0.7
0.5
0.8

0.7
0.6
0.6
0.8
0.4

0.5
0.2
0.1
0.1
0.2

0.6
0.7
0.8
0.4
0.3

0.4
0.5
0.7
0.6
0.7

0.4
0.7
0.9
0.7
0.6

0.4
0.7
0.6
0.9
0.7

0.1
0.0
0.6
0.5
0.5

0.5
0.5
0.6
0.9
0.6

0.1
0.2
0.5
0.4
0.3

0.7
0.5
0.3
0.4
0.5

0.6
0.3
0.5
0.7
0.4

Table 2: Disagree-generalized fuzzy soft expert set.

λ

0.4
0.8
0.9
0.7
0.9

0.5
0.6
0.7
0.4
0.3

0.3
0.2
0.4
0.3
0.8

0.9
0.4
0.2
0.6
0.7

0.3
0.5
0.3
0.4
0.2

0.3
0.8
0.6
0.9
0.5

0.2
0.1
0.3
0.7
0.2

0.7
0.8
0.3
0.4
0.9

0.7
0.4
0.3
0.2
0.6

0.8
0.6
0.7
0.5
0.1

0.5
0.8
0.3
0.7
0.4

0.3
0.2
0.2
0.4
0.5

0.8
0.6
0.5
0.7
0.6

0.1
0.3
0.6
0.8
0.5

0.2
0.6
0.5
0.3
0.5

Now, we compute the score of by using the data in Table 3:

 Score , Score , Score  , Score  .

Table 3: Grade table of agree-generalized fuzzy soft expert set.

Highest numerical grade
λ

0.4
0.3

0.7
0.4

0.9
0.1

0.7
0.8

0.8
0.4

0.5
0.2

0.8
0.3

0.7
0.7

0.9
0.6

0.9
0.7

0.6
0.5

0.9
0.6

0.5
0.3

0.7
0.5

0.7
0.4

Now, we compute the score of by using the data in Table 4:

 Score , Score , Score , Score .

Table 4: Grade table of disagree-generalized fuzzy soft expert set.

Highest numerical grade
λ

0.9
0.9

0.7
0.3

0.4
0.8

0.9
0.7

0.5
0.2

0.9
0.5

0.7
0.2

0.8
0.9

0.7
0.6

0.8
0.1

0.7
0.4

0.4
0.5

0.8
0.6

0.8
0.5

0.6
0.5

The final score of is as follows:

 Score , Score , Score , Score .

Then the decision is .

7. An Application of Generalized Fuzzy Soft Expert Set with Multiopinions in Decision Making

In this section, we present an application of the generalized fuzzy soft expert set theory in a decision making problem. In this section, we present an application of generalized fuzzy soft expert set with multiopinions in decision making problem. Let be a universe set, a set of parameters, a set of experts (agents), and a set of opinions. Let and .

Consider the previous application 6. Suppose that the management of broadcasting channels takes the opinion of the committee once again with multiopinions. Let where is the rest of the statement and so forth:

7.1. Algorithm

Our goal is to change the generalised Fuzzy soft expert set with multi opinions to generalised fuzzy soft set to find the optimal decision. Note: here we say that the term accept means agree and strongly agree and the term reject means disagree and strongly disagree.

Now we find the simplified for Table 5 where each entries , for all and the last entry , for all .

Table 5: Accepted opinions of generalized fuzzy soft expert set.

λ

0.2
0.4
0.3
0.6
0.2

0.0
0.5
0.1
0.4
0.4

0.4
0.3
0.3
0.5
0.5

0.4
0.5
0.4
0.7
0.2

0.1
0.6
0.3
0.5
0.4

0.5
0.4
0.5
0.6
0.5

0.5
0.7
0.7
0.8
0.5

0.5
0.6
0.5
0.6
0.3

0.3
0.5
0.5
0.7
0.4

0.6
0.7
0.8
0.8
0.5

0.7
0.8
0.6
0.9
0.3

0.5
0.7
0.6
0.8
0.4

0.8
0.9
0.6
0.7
0.7

0.9
0.7
0.7
0.6
0.3

0.6
0.6
0.8
0.7
0.2

0.9
1.0
0.8
0.9
0.7

0.9
0.8
0.7
0.8
0.3

0.7
0.8
0.8
0.9
0.2

0.3
0.6
0.8
0.5
0.2

0.1
0.5
0.6
0.4
0.8

0.2
0.4
0.6
0.4
0.6

0.5
0.7
0.8
0.6
0.2

0.3
0.7
0.8
0.5
0.8

0.4
0.6
0.7
0.7
0.6

0.7
0.5
0.9
0.4
0.9

0.8
0.6
0.7
0.3
0.5

0.6
0.4
0.7
0.5
0.6

0.8
0.6
0.9
0.6
0.9

0.9
0.8
0.9
0.5
0.5

0.7
0.6
0.8
0.6
0.6

Now we find the simplified for Table 6 where each entries , for all and the last entry , for all .

Table 6: 1-rejected opinions of generalized fuzzy soft expert set.

λ

0.4
0.5
0.4
0.7
0.2

0.2
0.3
0.5
0.4
0.4

0.4
0.3
0.4
0.3
0.5

0.2
0.3
0.2
0.5
0.2

0.1
0.2
0.3
0.3
0.4

0.3
0.2
0.3
0.2
0.5

0.7
0.8
0.9
0.9
0.5

0.5
0.6
0.7
0.9
0.3

0.4
0.7
0.6
0.9
0.4

0.6
0.7
0.8
0.9
0.5

0.4
0.5
0.4
0.8
0.3

0.3
0.4
0.5
0.7
0.4

0.9
0.9
0.8
0.8
0.7

0.8
0.7
0.9
0.7
0.3

0.7
0.8
0.9
0.8
0.2

0.8
1.0
0.7
0.8
0.7

0.7
0.6
0.8
0.6
0.3

0.6
0.7
0.8
0.6
0.2

0.4
0.8
0.9
0.6
0.2

0.3
0.7
0.6
0.5
0.8

0.4
0.6
0.7
0.5
0.6

0.3
0.7
0.7
0.5
0.2

0.1
0.6
0.5
0.4
0.8

0.3
0.4
0.5
0.4
0.6

0.8
0.7
1.0
0.6
0.9

0.8
0.6
0.7
0.5
0.5

0.6
0.5
0.8
0.6
0.6

0.6
0.5
0.9
0.5
0.9

0.6
0.5
0.6
0.3
0.5

0.5
0.5
0.7
0.4
0.6

Now to determine the best choice, we first mark the highest numerical grade (indicated in underline) in each row excluding the last column which is the grade of such belongingness of the expert against each pair of parameters. Now the score of each of the expert is calculated by taking the sum of the products of these numerical grades with the corresponding values of . The expert with the highest score is the desired expert.

The committee can use the following algorithm.

(1)Input the generalised fuzzy soft expert set with multiopinions .(2)Find the accept opinions of generalised fuzzy soft expert set (Table 5).(3)Find the 1-reject opinion of generalised fuzzy soft expert set (Table 6).(4)Find the simplified accept opinion of GFSES and the simplified 1-reject opinion of GFSES (Tables 7 and 8).(5)Find where defined as follows: for all and for all ,

where , Table 9.(6)Apply Majumdar and Samanta algorithm on GFSS.

(i)Construct the grade table of GFSS, Table 10.(ii)Compute the score of .(iii)The decision will select with highest score.

Table 7: Simplified accepted opinions of GFSES.

λ

1.6
2.7
1.9
3.3
2.2

3.1
4.0
3.7
4.6
2.4

4.8
4.8
4.4
4.6
2.4

1.8
3.5
4.3
3.1
3.2

4.5
3.5
4.9
2.9
4.0

Table 8: Simplified 1-rejected opinions of GFSES.

λ

1.6
1.8
2.1
2.4
2.2

2.9
3.7
3.9
5.1
2.4

4.5
4.7
4.9
4.3
2.4

1.8
3.8
3.9
2.9
3.2

3.9
3.3
4.7
2.9
4.0

Table 9: Translation from GFSES—with multiopinion to GFSS.

λ

0.266
0.375
0.333
0.475
0.366

0.500
0.641
0.633
0.808
0.400

0.775
0.791
0.775
0.741
0.400

0.300
0.608
0.683
0.500
0.533

0.700
0.566
0.800
0.483
0.666

Table 10: Grade table of GFSS.

Highest numerical grade
λ

0.475
0.366

0.808
0.400

0.791
0.400

0.683
0.533

0.800
0.666

Now, we compute the score of by using the data in Table 10:

 Score , Score , Score , Score .

Then the committee will select the expert with highest score. Hence they will select .

8. Conclusions

In this work we have introduced the concept of generalised fuzzy soft expert set and studied some of its properties. The complement, union, and intersection operations have been defined on the generalised fuzzy soft expert set. An application of this theory is given in solving a decision making problem.

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