`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 258361, 15 pageshttp://dx.doi.org/10.1155/2012/258361`
Research Article

## Fuzzy Parameterized Soft Expert Set

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Received 25 May 2012; Accepted 10 August 2012

Copyright © 2012 Maruah Bashir and Abdul Razak Salleh. 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 introduced the concept of soft expert set and gave an application in a decision-making problem. In this paper we introduce the concept of fuzzy parameterized soft expert set by giving an important degree for each element in the set of parameters. We also study its properties and define its basic operations, namely, complement, union, intersection, AND, and OR. Finally, we give an application in decision making.

#### 1. Introduction

Many fields deal with uncertain data that may not be successfully modeled by classical mathematics. Molodtsov [1] proposed a completely new approach for modeling vagueness and uncertainty. This so-called soft set theory has potential applications in many different fields. After Molodtsov’s work, some different operations and application of soft sets were studied by Chen et al. [2] and Maji et al. [3, 4]. Furthermore Maji et al. [5] presented the definition of fuzzy soft set as a generalization of Molodtsov’s soft set, and Roy and Maji [6] presented an application of fuzzy soft sets in a decision-making problem. Majumdar and Samanta [7] defined and studied the generalised fuzzy soft sets where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Zhou et al. [8] defined and studied generalised interval-valued fuzzy soft sets where the degree is attached with the parameterization of interval-valued fuzzy sets while defining an interval-valued fuzzy soft set. Alkhazaleh et al. [9] defined the concepts of possibility fuzzy soft set and gave their applications in decision making and medical diagnosis. They also introduced the concept of fuzzy parameterized interval-valued fuzzy soft set [10] where the mapping of approximate function is defined from the set of parameters to the interval-valued fuzzy subsets of the universal set and gave an application of this concept in decision making. Salleh et al. [11] introduced the concept of multiparameterized soft set and studied its properties and basic operations. In 2010 Çağman et al. introduced the concept of fuzzy parameterized fuzzy soft sets and their operations [12]. Also Çağman et al. [13] introduced the concept of fuzzy parameterized soft sets and their related properties. Alkhazaleh and Salleh [14] introduced the concept of a soft expert set, and Alkhazaleh [15] introduced fuzzy soft expert set, where the user can know the opinion of all experts in one model without any operations. In this paper we introduce the concept of fuzzy parameterized soft expert set which is a combination of fuzzy set and soft expert set. We also define its basic operations, namely, complement, union, intersection, and the operations AND and OR. Finally, we give an application of fuzzy parameterized soft expert set in decision-making problem.

#### 2. Preliminaries

In this section, we recall some basic notions in soft expert set theory. Alkhazaleh and Salleh [14] defined soft expert set in the following way. Let be a universe, let be a set of parameters, and let be a set of experts (agents). Let be a set of opinions, , and .

Definition 2.1. A pair is called a soft expert set over , where is a mapping , and denotes the power set of .

Definition 2.2. For two soft expert sets and over is called a soft expert subset of if(i), (ii)for all .

Definition 2.3. Two soft expert sets and over are said to be equal if is a soft expert subset of and is a soft expert subset of .

Definition 2.4. Let be a set of parameters and let be a set of experts. The NOT set of , denoted by , is defined by

Definition 2.5. The complement of a soft expert set is denoted by and is defined by where is a mapping given by , for all .

Definition 2.6. An agree-soft expert set over is a soft expert subset of defined as follows:

Definition 2.7. A disagree-soft expert set over is a soft expert subset of defined as follows:

Definition 2.8. The union of two soft expert sets and over , denoted by , is the soft expert set where , and for all ,

Definition 2.9. The intersection of two soft expert sets and over , denoted by , is the soft expert where for all ,

Definition 2.10. If and are two soft expert sets, then AND , denoted by , is defined by where for all .

Definition 2.11. If and are two soft expert sets, then OR , denoted by , is defined by where , for all .

#### 3. Fuzzy Parameterized Soft Expert Sets

In this section, we introduce the definition of fuzzy parameterized soft expert set and its basic operations, namely, complement, union, intersection, and the operations AND and OR. We give examples for these concepts.Basic properties of the operations are also given.

Definition 3.1. Let be a universe, let be a set of parameters, let denote the set of fuzzy subsets of , a set of experts (agents), and a set of opinions. Let and where . Then the cartesian product is defined as follows:

Definition 3.2. A pair is called a fuzzy parameterized soft expert set (FPSES) over , where is a mapping given by , and denotes the power set of .

Example 3.3. Suppose that a hotel chain is looking for a construction company to upgrade the hotels to keep pace with globalization and wishes to take the opinion of some experts concerning this matter. Let be a set of construction companies, let be a set of decision parameters where denotes the decision “good service,” “quality,” and “cheap,” respectively, and a fuzzy subset of , and let be a set of experts.
Suppose that the hotel chain has distributed a questionnaire to the three experts to make decisions on the construction companies, and we get the following information: Then we can view the FPSES as consisting of the following collection of approximations:

Definition 3.4. For two FPFESs and over is called an FPSE subset of , and we write , if(i), (ii)for all .

Definition 3.5. Two fuzzy FPSESs and over are said to be equal if is an FPSE subset of and is a FPSE subset of .

Example 3.6. Consider Example 3.3. Suppose that the hotel chain takes the opinion of the experts once again after the hotel chain has been opened.
Let be a fuzzy subset over , and let be another fuzzy subset over . Suppose Since is a fuzzy subset of , clearly . Let and be defined as follows: Therefore .

Definition 3.7. The complement of an FPSES is denoted by and is defined by where is a mapping given by , for all and .

Example 3.8. Consider Example 3.3. By using the basic fuzzy complement, we have

Definition 3.9. An agree-FPSES over is an FPSE subset of where the opinions of all experts are agree and is defined as follows:

Example 3.10. Consider Example 3.3. Then the agree-FPSES over is

Definition 3.11. A disagree-FPSES   over is an FPSE subset of where the opinions of all experts are disagree and is defined as follows:

Example 3.12. Consider Example 3.3. Then the disagree-FPSES over is

Proposition 3.13. If is an FPSES over , then .

Proof. By using Definition 3.7, we have is a mapping given by , for all and . Now, is a mapping given by , for all and , and since so the proof is complete.

Definition 3.14. The union of two FPSESs and over , denoted by , is the FPSES such that where and for all ,

Example 3.15. Consider Example 3.3. Let be a fuzzy subset over , and let be another fuzzy subset over : Suppose and are two FPSESs over the same given by By using the operator max which is the basic fuzzy union, we get

Proposition 3.16. If , , and are three FPSESs over , then(i), (ii).

Proof. (i) By using Definition 3.14 we have the union of two FPSESs , and is the FPSES such that where and for all , . Now, since union for fuzzy sets and crisp sets are commutative, then and , and this gives the result.
(ii) We use the fact that union for fuzzy sets and crisp sets is associative.

Definition 3.17. The intersection of two FPSESs and over , denoted by , is the FPSES such that where and for all ,

Example 3.18. Consider Example 3.15. Then by using the basic fuzzy intersection (minimum) we have

Proposition 3.19. If , and are three FPSESs over , then (i), (ii).

Proof. (i) By using Definition 3.17 we have the intersection of two FPSESs and is the FPSES such that where and for all . Now, since intersection for fuzzy sets and crisp sets is commutative, then and , and this gives the result.
(ii) We use the fact that intersection for fuzzy sets and crisp sets is associative.

Proposition 3.20. If , and are three FPSESs over , then (i), (ii).

Proof. We prove (i), and we can use the same method to prove (ii).

Let where , and let where . So

Definition 3.21. If and are two FPSESs over , then AND , denoted by , is defined by such that , for all , where .

Example 3.22. Consider Example 3.3. Let Suppose and are two FPSESs over the same such that By using the basic fuzzy intersection (minimum) we have where

Definition 3.23. If and are two FPSESs over , then OR , denoted by , is defined by such that , for all , where .

Example 3.24. Consider Example 3.22. By using the basic fuzzy union (maximum) we have

#### 4. An Application of Fuzzy Parameterized Soft Expert Set

Ahkhazaleh and Salleh [14] applied the theory of soft expert sets to solve a decision-making problem. In this section, we present an application of FPSES in a decision-making problem by generalizing Ahkhazaleh and Salleh’s Algorithm to be compatible with our work. We consider the following problem.

Example 4.1. Assume that a hotel chain wants to fill a position for the management of the chain. There are five candidates who form the universe . The hiring committee decided to have a set of parameters, , where the parameters stand for “computer knowledge,” “experience,” and “good speaking,” respectively. Let be a set of experts (committee members). Suppose In Tables 1 and 2 we present the agree-FPSES and disagree-FPSES, respectively, such thatif then , otherwise ,if then , otherwise ,where are the entries in  Tables 1 and 2.
The following Algorithm 4.2 may be followed by the hotel chain to fill the position.

Table 1: Agree-FPSES.
Table 2: Disagree-FPSES.

Algorithm 4.2. (1) Input the FPSES .
(2) Find an agree-FPSES and a disagree-FPSES.
(3) Find for agree-FPSES.
(4) Find for disagree-FPSES.
(5) Find .
(6) Find , for which .
Then is the optimal choice object. If has more than one value, then anyone of them could be chosen by the hotel chain using its option. Now we use Algorithm 4.2 to find the best choice for the hotel chain to fill the position.
Then , as shown in Table 3, so the committee will choose candidate for the job.

Table 3

#### 5. Weighted Fuzzy Parameterized Soft Expert Set

In this section we introduce the notion of weighted fuzzy parameterized soft expert sets and discuss its application to decision-making problem.

Definition 5.1. Let be the set of all fuzzy parameterized soft expert sets in the universe . Let be a set of parameters and . A weighted fuzzy parameterized soft expert set is a triple where is a fuzzy parameterized soft expert set over , and is a weight function specifying for each attribute .

By definition, every fuzzy parameterized soft expert set can be considered as a weighted fuzzy parameterized soft expert set. This is an extension of the weighted fuzzy soft sets discussed in [16]. The notion of weighted fuzzy parameterized soft expert set provides a mathematical framework for modeling and analyzing the decision-making problems in which all the choice experts may not be of equal importance. These differences between the importance of experts are characterized by the weight function in a weighted fuzzy parameterized soft expert set.

Example 5.2. Suppose that a hotel chain has imposed the following weights for the experts in Example 4.1. For the expert “,” ; for the expert “,” ; for the expert “,” . Thus we have a weight function , and the fuzzy parameterized soft expert set in Example 4.1 is changed into a weighted fuzzy parameterized soft expert set (WFPSES) .
In Tables 4 and 5 we present the agree-WFPSES and disagree-WFPSES respectively such thatif then , otherwise ,if then , otherwise ,where are the entries in  Tables 4 and 5.

Table 4: Agree-WFPSES.
Table 5: disagree-WFPSES.

The following Algorithm 5.3 may be followed by the hotel chain to fill the position.

Algorithm 5.3. Input the WFPSES .
(2) Find an agree-WFPSES and a disagree-WFPSES.
(3) Find for agree-WFPSES.
(4) Find for disagree-WFPSES.
(5) Find .
(6) Find , for which .
Then is the optimal choice object. If has more than one value, then any one of them could be chosen by the hotel chain using its option. Now we use Algorithm 5.3 to find the best choice for the hotel chain to fill the position.
Then , as shown in Table 6, so the committee will choose candidate for the job.

Table 6

Remark 5.4. By comparing the results obtained using Algorithms 4.2 and 5.3 we can see that giving more consideration to the expert (weight) might affect the result.

#### Acknowledgment

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant UKM-DLP-2011-038.

#### References

1. D. Molodtsov, “Soft set theory—first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.
2. D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757–763, 2005.
3. P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002.
4. P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003.
5. P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001.
6. A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007.
7. P. Majumdar and S. K. Samanta, “Generalised fuzzy soft sets,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1425–1432, 2010.
8. X. Zhou, Q. Li, and L. Guo, “On generalized interval-valued fuzzy soft sets,” Journal of Applied Mathematics, vol. 2012, Article ID 479783, 18 pages, 2012.
9. S. Alkhazaleh, A. R. Salleh, and N. Hassan, “Possibility fuzzy soft set,” Advances in Decision Sciences, vol. 2011, Article ID 479756, 18 pages, 2011.
10. S. Alkhazaleh, A. R. Salleh, and N. Hassan, “Fuzzy parameterized interval-valued fuzzy soft set,” Applied Mathematical Sciences, vol. 5, no. 67, pp. 3335–3346, 2011.
11. A. R. Salleh, S. Alkhazaleh, N. Hassan, and A. G. Ahmad, “Multiparameterized soft set,” Journal of Mathematics and Statistics, vol. 8, no. 1, pp. 92–97, 2012.
12. N. Çağman, F. Citak, and S. Enginoglu, “Fuzzy parameterized fuzzy soft set theory and its applications,” Turkish Journal of Fuzzy Systems, vol. 1, pp. 21–35, 2010.
13. N. Çağman, F. Çitak, and S. Enginoglu, “FP-soft set theory and its applications,” Annals of Fuzzy Mathematics and Informatics, vol. 2, no. 2, pp. 219–226, 2011.
14. S. Alkhazaleh and A. R. Salleh, “Soft expert sets,” Advances in Decision Sciences, Article ID 757868, 12 pages, 2011.
15. S. Alkhazaleh, Soft sets and fuzzy soft sets: some generalisations [Ph.D. thesis], Universiti Kebangsaan Malaysia, 2012.
16. F. Feng, Y. B. Jun, X. Liu, and L. Li, “An adjustable approach to fuzzy soft set based decision making,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 10–20, 2010.