Research Article  Open Access
Fuzzy Parameterized Soft Expert Set
Abstract
In 2011 Alkhazaleh and Salleh introduced the concept of soft expert set and gave an application in a decisionmaking 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 socalled 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 decisionmaking 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 intervalvalued fuzzy soft sets where the degree is attached with the parameterization of intervalvalued fuzzy sets while defining an intervalvalued 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 intervalvalued fuzzy soft set [10] where the mapping of approximate function is defined from the set of parameters to the intervalvalued 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 decisionmaking 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 agreesoft expert set over is a soft expert subset of defined as follows:
Definition 2.7. A disagreesoft 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 agreeFPSES 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 agreeFPSES over is
Definition 3.11. A disagreeFPSES 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 disagreeFPSES 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 decisionmaking problem. In this section, we present an application of FPSES in a decisionmaking 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 agreeFPSES and disagreeFPSES, respectively, such that if 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.


Algorithm 4.2. (1) Input the FPSES .
(2) Find an agreeFPSES and a disagreeFPSES.
(3) Find for agreeFPSES.
(4) Find for disagreeFPSES.
(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.