Fuzzy Parameterized Soft Expert Set

and Applied Analysis 3 Definition 2.8. The union of two soft expert sets F,A and G,B over U, denoted by F,A ∪̃ G,B , is the soft expert set H,C where C A ∪ B, and for all ε ∈ C,


Introduction
Many fields deal with uncertain data that may not be successfully modeled by classical math ematics.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 C ¸agman et al. introduced the concept of fuzzy parameterized fuzzy soft sets and their operations 12 .Also C ¸agman 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.

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 U be a universe, let E be a set of parameters, and let X be a set of experts agents .Let O {1 agree, 0 disagree} be a set of opinions, Z E × X × O, and A ⊆ Z. Definition 2.1.A pair F, A is called a soft expert set over U, where F is a mapping F : A → P U , and P U denotes the power set of U.

Definition 2.2. For two soft expert sets
Definition 2.3.Two soft expert sets F, A and G, B over U are said to be equal if F, A is a soft expert subset of G, B and G, B is a soft expert subset of F, A .Definition 2.4.Let E be a set of parameters and let X be a set of experts.The NOT set of Z E × X × O, denoted by ∼ Z, is defined by where ∼ e i is not e i .

2.1
Definition 2.5.The complement of a soft expert set F, A is denoted by F, A c and is defined by Definition 2.6.An agree-soft expert set F, A 1 over U is a soft expert subset of F, A defined as follows:

2.2
Definition 2.7.A disagree-soft expert set F, A 0 over U is a soft expert subset of F, A defined as follows:

2.4
Definition 2.9.The intersection of two soft expert sets F, A and G, B over U, denoted by

2.5
Definition 2.10.If F, A and G, B are two soft expert sets, then F, A AND G, B , denoted by F, A ∧ G, B , is defined by where

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.where F is a mapping given by F D : A → P U , and P U denotes the power set of U.
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 U {u 1 , u 2 , u 3 , u 4 } be a set of construction companies, let E {e 1 , e 2 , e 3 } be a set of decision parameters where e i i 1, 2, 3 denotes the decision "good service," "quality," and "cheap," respectively, and D {e 1 /0.3, e 2 /0.5, e 3 /0.8} a fuzzy subset of I E , and let X {p, q, r} 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:

3.2
Then we can view the FPSES F, Z as consisting of the following collection of approximations:

3.3
Abstract and Applied Analysis 5 Definition 3.4.For two FPFESs F, A D and G, B K over U, F, A D is called an FPSE subset of G, B K , and we write Definition 3.7.The complement of an FPSES F, A D is denoted by F, A c D and is defined by

3.6
Definition 3.9.An agree-FPSES F, A D 1 over U is an FPSE subset of F, A D where the opinions of all experts are agree and is defined as follows: Proof.i By using Definition 3.14 we have the union of two FPSESs F, A D , and since union for fuzzy sets and crisp sets are commutative, then R K ∪ D and H R ε F K ε ∪ G D ε , 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 F, A D and G, B K over U, denoted by Proof.i By using Definition 3.17 we have the intersection of two FPSESs . Now, since intersection for fuzzy sets and crisp sets is commutative, then , and this gives the result.ii We use the fact that intersection for fuzzy sets and crisp sets is associative.
Proof.We prove i , and we can use the same method to prove ii . 3.17

3.20
By using the basic fuzzy intersection minimum we have where Example 3.24.Consider Example 3.22.By using the basic fuzzy union maximum we have

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 U {u 1 , u 2 , u 3 , u 4 , u 5 }.The hiring committee decided to have a set of parameters, E {e 1 , e 2 , e 3 }, where the parameters e i i 1, 2, 3 stand for "computer knowledge," "experience," and "good speaking," respectively.Let X {p, q, r} be a set of experts committee members .Suppose

4.1
In Tables 1 and 2 we present the agree-FPSES and disagree-FPSES, respectively, such that where u ij are the entries in Tables 1 and 2.
The following Algorithm 4.2 may be followed by the hotel chain to fill the position.Then s m is the optimal choice object.If m 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 max s j s 3 , as shown in Table 3, so the committee will choose candidate u 3 for the job.

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 F U be the set of all fuzzy parameterized soft expert sets in the universe U. Let E be a set of parameters and A ⊆ X.A weighted fuzzy parameterized soft expert set is a triple ζ F, A, ω where F, A is a fuzzy parameterized soft expert set over U, and ω : X → 0, 1 is a weight function specifying w j ω ε j for each attribute ε j ∈ X.
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 "p," w 1 0.7; for the expert "q," w 2 0.6; for the expert "r," w 3 0.In Tables 4 and 5 we present the agree-WFPSES and disagree-WFPSES respectively such that if u i ∈ F 1 ε then u ij 1, otherwise u ij 0, if u i ∈ F 0 ε then u ij 1, otherwise u ij 0, where u ij are the entries in Tables 4 and 5.
The following Algorithm 5.3 may be followed by the hotel chain to fill the position.2 Find an agree-WFPSES and a disagree-WFPSES.3 Find c j x∈X i u ij μ E e i ω x for agree-WFPSES.4 Find k j x∈X i u ij μ E e i ω x for disagree-WFPSES.5 Find s j c j − k j .6 Find m, for which s m max s j .Then s m is the optimal choice object.If m 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 max s j s 4 , as shown in Table 6, so the committee will choose candidate u 4 for the job.
and since D c c D so the proof is complete.Definition 3.14.The union of two FPSESs F, A D and G, B k over U, denoted by

Definition 3 .
21.If F, A D and G, B K are two FPSESs over U, then F, A D AND G, B K , denoted by F, A D ∧ G, B K , is defined by

Algorithm 4 . 2 . 1
Input the FPSES F, A . 2 Find an agree-FPSES and a disagree-FPSES.3 Find c j x∈X i u ij μ E e i for agree-FPSES.4 Find k j x∈X i u ij μ E e i for disagree-FPSES.5 Find s j c j − k j .6 Find m, for which s m max s j .
5. Two fuzzy FPSESs F, A D and G, B K over U are said to be equal if F, A D is an FPSE subset of G, B K and G, B K is a FPSE subset of F, A D .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 D {e 1 /0.6, e 2 /0.3, e 3 /0.2}be a fuzzy subset over E, and let K {e 1 /0.3, e 2 /0.2, e 3 /0.1}be another fuzzy subset over E. Suppose Since K is a fuzzy subset of D, clearly B K ⊂ A D .Let F, A D and F, B K be defined as follows: Example 3.8.Consider Example 3.3.By using the basic fuzzy complement, we have , r, 0 , {u 2 , u 4 } .
Definition 3.11.A disagree-FPSES F, A D 0 over U is an FPSE subset of F, A D where the opinions of all experts are disagree and is defined as follows:Example 3.12.Consider Example 3.3.Then the disagree-FPSES F, A D 0 over U is .7 Example 3.10.Consider Example 3.3.Then the agree-FPSES F, A D 1 over U isF, A D 0 {F D α : α ∈ D × X × {0}}.3.9 15 If F, A D , G, B K and H, C R are three FPSESs over U, If F, A D and G, B K are two FPSESs over U, then F, A D OR G, B K , denoted by F, A D ∨ G, B K , is defined by , {u 1 , u 2 , u 5 } .

Table 6
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.