Generalized Neutrosophic Soft Expert Set for Multiple-Criteria Decision-Making

: Smarandache deﬁned a neutrosophic set to handle problems involving incompleteness, indeterminacy, and awareness of inconsistency knowledge, and have further developed it neutrosophic soft expert sets. In this paper, this concept is further expanded to generalized neutrosophic soft expert set (GNSES). We then deﬁne its basic operations of complement, union, intersection, AND, OR, and study some related properties, with supporting proofs. Subsequently, we deﬁne a GNSES-aggregation operator to construct an algorithm for a GNSES decision-making method, which allows for a more efﬁcient decision process. Finally, we apply the algorithm to a decision-making problem, to illustrate the effectiveness and practicality of the proposed concept. A comparative analysis with existing methods is done and the result afﬁrms the ﬂexibility and precision of our proposed method.


Introduction
For a proper description of objects in an uncertain and ambiguous environment, indeterminate and incomplete information has to be properly handled. Intuitionistic fuzzy sets were introduced by Atanassov [1], followed by Molodtsov on soft sets [2] and neutrosophy logic [3] and neutrosophic sets [4] were introduced by Smarandache. The term neutro-sophy means knowledge of neutral thought and this neutral represents the main distinction between fuzzy and intuitionistic fuzzy logic and a set. At present, work on the soft set theory is progressing rapidly. Various operations and applications of soft sets have been developed rapidly, including the possibility of fuzzy soft set [5], soft multiset theory [6], multiparameterized soft set [7], soft intuitionistic fuzzy sets [8], Q-fuzzy soft sets [9][10][11], multi Q-fuzzy sets [12][13][14], N-soft set [15], Hesitant N-soft set [16], and Fuzzy N-soft set [17], thereby, opening avenues to genetic applications [18,19]. Later, Maji [20] have introduced a more generalized concept-which is a combination of neutrosophic sets and soft sets-and have studied its properties. Alhazaymeh and Hassan [21,22] have studied the concept of vague soft set, which were later extended to vague soft expert set theory [23,24], bipolar fuzzy soft expert set [25], and multi Q-fuzzy soft expert set [26].Şahin et al. [27] introduced neutrosophic soft expert sets, while Al-Quran and Hassan [28,29] extended it further to neutrosophic vague soft expert set. Neutrosophic set theory has also been applied to multiple attribute decision-making [30][31][32]. Fuzzy modelling has long been widely applied to physical problems, which include intuitionistic hesitant fuzzy [33], t-concept lattices [34], fuzzy operators [35], medical image retrieval [36], and artificial bee colony [37] and multi criteria decision making [38,39]. Neutrosophic sets have also gained traction with recent publications Definition 10. [45] Let Ψ K ∈ NP-soft set. Then an NP-aggregation operator of Ψ K , denoted by Ψ agg K , is defined by which is a neutrosophic set over U, and where, such that |U| is the cardinality of U.

Generalized Neutrosophic Soft Expert Set
In this section, we introduce the concept of generalized neutrosophic soft expert set (GNSES) and define some of its properties. Throughout this paper, U is an initial universe, E is a set of parameters, X is a set of experts (agents), and O = {agree = 1, disagree = 0} a set of opinions. Let Z = E × X × O and A ⊆ Z and u is a fuzzy set of A; that is, u : A →= [0, 1] .

Definition 11.
A pair (F u , A) is called a generalized neutrosophic soft expert set (GNSES) over U, where F u is a mapping given by F u : A → N (U)×, with N (U) being the set of all neutrosophic soft expert subsets of U. For any parameter e ∈ A, F(e) is referred as the neutrosophic value set of parameter e, i.e., where T, , F : U → ] − 0, 1 + [ are the membership function of truth, indeterminacy, and falsity, respectively, of the element u ∈ U. For any u ∈ U and e ∈ A In fact, F u is a parameterized family of neutrosophic soft expert sets on U, which has the degree of possibility of the approximate value set which is prepresented by u(e) for each parameter e, which can be written as follows: Definition 13. Two GNSESs (F u , A) and (G η , B) over U are said to be equal if (F u , A) is a GNSES subset of (G η , B) and (G η , B) is a GNSES subset of (F u , A).

Definition 14.
An agree-GNSESs (F u , A) 1 over U is a GNSES subset of (F u , A) defined as follows.

Definition 15.
A disagree-GNSESs (F u , A) 0 over U is a GNSES subset of (F u , A) is defined as follows:   ( . The proofs of assertions (2) and (3) are obvious.

Definition 17. The union of two GNSESs
, is the GNSESs H Ω , C , where C = A ∪ B and the truth-membership, indeterminacy-membership, and falsity-membership of H Ω , C are as follows: where Ω(m) = max u (e) (m), η (e) (m) . ( Proof. (1) We want to prove that By using Definition 17, we consider the case when e ∈ A ∩ B, as other cases are trivial. We will have Also consider the case when e ∈ H, as the other cases are trivial. We will have (2) The proof is straightforward. (F u , A) and (G η , B) be two GNSESs over a common universe U. Then the intersection of (F u , A) and (G η , B) is denoted by (F u , A) ∼ ∩ (G η , B) = K δ , C , where C = A ∩ B and the truth-membership, indeterminacy-membership, and falsity-membership of K δ , C are as follows:

Proof. (1) We want to prove that
By using Definition 18, consider the case when e ∈ A ∩ B, since other cases are trivial. We have Also consider the case when e ∈ K, as the other cases are trivial. Then we have (2) The proof is straightforward. ( Proof. The proofs can be easily obtained from Definitions 17 and 18. T H Ω (α,β) (m) = min T F u (α) (m), T G η (β) (m) , and Ω(m) = min u (e) (m), η (e) (m) , ∀α ∈ A, ∀β ∈ B.  (F u , A) and (G η , B) are two GNSESs over U, then "(F u , A) OR (G η , B)" denoted by (F u , A) ∨ (G η , B), is defined by and the truth-membership, indeterminacy-membership, and falsity-membership of K δ , A × B are as follows.  ( Proof. The proofs can be easily obtained from Definitions 16, 19 and 20.

GNSES-Aggregation Operator
In this section, we define a GNSES-aggregation operator of a GNSES to construct a decision method by which approximate functions of a soft expert set are combined to produce a neutrosophic set that can be used to evaluate each alternative.

Definition 21.
Let Υ A ∈ GNSESs. Then a GNSES-aggregation operator of Υ A , denoted by Υ agg A , is defined by which is a GNSES over U, where |U| is the cardinality of U and µ is defined below (e i , i = 1, 2, 3, . . . , n).
Definition 22. Let Υ A ∈ GNSESs, Υ agg A be the corresponding GNSES aggregation operator. Then a reduced fuzzy set of Υ agg A is a fuzzy set over U, denoted by where τΥ

An Application of Generalized Neutrosophic Soft Expert Set
In this section, we present an application of generalized neutrosophic soft expert set theory in a decision-making problem. Based on Definitions 21 and 22, we constructed an algorithm for the GNSES decision-making method as follows.
Step 1-Choose a feasible subset of the set of parameters.
Step 2-Construct the GNSES tables for each opinion (agree, disagree) of experts.
Step 3-Compute the aggregation operator GNSES Υ agg A of Υ A and the reduced fuzzy set T Step 4-Score(u I ) = maxagree(u i ) − mindisagree(u i ).
Step 5-Choose the element of u i that has maximum score. This will be the optimal solution. Example 10. Suppose a company needs to employ a worker, which is to be decided by a few experts. The employee has to be chosen from five potential workers, U = {u 1 , u 2 , u 3 , u 4 , u 5 }. Suppose there are four parameters E = {e 1 , e 2 , e 3 , e 4 } where the parameters e i (i = 1, 2, 3, 4) stand for "education," "age," "capability" and "experience", respectively. Let X = {p, q, r} be a set of experts. After a serious discussion, the experts construct the following generalized neutrosophic soft expert set.
Step 1-Choose a feasible subset of the set of parameters Step 2-Construct the GNSES tables for each opinion (agree, disagree) of experts, as shown in Tables 1 and 2.   Table 2. Disagree-GNSES. Step 3-Now calculate the scores of agree (u i ) by using the data in Table 1, to obtain values in Table 3.  Now calculate the score of disagree (u i ) by using the data in Table 2, to obtain values in Table 4.  Step 4-The final score of u i is computed as follows.
Step 5-Score(u 1 ) = 0.2325 is the maximum. Hence, the best decision for the experts is to select worker u 1 as the company's employee.

Comparison Analysis
A generalized neutrosophic soft expert model gives more precision, flexibility, and compatibility than the existing neutrosophic models. These are verified by a comparison analysis, using neutrosophic soft expert decision method, with those methods used by Sahin et al. [27], Hassan [44], and Maji [20], as given in Table 5. The comparison is done based on the same example as in Section 5. The ranking order results obtained are consistent with those in [20,27,44]. Ranking

Conclusions
We have established the concept of generalized neutrosophic soft expert set (GNSES) as a generalization of NSES. The basic operations of GNSES of complement, union, intersection AND, and OR were defined. Subsequently, a definition of GNSES-aggregation operator was proposed to construct an algorithm of a GNSES decision method. Finally, an application of the constructed algorithm, to solve a decision-making, was provided. This new extension provides a significant contribution to current theories for handling indeterminacy, and it spurs the development of further research and pertinent applications.