Entropy, Measures of Distance and Similarity of Q-Neutrosophic Soft Sets and Some Applications

The idea of the Q-neutrosophic soft set emerges from the neutrosophic soft set by upgrading the membership functions to a two-dimensional entity which indicate uncertainty, indeterminacy and falsity. Hence, it is able to deal with two-dimensional inconsistent, imprecise, and indeterminate information appearing in real life situations. In this study, the tools that measure the similarity, distance and the degree of fuzziness of Q-neutrosophic soft sets are presented. The definitions of distance, similarity and measures of entropy are introduced. Some formulas for Q-neutrosophic soft entropy were presented. The known Hamming, Euclidean and their normalized distances are generalized to make them well matched with the idea of Q-neutrosophic soft set. The distance measure is subsequently used to define the measure of similarity. Lastly, we expound three applications of the measures of Q-neutrosophic soft sets by applying entropy and the similarity measure to a medical diagnosis and decision making problems.


Introduction
The idea of fuzzy set theory established by Zadeh [1] is an important aspect in the study of uncertainty. The massive success of this theory has brought about the creation of many extensions of fuzzy sets such as the intuitionistic fuzzy set [2], interval-valued fuzzy set [3], vague set [4], and hesitant fuzzy set [5]. Smarandache [6,7] introduced a new model called neutrosophic set theory which refers to neutral knowledge. The innovative concept of neutrosophic set was presented to cater for indeterminate information which were conspicuously absent in the realm of fuzzy set and intuitionistic fuzzy set. Neutrosophic set (NS) is identified by three independent membership functions which describe the degree of truth (T), the degree of indeterminacy (I), and the degree of falsity (F), whose values are real standard or non-standard subset of unit interval ] − 0, 1 + [ where − 0 = 0 − , 1 + = 1 + , is an infinitesimal number. The truth and falsity membership functions in a NS are analogous to the membership and nonmembership functions in an intuitionistic fuzzy set, and expresses the degree of belongingness and non-belongingness of the elements, whereas the indeterminacy membership function expresses the degree of neutrality in the information. The tri-membership structure of NSs enables it to handle uncertain, inconsistent and indeterminate data using truth, indeterminacy and falsity memberships. Indeterminacy membership function enables NSs to handle the neutrality aspects of the data, which cannot be handled by fuzzy sets and its extensions. The independency of the membership functions makes NSs more applicable than intuitionistic fuzzy set or other fuzzy-based models in which values of the membership and non-membership functions are dependent on one another.
x ∈ X, q ∈ Q for all i = 1, 2, . . . , l}, where T Γ Q i , I Γ Q i , F Γ Q i : X × Q → I l for all i = 1, 2, . . . , l are respectively, truth membership function, indeterminacy membership function and falsity membership function for each x ∈ X and q ∈ Q and satisfy the condition where l is called the dimension of Γ Q .
Definition 7 ([16]). Let X be a universal set, E be a set of parameters, and Q be a nonempty set. Let µ l QNS(X) denote the set of all multi Q-neutrosophic sets on X with dimension l = 1. Let A ⊆ E. A pair (Γ Q , A) is called a Q-neutrosophic soft set (Q-NSS) over X, where Γ Q is a mapping given by such that Γ Q (e) = φ if e / ∈ A. A Q-neutrosophic soft set (Q-NSS) can be represented by the set of ordered pairs (Γ Q , A) = {(e, Γ Q (e)) : e ∈ A, Γ Q ∈ µ l QNS(X)}.
The set of all Q-neutrosophic soft sets (Q-NSSs) in X and Q is denoted by QNSS(X).

Entropy of Q-Neutrosophic Soft Sets
We will propose in this section, the entropy of Q-NSS which measures the degree of fuzziness of a Q-NSS. This idea can be concreted in the following conditions, required for a Q-neutrosophic entropy: (i) Will be null when the set is a Q-intuitionistic fuzzy soft set. (ii) Will be maximum if the set is completely Q-neutrosophic soft set. (iii) The Q-neutrosophic entropy of a Q-NSS and its complement is equal. (iv) If the degree of membership, indeterminacy membership and non-membership of each element decrease, the sum will do so as well, and therefore, this set becomes fuzzier, and therefore the entropy should increase.
In view of the conditions stated above, we propose the following axiomatic definition for the Q-neutrosophic entropy of a Q-NSS. Definition 9. A function ε : Q-NSS(X) → R + ∪ {0} is an entropy on (Γ Q , A) ∈ Q-NSS(X) if the following properties are satisfied: Now we prove that the entropy of a Q-NSS has its maximum value if the Q-NSS is completely Q-neutrosophic soft set. Theorem 1. The entropy of (Γ Q , A), ε((Γ Q , A)) is maximum if and only if (Γ Q , A) is completely Q-neutrosophic soft set.
Next, we give definition of an expression which allows us to create entropies of Q-NSS(X).
is a mapping, which satisfies the following conditions: Examples of Φ functions which verify the previous conditions are: The following theorem gives an expression which will allow us to construct different Q-neutrosophic entropies using a mapping satisfies the conditions of Definition 10.
is a Q-neutrosophic entropy for (Γ Q , A).

Proof.
We shall prove all conditions of Definition 9.

1.
Let Hence, In the following, we give some examples of Q-neutrosophic entropy for Q-NSS.
In the following, we shall compare the performance of the proposed entropy measures by the following example: . , x m } with the nonempty set Q = {q 1 , q 2 , . . . , q l }. Then we define the Q-NSS, (Γ Q , A) n for any positive real number, n, as follows: When we consider the Q-NSS, (Γ Q , A), in the finite universe X = {x 1 , x 2 } with Q = {q 1 , q 2 } and the set of parameters A = {e 1 , e 2 } as Using the above operation, we can generate the following Q-NSSs: By taking into account the characterization of linguistic variable, (Γ Q , A) may be viewed as "Large" in X × Q. Using the above operations, (Γ Q , A) 2 may be viewed as "Very Large", (Γ Q , A) 3 may be viewed as "Quite Very Large" and (Γ Q , A) 4 may be viewed as "Very Very Large". From logical consideration and human intuition, the proposed entropy measures shown in Equations (1) and (2) of these Q-NSSs is required to satisfy the ranking Therefore, from calculated entropy values listed in Table 1, we can easily see that they satisfy the required ranking order.

Measures of Distance and Similarity of Q-Neutrosophic Soft Sets
The axiomatic definitions of the distance measure and similarity measure will be introduced. We present several distance measures between Q-NSSs, specifically the distances of Hamming, Euclidean, normalized Hamming and normalized Euclidean. These distances are obtained through generalizing the corresponding measures of distances for NSSs characterized in [27], to make them appropriate to the basis of Q-NSSs.
We start with defining the axiomatic definition for the measure of distance between two Q-NSSs. Next the distances of Hamming, Euclidean, and their normalized forms are defined.

Definition 11.
A real valued non-negative function d : Q − NSS(X) × Q − NSS(X) → R + ∪ 0 is a distance function between Q-NSSs if the conditions below are satisfied for any ( A)).

Definition 12.
If X = {x 1 , x 2 , . . . , x m } is a universal set, Q = {q 1 , q 2 , . . . , q l } is a nonempty set, A = {e 1 , e 2 , . . . , e n } being a set of parameters, (Γ Q , A) and (Ψ Q , A) are a Q-NSSs(X) and d a distance measure between Q-NSSs for all e j ∈ A and (x, q) i ∈ X × Q which are given below: (1) The Hamming distance: (2) The normalized Hamming distance: (3) The Euclidean distance: The normalized Euclidean distance: The following properties clearly hold: Another main concept in the investigation of imprecise data is the measure of similarity. It shows the level of similarity between sets. Next, we define the similarity measure between Q-NSSs.

Definition 13.
A real non-negative function S : Q − NSS(X) × Q − NSS(X) → [0, 1] is a similarity measure between Q-NSSs if the following conditions are satisfied for any (Γ Q , A), (Ψ Q , A), (Υ Q , A) ∈ Q − NSS(X): In mathematics, the distance measures and similarity measures are connected ideas. Consequently, we can use the proposed distance measure to characterize similarity measures between Q-NSSs. Hence, we can characterize several similarity measures between Q-NSSs (Γ Q , A) and (Ψ Q , A) as follows: The following example illustrates the proposed Hamming, Euclidean, normalized Hamming and normalized Euclidean distances between Q-NSSs and their corresponding similarity measures.

Applications
Q-NSSs is a suitable tool to better model and process imperfect two-dimensional information. In this section, we present a practical examples of Q-NSSs to show that the proposed entropy and similarity measures play a significant role in solving medical diagnosis and decision making problems. To facilitate the discussion, this section will be divided into two subsections. The first subsection will illustrate an application of the proposed entropy measure of Q-NSSs in decision making while the second subsection illustrates two applications of the proposed similarity measures of Q-NSSs in medical diagnosis and decision making.

Application of Entropy of Q-Neutrosophic Soft Sets
In the application of the proposed entropy measures in decision-making problems, if the entropy value for an alternative throughout all the attributes is smaller, the decision-maker can provide more useful information from this alternative. Hence, the alternative with the least entropy value should be assigned a significant preference and priority.
The following algorithm and subsequent example will illustrate the application of Q-neutrosophic soft entropy in decision making. Using the entropy defined in Theorem 2, we introduce the following approach to solve decision making problem followed by a real life example to show the validity of this approach.
Step 1: Suppose there exists P alternatives represented by Q-NSSs.
Step 3: Select the smallest entropy such that the corresponding Q-NSS is the best choice. As known, the less uncertainty information each attractiveness has, the larger possibility they will buy a house. Hence, we can rank buyers according to the entropy values of the corresponding Q-NSSs.

Applications of Similarity Measure of Q-Neutrosophic Soft Sets
We will now construct two algorithms to implement the defined Hamming similarity measure in medical diagnosis and decision making.

Similarity Measures of Q-Neutrosophic Soft Sets Applied to Medical Diagnosis
We will now show the implementation of the characterized Hamming similarity measure of Q-NSS in medical diagnosis.
The following algorithm to a medical diagnosis problem is used to determine if a patient suffers from cerebrum malignancy as we discuss in the next example.
Step 1: Build a model Q-NSS for illness, which may be developed with the assistance of medicinal master person.
Step 2: Construct Q-NSS for the patient.
Step 3: Calculate distance between the model Q-NSS for illness and the Q-NSS for the patient.
Step 4: Calculate similarity measure between the model Q-NSS for illness and the Q-NSS for the patient.
Step 5: If the similarity measure is greater than or equal 0.5 then the patient may possibly suffer from the disease and if the similarity measure is less than 0.5 then the patient may not possibly suffer from the disease.
Below is a conceivable use of similarity measure of Q-NSS in a medical diagnosis to determine whether a patient having a few symptoms is experiencing cerebrum malignancy. Example 4. Let X = {x 1 = severe, x 2 = mild} be a universal set which describes the intensity, Q = {q 1 = frequently, q 2 = rarely} describes the frequency of the symptoms. Let A = {e 1 = headache, e 2 = nausea, e 3 = weakness} be a set of certain symptoms. Step 2: Construct Q-NSSs for patients Y and Z respectively as: Step 3: By Definition 12 the Hamming distance between (Π Q , A) and (Γ Q , A) is 0.867 while between (Π Q , A) and (Ψ Q , A) is 3.567.

Similarity Measures of Q-Neutrosophic Soft Sets Applied to Multicriteria Decision Making
We consider a method to show how to carry out the multicriteria decision-making problem by the defined similarity measure for Q-NSSs. Q-neutrosophic sets with a parametrization tool are favorable for decision making problems in the neutrosophic environment because of their ability to handle two-dimensional indeterminate problems. To solve a multicriteria decision-making problem which is based on the concept of the Q-neutrosophic soft set, we will consider the method introduced in [27] to show how to carry out a multicriteria decision making problem by the defined similarity measure for Q-NSSs. The two assessment criteria under consideration are the benefit and cost. To decide the best estimation of every criterion among every conceivable option, an ideal option can be distinguished by using a max-min-min operator and a min-max-max operator for the benefit and cost criteria, respectively. Then, for Γ Q(e j ) (x, q) i = (T Γ Q (e j ) (x, q) i , I Γ Q (e j ) (x, q) i , F Γ Q (e j ) (x, q) i ), we define an ideal Q-neutrosophic value for a benefit criterion and a cost criterion, respectively, as follows: We can use the following algorithm to use the defined measures of similarity in decision making.
Step 1: Construct Q-NSSs for the alternatives.
Step 2: Construct the ideal Q-NSS from the Q-NSSs of the alternatives taking into account the benefit and the cost criteria.
Step 3: Calculate the Hamming distance between the Q-NSS of the alternatives and the ideal Q-NSS.
Step 4: Calculate the Hamming similarity measure between the Q-NSS of the alternatives and the ideal Q-NSS.
Step 5: The decision is to select the Q-NSS which has the highest similarity to the ideal Q-NSS.

Example 5.
Suppose that there is an investment company which has a downtrend in its returns. To overcome this situation, the administration need to put a rescue package into action. Four committees which are independent of one another, and an evaluation board are established by the administration. Each of these committees has prepared a contingency plan according to the following three criteria: e 1 is the hazard, e 2 is the development and e 3 is the ecological effect, where e 1 and e 3 are the cost criteria and e 2 is the benefit criterion. According to the reports offered separately by the evaluation board, the four possible alternatives are to be obtained under the above parameters by the form of Q-NSSs as follows: Using the approach created to acquire the most appropriate option, from the ideal Q-neutrosophic value, we can get the subsequent ideal Q-NSS: Thus, the four alternatives are ranked orderly as (Ψ Q , A), (Υ Q , A), (Λ Q , A) and (Γ Q , A). The administration should select the rescue with highest score, which is (Ψ Q , A).

Conclusions
We have introduced the entropy of Q-neutrosophic soft set which measures the degree of fuzziness of a Q-neutrosophic soft set , taking into account the two-dimensionality and indeterminacy of a data set, and have proposed several distance measures for Q-neutrosophic soft sets. Using the relations of distance measure to similarity measure, the corresponding similarity measures for Q-NSSs have been obtained. Finally, three different algorithms were constructed to implement the proposed measures in real life decision making issues to show the validity of these measures in a Q-neutrosophic soft environment. The advantages of the proposed measures are that they are defined over the set where the membership, indeterminacy membership and non-membership degrees are defined as a two-dimensional functions, and also complement the Q-NSS model in representing and modeling two-dimensional uncertain, indeterminate and inconsistent data. The measures proposed in this article may be extended further to different entropy, distance and similarity measures such as exponential entropy, cross entropy, Jaccard Measure, Cotangent Measure and others. Furthermore, the work presented in this paper can be used as a foundation to further extend the study of the information measures for structures obtained by incorporating the idea of Q-neutrosophic soft set to other concepts such as the refined neutrosophic set, soft expert set, bipolar/tripolar/multipolar neutrosophic, and many other structures.
Author Contributions: M.A.Q. and N.H. worked together to achieve this work.