In this section, several novel distance measures for (m,a,n)-FNSs are introduced. Their fundamental properties are discussed, and a comparative analysis is carried out with existing distance measures for PyNSs and FNSs.
3.1. Definitions and Properties
Definition 7. Let . Then, the DM is a function that satisfies the following conditions:
- (D1)
.
- (D2)
.
- (D3)
.
- (D4)
If , then and .
Definition 8. Let be a universe of discourse and . Then, the following apply:
- (i)
The Hamming distance between and is defined as follows: - (ii)
The Euclidean distance between and is defined by the following expression: - (iii)
The generalized distance between and is defined by the following expression:
Remark 2. The measures defined in Equations (7), (8), and (9) are not valid DMs because they do not satisfy all the conditions of Definition 7. Example 1. Let and and be two (2,1,3)-FNSs over , where Then, from the definitions of , and , we obtain that It is clear that the Hamming distance, Euclidean distance, and generalized distance do not satisfy condition of Definition 7.
Now, we define the (m,a,n)-FNDMs by considering the PMD, IMD, and NMD of the (m,a,n)-FNSs.
Definition 9. Let and . Then, the following apply:
- (i)
The normalized Hamming distance between and is defined as follows: - (ii)
The normalized Euclidean distance between and is defined by the following expression: - (iii)
The generalized normalized distance between and is defined by the following expression: - (iv)
The normalized Hamming–Hausdorff distance between and is defined as follows: - (v)
The normalized Euclidean–Hausdorff distance between and is defined as follows: - (vi)
The generalized normalized Hausdorff distance between and is defined as follows: - (vii)
The hybrid normalized Hamming distance between and is defined as follows: - (viii)
The hybrid normalized Euclidean distance between and is defined as follows: - (ix)
The generalized hybrid normalized distance between and is defined as follows: - (x)
The tangent inverse distance between and is defined as follows:
Remark 3. For any two (m,a,n)-FNSs and over a universe of discourse , the following apply:
- (i)
The generalized normalized distance between and is reduced to the normalized Hamming distance between and for and the normalized Euclidean distance between and for .
- (ii)
The generalized normalized Hausdorff distance between and is reduced to the normalized Hamming–Hausdorff distance between and for and the normalized Euclidean–Hausdorff distance between and for .
- (iii)
The generalized hybrid normalized distance between and is reduced to the hybrid normalized Hamming distance between and for and the hybrid normalized Euclidean distance between and for .
Example 2. Let and and be two (2,3,4)-FNSs over , where Then, from Definition 9, we obtain that Theorem 1. Let and . Then, is a DM.
Proof. We prove that satisfies the axioms (D1)–(D4).
- (D1)
Since
, and
, we have
,
, and
. It follows that
Again, since , and , we have .
Consequently, .
- (D2)
- (D3)
For any two (m,a,n)-FNSs
and
, we have
- (D4)
Let
be an (m,a,n)-FNS such that
. Then, we have
, and
. It follows that
,
,
,
,
, and
. Therefore, for
, we have
Consequently, .
Similarly, .
□
Theorem 2. Let and . Then, and are DMs.
Proof. Follows from Theorem 1 and Remark 3(i). □
Theorem 3. Let and . Then, is a DM.
Proof. We prove that satisfies axioms (D1)–(D4).
- (D1)
Since , and , we have .
Again, since , and , we have .
Consequently, .
- (D2)
- (D3)
For any two (m,a,n)-FNSs
and
, we have
- (D4)
Let
be an (m,a,n)-FNS such that
. Then, we have
, and
. It follows that
,
,
,
,
, and
. Therefore,
, and we have
Hence, . Similarly, .
□
Theorem 4. Let . Then, and are valid DMs.
Proof. Follows from Theorem 3 and Remark 3(ii). □
Theorem 5. Let . Then, is a valid distance measure.
Proof. We prove that satisfies the axioms (D1)–(D4).
- (D1)
Since
, we have
,
, and i=1…r. Similarly,
and
. It follows that
and
Hence, .
- (D2)
- (D3)
For any two (m,a,n)-FNSs
and
, we have
- (D4)
Let
be an (m,a,n)-FNS such that
. Then, we have
and
. It follows that
,
,
and
,
,
. Therefore,
, and we have
Consequently, . Similarly, .
□
Theorem 6. Let . Then, and are valid distance measures.
Proof. Follows from Theorem 5 and Remark 3(iii). □
Theorem 7. Let . Then, is a valid distance measure.
Proof. We prove that satisfies the axioms (D1)–(D4).
- (D1)
Since , for i…r, we have , and thus . Similarly, we have and . Therefore, we obtain .
- (D2)
- (D3)
For any two (m,a,n)-FNSs
and
, we have
- (D4)
Let
be an (m,a,n)-FNS such that
. Then, we have
and
. It follows that
,
,
,
,
, and
. Therefore,
, and we have
Thus, . Similarly, we have .
□
Now, we define the (m,a,n)-FNWDMs between two (m,a,n)-FNSs and by considering the weighting vector of the elements in the (m,a,n)-FNSs.
Definition 10. Let and . Assume that = is the weighting vector of the elements (i = 1, 2, …, r), satisfying the conditions , ∀∈ [0, 1], and i = 1, 2, …, r. Then, the following statements apply:
- (i)
The weighted normalized Hamming distance between and is defined as follows: - (ii)
The weighted normalized Euclidean distance between and is defined by the following expression: - (iii)
The generalized weighted normalized distance between and is defined by the following expression: - (iv)
The weighted normalized Hamming–Hausdorff distance between and is defined as follows: - (v)
The weighted normalized Euclidean–Hausdorff distance between and is defined as follows: - (vi)
The generalized weighted normalized Hausdorff distance between and is defined as follows: - (vii)
The hybrid weighted normalized Hamming distance between and is defined as follows: - (viii)
The hybrid weighted normalized Euclidean distance between and is defined as follows: - (ix)
The generalized hybrid weighted normalized distance between and is defined as follows: - (x)
The weighted tangent inverse distance between and is defined as follows:
Remark 4. For any two (m,a,n)-FNSs and over a universe of discourse , the following apply:
- (i)
The generalized weighted normalized distance between and is reduced to the weighted normalized Hamming distance between and for and the weighted normalized Euclidean distance between and for .
- (ii)
The generalized weighted normalized Hausdorff distance between and is reduced to the weighted normalized Hamming–Hausdorff distance between and for and the weighted normalized Euclidean–Hausdroff distance between and for .
- (iii)
The generalized hybrid weighted normalized distance between and is reduced to the hybrid weighted normalized Hamming distance between and for and the hybrid weighted normalized Euclidean distance between and for .
Remark 5. When we take the weighting vector ω = , then, in the (m,a,n)-FNWDM, the following occur:
- (i)
reduce to .
- (ii)
reduce to .
- (iii)
reduce to .
- (iv)
reduce to .
- (v)
reduce to .
- (vi)
reduce to .
- (vii)
reduce to .
- (viii)
reduce to .
- (ix)
reduce to .
- (x)
reduce to .
Example 3. Let and be two (m,a,n)-FNSs over , defined in Example 2, and consider the weighting vector ω = . Then, Theorem 8. Let and . Assume that = is the weighting vector of the elements (i = , r), satisfying the conditions , ∀, and i = , r. Then, the (m,a,n)-FNWDMs proposed in Definition 10 are the valid DMs for the (m,a,n)-FNSs.
Proof. This follows by considering a process similar to that used in the corresponding cases of (m,a,n)-FNDMs. □
3.2. Comparison of the Proposed DMs with the Existing DMs in a Neutrosophic Environment
This section presents a comparative analysis of the proposed (m,a,n)-FNDMs and (m,a,n)-FNWDMs and the existing DMs for PyNSs and FNSs using remarks and numerical examples.
Remark 6. The weighted DMs , , and proposed by Rajan and Krishnaswamy [46] for PyNSs are, respectively, the special cases of , , and for m = a = n = 2. Remark 7. The weighted DMs , , and proposed by Saeed and his coworkers [47] for FNSs are, respectively, the special cases of , , and for m = a = n = 3. Remark 8. The following example shows the that the weighted DMs , , and proposed by Rajan and Krishnaswamy [46] for PyNSs and the weighted DMs , , and proposed by Saeed and his coworkers [47] for FNSs are inconsistent and irrational. Example 4. Let , and (m,a,n)-FNS , (i=1..4) be defined as follows:
Assume that is the weight for the elements of . Table 2 displays the established (m,a,n)-FNWDMs and the existing weighted DMs for PyNSs and FNSs. In the considered cases, we observe that , , , and . Upon examining Table 2, it becomes apparent that the results produced by the Fermatean neutrosophic weighted Hamming distance for the pairs () and () is identical, which is contradictory and logically inconsistent given the differences in and . A similar anomaly is observed in the case of the Fermatean neutrosophic weighted Euclidean distance , where the distance values for the pairs () and () are also the same, despite the dissimilarity between and . Furthermore, the distance measures , , and fail to compute valid results for the pair ( ,) because the sum of the squares of the Positive Membership Degree (PMD) and the Negative Membership Degree (NMD) at point of exceeds 1, violating the required constraints. These inconsistencies highlight the limitations of existing distance measures for orthopair fuzzy sets within the neutrosophic environment. Remark 9. The (m,a,n)-FNDMs proposed in Definition 9 will be reduced to DMs for q-RONSs if we take m = a = n = q.
Remark 10. The (m,a,n)-FNWDMs proposed in Definition 10 will be reduced to weighted DMs for q-RONSs if we take m = a = n = q.
Remarks 6–10 and Examples 2–4 reveal that the proposed distance measures for (m,a,n)-FNSs exhibit enhanced sensitivity and discriminatory power, enabling more reliable and meaningful differentiation among data points. Moreover, the developed (m,a,n)-fuzzy neutrosophic-based distance measures (FNDMs) and weighted distance measures (FNWDMs) are versatile enough to be applied in decision-making problems involving PyNSs, FNSs, and q-RONSs. In contrast, the distance measures formulated under these existing frameworks are generally not applicable to the broader structure of (m,a,n)-FNSs. It is worth noting that the effectiveness of the proposed methods may depend on the selection of the parameters m, a, and n, highlighting the need for a comprehensive analysis of their robustness across varying parameter values. Future research may explore the practical implementation of this framework in real-world decision-making scenarios to assess its applicability and effectiveness.