Nonlinear strict distance and similarity measures for intuitionistic fuzzy sets with applications to pattern classification and medical diagnosis

In this paper, we propose a new type of nonlinear strict distance and similarity measures for intuitionistic fuzzy sets (IFSs). Our proposed methods not only have good properties, but also improve the drawbacks proposed by Mahanta and Panda (Int J Intell Syst 36(2):615–627, 2021) in which, for example, their distance value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{_{\textrm{MP}}}(\langle \mu , \nu \rangle , \langle 0, 0\rangle )$$\end{document}dMP(⟨μ,ν⟩,⟨0,0⟩) is always equal to the maximum value 1 for any intuitionistic fuzzy number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \mu , \nu \rangle \ne \langle 0, 0\rangle $$\end{document}⟨μ,ν⟩≠⟨0,0⟩. To resolve these problems in Mahanta and Panda (Int J Intell Syst 36(2):615–627, 2021), we establish a nonlinear parametric distance measure for IFSs and prove that it satisfies the axiomatic definition of strict intuitionistic fuzzy distances and preserves all advantages of distance measures. In particular, our proposed distance measure can effectively distinguish different IFSs with high hesitancy. Meanwhile, we obtain that the dual similarity measure and the induced entropy of our proposed distance measure satisfy the axiomatic definitions of strict intuitionistic fuzzy similarity measure and intuitionistic fuzzy entropy. Finally, we apply our proposed distance and similarity measures to pattern classification, decision making on the choice of a proper antivirus face mask for COVID-19, and medical diagnosis problems, to illustrate the effectiveness of the new methods.

To more effectively compare and distinguish IFNs and IFSs, the concept of strict intuitionistic fuzzy similarity/ distance measures was introduced by Wu et al. 37 as follows.

The proposed nonlinear strict distance, similarity and entropy measures for IFSs
After we investigate the distance measure for IFSs proposed by Mahanta and Panda 38 , we find that Mahanta and Panda's 38 distance gave serious drawbacks.We present these drawbacks in next subsection.
To overcome the drawbacks of Mahanta and Panda's distance measure mentioned above, we propose a new nonlinear strict distance measure for IFNs and IFSs in next subsection, which is proved to satisfy the axiomatic definition of IFDisM.
Similarly, we can define a new measure E on based on the parametric distance d ( ) pd as follows: The measure E defined by Eq. ( 8) is an entropy on .
Mahanta and Panda 38 claimed that the IFDisM d MP can deal adequately with the IF information having high uncertainty, i.e., having low values of membership and nonmembership grades.To close this section, it is shown that our proposed parametric distance d ( ) pd can effectively distinguish different IFVs with high hesitancy.Fix > 0 and give two different IFVs α = �µ α , ν α � and β = �µ β , ν β � with µ α + ν α ≤ 2 and µ β + ν β ≤ 2 .By differential mean value theorem, it can be verified that ( 5) According to the above theoretical analysis and also the presentation in Fig. 1, we can find that, when the parameter is sufficiently small, the distance d ( ) pd can reach very large numbers and is sensitive to small perturbations, even if the degrees of membership and nonmembership are very small.Thus, the smaller the parameter is, the stronger the sensitivity is.Therefore, the proposed parametric distance d ( ) pd can better distinguish IFVs with small degrees of membership and nonmembership.And so, throughout this paper, the values of the parameter are chosen smaller.Meanwhile, according to Eq. ( 4), it is clear that the value of d ( ) pd will be sufficiently close to , when the parameter is sufficiently higher.In this case, the distance measure d ( ) pd cannot distinguish different IFSs with high hesitancy, when the parameter is sufficiently higher.In this sense, the values of the parameter will not be chosen too high, but better with smaller values.
The proposed IFDisM, IFSisM and IFEM for IFSs.Following the newly defined function d ( ) pd on in "A new parametric distance on ", we now propose the new IFDisM (distance), IFSisM (similarity) and IFEM (entropy) for IFSs as follows.Let � = {ϑ 1 , ϑ 2 , . . ., ϑ ℓ } and > 0 .Define the function d ( ) New : Similarly, a new entropy measure for IFSs can be defined according to the defined IFDisM d ( ) New (I 1 , I 2 ) as follows: According to Theorems 3.1 and 3.2, we can directly obtain the following theorems.

Comparative analysis with
, by direct calculation and Eq. ( 9), we have By varying IFS I 2 within IFS(�) , Fig. 2 shows the changing trend of distances between I 1 and I 2 by using our proposed formula (9) for = 0.02, 0.04, 0.06, 0.08, 0.1 .Observing from Example 3.1, Proposition 3.4, and Fig. 2, it is revealed that the distance d ( ) New (I 1 , I 2 ) between I 1 and I 2 is always less than 1, and changed with the change of I 2 , which are reasonable, and significantly better than the result obtained by Mahanta and Panda's distance measure in Example 3.1.
(10) S ( ) .By direct calculation and Eq. ( 9), we have that, for 0 < µ, ν ≤ 1 , d ( ) New (I ′ 1 , By varying µ and ν from 0 to 1, Fig. 3 shows the changing trend of distances between I ′ 1 and I ′ 2 by using our proposed formula (9) for = 0.02, 0.04, 0.06, 0.08, 0.1 .Observing from Example 3.2, Proposition 3.4, and Fig. 3, it is revealed that the distance d ( ) New (I ′ 1 , I ′ 2 ) between I ′ 1 and I ′ 2 is always less than 1 except for µ = ν = 1 , and changes with the changed of I ′ 1 and I ′ 2 , which are reasonable, and significantly better than the result obtained by Mahanta and Panda's distance measure in Example 3.2.

Applications
A pattern classification problem.Example 4.1 ( 32 Application 2, 14 Example 4.3) Consider a pattern classification problem with three classes and three attributes A = {x 1 , x 2 , x 3 } , described by three patterns P = {P 1 , P 2 , P 3 } and a test sample S 1 expressed by the IFSs listed in Table 1.
By taking the weight vector ω of three attributes as ω = ( 1 3 , 1 3 , 1 3 ) ⊤ , based on the principle of the maximum degree of SimMs, the pattern classification results obtained by using different distance measures are listed in Table 2 and Fig. 4. Observing from Table 2 and Fig. 4, we know that the test sample S 1 is classified to the pattern P 3 by our proposed DisM with = 0.14, 0.16, 0.18 , which is consistent with the results obtained by the DisMs   A TOPSIS method based on our proposed strict IFDisM and IFSimM.Suppose that there are n alternatives A i ( i = 1, 2, . . ., n ) evaluated with respect to m attributes A j ( j = 1, 2, . . ., m ).The sets of the alter- natives and attributes are denoted by A = {A 1 , A 2 , . . ., A n } and A = {A 1 , A 2 , . . ., A m } , respectively.The rating (or evaluation) of each alternative , denoted by r ij = �µ ij , ν ij � for short, where µ ij ∈ [0, 1] and ν ij ∈ [0, 1] are respectively the satisfaction (or membership) degree and dissatisfaction (or non-membership) degree of the alternative A i ∈ A on the attribute o j satisfying the condition 0 ≤ µ ij + ν ij ≤ 1 .A multi-attribute decision-making (MADM) prob- lem with IFSs is expressed in matrix form shown in Table 3.
For the MADM problem with IFSs, by using our proposed IFDisM d ( ) New of Eq. ( 9), we construct a new IF TOPSIS method as follows: Step 1: (Construct the decision matrix) Supposing that the decision-maker gave the rating (or evaluation) of each alternative A i ∈ A ( i = 1, 2, . . ., n ) on each attribute A j ( j = 1, 2, . . ., m ) in the form of IFNs r ij = �µ ij , η ij � , construct an IF decision matrix R = (r ij ) m×n as shown in Table 3.
Step 2: (Normalize the decision matrix) Transform the IF matrix R = (r ij ) m×n to the normalized IF decision matrix R = (r ij ) m×n = (� μij , νij �) m×n as follows: where r ∁ ij is the complement of γ ij .
Step 3: (Determine the positive and negative ideal-points) Determine the IF positive ideal-point Step 4: (Compute the weighted similarity measures) Compute the weighted similarity measures between the alternatives A i ( i = 1, 2, . . ., n ) and the IF positive ideal-point I + , and between the alternatives A i ( i = 1, 2, . . ., n ) and the IF negative ideal-point I − , by using the following formulas: and Step 5: (Compute the relative similarity degrees) Calculate the relative similarity degrees C i of the alternatives A i ( i = 1, 2, . . ., n ) to the IF positive ideal-point I + by using the following formula: Step 6: (Rank the alternative) Rank the alternatives A i ( i = 1, 2, . . ., n ) according to the nonincreasing order of the relative closeness degrees C i and select the most desirable alternative.rij = r ij , for benefit attribute A j , r ∁ ij , for cost attribute A j , 3) After the outbreak of COVID-19 disease, the demand for masks has increased rapidly.There are six common types of masks on the market as follows: M 1 -disposable medical masks, M 2 -medical-surgical masks, M 3 -particulate respirators (N95), M 4 -ordinary nonmedical masks, M 5 -medical protective masks, and M 6 -gas masks.A citizen wants to buy a suitable mask from the above six types of masks by considering the following four attributes: A 1 -leakage rate, A 2 -recyclability, A 3 -quality of raw material, A 4 -filtration capability.
Step 1: (Construct the decision matrix) Through the market survey, the evaluations of each type of mask M i ( i = 1, 2, 3, 4, 5, 6 ) on each attribute A j ( j = 1, 2, 3, 4 ) in the form of IFNs are summarized in Table 4.
Step 2: (Normalize the decision matrix) Because A 1 is a cost attribute and A 2 -A 4 are the benefit attributes, the normalized IF decision matrix is formed as shown in Table 5.
From Table 7, which shows a comparison of the preference orders of the alternatives in Example 4.2 for different TOPSIS methods, we observe that although our ranking result is different from these obtained by the TOPSIS method in 25,35,43 , the most desirable mask type is always M 3 -particulate respirators (N95).Note that the scores of M 3 on the attributes A 2 , A 3 , and A 4 (by Table 5) are much greater than that of M 1 .This gives a reason to support the conclusion that M 3 is better than M 1 .Therefore, our method is more reasonable than that of Mahanta and Panda 38 .
To study the changing tendency of the relative similarity degrees and the rankings for M 1 , M 2 , . . ., M 6 with the variation of the parameter from 0 to 1, Fig. 5 is used for illustration.Observing from Fig. 5, it is revealed that the rankings for M 1 , M 2 , . . ., M 6 remain unchange with the variation of the parameter from 0 to 1.As a result, N95 is always the most desirable type of marks.
In the above analysis, we assume that four attributes A 1 -A 4 have the same weight.To study the impact of the weights of attributes on the decision process, Fig. 6 is used for illustration.Observing from Fig. 6, it is revealed that although the most desirable mask type is always M 3 -particulate respirators (N95), the rankings of M 1 and M 2 may change, when changing the weights of attributes and the parameter .

Figure 1 .
Figure 1.Entropy measure E for different values of .

Example 3 . 3 (
Mahanta and Panda's distance measure.This subsection illustrates that our proposed distance measure can completely overcome Mahanta and Panda's drawbacks mentioned in "The drawbacks of distance measure of Mahanta and Panda 38 ".Continuation of Example 3.1) Take the IFSs I 1 on � = {ϑ} as given in Example 3.1.For any

Table 1 .
Pattern classification in Example 4.1.

Table 2 .
38ttern recognition results by different similarity measures in Example 4.1.✕denotesthatitcannotbedetermined.The details for distance measures in Table2can be found in32, Section III.MP by Mahanta and Panda 38 have 1 − d MP (P 1 , S 1 ) = 0.8354 and 1 − d MP (P 3 , S 1 ) = 0.8383 .This means that 1 − d MP (P 3 , S 1 ) > 1 − d MP (P 1 , S 1 ) , and so it is able to distinguish between the patterns, but only a little.However, if we retain 2 digits after the decimal point, we have 1 − d MP (P 3 , S 1 ) = 0.84 = 1 − d MP (P 1 , S 1 ) , and so d MP by Mahanta and Panda38can not distinguish between the patterns.

Table 4 .
IFN evaluation of different types of masks.

Table 5 .
Normalized IFN evaluation of different types of masks.

problem. Example 4.3
( 38Example 144,14) Consider a medical diagnosis problem for 4 patients P = {P 1 , P 2 , P 3 , P 4 } with the symptoms S = {Temperature, Headache, Stomach pain, Cough, Chest pain} represented by using IFNs, as listed in Table8.The symptom characteristics for diagnosis D = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem} are represented by using IFNs, as shown in Table9.