1. Introduction

2. Preliminaries

3. Some novel tanimoto similarity and distance measures for PFSs

4. Applications

4.1. A model for pattern recognition and medical diagnosis

There are some known patterns $F=\{F_1,F_2,\dots ,F_n\}$, denoted by PFSs $F_i=\{〈z_l,{\rho }_{F_i},{\sigma }_{F_i}〉:z_l\in Z\}(i=1,2,\dots ,n)$ represented by a finite universe of discourse $Z=\{z_1,z_2,\dots ,z_k\}$. There are some samples of unknown categories $S=\{S_1,S_2,\dots ,S_m\}$ , denoted by $S_j=\{〈z_l,{\rho }_{S_j},{\sigma }_{S_j}〉:z_l\in Z\}(j=1,2,\dots ,m)$. Our goal is to classify $S_j$ into the given patterns $F_i$. The recognition process is described in detail below:

Step 1

Calculate the Tanimoto similarity(or distance) between $S_j$ and $F_i$.

Step 2

Obtain the maximum Tanimoto similarity $sm\left(F_o,S_j\right)$ using equation 17 or the minimum Tanimoto distance $d\left(F_o,S_j\right)$ using equation 18:

$$sm\left(F_o,S_j\right)=max\left\{T_{PFS}(F_i,S_j)\right\}$$

(17)

$$d\left(F_o,S_j\right)=min\left\{D{T}_{PFS}(F_i,S_j)\right\}$$

(18)

Step 3

If any pattern $F_o$ has the highest Tanimoto similarity between $S_i$, then, $S_i$ and $F_o$ belong to the same category:

$$o=argmax\left\{sm\left(F_o,S_j\right)\right\},S_j\to F_o$$

(19)

If distance measure is used as the standard of measure, then the following form would be applied:

$$o=argmin\left\{d\left(F_o,S_j\right)\right\},S_j\to F_o$$

(20)

The pseudo code description of the model is as Algorithm 1.

Algorithm 1 Algorithm for pattern recognition and medical diagnosis

Example 6. 

This example is centered on the recognition of patterns. It leverages PFSs for the delineation of three instances whose categories have been confirmed, denoted as $F=\{F_1,F_2,F_3\}$, as well as a solitary sample from an as yet unrecognized category, labeled S. The parameters $z_i(i=1,2,3,4,5)$ that govern these samples are detailed in Table 10.

In order to ascertain the categorical ascription of the unknown sample S, we employed the Tanimoto similarity measures as delineated in our methodological framework:

Step 1

Calculate the Tanimoto similarity(or distance) between $S_j$ and $F_i$, the results are presented in Table 11 and Table 12.

Step 2

According to the equation 17 and equation 18, the maximum similarity and minimum distance are shown as below:

$$sm\left(F_2,S\right)=0.804$$

$$d\left(F_2,S\right)=0.196$$

Step 3

According to the principle of maximum similarity and minimum distance, the pattern recognition result of S are as follows:

$$S\to F_2$$

Furthermore, the measurement results derived from the extant methods are detailed in Table 13.

Utilizing the data shown in Table 13, we can identify that S and $F_2$ demonstrate the highest degree of similarity. This finding is consistently confirmed across the Tanimoto similarity measures before, validating the effectiveness of the proposed similarity measures. It is noteworthy that the similarity scores derived from the Tanimoto measures exhibit significant variability. We computed the Tanimoto similarity values for all PFSs with respect to S and subtracted the second highest value from the maximum one, denoted as $D_d$, and a similar analysis was conducted for other measures, as depicted in Figure 4. We can find that the Tanimoto similarity measure yields the highest $D_d$ and ranks third in terms of indeterminacy. The similarity scores between S and $F_2$ as determined by our proposed metrics are substantially different from those between S and other PFSs. Consequently, we can assert with confidence that $F_2$ is a more suitable pattern. However, the similarity scores between S and the known samples derived from other measures are more closely aligned. Hence, relying on these measures for decision-making may result in greater hesitation. These findings align with the conclusions drawn in Example 5, suggesting that our proposed measures are superior in distinguishing between samples.

Example 7. 

In this example, the aim is to recognize mineral categories through pattern recognition. Specifically, we consider six common mixtures of minerals that are represented by PFSs $F_i(i=1,2,3,4,5,6)$, with each mixture consisting of six fundamental minerals which collectively form the universe of discourse $z_i(i=1,2,3,4,5,6)$. Our primary goal is to employ the proposed measure in order to determine the category to which an unknown mixed mineral S pertains. Table 14 displays the known PFSs and the unidentified mineral S, whereas Table 15 summarizes the resulting outcomes.

Table 15 displays the similarity measures for mineral S and $F_4$, with the Tanimoto measure producing the highest similarity score. This result confirms that mineral S belongs to $F_4$, which is consistent with findings from other studies that have used various similarity measures. Likewise, Figure 5 illustrates the $D_d$ among different measures, and the Tanimoto similarity measures rank the first and the second among the measures shown in the figure.

Example 8 

([55]). Assuming the presence of four patients, namely Ragu, Mathi, Velu and Karthi, denoted by $I=\{I_1,I_2,I_3,I_4\}$, and exhibiting symptoms including Headache, Acidity, Burning eyes, Back pain and Depression, represented as $Z=\{Z_1,Z_2,Z_3,Z_4,Z_5\}$. The set of possible diagnoses is denoted by $D=\{D_1,D_2,D_3,D_4,D_5\}$, and includes: $D_1$: Stress; $D_2$: Ulcer; $D_3$: Vision problem; $D_4$: Spinal problem; $D_5$: Blood pressure. The Pythagorean fuzzy relation $I\to Z$ is expressed by PFS, as shown in Table 16, while the Pythagorean fuzzy relation $Z\to D$ is represented by PFSs and listed in Table 17. Every entry in both tables is defined by the PFS, with the values indicating membership degree and nonmembership degree, respectively. The proposed similarity and distance measures are employed to evaluate the similarity and distance between each patient and potential diagnosis. Based on the principle of maximum similarity or minimum distance, each patient is diagnosed accordingly. Table 18 and Table 19 present the similarity measure outcomes and distance measure outcomes of each patient I towards each diagnosis D, alongside the ultimate diagnosis results.

Based on the findings presented in Table 18 and Table 19, it is observed that $I_1$ exhibits the highest Tanimoto similarity measure and the lowest Tanimoto distance measure towards $D_1$; $I_2$ displays the highest Tanimoto similarity measure and the lowest Tanimoto distance measure towards $D_4$; $I_3$ demonstrates the highest Tanimoto similarity measure and the lowest Tanimoto distance measure towards $D_3$; and $I_4$ showcases the highest Tanimoto similarity measure and the lowest Tanimoto distance measure towards $D_1$. Thus, we can conclude that Ragu is diagnosed with stress, Mathi with spinal problems, Velu with vision problems and Karthi with stress.

In order to validate the effectiveness of our proposed measures, a comparison was conducted against other methods, and the outcomes have been summarized in Table 20. It is observed from Table 20 that our proposed measures provide diagnostic outcomes that are consistent with those obtained using methods proposed by Xiao and Ding [55], Zhou [56] and Deng [57]. The experimental results lend support to the practicability of our proposed similarity and distance measures.

4.2. The model for MADM

Suppose that $I=\{I_1,I_2,\dots ,I_m\}$ is a discrete set of alternatives, and $A=\{A_1,A_2,\dots ,A_n\}$ is the set of attributes, $\omega =\{{\omega }_1,{\omega }_2,\dots ,{\omega }_n\}$ is the weighting vector of the attribute $A_j(j=1,2,\dots ,n)$, where $\omega >0$, $\sum _{j=1}^n{w}_j=1$. $T=\left\{t_1,t_2,\dots ,t_n\right\}$ is the experts evaluate, representing the type of each attribute, where 1 represents the benefit type and 0 represents the cost type. Suppose that $R={({\rho }_{ij},{\sigma }_{ij})}_{m\times n}$ is the Pythagorean fuzzy matrix, where ${\rho }_{ij}$ indicates the degree that the alternative $I_i$ satisfies the attribute $A_j$ and ${\sigma }_{ij}$ indicates the degree that the alternative $I_i$ does not satisfy the attribute $A_j$, ${\rho }_{ij}\subset [0,1]$, ${\sigma }_{ij}\subset [0,1]$, ${\left({\rho }_{ij}\right)}^2+{\left({\sigma }_{ij}\right)}^2\le 1$, $i=1,2,\dots ,m$, $j=1,2,\dots ,n$. The proposed model is described below:

Step 1

Defining the Pythagorean fuzzy positive ideal solution $I^{+}$:

$$({\rho }^{+},{\sigma }^{+})=(({\rho }_1^{+},{\sigma }_1^{+}),({\rho }_2^{+},{\sigma }_2^{+}),\dots ,({\rho }_n^{+},{\sigma }_n^{+}))=((1,0),(1,0),\dots ,(1,0))$$

(21)

The equation for correction is as follows:

$$I^{+}=\left(\left({\rho }_j,{\sigma }_j\right)\right)\left\{\begin{array}{c}\begin{array}{cc}\left({\rho }_j,{\sigma }_j\right)& for{t}_j=1\end{array}\\ \begin{array}{cc}\left({\sigma }_j,{\rho }_j\right)& for{t}_j=0\end{array}\end{array}\right.$$

(22)

Step 2

Calculating the weighted Tanimoto similarity measure between $I_i(i=1,2,\dots ,m)$ and $I^{+}$ as follows:

$$T_{PFS}^{\omega }(I_{\mathrm{i}},I^{+})=\frac{\sum _{i=1}^n{\omega }_{\mathrm{i}}^4({\rho }_{ij}^2{\left({\rho }_j^{+}\right)}^2+{\sigma }_{ij}^2{\left({\sigma }_j^{+}\right)}^2)}{(\sum _{i=1}^n{\omega }_{\mathrm{i}}^4({\rho }_{ij}^4+{\left({\rho }_j^{+}\right)}^4-{\rho }_{ij}^2{\left({\rho }_j^{+}\right)}^2)+\sum _{i=1}^n{\omega }_{\mathrm{i}}^4({\sigma }_{ij}^4+{\left({\sigma }_j^{+}\right)}^4-{\sigma }_{ij}^2{\left({\sigma }_j^{+}\right)}^2))}$$

(23)

If the standard of measure is distance, then additional distance measures must be calculated in the following way:

$$D{T}_{PFS}^{\omega }(I_i,I^{+})=1-T_{PFS}^{\omega }(I_i,I^{+})$$

(24)

Step 3

Obtain the maximum Tanimoto similarity $sm\left(I_o,I^{+}\right)$ using equation 25 or the minimum Tanimoto distance $d\left(I_o,I^{+}\right)$ using equation 26:

$$sm\left(I_o,I^{+}\right)=max\left\{T_{PFS}(I_i,I^{+})\right\}$$

(25)

$$d\left(I_o,I^{+}\right)=min\left\{D{T}_{PFS}(I_i,I^{+})\right\}$$

(26)

Step 4

If any alternative $I_o$ has the highest Tanimoto similarity between $I^{+}$, then, $I_o$ is the most important alternative:

$$o=argmax\left\{sm\left(I_o,I^{+}\right)\right\},I_o\to Result$$

(27)

If distance measure is used as the standard of measure, then the following form would be applied:

$$o=argmin\left\{d\left(I_o,I^{+}\right)\right\},I_o\to Result$$

(28)

The pseudo code description of the model is as Algorithm 2.

Algorithm 1 Algorithm for MADM

Example 9. 

A company is planning to purchase a batch of computers, with six alternative model schemes $I_{\mathrm{i}}(i=1,2,3,4,5,6)$ to choose from, and has selected five attributes as selection criteria, including the materials used, reputation, response time, service life and price $A_j(j=1,2,3,4,5)$. The weights of these attributes are denoted by ${\omega }_j(j=1,2,3,4,5)$, where $\omega =(0.2,0.4,0.1,0.1,0.2)$. After expert evaluation, all attribute types are provided: $T=\left\{1,1,0,1,0\right\}$. Forming a Pythagorean fuzzy decision matrix $R_{6\times 5}$:

$$R_{6\times 5}=\left[\begin{array}{ccccc}\left(0.287,0.955\right)& \left(0.846,0.123\right)& \left(0.622,0.776\right)& \left(0.214,0.791\right)& \left(0.808,0.524\right)\\ \left(0.232,0.364\right)& \left(0.940,0.241\right)& \left(0.564,0.366\right)& \left(0.019,0.805\right)& \left(0.207,0.834\right)\\ \left(0.081,0.645\right)& \left(0.026,0.584\right)& \left(0.801,0.266\right)& \left(0.870,0.030\right)& \left(0.598,0.554\right)\\ \left(0.565,0.459\right)& \left(0.759,0.416\right)& \left(0.860,0.093\right)& \left(0.480,0.145\right)& \left(0.622,0.743\right)\\ \left(0.130,0.768\right)& \left(0.613,0.584\right)& \left(0.114,0.653\right)& \left(0.048,0.531\right)& \left(0.050,0.440\right)\\ \left(0.210,0.300\right)& \left(0.634,0.176\right)& \left(0.146,0.919\right)& \left(0.323,0.702\right)& \left(0.812,0.575\right)\end{array}\right]$$

We will utilize the proposed MADM model to select the most appropriate solution.

Step 1

Defining the Pythagorean fuzzy positive ideal solution $I^{+}$. As we want the response time and price to be as low as possible, these two attributes are defined as cost types and set to $\left(0,1\right)$:

$$\begin{array}{cc}\hfill ({\rho }^{+},{\sigma }^{+})& =(({\rho }_1^{+},{\sigma }_1^{+}),({\rho }_2^{+},{\sigma }_2^{+}),({\rho }_3^{+},{\sigma }_3^{+}),({\rho }_4^{+},{\sigma }_4^{+}),({\rho }_5^{+},{\sigma }_5^{+}))\hfill \\ & =\left(\right(1,0),(1,0),(0,1),(1,0),(0,1\left)\right).\hfill \end{array}$$

Step 2

Calculating the weighted Tanimoto measure between $I_i(i=1,2,\dots ,6)$ and $I^{+}$ as Table 21 and Table 22.

Step 3

According to the equation 25 and equation 26, the maximum similarity and minimum distance are shown as below:

$$\begin{array}{c}sm\left(I_2,I^{+}\right)=0.913\\ d\left(I_2,I^{+}\right)=0.087\end{array}$$

Step 4

According to the equation 27 and equation 28, the most suitable computer type is $I_2$.

Furthermore, we have obtained congruent outcomes by employing the MADM models proposed by Wang [29] and Zhang [30], respectively, which proves the effectiveness of the proposed model. The corresponding calculation outcomes have been delineated in Table 23 and Table 24, respectively. The parameter α is a positive constant and $U^{+}$ represents the positive ideal solution, which are calculated using Wang’s method.

5. Conclusions

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