A framework for choosing an appropriate fuzzy set extension in modeling

Abstract

Real-world problems contain uncertainties. Fuzzy Set Theory (FST) is a popular approach to model these uncertainties. FST extensions (FSTEs) have been offered for better modeling of the uncertainties having different natures. It is essential to use the most suitable FSTE in modeling to achieve reasonable, reliable, and realistic results. However, FSTEs are preferred without stating a clear reason in most of the studies. This makes the quality and the reliability of the results of these studies questionable. Because, to obtain reliable models, the dynamics of the problem and environment should be well understood, the scenario should be well analyzed, and the assumptions and limitations of FSTE theories should be well known. In this study, a guiding framework for choosing the most suitable FSTE in modeling to obtain reliable, applicable, and efficient results is proposed. The framework consists of two parts: (i) conceptual analysis of the uncertainty types and FSTEs, (ii) a guiding procedure fed by the first step for deciding the most suitable FSTE. The procedure is illustrated by multiple numerical examples to make its benefits clear. Conceptual analysis and numerical examples show that some FSTEs have some advantages over the others for specific scenarios and problem types. For example, NS is more suitable than PFS for modeling the problems other than Multi-Criteria Decision-Making (MCDM). Another contribution of this study is showing that it is very important to choose the simplest possible FSTE to obtain reliable and applicable models.

Appendix 1. Aggregation Operators and Score Functions

The aggregation operators and the score functions used in numerical example (Section 5.2) are presented in the following subsections. Traditional FSs can be ranked according to their membership degree. FSTEs be ranked according to their score and accuracy function values. The FSTE having the biggest score value is a bigger FSTE. If the score value for two FSTEs are equal, then the FSTE having bigger accuracy value is the bigger one.

1.1 Traditional fuzzy set

Definition 15

Let${\overset{\sim }{A}}_j=\left\langle {\mu}_j\right\rangle$ be a traditional FS, ${w}_j\in \left[0,1\right],{\sum}_{j=1,\dots, n}{w}_j=1$ be weight for ${\overset{\sim }{A}}_j$. Weighted averaging operator for IFSs is defined as in Eq. (27) [14]:

$$ WA={\sum}_{j=1}^n{w}_j{\mu}_j $$

(27)

1.2 Intuitionistic fuzzy set

Definition 16

Let${\overset{\sim }{A}}_j=\left\langle {\mu}_j,{\vartheta}_j\right\rangle$ be IFSs, ${w}_j\in \left[0,1\right],\sum \limits_{j=1,\dots, n}{w}_j=1$ be weight for ${\overset{\sim }{A}}_j$. Weighted averaging operator for IFSs is defined as in Eq. (28) [14]:

$$ IWA={\sum}_{j=1}^n{w}_j{\overset{\sim }{A}}_j=\left\langle {\sum}_{j=1}^n{w}_j{\mu}_j,\kern0.5em {\sum}_{j=1}^n{w}_j{\vartheta}_j\right\rangle $$

(28)

Definition 17

Let$\overset{\sim }{A}=\left\langle \mu, \vartheta \right\rangle$ be an IFS. Score $\left({S}_{\overset{\sim }{A}}\right)$and accuracy $\left({H}_{\overset{\sim }{A}}\right)$values can be calculated as shown in Eqs. (29–30) [62]:

$$ {S}_{\overset{\sim }{A}}=\upmu -\vartheta $$

(29)

$$ {H}_{\overset{\sim }{A}}=\mu +\vartheta $$

(30)

1.3 Pythagorean fuzzy set

Definition 18

Let${\overset{\sim }{A}}_j=\left\langle {\mu}_j,{\vartheta}_j\right\rangle$ be PFSs, ${w}_j\in \left[0,1\right],\sum \limits_{j=1,\dots, n}{w}_j=1$ be weight for ${\overset{\sim }{A}}_j$. Weighted averaging operator for PFSs is defined as in Eq. (31) [42]:

$$ {\displaystyle \begin{array}{c} PWA={\sum}_{j=1}^n{w}_j{\overset{\sim }{A}}_j\\ {}=\left\langle \sqrt{1-{\prod}_{j=1}^n\left[{\left(1-{\mu}_j^2\right)}^{w_j}\right]},\right.\ \left.{\prod}_{j=1}^n\left[{\vartheta}_j^{w_j}\right]\right\rangle \kern1em \end{array}} $$

(31)

Definition 19

Let$\overset{\sim }{A}=\left\langle \mu, \vartheta \right\rangle$ be an PFS. Score $\left({S}_{\overset{\sim }{A}}\right)$and accuracy $\left({H}_{\overset{\sim }{A}}\right)$values can be calculated as shown in Eqs. (32–33) [15]:

$$ {S}_{\overset{\sim }{A}}={\mu}^2-{\vartheta}^2 $$

(32)

$$ {H}_{\overset{\sim }{A}}={\mu}^2+{\vartheta}^2 $$

(33)

1.4 Farmatean fuzzy set

Definition 20

Let${\overset{\sim }{A}}_j=\left\langle {\mu}_j,{\vartheta}_j\right\rangle$ be FFSs, ${w}_j\in \left[0,1\right],{\sum}_{j=1,\dots, n}{w}_j=1$ be weight for ${\overset{\sim }{A}}_j$. Weighted averaging operator for FFSs is defined as in Eq. (34) [61]:

$$ {\displaystyle \begin{array}{c}F\mathrm{W}A={\sum}_{j=1}^n{w}_j{\overset{\sim }{A}}_j\\ {}=\left\langle \sqrt[3]{1-{\prod}_{j=1}^n\left[{\left(1-{\mu}_j^3\right)}^{w_j}\right]},\right.\ \left.{\prod}_{j=1}^n\left[{\vartheta}_j^{w_j}\right]\right\rangle \kern1.75em \end{array}} $$

(34)

Definition 21

Let$\overset{\sim }{A}=\left\langle \mu, \vartheta \right\rangle$ be an FFS. Score $\left({S}_{\overset{\sim }{A}}\right)$and accuracy $\left({H}_{\overset{\sim }{A}}\right)$values can be calculated as shown in Eqs. (35–36) [61]:

$$ {S}_{\overset{\sim }{A}}={\mu}^3-{\vartheta}^3 $$

(35)

$$ {S}_{\overset{\sim }{A}}={\mu}^3+{\vartheta}^3 $$

(36)

1.5 Picture fuzzy set

Definition 25

Let${\overset{\sim }{A}}_{j=1,2}=\left\langle {\mu}_j,{\vartheta}_j,{\eta}_j\right\rangle$ and be two PiFSs. PiFSs are ordered by using the ranking rule presented in Eq. (37) [65]:

$$ {\displaystyle \begin{array}{c} if\kern0.5em {h}_1\left({\overset{\sim }{A}}_1\right)\ge {h}_1\left({\overset{\sim }{A}}_2\right)\Rightarrow {\overset{\sim }{A}}_1\ge {\overset{\sim }{A}}_2\\ {} if\ {h}_1\left({\overset{\sim }{A}}_1\right)={h}_1\left({\overset{\sim }{A}}_2\right)\\ {}\begin{array}{c} and\ {h}_2\left({\overset{\sim }{A}}_1\right)\ge {h}_2\left({\overset{\sim }{A}}_2\right)\Rightarrow {\overset{\sim }{A}}_1\ge {\overset{\sim }{A}}_2\\ {} if\ {h}_1\left({\overset{\sim }{A}}_1\right)={h}_1\left({\overset{\sim }{A}}_2\right)\ and\ {h}_2\left({\overset{\sim }{A}}_1\right)={h}_2\left({\overset{\sim }{A}}_2\right)\\ {} and\ {h}_3\left({\overset{\sim }{A}}_1\right)\ge {h}_3\left({\overset{\sim }{A}}_2\right)\Rightarrow {\overset{\sim }{A}}_1\ge {\overset{\sim }{A}}_2\end{array}\end{array}} $$

(37)

where ${h}_1\left({\overset{\sim }{A}}_j\right)={\mu}_j$, ${h}_2\left({\overset{\sim }{A}}_j\right)={\eta}_j$, ${h}_3\left({\overset{\sim }{A}}_j\right)={\mu}_j+{\eta}_j-{\vartheta}_j$.

1.6 Spherical fuzzy set

Definition 26

Let$\overset{\sim }{A}=\left\langle \mu, \vartheta, \eta \right\rangle$ and be an SFS. Score $\left({S}_{\overset{\sim }{A}}\right)$and accuracy $\left({H}_{\overset{\sim }{A}}\right)$values can be calculated as shown in Eqs. (38–39) [64]:

$$ {S}_{\overset{\sim }{A}}={\left(\mu -\eta \right)}^2-{\left(\vartheta -\eta \right)}^2 $$

(38)

$$ {H}_{\overset{\sim }{A}}={\mu}^2+{\vartheta}^2+{\eta}^2 $$

(39)

1.7 Neutrosophic set

Definition 22

Let${\overset{\sim }{\overset{\cdots }{A}}}_1=\left\langle {t}_1,{i}_1,{f}_1\right\rangle$ and ${\overset{\sim }{\overset{\cdots }{A}}}_2=\left\langle {t}_2,{i}_2,{f}_2\right\rangle$ be two NSs, and w₁ ∈ [0, 1], w₂ ∈ [0, 1], w₁ + w₂ = 1 be scalar weights. Additional operating between two NSs is defined as in Eq. (40) [66]:

$$ {\displaystyle \begin{array}{c}{w}_1{\overset{\sim }{\overset{\cdots }{A}}}_1\oplus {w}_2{\overset{\sim }{\overset{\cdots }{A}}}_2=\left\langle 1-{\left(1-{t}_1\right)}^{w_1}\times {\left(1-{t}_2\right)}^{w_2},\right.\ \\ {}1-{\left(1-{i}_1\right)}^{w_1}\times {\left(1-{i}_2\right)}^{w_2},\\ {}\left.1-{\left(1-{f}_1\right)}^{w_1}\times {\left(1-{f}_2\right)}^{w_2}\right\rangle \kern2.75em \end{array}} $$

(40)

Definition 23

Let${\overset{\sim }{\overset{\cdots }{A}}}_j=\left\langle {t}_j,{i}_j,{f}_j\right\rangle$ be an NS, ${w}_j\in \left[0,1\right],{\sum}_{j=1,\dots, n}{w}_j=1$ be weight for ${\overset{\sim }{\overset{\cdots }{A}}}_j$. Weighted averaging operator for NSs is defined as in Eq. (41) [66]:

$$ {\displaystyle \begin{array}{c} NWA={\sum}_{j=1}^n{w}_j{\overset{\sim }{\overset{\cdots }{A}}}_j=\left\langle 1-{\prod}_{j=1}^n{\left(1-{t}_j\right)}^{w_j},\right.\\ {}1-{\prod}_{j=1}^n{\left(1-{i}_j\right)}^{w_j},\\ {}\left.1-{\prod}_{j=1}^n{\left(1-{f}_j\right)}^{w_j}\right\rangle \end{array}} $$

(41)

Definition 24

Let$\overset{\sim }{\overset{\cdots }{A}}=\left\langle t,i,f\right\rangle$ be a NS. Score (${S}_{\overset{\sim }{\overset{\cdots }{A}}}\Big)$and accuracy $\left({H}_{\overset{\sim }{\overset{\dddot{}}{A}}}\right)$ values can be calculated as shown in Eqs. (42–43) [63]:

$$ {S}_{\overset{\sim }{\overset{\cdots }{A}}}=\frac{1+\left(t-2i-f\right)}{2} $$

(42)

$$ {H}_{\overset{\sim }{\overset{\cdots }{A}}}=t-i\times \left(1-t\right)-f\times \left(1-i\right) $$

(43)

NSs are ranked based on the score values. The NS having bigger score value is agreed as a bigger NS. There are several score functions available in the literature. Some examples are presented below:

Score and accuracy functions proposed by Nancy & Garg [55] is defined as in Eqs. (44–45):

$$ {S}_{\overset{\sim }{\overset{\cdots }{A}}}=\frac{1+\left(t-2i-f\right)\times \left(2-t-f\right)}{2} $$

(44)

$$ {H}_{\overset{\sim }{\overset{\cdots }{A}}}=t-2i-f $$

(45)

Score and accuracy functions proposed by Singh & Bhat [56] is defined as in Eqs. (46–47):

$$ {S}_{\overset{\sim }{\overset{\cdots }{A}}}=\frac{1+\left(t-2i-f\right)}{2\times \left(2-t-f\right)} $$

(46)

$$ {H}_{\overset{\sim }{\overset{\cdots }{A}}}=t-i-2f $$

(47)

The available score functions are not limited with the presented ones. The score functions may yield different results in ranking operations. Appendix Table 12 shows the ranking results of the mentioned score functions for the numerical example presented in Section 5.2.

Table 12 Ranking results for various score functions for the numerical example presented in section 5.2