T-Norms and T-Conorms of Symmetrical Linear Orthopair Fuzzy Sets and Their Cognitive Applications in Multiple-Criteria Decision-Making

Abstract

Orthopair fuzzy sets (OFSs) generally include q-rung orthopair fuzzy sets (q-ROFSs) and symmetrical linear orthopair fuzzy sets (SLOFSs), and the latter two models have a common element: intuitionistic fuzzy sets (IFSs). T-norms, t-conorms, and multiple-criteria decision-making (MCDM) are applied to q-ROFSs, but they have not been applied to SLOFSs. These valuable parts of SLOFSs are investigated by extending and simulating relevant results in IFSs, and their operational connections to addition and scalar multiplication are addressed in detail. For SLOFSs, axiomatic definitions, general properties, and concrete constructions for t-norms and t-conorms are first given. Then, special types of t-norms and t-conorms are used to motivate the addition and scalar multiplication operations, and related properties of the operations are obtained. Finally, addition and scalar multiplication are linearly combined with aggregation, and a relevant technique for order preference by similarity to an ideal solution (TOPSIS) method is designed for decision cognition. In this way, a new MCDM method based on SLOFSs is established, and its high reliability is validated by comparing the corresponding method based on q-ROFSs in two practical examples. This study advances work on SLOFSs and linearly extends IFS results, thus enriching OFSs, especially for cognitive computations and applications.

Appendix. Theorem 4’s proof regarding the second case (i.e., SLOFSs (2))

Proof

For SLOFSs (2), we mainly adopt Tables 27, 28, 29, 30 for T-operation verification. At first, we prove T is a mapping on $(L^{**})^2\rightarrow L^{**}$. If $\alpha _{1},\alpha _{2}\in D_{3}$, then $0<\frac{\omega }{2\omega -1}(\mu _{i}-\frac{1-\omega }{\omega })\le 1, 0<\frac{\omega }{2\omega -1}(\nu _{i}-\frac{1-\omega }{\omega })\le 1$ ($i=1,2$), and we have

$$\begin{aligned} 0\le t(\frac{\omega }{2\omega -1}(\mu _{1}-\frac{1-\omega }{\omega }),\frac{\omega }{2\omega -1}(\mu _{2}-\frac{1-\omega }{\omega }))\le 1,\\ 0\le s(\frac{\omega }{2\omega -1}(\nu _{1}-\frac{1-\omega }{\omega }),\frac{\omega }{2\omega -1}(\nu _{2}-\frac{1-\omega }{\omega }))\le 1. \end{aligned}$$

Due to the strong dual pair (t, s) and Theorem 3 (3)’s proof, we have

$$\begin{aligned} \begin{aligned}&(t(\frac{\omega }{2\omega -1}(\mu _{1}-\frac{1-\omega }{\omega }),\frac{\omega }{2\omega -1}(\mu _{2}-\frac{1-\omega }{\omega })),\\&s(\frac{\omega }{2\omega -1}(\nu _{1}-\frac{1-\omega }{\omega }),\frac{\omega }{2\omega -1}(\nu _{2}-\frac{1-\omega }{\omega })))\\ \ne&(1,t), (k,1)~(0<t,k\le 1). \end{aligned} \end{aligned}$$

So,

$$\begin{aligned}&\frac{1-\omega }{\omega }\le \frac{2\omega -1}{\omega } t(\frac{\omega }{2\omega -1}(\mu _{1}-\frac{1-\omega }{\omega }),\\&\frac{\omega }{2\omega -1}(\mu _{2}-\frac{1-\omega }{\omega }))+\frac{1-\omega }{\omega }\le 1,\\&\frac{1-\omega }{\omega }\le \frac{2\omega -1}{\omega } s(\frac{\omega }{2\omega -1}(\nu _{1}-\frac{1-\omega }{\omega }),\\&\frac{\omega }{2\omega -1}(\nu _{2}-\frac{1-\omega }{\omega }))+\frac{1-\omega }{\omega }\le 1, \end{aligned}$$

and

$$\begin{aligned}&(\frac{2\omega -1}{\omega } t(\frac{\omega }{2\omega -1}(\mu _{1}-\frac{1-\omega }{\omega }),\\&\frac{\omega }{2\omega -1}(\mu _{2} -\frac{1-\omega }{\omega }))+\frac{1-\omega }{\omega },\\&\frac{2\omega -1}{\omega } s(\frac{\omega }{2\omega -1}(\mu _{1}-\frac{1-\omega }{\omega }),\\&\frac{\omega }{2\omega -1}(\mu _{2}-\frac{1-\omega }{\omega }))+\frac{1-\omega }{\omega })\\ \ne&(1,t), (k,1)~(0<t,k\le 1). \end{aligned}$$

$$\begin{aligned}&\omega (\frac{2\omega -1}{\omega }t(\frac{\omega }{2\omega -1}(\mu _{1}-\frac{1-\omega }{\omega }), \frac{\omega }{2\omega \!-\!1}(\mu _{2}-\frac{1\!-\!\omega }{\omega }))\\&+\frac{1-\omega }{\omega })\\ +&\omega (\frac{2\omega -1}{\omega }s(\frac{\omega }{2\omega -1}(\nu _{1}-\frac{1-\omega }{\omega }), \frac{\omega }{2\omega -1}(\nu _{2}\!-\!\frac{1\!-\!\omega }{\omega }))\\&+\frac{1-\omega }{\omega })\\ \le&(2\omega -1) (t(\frac{\omega }{2\omega -1}\mu _{1}-\frac{1-\omega }{2\omega -1},\frac{\omega }{2\omega -1}\mu _{2}-\frac{1-\omega }{2\omega -1})\\&+s(\frac{\omega }{2\omega -1}(\frac{1}{\omega }-\mu _{1})-\frac{1-\omega }{2\omega -1},\frac{\omega }{2\omega -1}(\frac{1}{\omega }-\mu _{2}) \\&-\frac{1-\omega }{2\omega -1}))+2(1-\omega )\\ =&(2\omega -1)(t(\frac{\omega }{2\omega -1}\mu _{1}-\frac{1-\omega }{2\omega -1}, \frac{\omega }{2\omega -1}\mu _{2}-\frac{1-\omega }{2\omega -1}) \\&+(1-t(\frac{\omega }{2\omega -1}\mu _{1}-\frac{1-\omega }{2\omega -1}, \frac{\omega }{2\omega -1}\mu _{2}-\frac{1-\omega }{2\omega -1})))\\&+2(1-\omega )\\ =&1. \end{aligned}$$

Hence, $T(\alpha _{1},\alpha _{2})\in D_{3}$ becomes a SLOFN. Other distributions regarding $\alpha _{1},\alpha _{2}$ have similar checks, and relevant results are provided in Table 27. In other words, T submits to the operation closure for range $L^{**}$.

Table 27 Check table of T closure regarding SLOFSs (2)

Table 28 Check table of T associativity regarding SLOFSs (2) with $\alpha _{1}\in D_1$ (where $0.5<\omega <1$)

Table 29 Check table of T monotonicity regarding SLOFSs (2) with $\alpha _{1}\in D_1$ (where $0.5<\omega <1$)

Table 30 Check table of T bounder condition regarding SLOFSs (2) (where $0.5<\omega <1$)

Then, for operation T, its (T1) commutativity is directly obtained by defined formulas. When $\alpha _{1}\in D_1$, its (T2) associativity, (T3) monotonicity, and (SLOFT4) border can be proved from Tables 28, 29, and 30, respectively. Other cases, such as $\alpha _{1}\in D_2, D_3, D_4$, can be discussed similarly.

Therefore, T is a SLOF t-norm. Similarly, S is a SLOF t-conorm. $\square$