Single-Valued Neutrosophic Set: An Overview

Summary

The purpose of this chapter is to present an overview of neutrosophic sets. Professor Florentin Smarandache defined the neutrosophic set and helped popularize the concept to deal with uncertainty, inconsistency, and indeterminacy common in human existence. The chapter presents the basic definitions of neutrosophic sets, single-valued neutrosophic sets, single-valued neutrosophic numbers, score functions, accuracy and certainty functions, the ranking of neutrosophic numbers, and some extensions of neutrosophic sets. It describes different types of neutrosophic sets and a few examples of their applications in social sciences. The chapter also presents a critical discussion and the future scope of research.

The relation between classic set, fuzzy set, intuitionistic fuzzy set, and neutrosophic set.

The code of this chapter is 01001110 01100101 01110101 01110100 01110010 01101111 01110011 01101111 01110000 01101000 01111001.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein

1 Introduction

Since Zadeh [1] grounded the Fuzzy Set (FS) in 1965, a plethora of new theories dealing with imprecision and uncertainty has been reported in the fuzzy literature. Neutrosophic Sets (NSs) grounded by Smarandache [2,3,4,5,6] constitute a generalization of FS and Intuitionistic FS (IFS) [7]. While the FS offers the degree of Truth Membership (TM) of an element in a prescribed set, IFS offers both a degree of TM and a degree of False Membership (FM), whereas NS offers a degree of TM, a degree of Indeterminacy Membership (IM), and a degree of FM. In IFS, TM function and FM function are independent, but in FS, FM function is dependent on TM function. In NS, TM function, IM function, and FM function are independent. Smarandache [8] presented the differences between NSs [2] and various extensions of FSs. Single-Valued NS (SVNS) [9,10,11] is an instance of NS that has a root in the work of Smarandache [2]. Haibin Wang, the first author, has presented the SVNS [9] in the international seminar in Salt Lake City, USA. NSs [2,3,4,5,6] and SVNSs [9,10,11] have been presented in different seminars and published in different proceedings and journals to draw much attentions from the researchers.

The popularity of NSs [2] gains momentum after the publication of SVNS [11] and the international journal “Neutrosophic Sets and Systems”. NSs and SVNSs have been widely used in Multi-Attribute Decision-Making (MADM) [12,13,14,15,16,17,18,19,20,21] and Multi-Attribute Group Decision-Making (MAGDM) [22,23,24]. Interval Neutrosophic Set (INS) [25, 26] has been proposed as an instance of NS. INS is a subclass of NS, and it considers only subunitary intervals of [0, 1].

NSs have drawn much attention from the researchers. Various applications such as NS-based models have been introduced for options market [27], financial market [28], image denoising [29,30,31,32], cluster analysis [33], information retrieval [34, 35], love dynamics [36], video tracking [37], fault diagnosis [38], air surveillance [39, 40], and so on.

After 2010, various extensions of NS have been rapidly proposed in the literature. Ye [41] defined the Simplified NS (SNS) in terms of three numbers in [0, 1]. SNS is a subclass of NS. SNS includes of an INS and an SVNS. SNSs have been used in decision-making [42,43,44,45,46], medical diagnosis [47], and so on.

In 2003, Kandasamy and Smarandache [48] defined the  Neutrosophic Number (NN) by combining real numbers and indeterminate parts of the form u + Iv, where I is an indeterminate component, and u, v are real numbers. Several strategies for MAGDM have been proposed in NN environment [49,50,51,52,53,54]. Du et al. [55] combined SNS and NN to define the Simplified Neutrosophic Indeterminate Set (SNIS) and presented a new decision-making strategy. Köseoğlu et al. [56] extended SNS and defined the Simplified Neutrosophic Multiplicative Set (SNMS) and Simplified Neutrosophic Multiplicative Preference Relation (SNMPR).

Ye [57] defined the Single-Valued Neutrosophic Hesitant Fuzzy Set (SVNHFS) that encompasses the FS [1], IFS [7], hesitant FS [58], dual hesitant FS [59], and SVNS [10]. Theoretical developments and applications of SVNHFSs have been presented in several studies [60,61,62,63,64,65,66,67].

Neutrosophic Cubic Set (NCS) [68] was proposed by extending the cubic set [69]. Jun et al. [70] also proposed the NCS. Some theoretical developments and applications of NCSs have been made in MADM [71,72,73,74] and MAGDM [75, 76].

Bipolar NS (BNS) [77] was developed by extending bipolar FS [78] and NS [2]. BNSs have been utilized in dealing with MADM [79,80,81,82,83] and MAGDM [84].

Maji [85] grounded the Neutrosophic Soft Set (NSS) by combining Soft Set (SS) [86] and  NS [2]. The NSSs have been utilized in MADM [87,88,89,90,91,92] and MAGDM [93,94,95,96]. Ali et al. [97] defined the bipolar neutrosophic SS by combining SS [86] and BNS [77].

Many investigators paid wide attention to the research of hybridizing Rough Set (RS) [98] and NS [2]. Broumi et al. [99, 100] defined the Rough Neutrosophic Set (RNS). Broumi and Smarandache [101] grounded the Interval RNS (IRNS). Yang et al. [102] defined the Single-Valued Neutrosophic RS (SVNRS) by exploring constructive and axiomatic strategies. Single-Valued Neutrosophic Multi-Granulation RS (SVNMGRS) [103] was proposed by combining multi-granulation rough sets [104] with SVNS [10]. Jiao et al. [105] developed the three-way decision models utilizing Decision-Theoretic RS (DTRS) [106] with SVNS [10]. New theoretical developments and applications of the RNSs have been presented in the current neutrosophic literature [107,108,109,110,111,112,113,114,115,116,117,118,119]. Pramanik and Mondal [120] proposed the Rough BNS (RBNS). Recently, Pramanik [121] documented an overview of RNSs.

Yager [122] introduced the fuzzy bag or the Fuzzy Multi-Set (FMS) by extending the bag or multi-set [123, 124]. An element of a FMS can assume the same or different membership values more than once. In case of Intuitionistic Fuzzy Multi-Set (IFMS) [125], an element can assume membership and falsity values more than once. To overcome the shortcomings of FMS and IFMS, Smarandache [126] presented the n-valued refined neutrosophic logic. Smarandache [126] paved the way to define the Neutrosophic Refined (NR) set. Several investigators [127,128,129,130,131,132,133,134,135,136] studied the application of NR sets. Bao and Yang [137, 138] proposed the Single-Valued Neutrosophic Refined Rough Set (SVNRRS) by hybridizing NR sets with RSs.

Ye [139] extended the intuitionistic linguistic set [140] to Single-Valued Neutrosophic Linguistic (SVNL) set and SVNL number. Li et al. [141] defined the Neutrosophic Linguistic Set (NLS) and presented a comparison strategy for Neutrosophic Linguistic Numbers (NLNs). Ye [142] defined the Interval NLN (INLN) and employed it for MADM. Tian et al. [143] defined the Simplified NLS (SNLS) that combines the SNS and linguistic term set [144]. SNLS is capable of describing linguistic information to some extent. Tian et al. [143] employed the Simplified NLNs (SNLNs) for MADM.

Biswas et al. [145] and Ye [146] presented the Trapezoidal Neutrosophic Fuzzy Number (TrNFN) by extending the trapezoidal fuzzy numbers. TrNFNs have been utilized in MADM and MAGDM [147,148,149,150,151,152,153]. Biswas et al. [154] presented the ranking strategy for Single-Valued Neutrosophic Trapezoidal Number (SVNTrN) and employed it for MADM. Liang et al. [155] presented the Single-Valued Trapezoidal Neutrosophic Preference Relation (SVTrNPR) and the completely consistent SVTrNPR to solve MADM problems. Biswas et al. [156] extended the SVNTrN to interval trapezoidal neutrosophic number and employed it for MADM. Biswas et al. [157] defined the triangular fuzzy NSs and employed them for MADM. Abdel- Basset et al. [158] extended neutrosophic triangular number to propose the type-2 neutrosophic number. Chakraborty et al. [159] defined the neutrosophic pentagonal number and studied some of its properties. Karaaslan [160] presented the Gaussian  Single-Valued Neutrosophic Number (SVNN) and employed it for MADM.

Researchers extended NSs to different sets such as bipolar neutrosophic refined sets [161], tri-complex rough neutrosophic set [162], rough neutrosophic hyper-complex set [163], quadripartitioned SVNS [164], plithogenic set [165, 166]. Smarandache [167, 168] further proposed the neutrosophic off/under/over sets by extending NSs.

NSs, SVNSs, and their hybrid extensions and applications can be found in the studies [121, 169,170,171,172,173,174,175,176,177].

Rest of the chapter is designed as follows: Sect. 2 presents the basics of the NSs. Section 3 presents the triangular fuzzy number NS. Section 4 presents the trapezoidal fuzzy NS. Section 5 describes single-valued pentagonal neutrosophic numbers. Section 6 describes the cylindrical neutrosophic single-valued sets. Section 7 describes the neutrosophic numbers. Section 8 describes some applications of NSs. Section 9 describes the extensions of the NSs. Section 10 presents the direction of new research. Section 11 presents conclusions.

2 Basics of Neutrosophic Sets

2.1 Neutrosophic Set

Let Q be a space of points with a generic element ω in Q. An NS [2] $\phi$ in Q is characterized by a truth. Membership Function (MF) $\xi_{\phi }$, an indeterminacy MF $\psi_{\phi }$, a falsity MF $\zeta_{\phi }$ and is presented as:

$$\phi = \left\langle {\omega ,\xi_{\phi } (\omega ),\psi_{\phi } (\omega ),\zeta_{\phi } (\omega ) > /,\omega \in \Omega } \right\rangle .$$

Here, $\xi_{\phi } (\omega),\psi_{\phi } (\omega),\zeta_{\phi } (\omega)$ in Ω denote the subsets of ]⁻0, 1⁺[ such that $\xi_{\phi } (\omega)$:Q $\to$]⁻0, 1⁺[, $\psi_{\phi } (\omega)$: $Q$$\to$]⁻0, 1⁺[, and $\zeta_{\phi } (\omega)$: $\Omega$$\to$]⁻0, 1⁺[.

Then,

${}^{ - }0 \le \sup \xi_{\phi } (\omega) + \sup \,\psi_{\phi } (\omega) + \,\sup \,\zeta_{\phi } (\omega) \le 3^{ + }$.

2.2 Single-Valued Neutrosophic Set

An SVNS [10] $\chi$ in a universal set $\Omega$ is presented by a truth MF $\xi_{\chi } (\omega )$, an indeterminacy MF $\psi_{\chi } (\omega )$, and a falsity MF $\psi_{\chi } \,(\omega )$ such that.

$\mathop \xi \nolimits_{\chi } \left( \omega \right),\mathop \psi \nolimits_{\chi } \left( \omega \right),\mathop \zeta \nolimits_{\chi } \left( \omega \right)$ are in [0, 1] for all $\omega \in \Omega$.

If $\Omega$ is continuous, $\chi$ is presented as

$$\chi = \int\limits_{\omega } {{{\left\langle {\mathop \xi \nolimits_{\chi } \left( \omega \right),\mathop \psi \nolimits_{\chi } \left( \omega \right),\mathop \zeta \nolimits_{\chi } \left( \omega \right)\,\,} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {\mathop \xi \nolimits_{\chi } \left( \omega \right),\mathop \psi \nolimits_{\chi } \left( \omega \right),\mathop \zeta \nolimits_{\chi } \left( \omega \right)\,\,} \right\rangle } {\omega ,\,\forall \omega \in \Omega }}} \right. \kern-\nulldelimiterspace} {\omega ,\,\forall \omega \in \Omega }}} .$$

If $\Omega$ is discrete, $\chi$ is presented as

$$\chi = \sum {{{\left\langle {\mathop \xi \nolimits_{\chi } \left( \omega \right),\,\mathop \psi \nolimits_{\chi } \left( \omega \right),\mathop \zeta \nolimits_{\chi } \left( \omega \right)\,} \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {\mathop \xi \nolimits_{\chi } \left( \omega \right),\,\mathop \psi \nolimits_{\chi } \left( \omega \right),\mathop \zeta \nolimits_{\chi } \left( \omega \right)\,} \right\rangle } \omega }} \right. \kern-\nulldelimiterspace} \omega }} ,\,\forall \omega \in \Omega$$

with $0\mathop { \le \sup \xi }\nolimits_{\chi } \left( \omega \right) + \sup \,\mathop \psi \nolimits_{\chi } \left( \omega \right) + \mathop \zeta \nolimits_{\chi } \left( \omega \right)\,\, \le 3,\omega \forall \in \Omega .$

An SVNS $\chi$ is also presented as.

$\chi = \left\langle {\omega ,\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega ) > /,\,\omega \in \Omega } \right\rangle$, where $\xi_{\chi } (\omega ),\,\psi_{\xi } (\omega ),\zeta_{\xi } (\omega )$$\in$[0, 1], for each $\omega$ in $\Omega$. Therefore,

$$0 \le \sup \xi_{\chi } (\omega )\, + \sup \,\psi_{\chi } (\omega ) + \,\,\sup \zeta_{\chi } (\omega ) \le 3.$$

Note: Since SVNS is a subclass of NS, we use NS and SVNS equivalently throughout chapter.

For convenience, the triplet $\left( {\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega )} \right)$ is called as the SVNN and simply presented as $\left( {\xi_{\chi } ,\,\psi_{\chi } ,\zeta_{\chi } } \right)$.

2.2.1 Some Operations of SVNNs

Let $\eta_{1} = (\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} )$ and $\eta_{2} = (\alpha_{2} ,\,\beta_{2} ,\,\gamma_{2} )$ be any two SVNNs with $\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ,\alpha_{2} ,\,\beta_{2} ,\,\gamma_{2} \in [0,\,1]$, $(\alpha_{1} + \,\beta_{1} + \,\gamma_{1} ) \in [0,3]$ and $(\alpha_{2} + \,\beta_{2} + \,\gamma_{2} ) \in [0,3]$.

The following operations for SVNNs [171] hold

• i. $\eta_{1} \oplus \eta_{2} = (\alpha_{1} + \alpha_{2} - \alpha_{1} \alpha_{2} ,\,\beta_{1} ,\beta_{2} ,\,\gamma_{1} \gamma_{2} )$ [Summation]

• ii. $\eta_{1} \otimes \eta_{2} = (\alpha_{1} \alpha_{2} ,\beta_{1} + \beta_{2} - \beta_{1} \beta_{2} ,\,\gamma_{1} + \gamma_{2} - \gamma_{1} \gamma_{2} )$ [Multiplication]

• iii. $\lambda \eta_{1} = (1 - (1 - \alpha_{1} )^{\lambda } ,\beta_{1}^{\lambda } ,\gamma_{1}^{\lambda } )$ [Scalar multiplication]

• iv. $\eta_{1}^{\lambda } = (\alpha_{1}^{\lambda } ,1 - (1 - \beta_{1} )^{\lambda } ,1 - (1 - \gamma_{1} )^{\lambda } )),\,\lambda > 0$.

2.2.2 Score Function and Accuracy Function of SVNNs

Assume that $\eta_{1} = (\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} )$ is an SVNN. Score function denoted by $\Gamma (n_{1} )$, accuracy function denoted by $H(n_{1} )$ [178] of $\eta_{1}$ are, respectively, represented as.

• i. $\Gamma (n_{1} ) = \frac{{(1 + \alpha_{1} - 2\alpha_{2} - \alpha_{3} )}}{2},\,\,\,\,\,\,\,$ where $\Gamma (n_{1} ) \in [ - 1,\,1]$

• ii. $H(n_{1} ) = \alpha_{1} - \alpha_{2} (1 - \alpha_{1} ) - \alpha_{3} (1 - \alpha_{2} ),\,$ where $H(n_{1} ) \in [ - 1,1]$

  Nancy and Garg [179] presented the improved score function as follows:

• iii. $S(n_{1} ) = \frac{{(1 + (\alpha_{1} - 2\alpha_{2} - \alpha_{3} )(2 - \alpha_{1} - \alpha_{3} ))}}{2},$

Clearly, if $\alpha_{1} + \alpha_{3} = 1,$ $S(n_{1} )$ reduces to $\Gamma (n_{1} )$.

2.2.3 Comparison of SVNNs

Assume that $\chi_{1} = (\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} )$ and $\chi_{2} = (\beta_{1} ,\,\beta_{2} ,\,\beta_{3} )$ be any two SVNNs. Comparison strategy [179] between $\chi_{1}$ and $\chi_{2}$ is presented as.

• i. if $\Gamma (\chi_{1} ) < \Gamma (\chi_{2} )$), then $\chi_{1} \prec \chi_{2}$

• ii. if $\Gamma (\chi_{1} ) = \Gamma (\chi_{2} )$, then

  • • İf $S(\chi_{1} ) < S(\chi_{2} )$, then $\chi_{1} \prec \chi_{2}$

  • • $S(\chi_{1} ) > S(\chi_{2} )$, then $\chi_{1} \succ \chi_{2}$

  • • $S(\chi_{1} ) = S(\chi_{2} )$, then $\chi_{1} \approx \chi_{2}$.

2.3 Interval Neutrosophic Set

Let P be a space of points having generic element p in P.

An INS [25] $\varphi$ in P is presented as $\varphi = \left\langle {p,M_{\varphi } (p),\,N_{\varphi } (p),O_{\varphi } (p) > /,\,p \in P} \right\rangle .$

Here, $\,M_{\varphi } (p) = [M_{\varphi }^{L} (p),M_{\varphi }^{U} (p)],\,\,N_{\varphi } (p) = [N_{\varphi }^{L} (p),N_{\varphi }^{U} (p)],\,\,O_{\varphi } (p) = [O_{\varphi }^{L} (p),O_{\varphi }^{U} (p)]$ and for each $p \in P,\,M_{\varphi } (p),\,N_{\varphi } (p),O_{\varphi } (p) \subseteq \left[ {0, \, 1} \right]$.

For convenience, an Interval Neutrosophic Number (INN) $\eta_{1}$ is presented in the form: $\eta_{1} = \left\langle {\left[ {M_{1}^{L} ,M_{1}^{U} } \right],\left[ {N_{1}^{L} ,N_{1}^{U} } \right],\left[ {O_{1}^{L} ,O_{1}^{U} } \right]} \right\rangle$

2.3.1 Operations on INNs

Let $\eta_{1} = \left\langle {\left[ {M_{1}^{L} ,M_{1}^{U} } \right],\left[ {N_{1}^{L} ,N_{1}^{U} } \right],\left[ {O_{1}^{L} ,O_{1}^{U} } \right]} \right\rangle$ and $\eta_{2} = \left\langle {\left[ {M_{2}^{L} ,M_{2}^{U} } \right],\left[ {N_{2}^{L} ,N_{2}^{U} } \right],\left[ {O_{2}^{L} ,O_{2}^{U} } \right]} \right\rangle$ be any two INNs. The operations for INNs [180] are presented as:

• i. $\begin{aligned}\eta_{1} \oplus \eta_{2}& = \left\langle \left[ {M_{1}^{L} + M_{2}^{L} - M_{1}^{L} M_{2}^{L} ,M_{1}^{U} + M_{2}^{U} - M_{1}^{U} M_{2}^{U} } \right],\right.\\ &\quad \left. \left[ {N_{1}^{L} N_{2}^{L} ,N_{1}^{U} N_{2}^{U} } \right],\left[ {O_{1}^{L} O_{2}^{L} ,O_{1}^{U} O_{2}^{U} } \right] \right\rangle \end{aligned}$

• ii. $\begin{aligned}\eta_{1} \otimes \eta_{2} & = \left\langle \left[ {M_{1}^{L} M_{2}^{L} ,M_{1}^{U} M_{2}^{U} } \right],\left[ {N_{1}^{L} + N_{2}^{L} - N_{1}^{L} N_{2}^{L} ,N_{1}^{U} + N_{2}^{U} - N_{1}^{U} N_{2}^{U} } \right],\right.\\ & \left.\quad \left[ {O_{1}^{L} + O_{2}^{L} - O_{1}^{L} O_{2}^{L} ,O_{1}^{U} + O_{2}^{U} - O_{1}^{U} O_{2}^{U} } \right] \right\rangle \end{aligned}$

• iii. $\gamma \eta_{1} = \left\langle {\left[ {1 - (1 - M_{1}^{L} )^{\gamma } ,\!1 - (1 - M_{1}^{U} )^{\gamma } )} \right],\!\left[ {(N_{1}^{L} )^{\gamma },\!(N_{1}^{U} )^{\gamma } } \right],\!\left[ {(O_{1}^{L} )^{\gamma },\!(O_{1}^{U} )^{\gamma } } \right]} \right\rangle$

• iv. $\begin{aligned}\eta_{1}^{\gamma }& = \left\langle \left[ {(M_{1}^{L} )^{\gamma } ,(M_{1}^{U} )^{\gamma } )} \right],\left[ {1 - (1 - N_{1}^{L} )^{\gamma } ,1 - (1 - N_{1}^{U} )^{\gamma } )} \right], \right.\\ & \quad \left. \left[ {1 - (1 - O_{1}^{L} )^{\gamma } ,1 - (1 - O_{1}^{U} )^{\gamma } )} \right] \right\rangle \end{aligned}$ where $\gamma > 0$.

2.3.2 Score Function and Accuracy Functions of INNs

Assume that $\eta_{1} = \left\langle {\left[ {M_{1}^{L} ,M_{1}^{U} } \right],\left[ {N_{1}^{L} ,N_{1}^{U} } \right],\left[ {O_{1}^{L} ,O_{1}^{U} } \right]} \right\rangle$ be an INN. The score function $\Gamma (\eta_{1} )$, accuracy function $H(\eta_{1} )$ [178] of $\eta_{1}$ are, respectively, presented as:

• i. $\Gamma (\eta_{1} ) = \left( \frac{1}{4} \right) \times \left[ {2 + M_{1}^{L} + M_{1}^{U} - 2N_{1}^{L} - 2N_{1}^{U} - O_{1}^{L} - O_{1}^{U} } \right],\,\,\Gamma (\eta_{1} ) \in [ - 1,\,1]$

• ii. $H(\eta_{1} ) = \frac{{M_{1}^{L} + M_{1}^{U} - N_{1}^{U} (1 - M_{1}^{U} ) - N_{1}^{L} (1 - M_{1}^{L} ) - O_{1}^{U} (1 - N_{1}^{U} ) - \zeta_{1}^{L} (1 - N_{1}^{L} )}}{2},\,\,H(\eta_{1} ) \in [ - 1,\,1]$

2.3.3 Comparison INNs

The convenient strategy for comparing INNs [178] is described as follows:

Let $\eta_{1} = \left\langle {\left[ {M_{1}^{L} ,M_{1}^{U} } \right],\left[ {N_{1}^{L} ,N_{1}^{U} } \right],\left[ {O_{1}^{L} ,O_{1}^{U} } \right]} \right\rangle$ and $\eta_{2} = \left\langle {\left[ {M_{2}^{L} ,M_{2}^{U} } \right],\left[ {N_{2}^{L} ,N_{2}^{U} } \right],\left[ {O_{2}^{L} ,O_{2}^{U} } \right]} \right\rangle$ be any two INNs. Then,

• i. If $\Gamma (\eta_{1} ) \succ \Gamma (\eta_{2} )$, then $\eta_{1} \succ \eta_{2}$

If $\Gamma (\eta_{1} ) = \Gamma (\eta_{2} )$,and $H(\eta_{1} ) \succ H(\eta_{2} )$, then $\eta_{1} \succ \eta_{2}$.

2.4 Spherical Neutrosophic Set

Smarandache presented spherical NS [8], which is a generalization of spherical FS [181].

2.4.1 Single-Valued Spherical NS

A Single-Valued Spherical NS [8] of the universe of discourse $\Theta$ is presented as follows:

$$\chi = \left\langle {\omega ,\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega ) > /,\,\omega \in \Theta } \right\rangle .$$

Here, $\forall w \in \Theta ,$ the functions $\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega ):\Omega \to \left[ {0, \, \sqrt 3 } \right]$ indicate the degrees of truth, indeterminacy, and falsity MF, respectively, and $0 \le \xi_{\chi }^{2} (\omega )\, + \psi_{\chi }^{2} (\omega ) + \,\,\zeta_{\chi }^{2} (\omega ) \le 3$.

2.4.1.1 Single-Valued Spherical Neutrosophic Number (SVSpNN)

Smarandache [182] presented the SVSpNN having the form: $(q,r,s)$ where $q,r,s \in [0,\,\sqrt {3)}$ and $q^{2} + r^{2} + s^{2} \le 3.$

SVSrNN is the generalization of Single-Valued Pythagorean Fuzzy Number (SVPFN) having the form: $(q,r)$ with $q,\,r \in [0,\,2]$ and $q^{2} + r^{2} \le 2$.

2.4.1.2 Interval-Valued Spherical Neutrosophic Number (ISpNN)

Smarandache [182] presented the ISpNN that has the form:$(q,\,r,s)$ where the real intervals $q,\,r,s \subseteq [0,\,\sqrt 3 ]$ and $q^{2} + r^{2} + s^{2} \subseteq [0,\,3].$

2.4.2 n-Hyper Spherical Neutrosophic Set (n-HSpNS)

Single-valued n-HSpNS [8] is a generalization of the spherical NS in the universe of discourse $\Omega$, for n ≥ 1. It is defined as $\chi = \left\langle {\omega ,\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega ) > /,\,\omega \in \Omega } \right\rangle$, where, $\forall \omega \in \Omega$, the functions $\xi_{\chi } (\omega ),\,\psi_{A} (\omega ),\zeta_{A} (\omega ):\Omega \to [0,\,\sqrt[n]{3}\,]$ represent the degrees of truth, indeterminacy, and falsity MF, respectively, and $0 \le \xi_{\chi }^{n} (\omega ) + \,\psi_{\chi }^{n} (\omega ) + \zeta_{\chi }^{n} (\omega ) \le 3$.

3 Triangular Fuzzy Number Neutrosophic Set (TFNNS)

Biswas et al. [157] hybridized the Triangular Fuzzy Number (TFN) with SVNSs to define the TFNNS.

3.1 TFNNS

Let $\Omega$ be the finite universe of discourse and $\theta$[0, 1] be the set of all TFNs on [0, 1]. A TFNNS $\varphi$ in $\Omega$ is presented by

$$\varphi = \left\{ {\left\langle {\omega ,\xi_{\varphi } (\omega ),\psi_{\varphi } (\omega ),\zeta_{\varphi } (\omega )\,} \right\rangle |\omega \in \Omega } \right\},$$

where $\xi_{\varphi } (\omega ):\Omega \to \theta \left[ {0,1} \right]$,$\psi_{\varphi } (\omega ):\Omega \to \theta \left[ {0,1} \right]$, and $\zeta_{\varphi } (\omega ):\Omega \to \theta \left[ {0,1} \right].$

The TFNs $\xi_{\varphi } (\omega ) = \left( {\xi_{\varphi }^{1} (\omega ),\xi_{\varphi }^{2} (\omega ),\,\xi_{\varphi }^{3} (\omega )} \right)$,$\psi_{\varphi } (\omega ) = \left( {\psi_{\varphi }^{1} (\omega ),\psi_{\varphi }^{2} (\omega ),\,\psi_{\varphi }^{3} (\omega )} \right)$, and $\zeta_{\varphi } (\omega ) = \left( {\zeta_{\varphi }^{1} (\omega ),\zeta_{\varphi }^{2} (\omega ),\,\zeta_{\varphi }^{3} (\omega )} \right)$ present the degree of truth, indeterminacy, and falsity MF, respectively, of $\omega \in \varphi ,\,\forall \omega \in \Omega$ and $0 \le \xi_{\varphi } (\omega ) + \psi_{\varphi } (\omega ) + \zeta_{\varphi } (\omega ) \le 3$.

3.2 Triangular Fuzzy Neutrosophic Number (TFNN)

For notational convenience, we consider $\phi = \left\langle {(\alpha ,\beta ,\gamma ),(\rho ,\sigma ,\tau ),(l,m,n)} \right\rangle$ as a TFNN [157] where $\left( {\xi_{\varphi }^{1} (\omega ),\xi_{\varphi }^{2} (\omega ),\,\xi_{\varphi }^{3} (\omega )} \right) = (\alpha ,\beta ,\gamma )$, $\left( {\psi_{\varphi }^{1} (\omega ),\psi_{\varphi }^{2} (\omega ),\,\psi_{\varphi }^{3} (\omega )} \right) = (\rho ,\sigma ,\tau )$, and $\left( {\zeta_{\varphi }^{1} (\omega ),\zeta_{\varphi }^{2} (\omega ),\,\zeta_{\varphi }^{3} (\omega )} \right)$ = $(l,m,n)$.

3.3 Operations on TFNNs

Let $\chi_{1} = \left\langle {(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ),(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ),(l_{1} ,\,m_{{_{1} }} ,\,n_{1} )} \right\rangle$ and $\chi_{2} = \left\langle {(\alpha_{2} ,\,\beta_{2} ,\,\gamma_{2} ),(\rho_{2} ,\,\sigma_{2} ,\,\tau_{2} ),(l_{2} ,\,m_{{_{2} }} ,\,n_{2} )} \right\rangle$ be any two TFNNs in $\Re$. The basic operations for TFNNs [157] hold good:

• i. $\begin{aligned}\chi_{1} \oplus \chi_{2}& = \left\langle \left( {\alpha_{1} + \alpha_{2} - \alpha_{1} \alpha_{2} ,\beta_{1} + \beta_{2} - \beta_{1} \beta_{2} ,\,\,\gamma_{1} + \gamma_{2} - \gamma_{1} \gamma_{2} } \right),\right.\\ &\left. \quad \left( {\rho_{1} \rho_{2} ,\sigma_{1} \sigma_{2} ,\tau_{1} \tau_{2} } \right),\left( {l_{1} l_{2} ,m_{1} m_{2} ,n_{1} n_{2} } \right) \right\rangle ; \end{aligned}$

• ii. $\begin{aligned} & \chi_{1} \otimes \chi_{2}\\ & \quad \!= \!\left\langle \begin{gathered} \left( {\alpha_{1} \alpha_{2},\!\beta_{1} \beta_{2},\!\gamma_{1} \gamma_{2} } \right),\!\left( {\rho_{1} + \rho_{2} - \rho_{1} \rho_{2} ,\sigma_{1} + \sigma_{2} - \sigma_{1} \sigma_{2} ,\!\tau_{1} + \tau_{2} - \tau_{1} \tau_{2} } \right), \hfill \\ \left( {l_{1} + l_{2} - l_{1} l_{2} ,\!m_{1} + m_{2} - m_{1} m_{2} ,\!n_{1} + n_{2} - n_{1} n_{2} } \right) \hfill \\ \end{gathered} \right\rangle ; \end{aligned}$

• iii. $\lambda \chi_{1}\! = \!\left\langle {\left( {1 - (1 - \varepsilon_{1} )^{\lambda } ,\!1 - (1 - \beta_{1} )^{\lambda } ,\!1 - (1 - \gamma_{1} )^{\lambda } } \right),\left( {\rho_{1}^{\lambda } ,\!\sigma_{1}^{\lambda } ,\tau_{1}^{\lambda } } \right),\!\left( {l_{1}^{\lambda } ,\!m_{1}^{\lambda } ,\!n_{1}^{\lambda } } \right)} \right\rangle$ for $\lambda > 0$ and

• iv. $\chi_{1}^{\lambda } = \left\langle \begin{gathered} \left( {\alpha_{1}^{\lambda } ,\beta_{1}^{\lambda } ,\gamma_{1}^{\lambda } } \right),\left( {1 - (1 - \rho_{1} )^{\lambda } ,1 - (1 - \sigma_{1} )^{\lambda } ,1 - (1 - \tau_{1} )^{\lambda } } \right), \hfill \\ \left( {1 - (1 - l_{1} )^{\lambda } ,1 - (1 - m_{1} )^{\lambda } ,1 - (1 - n_{1} )^{\lambda } } \right) \hfill \\ \end{gathered} \right\rangle$ for $\lambda > 0$.

The operations presented in Sect. 3 satisfy the following properties:

• i. $\left( {{\text{Commutativity}}} \right):\,\chi_{1} \oplus \chi_{2} = \chi_{2} \oplus \chi_{1} ;\,\,\chi_{1} \otimes \chi_{2} = \chi_{2} \otimes \chi_{1} ;$

• ii. $\left( {{\text{Distributivity}}} \right):\,\,\lambda \left( {\chi_{1} \oplus \chi_{2} } \right) = \lambda \chi_{1} \oplus \lambda \chi_{2} ;\,\,\left( {\chi_{1} \otimes \chi_{2} } \right)^{\lambda } = \chi_{1}^{\lambda } \otimes \chi_{2}^{\lambda }$, $\lambda > 0$, and

• iii. $\left( {{\text{Associativity}}} \right):\,\lambda_{1} \chi_{1} \oplus \lambda_{2} \chi_{1} = \left( {\lambda_{1} + \lambda_{2} } \right)\chi_{1} ;\,\,\chi_{1}^{{\lambda_{1} }} \oplus \chi_{1}^{{\lambda_{2} }} = \chi_{1}^{{(\lambda_{1} + \lambda_{2} )}}$,$\lambda_{1} ,\,\lambda_{2} > 0$.

3.4 Score and Accuracy Function of TFNN

Let $\chi_{1} = \left\langle {(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ),(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ),(l_{1} ,\,m_{{_{1} }} ,\,n_{1} )} \right\rangle$ be a TFNN in $\Re$. The score function [157] $\Gamma (\chi_{1} )$, the accuracy function [157] $H\left( {\chi_{1} } \right)$ of $\chi_{1}$ are, respectively, presented as:

$$\begin{aligned}\Gamma \left( {\chi_{1} } \right)& = \frac{1}{12}\left[ {8 + (\alpha_{1} + 2\beta_{1} + \gamma_{1} ) - (\rho_{1} + 2\sigma_{1} + \tau_{1} ) - (l_{1} + 2m_{1} + n_{1} )} \right],\\ & \quad \Gamma (\chi_{1} ) \in [1,\,0] \end{aligned}$$

$$H\left( {\chi_{1} } \right) = \frac{1}{4}\left[ {(\alpha_{1} + 2\beta_{1} + \gamma_{1} ) - (l_{1} + 2m_{1} + n_{1} )} \right],\,\,H(\chi_{1} ) \in [ - 1,\,1]$$

For $\chi^{ + } = \left\langle {(1,1,1),(0,0,0),(0,0,0)} \right\rangle$ and $\chi^{ - } = \left\langle {(0,0,0),(1,1,1),(1,1,1)} \right\rangle$, $\Gamma (\chi^{ + } )$ = 1 and $\Gamma (\chi^{ - } ) = 0$.

For $\chi^{ + } = \left\langle {(1,1,1),(0,0,0),(0,0,0)} \right\rangle$ and $\chi^{ - } = \left\langle {(0,0,0),(1,1,1),(1,1,1)} \right\rangle$, $H(\chi^{ + } )$ = 1 and $H(\chi^{ - } ) = 0$.

3.5 Comparison of TFNNs

Assume that $\chi_{1} = \left\langle {(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ),(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ),(l_{1} ,\,m_{{_{1} }} ,\,n_{1} )} \right\rangle$ and $\chi_{2} = \left\langle {(\alpha_{2} ,\,\beta_{2} ,\,\gamma_{2} ),(\rho_{2} ,\,\sigma_{2} ,\,\tau_{2} ),(l_{2} ,\,m_{{_{2} }} ,\,n_{2} )} \right\rangle$ are any two TFNNs in $\Re$. If $\Gamma (\chi_{i} )$ and $H(\chi_{i} )$ denote, respectively, the score and accuracy function of $\chi_{i} (i = 1,2)$, then the ranking of TFNNs [157] is presented as:

• i. If $\Gamma (\chi_{1} ) > \Gamma (\chi_{2} )$, then $\chi_{1}$ is greater than $\chi_{2}$ that is $\chi_{1} \succ \chi_{2}$;

• ii. If $\Gamma (\chi_{1} ) = \Gamma (\chi_{2} )$ and $H(\chi_{1} ) > H(\chi_{2} )$, then $\chi_{1}$ is greater than $\chi_{2}$, that is,$\chi_{1} \succ \chi_{2}$;

• iii. If $\Gamma (\chi_{1} ) = \Gamma (\chi_{2} )$,$\Gamma (\chi_{1} ) = \Gamma (\chi_{2} )$, then $\chi_{1}$ is indifferent to $\chi_{2}$, that is, $\chi_{1} \approx \chi_{2}$.

3.6 Triangular Fuzzy Neutrosophic Number Arithmetic Averaging (TFNNAA) Operator

3.6.1 Triangular Fuzzy Neutrosophic Number Weighted Arithmetic Averaging (TFNNWAA) Operator

Let $\theta_{\iota } = \left\langle {(c_{\iota } ,\,d_{\iota } \,,e_{\iota } ),(f_{\iota } ,g_{\iota } ,h_{\iota } ),(o_{\iota } ,p_{\iota } ,q_{\iota } )} \right\rangle$$(\iota = 1,2,...,r)$ be a collection TFNNs in $\Re$ and let ${\text{TFNNWAA}}:\Sigma^{r} \to \Sigma$.

If ${\text{TFNNWAA}}(\theta_{1} ,\theta_{2} ,...,\theta_{r} ) = \omega_{1} \theta_{1} \oplus \omega_{2} \theta_{2} \oplus \cdots \oplus \omega_{r} \theta_{r} = \mathop \oplus \limits_{k = 1}^{r} (\omega_{k} \theta_{k} )$,

then the function ${\text{TFNNWAA}}(\theta_{1} ,\theta_{2} ,...,\theta_{r} )$ is called the TFNNWAA operator, where the weight of $\theta_{\iota } (\iota = 1,2,...,r)$ is denoted by $\omega_{\iota } \in [0,1]$ and $\sum\limits_{\iota = 1}^{r} {\omega_{\iota } = 1.}$

If $\omega = \left( {{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r},{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r},...,{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} \right)^{T}$, then the ${\text{TFNNWAA}}(\theta_{1} ,\theta_{2} ,...,\theta_{r} )$ operator reduces to TFNNAA operator:

${\text{TFNNAA}}(\theta_{1} ,\theta_{2} ,...,\theta_{r} ) = \frac{1}{r}\left( {\theta_{1} \oplus \theta_{2} \oplus \cdots \oplus \theta_{r} } \right)$

Theorem 3.6.1

Let $\theta_{\iota } = \left\langle {(c_{\iota } ,\,d_{\iota } \,,e_{\iota } ),(f_{\iota } ,g_{\iota } ,h_{\iota } ),(o_{\iota } ,p_{\iota } ,q_{\iota } )} \right\rangle$$(\iota = 1,2,...,r)$ be a collection of TFNNs in $\Re .$ Then, ${\text{TFNNWAA}}_{\omega } (\theta_{1} ,\theta_{2} ,...,\theta_{r} ) = \omega_{1} \theta_{1} \oplus \omega_{2} \theta_{2} \oplus \cdots \oplus \omega_{r} \theta_{r} ) = \mathop \oplus \limits_{\iota = 1}^{r} (\omega_{\iota } \theta_{\iota } )$.

$\begin{aligned}& = \left\langle \left( {1 - \prod\limits_{\iota = 1}^{r} {(1 - c_{\iota } )^{{\omega_{\iota } }} ,1 - \prod\limits_{\iota = 1}^{r} {(1 - d_{\iota } )^{{\omega_{\iota } }} ,} 1 - \prod\limits_{\iota = 1}^{r} {(1 - e_{\iota } )^{{\omega_{\iota } }} } } } \right), \right.\\ & \quad \left. \left( {\prod\limits_{\iota = 1}^{r} {f_{\iota }^{{\omega_{\iota } }} } ,\prod\limits_{\iota = 1}^{r} {g_{\iota }^{{\omega_{\iota } }} } ,\prod\limits_{\iota = 1}^{r} {h_{\iota }^{{\omega_{\iota } }} } } \right),\left( {\prod\limits_{\iota = 1}^{r} {o_{\iota }^{{\omega_{\iota } }} } ,\prod\limits_{\iota = 1}^{r} {p_{\iota }^{{\omega_{\iota } }} } ,\prod\limits_{\iota = 1}^{r} {q_{\iota }^{{\omega_{\iota } }} } } \right) \right\rangle , \end{aligned}$, where $\omega_{\iota } \in [0,1]$ denotes the weight of $\theta_{\iota } (\iota = 1,2,...,r)$ and $\sum\limits_{\iota = 1}^{r} {\omega_{\iota } = 1.}$

Proof

For proof, see Biswas et al. [157].

3.6.2 Triangular Fuzzy Number Neutrosophic Geometric Averaging (TFNNGA) Operator

3.6.2.1 Triangular Fuzzy Number Neutrosophic Weighted Geometric Averaging (TFNNWGA) Operator

Let $\theta_{\iota } = \left\langle {(c_{\iota } ,\,d_{\iota } \,,e_{\iota } ),(f_{\iota } ,g_{\iota } ,h_{\iota } ),(o_{\iota } ,p_{\iota } ,q_{\iota } )} \right\rangle$ $(\iota = 1,2,...,r)$ be a collection of TFNNs in $\Re$ and ${\text{TFNNWGA}}:\Pi^{r} \to \Pi$.

If

${\text{TFNNWGA}}_{\omega } (\theta_{1} ,\theta_{2} ,...,\theta_{r} ) = \theta_{1}^{{\omega_{1} }} \otimes \theta_{2}^{{\omega_{2} }} \otimes \cdots \otimes \theta_{r}^{{\omega_{r} }} = \mathop \otimes \limits_{\iota = 1}^{r} \left( {\theta_{\iota }^{{\omega_{\iota } }} } \right)$

then ${\text{TFNNWGA}}_{\omega } (\theta_{1} ,\theta_{2} ,...,\theta_{r} )$ is called the TFNNWGA operator where $\omega_{\iota } \in [0,1]$ is the exponential weight of $\theta_{k} (\iota = 1,2,...,r)$ such that $\sum\nolimits_{i = 1}^{r} {w_{k} = 1.}$

If $\omega = \left( {{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r},r,...,{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} \right)^{T}$, then the ${\text{TFNNWGA}}(\theta_{1} ,\theta_{2} ,...,\theta_{r} )$ operator reduces to TFNNGA operator:

${\text{TFNNGA}}(\theta_{1} ,\theta_{2} ,...,\theta_{r} ) = \left( {\theta_{1} \otimes \theta_{2} \otimes \cdots \otimes \theta_{r} } \right)^{\frac{1}{r}}$.

Theorem 3.6.2

Let $\theta_{\iota } = \left\langle {(c_{\iota } ,\,d_{\iota } \,,e_{\iota } ),(f_{\iota } ,g_{\iota } ,h_{\iota } ),(o_{\iota } ,p_{\iota } ,q_{\iota } )} \right\rangle$$(\iota = 1,2,...,r)$ is a collection of TFNNs in $\Re$. Then,${\text{TFNNWGA}}_{\omega } (\theta_{1} ,\theta_{2} ,...,\theta_{r} ) = \theta_{1}^{{\omega_{1} }} \otimes \theta_{2}^{{\omega_{2} }} \otimes \cdots \otimes \theta_{n}^{{\omega_{\iota } }} = \mathop \otimes \limits_{\iota = 1}^{r} \left( {\theta_{\iota }^{{\omega_{\iota } }} } \right)$

$= \left\langle \begin{array}{l} \left( {\prod\limits_{\iota = 1}^{r} {c_{\iota }^{{\omega_{\iota } }} } ,\prod\limits_{\iota = 1}^{r} {d_{\iota }^{{^{{\omega_{\iota } }} }} } ,\prod\limits_{\iota = 1}^{r} {e_{\iota }^{{\omega_{\iota } }} } } \right),\\ \left( {1 - \prod\limits_{\iota = 1}^{r} {(1 - f_{\iota } )^{{\omega_{\iota } }} ,1 - \prod\limits_{\iota = 1}^{r} {(1 - g_{\iota } )^{{\omega_{\iota } }} ,} 1 - \prod\limits_{\iota = 1}^{r} {(1 - h_{\iota } )^{{^{{\omega_{\iota } }} }} } } } \right), \hfill \\ \left( {1 - \prod\limits_{\iota = 1}^{r} {(1 - o_{\iota } )^{{^{{\omega_{\iota } }} }} ,1 - \prod\limits_{\iota = 1}^{r} {(1 - p_{\iota } )^{{\omega_{\iota } }} ,} 1 - \prod\limits_{\iota = 1}^{r} {(1 - q_{\iota } )^{{\omega_{\iota } }} } } } \right) \hfill \\ \end{array} \right\rangle$ is a TFNN, where $\omega_{k} \in [0,1]$ denotes the weight vector of TFNN $\theta (\iota = 1,2,...,r)$ such that $\sum_{\iota = 1}^{r} {\omega_{\iota } = 1.}$

Proof

For proof, see Biswas et al. [157].

4 Trapezoidal Fuzzy  Number NS

Ye [146] and Biswas et al. [145] combined Trapezoidal Fuzzy Number (TrFN) with SVNS to define Trapezoidal Fuzzy Number NS (TrFNNS).

4.1 TrFNNS

Assume that $\Theta$ is the finite universe of discourse. A TrFNNS $\theta$ in $\Theta$ is presented as:

$\theta = \left\{ {\left\langle {x,o_{\theta } (x),p_{\theta } (x),q_{\theta } (x)\,} \right\rangle |x \in \Theta } \right\},\,$ where $O_{\theta } (x)\subset[0,1]$,$p_{\theta } (x)\subset \left[ {0,1} \right]$ $q_{\theta } (x)\subset \left[ {0,1} \right]$ are trpezoidal fuzzy numbers and  $o_{\theta } (x) = \left( {o_{\theta }^{1} (x),o_{\theta }^{2} (x),\,o_{\theta }^{3} (x),\,o_{\theta }^{4} (x)} \right):{\Theta}{\rightarrow}{[0,1]}$,$p_{\theta } (x) = \left( {p_{\theta }^{1} (x),p_{\theta }^{2} (x),\,p_{\theta }^{3} (x),\,p_{\theta }^{4} (x)} \right):{\Theta}{\rightarrow}{[0,1]}$, and $q_{\theta } (x) = \left( {q_{\theta }^{1} (x),q_{\theta }^{2} (x),\,q_{\theta }^{3} (x),\,q_{\theta }^{4} (x)} \right):{\Theta}{\rightarrow}{[0,1]}$ present, respectively, the degrees of truth, indeterminacy, and falsity MF of $x$ in $\theta$ and for every $x \in \Theta$ and $0 \le o_{\theta }^4 (x) + p_{\theta }^4 (x) + q_{\theta }^4 (x) \le 3$.

4.2 Trapezoidal Fuzzy Neutrosophic Number (TrFNN)

A TrFNN [145] $\chi_{\varphi }$ is presented as:

$\chi_{\varphi } = \left\langle {\left( {\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} ,\,\alpha_{4} } \right),\,\left( {\beta_{1} ,\beta_{2} ,\beta_{3} ,\beta_{4} } \right),\,\left( {\gamma_{1} ,\,\gamma_{2} ,\,\gamma_{3} ,\,\gamma_{4} } \right)} \right\rangle$ in a universe of discourse $W$ with $\alpha_{1} \, \le \alpha_{2} \le \alpha_{3} \le \alpha_{4} ,$ $\beta_{1} \le \,\beta_{2} \le \beta_{3} \le \beta_{4}$ and $\gamma_{1} \le \,\gamma_{2} \le \gamma_{3} \le \gamma_{4}$. Here,$\chi_{\varphi }$ is defined as follows:

Its truth MF is presented as:

$$\xi_{{\chi_{\varphi } }} (\omega ) = \left\{ \begin{array}{lll} \frac{{\omega - \alpha_{1} }}{{\alpha_{2} - \alpha_{1} }}, & \alpha_{1} \le \omega \le \alpha_{2} \hfill \\ 1, & \,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{\alpha_{4} - \omega }}{{\alpha_{4} - \alpha_{3} }}, & \,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0, & \,{\text{otherwise}}\,. \hfill \\ \end{array} \right.$$

Its indeterminacy MF is presented as:

$$\psi_{{\chi_{\varphi } }} (\omega ) = \left\{ \begin{array}{lll} \frac{{\omega - \beta_{1} }}{{\beta_{2} - \beta_{1} }}, & \beta_{1} \le \omega \le \beta_{2} \hfill \\ 1, & \beta_{2} \le \omega \le \beta_{3} \hfill \\ \frac{{\beta_{4} - \omega }}{{\beta_{4} - \beta_{3} }}, & \beta_{3} \le \omega \le \beta_{4} \hfill \\ 0, & {\text{otherwise}}\,. \hfill \\ \end{array} \right.$$

and its falsity MF is presented as:

$$\zeta _{{\chi _{\varphi } }} (\omega ) = \left\{ \begin{gathered} \frac{{\omega - \gamma _{1} }}{{\gamma _{2} - \gamma _{1} }},\quad \gamma _{1} \le \omega \le \gamma _{2} \hfill \\ 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\quad \gamma _{2} \le \omega \le \gamma _{3} \hfill \\ \frac{{\gamma _{4} - \omega }}{{\gamma _{4} - \gamma _{3} }},\quad \gamma _{3} \le \omega \le \gamma _{4} \hfill \\ 0\quad \quad \quad \quad \text{otherwise}. \hfill \\ \end{gathered} \right.$$

Let $\chi_{1} = \left\langle {(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ,\delta_{1} ),(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ,\upsilon_{1} ),(l_{1} ,\,m_{{_{1} }} ,\,n_{1} ,o_{1} )} \right\rangle$ and $\chi_{2} = \left\langle {(\alpha_{2} ,\,\beta_{2} ,\,\gamma_{2} ,\,\delta_{2} ),(\rho_{2} ,\,\sigma_{2} ,\,\tau_{2} ,\,\upsilon_{2} ),(l_{2} ,\,m_{{_{2} }} ,\,n_{2} ,\,0_{2} )} \right\rangle$. be any two TrFNNs in $\Re$. Then, the operational rules for $\chi_{1}$ and $\chi_{2}$ are presented as.

• i. $\begin{aligned}&\chi_{1} \oplus \chi_{2}\\ & = \left\langle \begin{gathered} \left( {\alpha_{1} + \alpha_{2} - \alpha_{1} \alpha_{2} ,\beta_{1} + \beta_{2} - \beta_{1} \beta_{2} ,\gamma_{1} + \gamma_{2} - \gamma_{1} \gamma_{2} ,\delta_{1} + \delta_{2} - \delta_{1} \delta_{2} \,} \right),\\ \left( {\rho_{1} \rho_{2} + \sigma_{1} \sigma_{2} + \tau_{1} \tau_{2} + \upsilon_{1} \upsilon_{2} } \right), \hfill \left( {l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} + o_{1} o_{2} } \right) \hfill \\ \end{gathered} \right\rangle \end{aligned}$

• ii. $\begin{aligned} & \chi_{1} \otimes \chi_{2}\\ & = \left\langle \begin{gathered} \left( {\alpha_{1} \alpha_{2} ,\,\beta_{1} \beta_{2} ,\,\gamma_{1} \gamma_{2} ,\,\delta_{1} \delta_{2} } \right),\\ \left( {\rho_{1} + \rho_{2} - \rho_{1} \rho_{2} ,\,\,\sigma_{1} + \sigma_{2} - \sigma_{1} \sigma_{2} ,\tau_{1} + \tau_{2} - \tau_{1} \tau_{2} ,\,\upsilon_{1} + \upsilon_{2} - \upsilon_{1} \upsilon_{2} } \right), \hfill \\ \left( {l_{1} + l_{2} - l_{1} l_{2} ,m_{1} + m_{2} - m_{1} m_{2} ,n_{1} + n_{2} - n_{1} n_{2} ,\,\,o_{1} + o_{2} - o_{1} o_{2} } \right) \hfill \\ \end{gathered} \right\rangle ;\end{aligned}$

• iii. $\eta \chi_{1}$ = $\left\langle \begin{gathered} \left( \begin{gathered} 1 - (1 - \alpha_{1} )^{\eta } ,1 - (1 - \beta_{1} )^{\eta } ,\, \hfill \\ 1 - (1 - \gamma_{1} )^{\eta } ,1 - (1 - \delta_{1} )^{\eta } \, \hfill \\ \end{gathered} \right), \hfill \\ \left( {\rho_{1}^{\eta } ,\,\sigma_{1}^{\eta } ,\tau_{1}^{\eta } ,\upsilon_{1}^{\eta } } \right),\,\,\left( {l_{1}^{\eta } ,\,m_{1}^{\eta } ,n_{1}^{\eta } ,o_{1}^{\eta } } \right) \hfill \\ \end{gathered} \right\rangle$ for $\eta > 0$

• iv. $\left( {\chi_{1} } \right)^{\eta }$ = $\left\langle \begin{gathered} \left( {\alpha_{1}^{\eta } ,\,\beta_{1}^{\eta } ,\gamma_{1}^{\eta } ,\delta_{1}^{\eta } } \right), \\ \left( {1 - (1 - \rho_{1} )^{\eta } ,1 - (1 - \sigma_{1} )^{\eta } ,1 - (1 - \tau_{1} )^{\eta } ,1 - (1 - \upsilon_{1} )^{\eta } \,} \right), \hfill \\ \left( {1 - (1 - l_{1} )^{\eta } ,1 - (1 - m_{1} )^{\eta } ,1 - (1 - n_{1} )^{\eta } ,1 - (1 - o_{1} )^{\eta } \,} \right), \hfill \\ \end{gathered} \right\rangle$ for $\eta > 0$.

• v. $\chi_{1} = \chi_{2} ,$ if $(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ,\delta_{1} ) = (\alpha_{2} ,\,\beta_{2} ,\,\gamma_{2} ,\,\delta_{2} );\,(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ,\upsilon_{1} ) = (\rho_{2} ,\,\sigma_{2} ,\,\tau_{2} ,\,\upsilon_{2} );\,(l_{1} ,\,m_{{_{1} }} ,\,n_{1} ,o_{1} ) = (l_{2} ,\,m_{{_{2} }} ,\,n_{2} ,\,0_{2} )$

4.3 Expected Value (EV) and Expected Interval (EI) of TrFNN

The EI and the EV [145] of the truth MF $\xi_{{\chi_{1} }} (\omega ) = (\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} ,\alpha_{4} )$$= \left\{ \begin{array}{lll} \frac{{\omega - \alpha_{1} }}{{\alpha_{2} - \alpha_{1} }}, & \alpha_{1} \le \omega \le \alpha_{2} \hfill \\ 1, & \,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{\alpha_{4} - \omega }}{{\alpha_{4} - \alpha_{3} }}, & \,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0, & \,{\text{otherwise}}\,. \hfill \\ \end{array} \right.$ of $\chi_{1}$ in a universe of discourse $W$ are defined as follows:

$${\text{EI}}\xi_{{\chi_{1} }} (\omega ) = \left[ {\frac{{(\alpha_{1} + \alpha_{2} )}}{2},\frac{{(\alpha_{3} + \alpha_{4} )}}{2}} \right]$$

$${\text{EV}}\xi_{{\chi_{1} }} (\omega ) = \frac{{(\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} )}}{4}$$

In similar way, the EI and the EV of the indeterminacy MF $\psi_{{\chi_{1} }} (\omega ) = \left\{ \begin{array}{lll} \frac{{\omega - \beta_{1} }}{{\beta_{2} - \beta_{1} }}, & \beta_{1} \le \omega \le \beta_{2} \hfill \\ 1, & \beta_{2} \le \omega \le \beta_{3} \hfill \\ \frac{{\beta_{4} - \omega }}{{\beta_{4} - \beta_{3} }}, & \beta_{3} \le \omega \le \beta_{4} \hfill \\ 0, & {\text{otherwise}}\,. \hfill \\ \end{array} \right.$

of $\chi_{1}$ are defined as:

$${\text{EI}}\psi_{{\chi_{1} }} (\omega ) = \left[ {\frac{{(\beta_{1} + \beta_{2} )}}{2},\frac{{(\beta_{3} + \beta_{4} )}}{2}} \right]$$

$${\text{EV}}\psi_{{\chi_{1} }} (\omega ) = \frac{{(\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} )}}{4}$$

and the EI and the EV of the falsity MF

$$\zeta _{{\chi _{1} }} (\omega ) = \left\{ \begin{gathered} \frac{{\omega - \gamma _{1} }}{{\gamma _{2} - \gamma _{1} }},\quad \gamma _{1} \le \omega \le \gamma _{2} \hfill \\ 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\quad \gamma _{2} \le \omega \le \gamma _{3} \hfill \\ \frac{{\gamma _{4} - \omega }}{{\gamma _{4} - \gamma _{3} }},\quad \gamma _{3} \le \omega \le \gamma _{4} \hfill \\ 0\quad \quad \quad \quad \text{otherwise}. \hfill \\ \end{gathered} \right.$$

of $\chi_{1}$ are defined as follows:

$${\text{EI}}(\zeta_{{\chi_{1} }} (\omega )) = \left[ {\frac{{(\gamma_{1} + \gamma_{2} )}}{2},\frac{{(\gamma_{3} + \gamma_{4} )}}{2}} \right]$$

$${\text{EV}}(\zeta_{1} (\omega )) = \frac{{\left( {\gamma_{1} + \gamma_{2} + \gamma_{3} + \gamma_{4} } \right)}}{4}$$

4.4 Truth Favourite Relative Expected Value (TrFREV) of TrFNN

Let $a = \left\langle {(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ,\delta_{1} ),(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ,\upsilon_{1} ),(l_{1} ,\,m_{{_{1} }} ,\,n_{1} ,o_{1} )} \right\rangle$ be a TrFNN in $\Re$. Then TrFREV [145] of $a$ is defined as:

$${\text{EV}}^{{{\text{truth}}}} (a) = \frac{{3EV(\xi_{a} (\omega ))}}{{\text{EV}(\xi_{a} (\omega )) + \text{EV}(\psi_{a} (\omega )) + \text{EV}(\zeta_{a} (\omega ))}},$$

where ${\text{EV}}(\xi_{{a}} (w))$, ${\text{EV}}(\psi_{{a}} (\omega ))$, and ${\text{EV}}(\zeta_{{a}} (\omega ))$ denote, respectively, the EVs of truth, indeterminacy, and falsity MF of $a$.

4.5 Expected Value Theorem

Assume that $\phi_{1} = \left\langle {(\alpha_{1} ,\,\beta_{1} ,\,\gamma_{1} ,\delta_{1} ),(\rho_{1} ,\,\sigma_{1} ,\,\tau_{1} ,\upsilon_{1} ),(l_{1} ,\,m_{{_{1} }} ,\,n_{1} ,o_{1} )} \right\rangle$ is a TrFNN in $\Re$, then the TFREV of $\phi_{{1}}$ is

$${\text{EV}}^{{{\text{truth}}}} (a) = \frac{{3\sum\nolimits_{i = 1}^{4} {\alpha_{i} } }}{{\left( {\sum\nolimits_{i = 1}^{4} {(\alpha_{i} } + \beta_{i} + \gamma_{i} )} \right)}}$$

Proof

For proof, see [145].

4.6 Single-Valued Trapezoidal Neutrosophic Number (SVTrNN)

An SVTrNN [146] (SVTNN) $\alpha = \langle (\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } ) \rangle$ is a special NS on the real number set R, whose truth MF, indeterminacy MF, and a falsity MF are presented as:

$$\xi _{\alpha } (\omega ) = \left\{ \begin{gathered} {\raise0.7ex\hbox{${(\omega - \alpha _{1} )\xi _{\alpha } }$} \!\mathord{\left/ {\vphantom {{(\omega - \alpha _{1} )\xi _{\alpha } } {(\alpha _{2} - \alpha _{1} )}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(\alpha _{2} - \alpha _{1} )}$}},\quad (\alpha _{1} \le \omega \le \alpha _{2} ) \hfill \\ \xi _{\alpha } ,\quad \quad \quad \quad \quad \quad \quad \quad (\alpha _{2} \le \omega \le \alpha _{3} ) \hfill \\ {\raise0.7ex\hbox{${(\alpha _{4} - \omega )\xi _{\alpha } }$} \!\mathord{\left/ {\vphantom {{(\alpha _{4} - \omega )\xi _{\alpha } } {(\alpha _{4} - \alpha _{3} )}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(\alpha _{4} - \alpha _{3} )}$}},\quad (\alpha _{3} \le \omega \le \alpha _{4} ) \hfill \\ 0,\quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

$$\psi _{\alpha } (\omega ) = \left\{ \begin{gathered} {\raise0.7ex\hbox{${(\alpha _{2} - \omega + \psi _{a} (\omega - \alpha _{1} ))}$} \!\mathord{\left/ {\vphantom {{(\alpha _{2} - \omega + \psi _{a} (\omega - \alpha _{1} ))} {(\alpha _{2} - \alpha _{1} )}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(\alpha _{2} - \alpha _{1} )}$}},(\alpha _{1} \le \omega \le \alpha _{2} ) \hfill \\ \psi _{\alpha } ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\alpha _{2} \le \omega \le \alpha _{3} ) \hfill \\ {\raise0.7ex\hbox{${(\omega - \alpha _{3} + \psi _{\alpha } (\alpha _{4} - \omega ))}$} \!\mathord{\left/ {\vphantom {{(\omega - \alpha _{3} + \psi _{\alpha } (\alpha _{4} - \omega ))} {(\alpha _{4} - \alpha _{3} )}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(\alpha _{4} - \alpha _{3} )}$}},(\alpha _{3} \le x \le \alpha _{4} ) \hfill \\ 1,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

$$\zeta _{\alpha } (\omega ) = \left\{ \begin{gathered} {\raise0.7ex\hbox{${(\alpha _{2} - \omega + \zeta _{a} (\omega - \alpha _{1} ))}$} \!\mathord{\left/ {\vphantom {{(\alpha _{2} - \omega + \zeta _{a} (\omega - \alpha _{1} ))} {(\alpha _{2} - \alpha _{1} )}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(\alpha _{2} - \alpha _{1} )}$}},(\alpha _{1} \le \omega \le \alpha _{2} ) \hfill \\ \zeta _{\alpha } ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\alpha _{2} \le \omega \le \alpha _{3} ) \hfill \\ {\raise0.7ex\hbox{${(\omega - \alpha _{3} + \zeta _{\alpha } (\alpha _{4} - \omega ))}$} \!\mathord{\left/ {\vphantom {{(\omega - \alpha _{3} + \zeta _{\alpha } (\alpha _{4} - \omega ))} {(\alpha _{4} - \alpha _{3} )}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${(\alpha _{4} - \alpha _{3} )}$}},(\alpha _{3} \le x \le \alpha _{4} ) \hfill \\ 1,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

where $0 \le \,\,\xi_{\alpha } \le \,\,1,\,\,\,0 \le \,\,\psi_{\alpha } \le \,\,1,\,\,\,0 \le \,\,\zeta_{\alpha } \le \,\,1$; and $0 \le \xi_{\alpha } + \psi_{\alpha } + \zeta_{\alpha } \le 3$; $\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} \in \Re$.

Here, $\alpha = < (\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } ) >$ is called a positive (+ve) SVTrNN, if $\alpha_{1} > 0$.

Similarly, $\alpha = < (\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );\,(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } ) >$ reduces to a negative (–ve) SVTrNN, if $\alpha_{4} \le 0,$

When $0 \le \alpha_{1} \le \alpha_{2} \le \alpha_{3} \le \alpha_{4} \le 1),\,$ and $0 \le \,\,\xi_{\alpha } \le \,\,1,\,\,\,0 \le \,\,\psi_{\alpha } \le \,\,1,\,\,\,0 \le \,\,\zeta_{\alpha } \le \,\,1$, $\alpha$ is called a normalized SVTrNN.

4.6.1 Score Function of SVTrNNs

Let $\alpha = \langle (\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } ) \rangle$ be an SVTrNN. Then, the score and accuracy function [183] of $\alpha$ are denoted by $\Gamma (\alpha )$ and $H(\alpha )$, respectively, and are presented as:

• i. $\Gamma (\alpha ) = \left( \frac{1}{12} \right)\,(\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} ) \times (2 + \xi_{\alpha } - \psi_{\alpha } - \zeta_{\alpha } )$

• ii. $H(\alpha ) = \left( \frac{1}{12} \right)\,(\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} ) \times (2 + \xi_{\alpha } - \psi_{\alpha } + \zeta_{\alpha } )$

4.6.2 Ranking of SVTrNN

Let $\alpha = \left\langle {(\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } )} \right\rangle$ and $\beta = \left\langle {(\beta_{1} ,\beta_{2} ,\beta_{3} ,\beta_{4} );(\xi_{\beta } ,\psi_{\beta } ,\zeta_{\beta } )} \right\rangle$ be any two SVTrNNs. Ranking of SVTrNNs [183] is presented as follows:

• (i) When $\Gamma (\alpha ) < \Gamma (\beta )$, then $\alpha < \beta$

• (ii) When $\Gamma (\alpha ) = \Gamma (\beta )$ and if $H(\alpha ) < H(\beta )$, then $\alpha < \beta$

• (iii) When $\Gamma (\alpha ) = \Gamma (\beta )$ and $H(\alpha ) = H(\beta )$, then $\alpha = \beta$.

4.6.3 Centre of Gravity

Let $\alpha = [\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} ]$ be a TrFN on $\Re$, and $\alpha_{1} \le \alpha_{2} \le \alpha_{3} \le \alpha_{4}$; then, the centre of gravity (COG) of $\alpha$ [155, 184, 185] defined by

${\text{COG(}}\alpha {) = }\left\{ \begin{gathered} \alpha_{1} ,\quad {\text{if}}\quad \alpha_{1} = \alpha_{2} = \alpha_{3} = \alpha_{4} \hfill \\ \frac{1}{3}\left[ {\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} - \frac{{\alpha_{4} - \alpha_{3} - \alpha_{21} - \alpha_{1} }}{{\alpha_{4} + \alpha_{3} - \alpha_{2} - \alpha_{1} }}} \right],\quad {\text{otherwise}} \hfill \\ \end{gathered} \right\}$

4.6.4 Score function SVTrNN

Assume that $\alpha = \left\langle {(\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } )} \right\rangle$ is an SVTrNN. Then, the score, accuracy, and certainty function [155] of $\alpha$ are denoted by $\Gamma (\alpha )$, $H(\alpha )$, and $\kappa (\alpha )$), respectively, and presented as:

$$\begin{aligned} \Gamma (\alpha ) & = COG(\alpha ) \times \frac{{(2 + \xi_{\alpha } - \psi_{\alpha } - \zeta_{\alpha } )}}{3} \\ H(\alpha ) & = COG(\alpha ) \times (\xi_{\alpha } - \zeta_{\alpha } ) \\ \kappa (\alpha ) & = COG(\alpha ) \times \xi_{\alpha } \\ \end{aligned}$$

4.6.5 Comparison of SVTrNNs

Assume that $\alpha = \left\langle {(\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} );(\xi_{\alpha } ,\psi_{\alpha } ,\zeta_{\alpha } )} \right\rangle$, and $\beta = \left\langle {(\beta_{1} ,\beta_{2} ,\beta_{3} ,\beta_{4} );(\xi_{\beta } ,\psi_{\beta } ,\zeta_{\beta } )} \right\rangle$ are any two SVTRNNs. Comparison between SVTrNNs [155] is presented as follows:

• (i) When $\Gamma (\alpha ) > \Gamma (\beta )$, then $\alpha > \beta$

• (ii) When $\Gamma (\alpha ) = \Gamma (\beta )$ and if $H(\alpha ) > H(\beta )$, then $\alpha > \beta$

• (iii) When $\Gamma (\alpha ) = \Gamma (\beta )$ and $H(\alpha ) < H(\beta )$, then $\alpha < \beta$

• (iv) When $\Gamma (\alpha ) = \Gamma (\beta ),$ $H(\alpha ) = H(\beta )$, and

  • • $\kappa (\alpha ) > \kappa (\beta ),$ then $\alpha > \beta$

  • • $\kappa (\alpha ) < \kappa (\beta ),$ then $\alpha < \beta$

  • • $\kappa (\alpha ) = \kappa (\beta ),$ then $\alpha = \beta$.

4.7 Interval TrNN (ITrNN)

Suppose that $\chi$ is an SVTrNN [156]. Its truth MF, indeterminacy MF, and falsity MF are defined as

$$\xi_{\chi } (\omega ) = \left\{ \begin{gathered} \frac{{(\omega - \alpha_{1} )a^{\prime}_{\chi } }}{{(\alpha_{2} - \alpha_{1} )}},\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{1} \le \omega \le \alpha_{2} \hfill \\ a^{\prime}_{\chi } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{(\alpha_{4} - \omega )a^{\prime}_{\chi } }}{{(\alpha_{4} - \alpha_{3} )}},\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}}{.} \hfill \\ \end{gathered} \right.$$

$$\psi_{\chi } (\omega ) = \left\{ \begin{gathered} \frac{{(\alpha_{2} - \omega ) + (\omega - \alpha_{1} )b^{\prime}_{\chi } }}{{(\alpha_{2} - \alpha_{1} )}},\,\,\,\,\,\alpha_{1} \le \omega \le \alpha_{2} \hfill \\ b^{\prime}_{\chi } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{\omega - \alpha_{3} + (\alpha_{4} - \omega )b^{\prime}_{\chi } }}{{\alpha_{4} - \alpha_{3} }},\,\,\,\,\,\,\,\,\,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}}{.} \hfill \\ \end{gathered} \right.$$

$$\zeta_{\chi } (\omega ) = \left\{ \begin{gathered} \frac{{\alpha_{2} - \omega + (\omega - \,\alpha_{1} )c^{\prime}_{\chi } }}{{\alpha_{2} - \,\alpha_{1} }},\,\,\,\,\,\,\,\,\,\,\,\alpha_{1} \le \omega \le \alpha_{2} \hfill \\ c^{\prime}_{\chi } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{\omega - \alpha_{3} + (\alpha_{4} - \omega )c^{\prime}_{\chi } }}{{\alpha_{4} - \alpha_{3} }},\,\,\,\,\,\,\,\,\,\,\,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}}{.} \hfill \\ \end{gathered} \right.$$

where $0 \le \xi_{\chi } (\omega ) \le 1,\,\,0 \le \psi_{\chi } (\omega ) \le 1,0 \le \zeta_{\chi } (\omega ) \le 1,$ $0 \le \xi_{\chi } (\omega ) + \psi_{\chi } (\omega ) + \zeta_{\chi } (\omega ) \le 3,\,\,\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} ,\,\alpha_{4} \in \Re$ and $a^{\prime}_{\chi } ,b^{\prime}_{\chi } ,c^{\prime}_{\chi } \in [0,\,1]$. Then, neutrosophic trapezoidal number $\chi$ is presented as $\chi = (\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} ,\,\alpha_{4} ;a^{\prime}_{\chi } ,b^{\prime}_{\chi } ,c^{\prime}_{\chi } )$.

4.7.1 Defınıtıon of ITrNN

Assume that $a_{\chi }^{\prime } = [a_{\chi }^{\prime l} ,a_{\chi }^{\prime u} ],{\mkern 1mu} b_{\chi }^{\prime } = [b_{\chi }^{\prime l} ,b_{\chi }^{\prime u} ],{\mkern 1mu} c_{\chi }^{\prime } = [c_{\chi }^{\prime l} ,c_{\chi }^{\prime u} ,]$. Then, an ITrNN [156] $\chi$ denoted by $\chi = ([\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} ,\,\alpha_{4} ];a^{\prime}_{\chi } ,b^{\prime}_{\chi } ,c^{\prime}_{\chi } )$ is defined as

$$\xi_{\chi } (\omega ) = \left\{ \begin{gathered} \frac{{(\omega - \alpha_{1} )a^{\prime}_{\chi } }}{{(\alpha_{2} - \alpha_{1} )}},\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{1} \le \omega \le \alpha_{2} \hfill \\ a^{\prime}_{\chi } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{(\alpha_{4} - \omega )a^{\prime}_{\chi } }}{{(\alpha_{4} - \alpha_{3} )}},\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}}{.} \hfill \\ \end{gathered} \right.$$

$$\psi_{\chi } (\omega ) = \left\{ \begin{gathered} \frac{{(\alpha_{2} - \omega ) + (\omega - \alpha_{1} )b^{\prime}_{\chi } }}{{(\alpha_{2} - \alpha_{1} )}},\,\,\,\,\,\alpha_{1} \le \omega \le \alpha_{2} \hfill \\ b^{\prime}_{\chi } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{\omega - \alpha_{3} + (\alpha_{4} - \omega )b^{\prime}_{\chi } }}{{\alpha_{4} - \alpha_{3} }},\,\,\,\,\,\,\,\,\,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}}{.} \hfill \\ \end{gathered} \right.$$

$$\zeta_{\chi } (\omega ) = \left\{ \begin{gathered} \frac{{\alpha_{2} - \omega + (\omega - \,\alpha_{1} )c^{\prime}_{\chi } }}{{\alpha_{2} - \,\alpha_{1} }},\,\,\,\,\,\,\,\,\,\,\,\alpha_{1} \le \omega \le \alpha_{2} \hfill \\ c^{\prime}_{\chi } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha_{2} \le \omega \le \alpha_{3} \hfill \\ \frac{{\omega - \alpha_{3} + (\alpha_{4} - \omega )c^{\prime}_{\chi } }}{{\alpha_{4} - \alpha_{3} }},\,\,\,\,\,\,\,\,\,\,\,\alpha_{3} \le \omega \le \alpha_{4} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}}{.} \hfill \\ \end{gathered} \right.$$

where $a^{\prime}_{\chi } ,b^{\prime}_{\chi } ,\,c^{\prime}_{\chi } \subset [0,\,1]$ denote interval numbers, $0 \le \sup (a^{\prime}_{\chi } ) + \sup (b^{\prime}_{\chi } ) + \sup (c^{\prime}_{\chi } ) \le 3$ and $\chi = ([\alpha_{1} ,{\mkern 1mu} \alpha_{2} ,{\mkern 1mu} \alpha_{3} ,{\mkern 1mu} \alpha_{4} ];{\mkern 1mu} [a_{\chi }^{\prime l} ,a_{\chi }^{\prime u} ],{\mkern 1mu} [b_{\chi }^{\prime l} ,b_{\chi }^{\prime u} ],{\mkern 1mu} [c_{\chi }^{\prime l} ,c_{\chi }^{\prime u} ,]$ is said to be positive ITrNN if $\chi > 0$ and one of $\alpha_{1} ,\,\alpha_{2} ,\,\alpha_{3} ,\,\alpha_{4}$ is not equal to zero.

4.7.2 Operations on ITrNNs

Let $\chi_{1} = \left( {\left[ {\alpha_{11} ,\alpha_{12} ,\alpha_{13} ,\alpha_{14} } \right];[p_{1}^{l} ,p_{1}^{u} ],\,[q_{1}^{l} ,q_{1}^{u} ],[r_{1}^{l} ,r_{1}^{u} ]} \right)$ and $\,\chi_{2} = (\left[ {\alpha_{21} ,\,\alpha_{22} ,\alpha_{23} ,\alpha_{24} } \right];[p_{2}^{l} ,p_{2}^{u} ],[\,q_{2}^{l} ,\,q_{2}^{u} ],[r_{2}^{l} ,\,r_{2}^{u} ])$ be any two ITrNNs and $\lambda > 0$. Then, the following operations hold good [156].

• i. $\chi_{1} \oplus \chi_{2} = = (\left[ {\alpha_{11} + \alpha_{21} ,\alpha_{12} + \alpha_{22} ,\alpha_{13} + \alpha_{23} ,\alpha_{14} + \alpha_{24} } \right];[p_{1}^{l} + p_{2}^{l} - p_{1}^{l} p_{2}^{l} ,p_{1}^{u} + p_{2}^{u} - p_{1}^{u} p_{2}^{u} ],[q_{1}^{l} q_{2}^{l} ,q_{1}^{u} q_{2}^{u} ],[r_{1}^{l} r_{2}^{l} ,r_{1}^{u} r_{2}^{u} ])$

$\begin{aligned}\chi_{1} \otimes \chi_{2}& = (\left[ \alpha_{11} \,\alpha_{21} ,\alpha_{12} \alpha_{22} ,\alpha_{13} \,\alpha_{31} ,\alpha_{14} \alpha_{24} \right];[p_{1}^{l} p_{2}^{l} ,\,p_{1}^{u} p_{2}^{u} ], \\& \left[ {q_{1}^{l} + q_{2}^{l} - q_{1}^{l} q_{2}^{l} ,\,_{1}^{u} + q_{2}^{u} - q_{1}^{u} q_{2}^{u} } \right], \\& \left[ {r_{1}^{l} + r_{2}^{l} - r_{1}^{l} r_{2}^{l} ,\,r_{1}^{u} + r_{2}^{u} - r_{1}^{u} r_{2}^{u} } \right]) \end{aligned}$

$\lambda \chi_{1} = (\left[ {\lambda \alpha_{11} ,\lambda \,\alpha_{12} ,\lambda \alpha_{13} ,\lambda \alpha_{14} } \right];[1 - (1 - p_{1}^{l} )^{\lambda } ,1 - (1 - p_{2}^{u} )^{\lambda } ],[(q_{1}^{l} )^{\lambda } ,(q_{2}^{u} )^{\lambda } ],[(r_{1}^{l} )^{\lambda } ,(r_{2}^{\lambda } )^{\lambda } ])$

$(\chi_{1} )^{\lambda } = (\left[ {\left( {\alpha_{11} } \right)^{\lambda } ,\left( {\alpha_{12} } \right)^{\lambda } ,\left( {\alpha_{13} } \right)^{\alpha } ,\left( {\alpha_{14} } \right)^{\alpha } } \right];[(p_{1}^{l} )^{\lambda } ,(p_{1}^{u} )^{\lambda } ],[1 - (1 - q_{1}^{l} )^{\lambda } ,1 - (1 - q_{1}^{u} )^{\lambda } ],[1 - (1 - r_{1}^{l} )^{\lambda } ,1 - (1 - r_{1}^{u} )^{\lambda } ])$

5 A Single-Valued Pentagonal Neutrosophic Number (SVPNN)

An SVPNN [159] $\chi$ is defined as

$$\chi = \left\langle {\omega ,\xi_{\chi } (\omega ),\,\psi_{A} (\omega ),\zeta_{A} (\omega ) > /,\,\omega \in \Omega } \right\rangle .$$

The truth MF $\xi_{\chi } (\omega ):\Re \to [0,\alpha ]$, the indeterminacy MF $\psi_{\chi } (\omega ):\Re \to [\beta ,\,1]$, and the falsity MF $\zeta_{\chi } (\omega ):\Re \to [\gamma ,\,1]$ are presented as:

$\begin{aligned} \xi_{\chi } (\omega ) & = \left\{ \begin{gathered} \xi_{{\chi_{L1} }} (\omega ),\,p^{\prime}_{1} \le \omega \le q^{\prime}_{1} \hfill \\ \xi_{{\chi_{L2} }} (\omega ),q^{\prime}_{1} \le \omega \le r^{\prime}_{1} \hfill \\ \tau ,\,\,\,\,\,\,\,\,\omega = r^{\prime}_{1} \hfill \\ \xi_{{\chi_{U1} }} (\omega ),\,r^{\prime}_{1} \le \omega \le s^{\prime}_{1} \hfill \\ \xi_{{\chi_{U2} }} (\omega ),\,s^{\prime}_{1} \le \omega \le t^{\prime}_{1} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right., \\ \psi_{\chi } (\omega ) & = \left\{ \begin{gathered} \psi_{{\chi_{L1} }} (\omega ),\,p^{\prime}_{2} \le \omega \le q^{\prime}_{2} \hfill \\ \psi_{{\chi_{L2} }} (\omega ),q^{\prime}_{2} \le \omega \le r^{\prime}_{2} \hfill \\ \iota ,\,\,\,\,\,\,\,\,\omega = r^{\prime}_{2} \hfill \\ \psi_{{\chi_{U1} }} (\omega ),\,r^{\prime}_{2} \le \omega \le s^{\prime}_{2} \hfill \\ \psi_{{\chi_{U2} }} (\omega ),\,s^{\prime}_{2} \le \omega \le t^{\prime}_{2} \hfill \\ 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right. \\ \zeta_{\chi } (\omega ) & = \left\{ \begin{gathered} \zeta_{{\chi_{L1} }} (\omega ),\,p^{\prime}_{3} \le \omega \le q^{\prime}_{3} \hfill \\ \zeta_{{\chi_{L2} }} ,q^{\prime}_{3} \le \omega \le r^{\prime}_{3} \hfill \\ \kappa ,\,\,\,\,\,\,\,\,\omega = r^{\prime}_{3} \hfill \\ \zeta_{{\chi_{U1} }} (\omega ),\,r^{\prime}_{3} \le \omega \le s^{\prime}_{3} \hfill \\ \zeta_{{\chi_{U2} }} (\omega ),\,s^{\prime}_{3} \le \omega \le t^{\prime}_{3} \hfill \\ 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{otherwise}} \hfill \\ \end{gathered} \right. \\ \end{aligned}$,

5.1 Score Function of SVPNN

Assume that $\iota = \left\langle {(\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} ,\alpha_{5} );(\beta_{1} ,\beta_{2} ,\beta_{3} ,\beta_{4} ,\beta_{5} );(\gamma_{1} ,\gamma_{2} ,\gamma_{3} ,\gamma_{4} ,\gamma_{5} ))} \right\rangle$ is an SVPNN.

Beneficiary degree of truth indicator $= (\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} + \alpha_{5} )/5$.

Hesitation degree of indeterminacy indicator = $(\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} )/5$.

Non-beneficiary degree of falsity indicator = $(\gamma_{1} + \gamma_{2} + \gamma_{3} + \gamma_{4} + \gamma_{5} )/5$.

Chakraborty et al. [159] defined score function $\Gamma (\iota )$ and accuracy function $H(\iota )$ of $\iota$ as follows:

$$\begin{aligned}\Gamma (\iota )& = \frac{1}{3}\left( 2 + \frac{{(\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} + \alpha_{5} )}}{5} - \frac{{(\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} )}}{5} \right.\\ & \quad \left. - \frac{{(\gamma_{1} + \gamma_{2} + \gamma_{3} + \gamma_{4} + \gamma_{5} )}}{5} \right) \end{aligned}$$

Here, $\Gamma (\iota ) \in [0,\,1]$.

$H(\iota ) = \left( {\frac{{(\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} + \alpha_{5} )}}{5} - \frac{{(\gamma_{1} + \gamma_{2} + \gamma_{3} + \gamma_{4} + \gamma_{5} )}}{5}} \right),\,H(\iota ) \in [ - 1,1]$.

5.2 Comparison of SVPNNS

Assume that $\iota_{1} = \langle (\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} ,\alpha_{5} );(\beta_{1} ,\beta_{2} ,\beta_{3} ,\beta_{4} ,\beta_{5} );(\gamma_{1} ,\gamma_{2} ,\gamma_{3} ,\gamma_{4} ,\gamma_{5} )) \rangle$

and $\iota_{2} = \langle (c_{1} ,c_{2} ,c_{3} ,c_{4} ,c_{5} );(d_{1} ,d_{2} ,d_{3} ,d_{4} ,d_{5} );(e_{1} ,e_{2} ,e_{3} ,e_{4} ,e_{5} )) \rangle$ be any two SVPNNs.

Comparison between any two SVPNNs [159] $\iota_{1}$ and $\iota_{2}$ is presented as

• i. If $\Gamma (\iota_{1} ) > \Gamma (\iota_{2} )$, then $\iota_{1} > \iota_{2}$;

• ii. If $\Gamma (\iota_{1} ) < \Gamma (\iota_{2} )$, then $\iota_{1} < \iota_{2}$;

• iii. If $\Gamma (\iota_{1} ) = \Gamma (\iota_{2} )$, and

  • • if $H(\iota_{1} ) > H(\iota_{2} )$, then $\iota_{1} > \iota_{2}$;

  • • $H(\iota_{1} ) < H(\iota_{2} )$, then $\iota_{1} < \iota_{2}$;

  • • $H(\iota_{1} ) = H(\iota_{2} )$, then $\iota_{1} \approx \iota_{2}$.

6 Cylindrical Neutrosophic Single-Valued (CNSV) Set

Let $W$ be a space of objects with generic element ω in $W$. A CNSV set [8] $\chi$ in $W$ is presented as:  $\chi = \left\langle {\omega ,\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega ) > /,\,\omega \in W} \right\rangle$, where $\chi$ $\xi_{\chi } (\omega ),\,\psi_{\chi } (\omega ),\zeta_{\chi } (\omega )$ denote the truth MF, the indeterminacy MF, and the falsity MF, respectively.

Here, $\left( {\xi_{\chi } (\omega )} \right)^{2} + \,\left( {\psi_{\chi } (\omega )} \right)^{2} \le 1^{2} ,\,\zeta_{\chi } (\omega ) \le 1.$

For convenience, $\chi = \left( {\xi_{\chi } ,\,\psi_{\chi } ,\zeta_{\chi } } \right)$ is simply defined as a CNSV Number (CNSVN).

6.1 Score Function of CNSVN

For any CNSVN $\chi = \left( {\xi_{\chi } ,\,\psi_{\chi } ,\zeta_{\chi } } \right)$.

• Beneficiary degree of truth MF = $2\left( {\xi_{\chi } } \right)^{2}$

• Indeterminacy degree of indeterminacy MF = $2\left( \psi \right)^{2}$

• Non-beneficiary degree of falsity MF = $2\left( {\zeta_{\chi } } \right)^{2}$.

The score function and accuracy function are, respectively, denoted by $\Gamma (\chi )$ and $A(\chi )$ [186] and presented as:

• i. $\Gamma (\chi ) = \left( {2\left( {\xi_{\chi } } \right)^{2} - \left( {\psi_{\chi } } \right)^{2} - \left( {\zeta_{\chi } } \right)^{2} } \right)$, with $\Gamma (\chi ) = [ - 1,\,1]$ and the accuracy function $A(\chi )$ is defined as:

• ii. $A(\chi ) = \frac{{2\left( {\xi_{\chi } } \right)^{2} + \left( {\psi_{\chi } } \right)^{2} + \left( {\zeta_{\chi } } \right)^{2} }}{2},\,\,A(\chi ) \in [0,2]$

6.2 Comparıson of CNSVNs

Assume that $\iota_{1} = \left( {\xi_{{\tau_{1} }} ,\psi_{{_{{\tau_{1} }} }} ,\zeta_{{\tau_{1} }} } \right)$ and $\iota_{2} = \left( {\xi_{{\tau_{2} }} ,\psi_{{\tau_{2} }} ,\zeta_{{_{{\tau_{2} }} }} } \right)$ are any two CNSVNs. Then, comparison between $\iota_{1}$ and $\iota_{2}$ [186] is presented as:

• i. If $\Gamma (\iota_{1} ) > \Gamma (\iota_{2} )$, then $\iota_{1} > \iota_{2}$;

• ii. If $\Gamma (\iota_{1} ) < \Gamma (\iota_{2} )$, then $\iota_{1} < \iota_{2}$;

• iii. If $\Gamma (\iota_{1} ) = \Gamma (\iota )$, and

  • • if $A(\iota_{1} ) > A(\iota_{2} )$, then $\iota_{1} > \iota_{2}$;

  • • if $A(\iota_{1} ) < A(\iota_{2} )$, then $\iota_{1} < \iota_{2}$;

  • • if $A(\iota_{1} ) = A(\iota_{2} )$, then $\iota_{1} \approx \iota_{2}$.

7 Neutrosophic Number (NN)

Kandasamy and Smarandache [48, 54] introduced the NN of the structure $\eta = \alpha + \beta \iota$, where $\alpha ,\,\beta$ denote real or complex numbers, and “$\iota$” denotes the indeterminacy component of $\eta$.

An NN $\eta$ can be presented as $\eta = [\alpha + \beta \iota^{l} ,\,\alpha + \beta \iota^{u} ],\,$ $\eta \in$ N, N denotes the set having $\forall$ NNs and $\iota \in [\iota^{l} ,\iota^{u} ]$. The interval $\iota \in [\iota^{l} ,\iota^{u} ]$ is called an indeterminate interval.

• When $\beta$ = 0, $\eta$ reduces to crisp number $\eta = \alpha$

• When $\alpha = 0$, then $\eta$ reduces to the indeterminate number $\eta = \beta I$

• When $\iota^{l} = \iota^{u}$, then $\eta$ reduces to a crisp number.

Assume that $\eta_{1} = \alpha_{1} + \beta_{1} \iota$ and $\eta_{2} = \alpha_{2} + \beta_{2} \iota$ for $\eta_{1} ,\,\eta_{{_{2} }} \in$ N and $\iota \in [\iota^{l} ,\iota^{u} ]$ are two NNs. Some basic operational laws [187] for $\eta_{1}$ and $\eta_{2}$ are presented as:

• (1) $\iota^{2} = \iota$

• (2) $\iota .0 = 0$

• (3) $\frac{\iota }{\iota } =$ Undefined

• (4) $\eta_{1} + \eta_{2} = \alpha_{1} + \alpha_{2} + (\beta_{1} + \beta_{2} )\iota$ = $[\alpha_{1} + \alpha_{2} + (\beta_{1} + \beta_{2} )\iota^{l} ,\,\,\alpha_{1} + \alpha_{2} + (\beta_{1} + \beta_{2} )\iota^{u} ]$

• (5) $\eta_{1} - \eta_{2} = \alpha_{1} - \alpha_{2} + (\beta_{1} - \beta_{2} )\iota = [\alpha_{1} - \alpha_{2} + (\beta_{1} - \beta_{2} )\iota^{l} ,\alpha_{1} - \alpha_{2} + (\beta_{1} - \beta_{2} )\iota^{u} ]$

• (6) $\eta_{1} \times \eta_{2} = \alpha_{1} \alpha_{2} - \alpha_{2} + (\alpha_{1} \beta_{2} + \alpha_{2} \beta_{1} )\iota + \beta_{1} \beta_{2} \iota^{2} = \alpha_{1} \alpha_{2} - \alpha_{2} + (\alpha_{1} \beta_{2} + \alpha_{2} \beta_{1} + \beta_{1} \beta_{2} )\iota$

• (7) $\frac{{\alpha {}_{1}{ + }\beta {}_{1}\iota }}{{\alpha {}_{2}{ + }\beta {}_{2}\iota \, }}\,\, =$$\frac{{\alpha {}_{1}}}{{\alpha {}_{2}}} + \frac{{\alpha {}_{2}\beta {}_{1}\, - \,\alpha {}_{1}\beta {}_{2}}}{{\alpha {}_{2}(\alpha {}_{2}\, + \,\beta {}_{2})}}\iota \,;\,\,\alpha {}_{2}\, \ne \,0\,,\,\alpha {}_{{2}}\, \ne \, - \beta {}_{{2}}$

• (8) $\eta_{1}^{2} = \alpha_{1}^{2} \, + \,(2\alpha {}_{1}\beta {}_{1}\, + \beta_{1}^{2} )\iota$

• (9) $\lambda \eta_{1} = \,\,\lambda \alpha_{1} + \,\lambda \beta_{1} \iota$

Here, indeterminacy “$\iota$” and the imaginary i = √ − 1 are different concepts.

In general, $\iota^p$ = $\iota$ if p > 0, and it is undefined if p < 0.

7.1 Score Function of NNs

Assume that an NN is of the form: $\tau = c + \iota d = [c + d\iota^{l} ,\,c + d\iota^{u} ]$, where c and d are not simultaneously zeroes. The score function $\Xi (\chi )$ [54] and the accuracy function $A(\chi )$ [54] are, respectively, presented as

$$\Xi (\tau ) = \left| {\frac{{c + d(\iota^{u} - \iota^{l} )}}{{2\,\sqrt {c^{2} } + d^{2} }}} \right|,\Xi (\tau ) \in [0,\,1]$$

$$A(\tau ) = 1 - \exp ( - \left| {c + d(\iota^{l} - \iota^{u} )} \right|),$$

7.2 Comparison of NNs

Assume that $\eta_{1} = c_{1} + \iota d_{1}$ and $\eta_{2} = c_{2} + \iota d_{2}$ be any two NNs. Comparison between $\eta_{1}$ and $\eta_{2}$ [54] is presented as follows:

• i. If $\Xi (\eta_{1} ) > \Xi (\eta_{2} )$, then $\eta_{1} > \eta_{2}$;

• ii. $\Xi (\eta_{1} ) < \Xi (\eta_{2} )$, then $\eta_{1} < \eta_{2}$;

• iii. If $\Xi (\eta_{1} ) = \Xi (\eta_{2} )$, and

  • • $A(\eta_{1} ) > A(\eta_{2} )$, then $\eta_{1} > \eta_{2}$;

  • • $A(\eta_{1} ) < A(\eta_{2} )$, then $\eta_{1} < \eta_{2}$;

  • • $A(\eta_{1} ) = A(\eta_{2} )$, then $\eta_{1} \approx \eta_{2}$.

7.3 Neutrosophic Refined Number (NRN)

NRN [188] was defined as:$[p + q_{1} \iota_{1} + q_{2} \iota_{2} + ... + q_{m} \iota_{m} ]$, where $p,\,q_{1} ,\,q_{2} ,\,...,\,q_{m}$ denote a real or complex number, and $\iota_{1} ,\,\iota_{2} ,\,...,\iota_{m}$ denote sub-indeterminacies, for m ≥ 1.

8 Some Applications of Neutrosophic Sets

NSs have been applied in different fields. Few applications are depicted in Table 1.

Table 1 Some applications of NSs and neutrosophic logic

8.1 Sentiment Analysis

Smarandache et al. [189] studied the similarity measure by defining the words’ sentiment scores. In their study, Smarandache et al. [189] dealt with the sentiment characteristics of the words only. They developed a novel word-level similarity measure and found promising results.

8.2 Cosmology

Christianto and Smarandache [190] argued that neutrosophic logic resolved the dispute dealing with the “beginning and the eternity of the Universe”. In their study, Christianto and Smarandache [190] agreed that “the universe could have both a beginning and an eternal existence” leading to the paradox that “there might have been a time before time or a beginning of time in time”.

8.3 Neutrosophic Cognitive Map (NCM) for Social Problems

Using the NCM, Devadoss et al. [191] studied to evaluate the impact of playing violent video games among the teenagers (13–18 years) in Chennai. In their study, Devadoss et al. [191] considered nine concepts and presented the outcome of the study by comparing the results derived from Fuzzy Cognitive Map (FCM) and NCM.

8.4 Neutrosophic Strategy to Combat COVID-19

Yasser et al. [192] developed a novel health-fog framework in assisting diagnosis and treatment for COVID-19 patients efficiently based on neutrosophic classifier. The study [192] integrated the information scattered among different medical centres and health organizations to combat with COVID-19.

8.5 Social Network Analysis e-Learning Systems via NSs

Using NSs, Salama et al. [193] integrated social activities in the environment of e- learning and developed a social learning management system. Radwan [194] presented the current trends and challenges in e-learning processes in NS environment.

8.6 Raga Classification

Anitha and Gunavathi [195] presented the study of Carnatic raga by classifying all 72 melakarta ragas by utilizing neutrosophic logic and NCM.

8.7 Early Diagnosis of Leaf Ailments

Shadrach and Kandasamy [196] presented a new feature selection strategy in detecting leaf disease using a computer-based method to classify the leaf diseases. In their study, Shadrach and Kandasamy [196] compared the eight existing selection strategies with their developed strategy to demonstrate the capability of their strategy. Their developed strategy [196] obtained 99.8% classification accuracy in selecting 11 characteristics for leaf disease diagnosis.

8.8 Potential Energy Storage Options.

Pamucar et al. [197] developed the neutrosophic Multi-Criteria Decision-Making (MCDM) strategy to evaluate potential energy storage options and conducted a case study in Romania by identifying four criteria and thirteen sub-criteria.

8.9 Neutrosophic Computational Model

Albeanu [198] reviewed the principles of computing and presented some new neutrosophic computational models to identify requirements for software implementation.

8.10 Finance and Economics

Bencze [199] clearly stated that Sukanto Bhattacharya made a significant contribution in neutrosophic research by reflecting the applications of NS-based models dealing with financial economic and social science problems [27, 200,201,202,203,204].

8.11 Conflict Resolution

Bhattacharya et al. [205] presented the arguments in applying the principles of the neutrosophic game theory in order to reflect Israel–Palestine conflict with regard to the goals and strategies of each side. Bhattacharya et al. [205] extended the game theoretic explanation of Plessner [206] and developed the neutrosophic game theoretic model.

Pramanik and Roy [207] presented the arguments in applying the principles of the game theory in order to understand the Indo-Pak conflict over Jammu and Kashmir (J&K) with regard to the goals and strategies of either country. In their study, Pramanik and Roy [207] presented the 2 × 2 zero-sum game theoretic crisp model of Indo-Pak conflict over J&K by identifying the goals, strategies, and options of either country. Pramanik and Roy [208] also extended the game theoretic crisp model of Pramanik and Roy [207] to neutrosophic game theoretic model to obtain the optimal solution of the ongoing Indo-Pak conflict over J&K.

Deli [209] initiated to study neutrosophic game theory and presented matrix games with simplified neutrosophic payoffs. New research is very important in this area.

8.12 Air Surveillance

Fan et al. [39] proposed the neutrosophic Hough transform (NHT) strategy to deal with the complex surveillance issues. NS is used to characterize the different targets, namely real, false, and uncertain (indeterminate) targets in surveillance environments. They proposed a new neutrosophic Hough transform-based track initiation (NHT-TI) strategy that performs better than modified HT-TI strategy and improved HT-TI strategy.

Air surveillance involves various uncertain factors. In sensor network, uncertain factors may result from unknown target dynamic models, unknown environmental disturbances, the imprecise data processing, and limited performance of sensors [39]. Fan et al. [40] reviewed the NS-based multiple target tracking (MTT) strategies and presented the NS-based MTT strategies and that help in improving the performance.

9 The Extensions of Neutrosophic Sets

NS generalizes the classic set, FS [1] and IFS [7] (see Graphical abstract). NSs have been widely studied, and many extensions have been proposed in the literature. Different applications of NSs are presented in Table 1. The hybrid and extensions of NSs and the contributing authors are shown in Table 2. Currently, there are 82 neutrosophic-related sets derived from neutrosophics in the literature.

Table 2 The various hybrid and extensions of NSs

10 New Directions

The management of neutrosophic information in human controlled real-world issue appears to be a complex and difficult task. NSs facilitate in handling inconsistency and indeterminacy caused by limited knowledge of the domain experts. This chapter has presented various neutrosophic concepts and tools to deal with neutrosophic information.

To manage neutrosophic information in complex problems, different theoretical strategies [169, 171] have been presented in the current neutrosophic literature. Some weaknesses of the neutrosophic studies are highlighted.

• Since the introduction of SVNS [11], many extensions and versions of NSs have been proposed. But some of the extensions of NSs are debatable with respect to their usefulness. These extensions should be able to solve real problems with uncertainty, inconsistency, and indeterminacy. Theoretical or practical dimensions of the extensions must be justified.

• Some papers with the same title and almost same content [2,3,4,5,6, 9,10,11, 99, 100] have been published in different journals leading to the confusion, self-plagiarism and the waste of time to find differences between them.

• Too many NSs and decision-making strategies based on these sets have been proposed in the literature without presenting a commanding justification of their applications in real-life problems. It appears that a commanding justification of their necessity and applicability are necessary.

• Researchers utilize different notations for presenting concepts, tools, and extensions to deal with neutrosophic information.

• The increase in volume, uncertainty, inconsistency, falsity, indeterminacy, and incompleteness in data reflects currently a major challenge. The current scenario demands the scientific analysis and new development of neutrosophic frameworks that are capable of modelling, clustering, and data fusion problems in different fields of scientific study. Uncertainty, falsity and indeterminacy in prices are the key aspects in economic activities. So, new neutrosophic research should address the diverse topics such as stock trending analysis [276], performance of stock market [277], finance, economics and politics [27, 199,200,201,202,203,204], conflict resolution [208], air surveillance, and multiple target tracking strategy [37, 39].

• Neutrosophic game theoretic model [209] must be further studied.

• A new trend in research appears as the utilization of the neutrosophic theoretical models to realistic problems. Neutrosophic models should be a new and provide solutions to the problems, which cannot be solved by developed strategies in the literature.

• Since vague sets [278] are IFSs, and IFS is equivalent interval FS [279], the relations between neutrosophic FS [258], vague NS [259], and intuitionistic NS [267] should be deeply investigated.

• In upcoming 30 years, NS and its hybrid sets will highly be useful in artificial intelligence, automation, cybernetics, data analysis, engineering management science, mobile ad hoc network, neurosciences, operations research, interdisciplinary applications, multidisciplinary science, weather forecasting, etc.

11 Conclusion

Uncertainty, indeterminacy and inconsistency usually get involved in many human controlled complex real-world problems. NSs and their various extensions offer successful results in dealing with different neutrosophic decision-making problems. Much attentions have been given to some of them that manage neutrosophic situations, which often appear when indeterminacy must be dealt with. These new strategies have attracted the great attention of the investigators who deeply studied the diverse neutrosophic concepts, diverse hybrid extensions, similarity measures, and aggregation operators to deal with neutrosophic information.

The chapter  has presented some directions and considerations of future researches that should be considered in the coming NS-based proposals. It  has also recognizez that there exist many avenues of research in NSs and their extensions. It is to be noted that the results of this chapter offer a current overview of NSs and SVNSs. However, NSs will evolve and possibly develop in the future according to new ideas and topics that will dominate the current neutrosophic research arena.

Core Messages

• NS offers a natural foundation for dealing with the neutrosophic phenomena mathematically that exist widely in the real world.

• For realistic decision-making problems, the neutrosophic set is a very promising mathematical tool that can dominate the other mathematical tools for making pragmatic and rational decision-makıng in a complex and uncertain environment.

• NSs are capable of handling uncertainty, inconsistency, indeterminacy, and incompleteness and that is why they attract all branches of knowledge.

• NS is a promising mathematical tool for integrated science.