Interval-Valued Linear Diophantine Fuzzy Frank Aggregation Operators with Multi-Criteria Decision-Making

Abstract

We introduce the notion of the interval-valued linear Diophantine fuzzy set, which is a generalized fuzzy model for providing more accurate information, particularly in emergency decision-making, with the help of intervals of membership grades and non-membership grades, as well as reference parameters that provide freedom to the decision makers to analyze multiple objects and alternatives in the universe. The accuracy of interval-valued linear Diophantine fuzzy numbers is analyzed using Frank operations. We first extend the Frank t-conorm and t-norm (FT${}_c$T${}_n$) to interval-valued linear Diophantine fuzzy information and then offer new operations such as the Frank product, Frank sum, Frank exponentiation, and Frank scalar multiplication. Based on these operations, we develop novel interval-valued linear Diophantine fuzzy aggregation operators (AOs), including the “interval-valued linear Diophantine fuzzy Frank weighted averaging operator and the interval-valued linear Diophantine fuzzy Frank weighted geometric operator”. We also demonstrate various features of these AOs and examine the interactions between the proposed AOs. FT${}_c$T${}_n$s offer two significant advantages. Firstly, they function in the same way as algebraic, Einstein, and Hamacher t-conorms and t-norms. Secondly, they have an additional parameter that results in a more dynamic and reliable aggregation process, making them more effective than other general t-conorm and t-norm approaches. Furthermore, we use these operators to design a method for dealing with multi-criteria decision-making with IVLDFNs. Finally, a numerical case study of the novel carnivorous issue is shown as an application for emergency decision-making based on the proposed AOs. The purpose of this numerical example is to demonstrate the practicality and viability of the provided AOs.

1. Introduction

Multi-criteria decision-making (MCDM) is an important part of decision science in which the primary goal is to seek a best alternative from a universe of discourse that involves multiple alternatives. In MCDM, a decision maker or group of decision makers (DMs) analyze given alternatives under a single criterion or multiple criteria, and their judgments are expressed either as a crisp value or a linguistic value. However, in today’s complex real-world environment, vague and ambiguous information frequently appears in MCDM processes in various fields such as clustering, classification, decision-making, supplier selection, medical diagnosis, etc. The uncertainties play a prominent role in the MCDM process, and it is a difficult task for DMs to achieve accurate conclusions without coping with imprecise, unclear, or uncertain data. Zadeh [1] presented the perception of a fuzzy set (FS) as a generalization of a crisp set. The membership function is used to calculate the degrees of membership of elements in an FS. Zadeh’s FS theory has been utilized in several fields and several extensions of FS theory have been developed by researchers. The concept of an interval-valued fuzzy set (IVFS) was originally proposed by Zadeh [2] as an extension of FS. IVFSs treat the FS’s membership degree as an interval value rather than a single value. Atanassov [3] extended FS theory by introducing the idea of the intuitionistic fuzzy set (IFS), based on a membership degree (MSD) and a non-membership degree (NMSD) such that their sum is less than or equal to 1. There is only one MSD in the FS, which is insufficient for dealing with vague and uncertain information. By incorporating MSD as well as NMSD, the IFS has become one of the most effective and powerful tools for dealing with and handling vague and uncertain information. In decision support systems, an MSD is used to express a positive membership grade in the range $[0,1]$ to express the merits of the objects, and an NMSD gives a negative membership grade in the range [0,1] to express the demerits of the objects.

Atanassov [4] further proposed the concept of an interval-valued intuitionistic fuzzy set (IVIFS). Riaz and Hashmi [5] extensively tested the restrictions related to MSDs and NMSDs in the structures of the FS, IFS, Pythagorean fuzzy set (PFS) [6], and q-rung orthopair fuzzy set (q-ROFS) [7] in 2019, and these limitations were discussed quantitatively. To overcome these challenges, they developed the notion of a linear Diophantine fuzzy set (LDFS) by incorporating reference parameters that provide freedom to DMs in choosing the MSD and NMSD. The concept of LDFS eliminates the limitations of MSD and NMSD as well as increasing the space of ordered pairs of MSD and NMSD. The LDFS is applicable when the constraints of existing models fail to hold, so that this model is superior to FS, IFS, PFS, and q-ROFS.

Table 1. Comparison of LDFS with some existing fuzzy sets.

Concept	Remarks
Fuzzy set [1]	It considers MSDs but it does not consider NMSDs.
IFS [3]	It fails if $MSD+NMSD>1$ for some $\breve{\rho }$.
PFS [6]	It fails if ${\left(MSD\right)}^2+{\left(NMSD\right)}^2>1$ for some $\breve{\rho }$.
q-ROFS [7]	It fails for smaller values of “q” with ${\left(MSD\right)}^q+{\left(NMSD\right)}^q>1$, or if $MSD=NMSD=1$ for some $\breve{\rho }$.
LDFS [5]	(1) Deals with situations when IFS, PFS, and q-ROFS cannot be applied; (2) The reference parameters $\langle {\beta }_{\breve{\rho }},{\aleph }_{\breve{\rho }}\rangle$ are used as a weight vector such that their sum cannot exceed unity; (3) MDs and NMDs $\langle {\mu }_{\breve{\rho }},{\nu }_{\breve{\rho }}\rangle$ can be chosen freely from $[0,1]$; (4) The real value of the linear combination ${\beta }_{\breve{\rho }}{\mu }_{\breve{\rho }}+{\aleph }_{\breve{\rho }}{\nu }_{\breve{\rho }}$ always lies in $[0,1]$.

gives a comparison of LDFS with existing concepts.

Aggregation operators (AOs) are essential components of information fusion. Many AOs have been developed to aggregate various types of fuzzy numbers. Xu [8] initiated the idea of IVIF information aggregation and suggested two fundamental AOs: IVIF weighted average and geometric AOs. Following this, other types of extended AOs were developed. Yu [9] enhanced the existing AOs with a parameter and proposed the generalized geometric operator for aggregating IVIF data. Meng et al. [10] used the Banzhaf index to introduce certain IVIF geometric Choquet AOs. Zhao and Xu [11] investigated density-based operators in the IVIF context. Zhang [12] developed Frank AOs for aggregating IVIF data. Depending on some developed precise AOs, Zhou and He [13] investigated MCDM approaches in the IVIF context. Meng et al. [14] developed a hybrid Shapley averaging AO by combining the Shapley function with existing AOs for the IVIFS. Liu and Teng [15] proposed a number of AOs for normal IVIF data. Wu and Su [16] proposed prioritized hybrid weighted AOs for IVIFNs. Ashraf and Abdullah [17] initiated the idea of emergency decision support modeling for COVID-19 based on AOs for spherical fuzzy sets. Ashraf et al. [18] pioneered a more accurate production evaluation in gold mines by employing a revolutionary distance measure method using cubic picture fuzzy numbers. Ashraf and Abdullah [19] introduced spherical AOs and their applications in MCDM. Saha et al. [20] developed many hybrid hesitant fuzzy weighted AOs. Saha et al. [21] also developed an MCDM using q-ROF fairly weighted AOs. Farid and Riaz [22] also introduced Einstein interactive geometric AOs with applications in MCDM for q-ROFNs. Riaz and Tehrim [23] proposed a robust “VIseKriterijumska Optimizacija I Kompromisno Rasenje (VIKOR)” method for bipolar fuzzy sets using connection numbers. Riaz and Hashmi [24] proposed m-polar neutrosophic soft mapping with applications in MCDM in personality disorders. Munir et al. [25] proposed probability interactive geometric AOs for T-spherical fuzzy sets. Peng and Selvachandran [26] gave the state of the art and future directions for PFSs. Liu and Wang [27] presented some basic geometric and averaging AOs related to q-ROFNs. Garg [28] introduced the novel concept of confection-number-based q-ROFS and their applications in MCDM. Peng et al. [29] introduced a novel exponential operational law for q-ROFNs and their AOs. They also defined a new score function for the ranking of q-ROFNs. Jana et al. [30] developed the famous Dombi AOs for q-ROFNs and their applications in MCDM. Wei et al. [31] presented the idea of Heronian mean AOs for the aggregation of q-ROFNs. Liu and Liu [32] developed the Bonferroni mean AOs for q-ROFNs. Kamaci [33] proposed the idea of linguistic single-valued neutrosophic soft sets. Karaaslan and Ozlu [34] introduced some correlation coefficients for dual type-2 hesitant fuzzy sets.

Many researchers have contributed work on different extensions of fuzzy sets. Sitara et al. [35] introduced q-rung picture fuzzy graph structures, Chakraborty et al. [36] developed an intelligent decision model and Alosta et al. [37] solved the location selection problem by means of an integrated AHP-RAFSI technique. Akram et al. [38] also introduced Dombi AOs for decision-making under m-polar fuzzy information. Wang et al. [39] proposed Pythagorean fuzzy interactive Hamacher power AOs. Alcantud proposed the novel idea of softarisons [40] and the characterization of soft topologies by crisp topologies [41]. Many excellent studies related to AOs can be seen in [42,43,44,45].

Iampan et al. [46] introduced linear Diophantine fuzzy Einstein AOs and their medical applications. Riaz et al. [47] developed prioritized AOs for linear Diophantine fuzzy numbers (LDFNs), with application to third-party logistic provider selection. Farid et al. [48] proposed Einstein prioritized linear Diophantine fuzzy AOs with applications. However, all the existing approaches listed above are based on the t-conorm and t-norm (T${}_c$T${}_n$), which lack flexibility in the information fusion process. FT${}_c$T${}_n$ represents the only class of T${}_c$T${}_n$s that fulfils the compatibility requirement [49]. FT${}_c$T${}_n$s have two primary benefits. Firstly, they have the same benefits as the algebraic, Einstein, and Hamacher T${}_c$T${}_n$s. Secondly, they have an extra parameter that leads to a more flexible and robust aggregation process, making them more powerful than other general T${}_c$T${}_n$s. As a result, it is worthwhile to examine AOs based on the FT${}_c$T${}_n$. In this study, we developed Frank AOs for interval-valued linear Diophantine fuzzy numbers (IVLDFNs).

The rest of the paper is organized as follows. In Section 2, we give the introductory notions and principal properties regarding the fuzzy set and its extended models. In Section 3, we present a framework for linear Diophantine fuzzy set theory in which the grades of membership, non-membership, and reference parameters are interval values. Section 4 presents several interval-valued linear Diophantine fuzzy Frank AOs. Section 5 presents a novel technique for solving MCDM problems based on the proposed AOs. Section 6 provides a discussion of an emergency induced by the nCOVID-19 pandemic in Pakistan, along with numerical illustrations based on the proposed AOs. Section 7 provides sensitivity analysis and in Section 8 comparison analysis is given. The study’s conclusions are summarized in Section 9.

2. Certain Fundamental Concepts

This section reviews some basic concepts and results related to the fuzzy set and its extended models.

We start this section with Zadeh’s definition of a fuzzy set.

Definition 1

([1]). A fuzzy set (FS) $\mathcal{F}$ in the universal discourse set $\mathfrak{X}$ is defined as follows:

$$\mathcal{F}=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\mathcal{F}}\left({\breve{\Phi }}_i\right)\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$$

where ${\mathcal{M}}_{\mathcal{F}}:\mathfrak{X}\to [0,1]$ represents the membership function of $\mathcal{F}$ and the value ${\mathcal{M}}_{\mathcal{F}}\left({\breve{\xi }}_i\right)$ denotes the degree of membership of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\mathcal{F}$. The set of all FSs over the universal discourse set $\mathfrak{X}$ is denoted by $FS\left(\mathfrak{X}\right)$.

Definition 2

([2]). Let ${\mathfrak{I}}^{[0,1]}$ be the set of all closed subintervals of the interval $[0,1]$. Then, an IVFS $\ddot{\mathcal{F}}$ in the universal discourse set $\mathfrak{X}$ can be described as follows:

$$\ddot{\mathcal{F}}=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\ddot{\mathcal{F}}}\left({\breve{\xi }}_i\right)\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$$

where ${\mathcal{M}}_{\ddot{\mathcal{F}}}:\mathfrak{X}\to {\mathfrak{I}}^{[0,1]}$ represents the membership function of $\ddot{\mathcal{F}}$ such that ${\mathcal{M}}_{\ddot{\mathcal{F}}}\left({\breve{\xi }}_i\right)$ = $[{\mathcal{M}}_{\ddot{\mathcal{F}}}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\ddot{\mathcal{F}}}^U\left({\breve{\xi }}_i\right)]$ and $0\le {\mathcal{M}}_{\ddot{\mathcal{F}}}^L\left({\breve{\xi }}_i\right)\le {\mathcal{M}}_{\ddot{\mathcal{F}}}^U\left({\breve{\xi }}_i\right)\le 1$. The values ${\mathcal{M}}_{\ddot{\mathcal{F}}}^L\left({\breve{\xi }}_i\right)$ and ${\mathcal{M}}_{\ddot{\mathcal{F}}}^U\left({\breve{\xi }}_i\right)$ denote the lower and upper degrees of membership of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\ddot{\mathcal{F}}$, respectively. The set of all IVFSs over the universal discourse set $\mathfrak{X}$ is denoted by $IVFS\left(\mathfrak{X}\right)$.

Definition 3

([3]). A IFS $\mathcal{I}$ in the universal discourse set $\mathfrak{X}$ is defined as follows:

$$\mathcal{I}=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\mathcal{I}}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$$

where ${\mathcal{M}}_{\mathcal{I}}:\mathfrak{X}\to [0,1]$ and ${\mathcal{N}}_{\mathcal{I}}:\mathfrak{X}\to [0,1]$ represent the membership function and non-membership function of $\mathcal{I}$, respectively. The values ${\mathcal{M}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)$ and ${\mathcal{N}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)$ denote the MSD and NMSD of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\mathcal{I}$ with the condition $0\le {\mathcal{M}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)+{\mathcal{N}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)\le 1$ for each ${\breve{\xi }}_i\in \mathfrak{X}$. The hesitation margin, which is the degree of non-determinacy of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\mathcal{I}$, is described as ${\mathfrak{H}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)=1-({\mathcal{M}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)+{\mathcal{N}}_{\mathcal{I}}\left({\breve{\xi }}_i\right))$ for each ${\breve{\xi }}_i\in \mathfrak{X}$. The set of all IFSs over the universal discourse set $\mathfrak{X}$ is denoted by $IFS\left(\mathfrak{X}\right)$.

Definition 4

([4]). A IVIFS $\ddot{\mathcal{I}}$ in the universal discourse set $\mathfrak{X}$ is described as follows:

$$\ddot{\mathcal{I}}=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\ddot{\mathcal{I}}}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\ddot{\mathcal{I}}}\left({\breve{\xi }}_i\right)\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$$

where ${\mathcal{M}}_{\ddot{\mathcal{I}}}=[{\mathcal{M}}_{\mathcal{I}}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\mathcal{I}}^U\left({\breve{\xi }}_i\right)]:\mathfrak{X}\to {\mathfrak{I}}^{[0,1]}$ and ${\mathcal{N}}_{\ddot{\mathcal{I}}}=[{\mathcal{N}}_{\mathcal{I}}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\mathcal{I}}^U\left({\breve{\xi }}_i\right)]:\mathfrak{X}\to {\mathfrak{I}}^{[0,1]}$ represent the membership function and non-membership function of $\ddot{\mathcal{I}}$, respectively. The values ${\mathcal{M}}_{\mathcal{I}}^L\left({\breve{\xi }}_i\right)$, ${\mathcal{M}}_{\mathcal{I}}^U\left({\breve{\xi }}_i\right)$ denote the lower and upper MSDs of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\ddot{\mathcal{I}}$, and the values ${\mathcal{N}}_{\mathcal{I}}^L\left({\breve{\xi }}_i\right)$, ${\mathcal{N}}_{\mathcal{I}}^U\left({\breve{\xi }}_i\right)$ denote the lower and upper NMSDs of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\ddot{\mathcal{I}}$. In addition, the following condition is provided: $0\le {\mathcal{M}}_{\mathcal{I}}^U\left({\breve{\xi }}_i\right)+{\mathcal{N}}_{\mathcal{I}}^U\left({\breve{\xi }}_i\right)\le 1$ for each ${\breve{\xi }}_i\in \mathfrak{X}$. The hesitation margin is evaluated as ${\mathfrak{H}}_{\ddot{\mathcal{I}}}\left({\breve{\xi }}_i\right)=[1-({\mathcal{M}}_{\ddot{\mathcal{I}}}^L\left({\breve{\xi }}_i\right)+{\mathcal{N}}_{\ddot{\mathcal{I}}}^L\left({\breve{\xi }}_i\right)),1-({\mathcal{M}}_{\ddot{\mathcal{I}}}^U\left({\breve{\xi }}_i\right)+{\mathcal{N}}_{\ddot{\mathcal{I}}}^U\left({\breve{\xi }}_i\right))]$ for each ${\breve{\xi }}_i\in \mathfrak{X}$. The set of all IVIFSs over the universal discourse set $\mathfrak{X}$ is denoted by $IVIFS\left(\mathfrak{X}\right)$.

IFSs, PFSs, and q-ROFSs each have their own MSD and NMSD limitations. To circumvent these limitations, a linear Diophantine fuzzy set (LDFS) with reference parameters related to the nature of the IFS was used. In practice, the LDFS abolishes the limitations of current data sets and allows for the free selection of data using the reference parameters’ arbitrary property. LDFS has a larger space than FS, IFS, PFS, or q-ROFS for the selection of MSD and NMSD.

Definition 5

([5]). An LDFS $\tilde{\partial }$ in the universal discourse set $\mathfrak{X}$ is defined as follows:

$$\tilde{\partial }=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right)\rangle ,\langle {\aleph }_{\tilde{\partial }},{\breve{\wp }}_{\tilde{\partial }}\rangle ):r_k\in \Re \}$$

where ${\mathcal{M}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right),{\aleph }_{\tilde{\partial }},{\breve{\wp }}_{\tilde{\partial }}\in [0,1]$, respectively, represent the MSD, NMSD, and references parameters of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\tilde{\partial }$ with the conditions $0\le {\aleph }_{\tilde{\partial }}+{\breve{\wp }}_{\tilde{\partial }}\le 1$ and $0\le {\aleph }_{\tilde{\partial }}{\mathcal{M}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right)+{\breve{\wp }}_{\tilde{\partial }}{\mathcal{N}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right)\le 1$. The hesitation margin for each ${\breve{\xi }}_i\in \mathfrak{X}$ is ${\theta }_{\tilde{\partial }}{\mathfrak{H}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right)=1-({\aleph }_{\tilde{\partial }}{\mathcal{M}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right)+{\breve{\wp }}_{\tilde{\partial }}{\mathcal{N}}_{\tilde{\partial }}\left({\breve{\xi }}_i\right))$. The set of all LDFSs over the universal discourse set $\mathfrak{X}$ is denoted by $LDFS\left(\mathfrak{X}\right)$.

We believe that assigning an exact number to an expert’s opinion is too restrictive, and it is more realistic to assign an interval of values. However, the LDFS is quite an extensive theory of FS, IFS, PFS, and q-ROFS.

3. Interval-Valued Linear Diophantine Fuzzy Sets

In this section, we present a framework for linear Diophantine fuzzy set theory in which the values of MSD, NMSD, and the reference parameters are intervals.

3.1. The Construction of Interval-Valued Linear Diophantine Fuzzy Set

Definition 6.

Let ${\mathfrak{I}}^{[0,1]}$ be the set of all closed subintervals of the interval $[0,1]$. Then, an interval-valued linear Diophantine fuzzy set (IVLDFS) $\breve{⋏}$ in the universal discourse set $\mathfrak{X}$ is described as follows:

$$\breve{⋏}=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\breve{⋏}}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\breve{⋏}}\left({\breve{\xi }}_i\right)\rangle ,\langle {\aleph }_{\breve{⋏}},{\breve{\wp }}_{\breve{⋏}}\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$$

where ${\mathcal{M}}_{\breve{⋏}}\left({\breve{\xi }}_i\right)=[{\mathcal{M}}_{\breve{⋏}}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\breve{⋏}}^U\left({\breve{\xi }}_i\right)],{\mathcal{N}}_{\breve{⋏}}\left({\breve{\xi }}_i\right)=[{\mathcal{N}}_{\breve{⋏}}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\breve{⋏}}^U\left({\breve{\xi }}_i\right)],{\aleph }_{\breve{⋏}}=[{\aleph }_{\breve{⋏}}^L,{\aleph }_{\breve{⋏}}^U],{\breve{\wp }}_{\breve{⋏}}=[{\breve{\wp }}_{\breve{⋏}}^L,{\breve{\wp }}_{\breve{⋏}}^U]\in {\mathfrak{I}}^{[0,1]}$, respectively, represent the lower and upper MSD, NMSD, and reference parameters of ${\breve{\xi }}_i\in \mathfrak{X}$ into the set $\breve{⋏}$ with the conditions $0\le {\aleph }_{\breve{⋏}}^U+{\breve{\wp }}_{\breve{⋏}}^U\le 1$ and $0\le {\aleph }_{\breve{⋏}}^U{\mathcal{M}}_{\breve{⋏}}^U\left({\breve{\xi }}_i\right)+{\breve{\wp }}_{\breve{⋏}}^U{\mathcal{N}}_{\breve{⋏}}^U\left({\breve{\xi }}_i\right)\le 1$. The hesitation margin of ${\breve{\xi }}_i\in \mathfrak{X}$ into $\breve{⋏}$ is described as ${\theta }_{\breve{⋏}}{\mathfrak{H}}_{\breve{⋏}}\left({\breve{\xi }}_i\right)=[1-({\aleph }_{\breve{⋏}}^L{\mathcal{M}}_{\breve{⋏}}^L\left({\breve{\xi }}_i\right)+{\breve{\wp }}_{\breve{⋏}}^L{\mathcal{N}}_{\breve{⋏}}^L\left({\breve{\xi }}_i\right)),1-({\aleph }_{\breve{⋏}}^U{\mathcal{M}}_{\breve{⋏}}^U\left({\breve{\xi }}_i\right)+{\breve{\wp }}_{\breve{⋏}}^U{\mathcal{N}}_{\breve{⋏}}^U\left({\breve{\xi }}_i\right))]$. Any element of IVLDFS is said to be an interval-valued linear Diophantine fuzzy number (IVLDFN) and is briefly denoted by ${\overline{⨿}}^{\gimel }=(\langle [{\mu }^{\gimel },{\nu }^{\gimel }],[{\alpha }^{\gimel },{\beta }^{\gimel }]\rangle ,\langle [{\gamma }^{\gimel },{\delta }^{\gimel }],[{\psi }^{\gimel },{\eta }^{\gimel }]\rangle )$. The set of all IVLDFSs over the universal discourse set $\mathfrak{X}$ is denoted by $IVLDFS\left(\mathfrak{X}\right)$.

Example 1.

Suppose that $X=\{x_1,x_2,x_3,x_4\}$ is the set of four mobile phones that one is considering purchasing. One might want to determine the best mobile phone having eligible functionality-quality based on the price level. Then, “cheap” and “not cheap or expensive” can be considered as reference parameters. For these reference parameters, the following IVLDFS can be created.

$$\begin{array}{c}\hfill \breve{⋏}=\left\{\begin{array}{c}(x_1,\langle [0.7,0.8],[0.3,1]\rangle ,\langle [0.2,0.3],[0.5,0.6]\rangle ),(x_2,\langle [0.4,0.7],[0.5,0.5]\rangle ,\hfill \\ \langle [0.2,0.5],[0.3,0.5]\rangle ),(x_3,\langle [0.8,0.9],[0,0.9]\rangle ,\langle [0.4,0.55],[0.1,0.3]\rangle ),\hfill \\ (x_4,\langle [0.25,0.4],[0.7,0.9]\rangle ,\langle [0,0.9],[0.1,0.1]\rangle )\hfill \end{array}\right\}.\end{array}$$

The IVLDF data is shown in

Table 2. Functionality-quality-assessed mobile phones in Example 1.

$\breve{⋏}$	$(\langle {\mathcal{M}}_{\breve{⋏}}\left({\breve{\mathit{\xi }}}_{\mathit{i}}\right),{\mathcal{N}}_{\breve{⋏}}\left({\breve{\mathit{\xi }}}_{\mathit{i}}\right)\rangle ,\langle {\mathit{\aleph }}_{\breve{⋏}},{\breve{\mathit{\wp }}}_{\breve{⋏}}\rangle )$
$x_1$	$(\langle [0.7,0.8],[0.3,1]\rangle ,\langle [0.2,0.3],[0.5,0.6]\rangle )$
$x_2$	$(\langle [0.4,0.7],[0.5,0.5]\rangle ,\langle [0.2,0.5],[0.3,0.5]\rangle )$
$x_3$	$(\langle [0.8,0.9],[0,0.9]\rangle ,\langle [0.4,0.55],[0.1,0.3]\rangle )$
$x_4$	$(\langle [0.25,0.4],[0.7,0.9]\rangle ,\langle [0,0.9],[0.1,0.1]\rangle )$

.

For the $x_1$ IVLDFS in the preceding example, $x_1$ has functionality-quality within the interval [0.7, 0.8] but not within the interval [0.3, 0.1], $x_1$ is cheap in price within the range [0.2, 0.3], and $x_1$ is not cheap or expensive in price within the interval [0.2, 0.3].

The optimal screen resolution and battery for a mobile phone may vary depending on the screen size. In other words, to watch a video in sufficient quality, the ideal pixel size for a mobile phone with a screen size of 6 inches is 1440 × 2560, while 1080 × 1920 is a good option for a mobile phone with a screen size of 5 inches. However, for a mobile phone with a screen size of 6 inches, if the pixel size is 1080 × 1920 this may not be satisfactory. One might want to determine the best mobile phone having eligible functionality-quality based on screen size. Then, the reference parameters “ideal screen size” and “not ideal screen size” can be considered. According to these reference parameters, the IVLDF data can be take the form shown in

Table 3. IVLDFS for screen size.

$\breve{⋏}$	$(\langle {\mathcal{M}}_{\breve{⋏}}\left({\breve{\mathit{\xi }}}_{\mathit{i}}\right),{\mathcal{N}}_{\breve{⋏}}\left({\breve{\mathit{\xi }}}_{\mathit{i}}\right)\rangle ,\langle {\mathit{\aleph }}_{\breve{⋏}},{\breve{\mathit{\wp }}}_{\breve{⋏}}\rangle )$
$x_1$	$(\langle [0.4,0.5],[0.7,0.8]\rangle ,\langle [0.4,0.45],[0.1,0.3]\rangle )$
$x_2$	$(\langle [0.7,0.9],[0.2,0.8]\rangle ,\langle [0.5,0.6],[0.3,0.3]\rangle )$
$x_3$	$(\langle [0.3,0.7],[0.4,0.7]\rangle ,\langle [0.2,0.25],[0.6,0.7]\rangle )$
$x_4$	$(\langle [0.3,1],[0,0.6]\rangle ,\langle [0.35,0.55],[0.3,0.4]\rangle )$

.

These new notations will contribute to a better understanding of the proposed concepts and operations for IVLDFS.

Definition 7.

An IVLDFS in the universal discourse set $\mathfrak{X}$ of the form ${\breve{⋏}}^1=\left\{\right({\breve{\xi }}_i,$$\langle [1,1],[0,0]\rangle ,$$\langle [1,1],[0,0]\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ is termed an absolute IVLDFS and ${\breve{⋏}}^0=\left\{\right({\breve{\xi }}_i,\langle [0,0],[1,1]\rangle ,$$\langle [0,0],[1,1]\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ is termed a null or empty IVLDFS.

Definition 8.

Let ${\breve{⋏}}_1,{\breve{⋏}}_2\in IVLDFS\left(\mathfrak{X}\right)$. Then,

(a)${\breve{⋏}}_1$ is a subset of ${\breve{⋏}}_2$, symbolized by ${\breve{⋏}}_1\subseteq {\breve{⋏}}_2$, if for each ${\breve{\xi }}_i\in \mathfrak{X}$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{M}}_{{\breve{⋏}}_1}^L\left({\breve{\xi }}_i\right)\le {\mathcal{M}}_{{\breve{⋏}}_2}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{{\breve{⋏}}_1}^U\left({\breve{\xi }}_i\right)\le {\mathcal{M}}_{{\breve{⋏}}_2}^U\left({\breve{\xi }}_i\right),\hfill \\ {\mathcal{N}}_{{\breve{⋏}}_1}^L\left({\breve{\xi }}_i\right)\ge {\mathcal{N}}_{{\breve{⋏}}_2}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{{\breve{⋏}}_1}^U\left({\breve{\xi }}_i\right)\ge {\mathcal{N}}_{{\breve{⋏}}_2}^U\left({\breve{\xi }}_i\right),\hfill \\ {}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_1}^L\le {}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_2}^L,{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_1}^U\le {}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_2}^U,\hfill \\ {}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_1}^L\ge {}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_2}^L,{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_1}^U\ge {}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_2}^U\hfill \end{array}\right\}\end{array}$$

(b)${\breve{⋏}}_1$ and ${\breve{⋏}}_2$ are equal, symbolized by ${\breve{⋏}}_1={\breve{⋏}}_2$, if for each ${\breve{\xi }}_i\in \mathfrak{X}$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{M}}_{{\breve{⋏}}_1}^L\left({\breve{\xi }}_i\right)={\mathcal{M}}_{{\breve{⋏}}_2}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{{\breve{⋏}}_1}^U\left({\breve{\xi }}_i\right)={\mathcal{M}}_{{\breve{⋏}}_2}^U\left({\breve{\xi }}_i\right),\hfill \\ {\mathcal{N}}_{{\breve{⋏}}_1}^L\left({\breve{\xi }}_i\right)={\mathcal{N}}_{{\breve{⋏}}_2}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{{\breve{⋏}}_1}^U\left({\breve{\xi }}_i\right)={\mathcal{N}}_{{\breve{⋏}}_2}^U\left({\breve{\xi }}_i\right),\hfill \\ {}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_1}^L={}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_2}^L,{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_1}^U={}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_2}^U,\hfill \\ {}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_1}^L={}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_2}^L,{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_1}^U={}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_2}^U\hfill \end{array}\right\}\end{array}$$

From Definition 8 (a) and (b), we can say that ${\breve{⋏}}_1={\breve{⋏}}_2$, if and only if ${\breve{⋏}}_1\subseteq {\breve{⋏}}_2$ and ${\breve{⋏}}_2\subseteq {\breve{⋏}}_1$.

3.2. Some Operations on Interval-Valued Linear Diophantine Fuzzy Sets

Definition 9.

Let ${\breve{⋏}}_k$=$\left\{\right({\breve{\xi }}_i,\langle [{\mathcal{M}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)],[{\mathcal{N}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)]\rangle ,\langle {[}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^U],$${[}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^U]\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ for $k=1,2,\cdots ,s$. Then, we derive the following operations:

(a)(Complement of IVLDFS)

${\breve{⋏}}_k^c=\left\{\right({\breve{\xi }}_i,\langle [{\mathcal{N}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)],[{\mathcal{M}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)]\rangle ,\langle {[}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^U],$

${[}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^L,{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^U]\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$.

(b)(Union of IVLDFSs)

${\displaystyle \bigcup _{k=1}^s}{\breve{⋏}}_k=\left\{\right\{({\breve{\xi }}_i,\langle [{\mathcal{M}}_{\tilde{\cup }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\tilde{\cup }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)],[{\mathcal{N}}_{\tilde{\cup }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\cup }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)]\rangle ,\langle {[}^{\left(i\right)}{\aleph }_{\tilde{\cup }{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\aleph }_{\tilde{\cup }{\breve{⋏}}_k}^U],$

${[}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cup }{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cup }{\breve{⋏}}_k}^U]\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{M}}_{\tilde{\cup }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{max}{\mathcal{M}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\tilde{\cup }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{max}{\mathcal{M}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right),\hfill \\ {\mathcal{N}}_{\tilde{\cup }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{min}{\mathcal{N}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\cup }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{min}{\mathcal{N}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right),\hfill \\ {}^{\left(i\right)}{\aleph }_{\tilde{\cup }{\breve{⋏}}_k}^L=\underset{k\in \{1,2,\dots ,n\}}{max}{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^L,{}^{\left(i\right)}{\aleph }_{\tilde{\cup }{\breve{⋏}}_k}^U=\underset{k\in \{1,2,\dots ,n\}}{max}{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^U,\hfill \\ {}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cup }{\breve{⋏}}_k}^L=\underset{k\in \{1,2,\dots ,n\}}{min}{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^L,{}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cup }{\breve{⋏}}_k}^U=\underset{k\in \{1,2,\dots ,n\}}{min}{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^U,\hfill \end{array}\right\}\end{array}$$

for each $x_i\in \mathfrak{X}$.

(c)(Intersection of IVLDFSs)

${\displaystyle \bigcap _{k=1}^s}{\breve{⋏}}_k=\left\{\right\{({\breve{\xi }}_i,\langle [{\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)],[{\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)]\rangle ,\langle {[}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U],$

${[}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^L{,}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U]\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{min}{\mathcal{M}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{min}{\mathcal{M}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right),\hfill \\ {\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{max}{\mathcal{N}}_{{\breve{⋏}}_k}^L\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)=\underset{k\in \{1,2,\dots ,n\}}{max}{\mathcal{N}}_{{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right),\hfill \\ {}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^L=\underset{k\in \{1,2,\dots ,n\}}{min}{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^L,{}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U=\underset{k\in \{1,2,\dots ,n\}}{min}{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^U,\hfill \\ {}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^L=\underset{k\in \{1,2,\dots ,n\}}{max}{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^L,{}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U=\underset{k\in \{1,2,\dots ,n\}}{max}{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_k}^U,\hfill \end{array}\right\}\end{array}$$

for each $x_i\in \mathfrak{X}$.

Example 2.

Let us consider the IVLDFSs in the universal discourse set $\mathfrak{X}=\{x_1,x_2\}$ as follows:

$$\begin{array}{c}\hfill {\breve{⋏}}_k=\left\{\begin{array}{c}(x_1,\langle [\frac{1}{2+k},\frac{1}{1+k}],[\frac{1}{4+k},\frac{1}{k}]\rangle ,\langle [\frac{1}{3+k},\frac{1}{3+k}],[\frac{1}{4+k},\frac{1}{3+k}]\rangle ),\hfill \\ (x_2,\langle [\frac{1}{3+k},\frac{1}{2+k}],[\frac{1}{3+k},\frac{1}{1+k}]\rangle ,\langle [\frac{1}{3+k},\frac{1}{2+k}],[\frac{1}{4+k},\frac{1}{3+k}]\rangle )\hfill \end{array}\right\}\end{array}$$

for $k=1,2,\cdots ,100$. Then, we obtain

i.The complement of the IVLDFS ${\breve{⋏}}_k$ for each $k\in \{1,2,\cdots ,100\}$ is

$$\begin{array}{c}\hfill {\breve{⋏}}_k^c=\left\{\begin{array}{c}(x_1,\langle [\frac{1}{4+k},\frac{1}{k}],[\frac{1}{2+k},\frac{1}{1+k}]\rangle ,\langle [\frac{1}{4+k},\frac{1}{3+k}],[\frac{1}{3+k},\frac{1}{3+k}]\rangle ),\hfill \\ (x_2,\langle [\frac{1}{3+k},\frac{1}{1+k}],[\frac{1}{3+k},\frac{1}{2+k}]\rangle ,\langle [\frac{1}{4+k},\frac{1}{3+k}],[\frac{1}{3+k},\frac{1}{2+k}]\rangle )\hfill \end{array}\right\}.\end{array}$$

ii.The union of IVLDFSs ${\breve{⋏}}_k$ for all $k\in \{1,2,\cdots ,100\}$ is

$$\begin{array}{c}\hfill {\displaystyle \bigcup _{k=1}^{100}}{\breve{⋏}}_k=\left\{\begin{array}{c}(x_1,\langle [\frac{1}{3},\frac{1}{2}],[\frac{1}{104},\frac{1}{100}]\rangle ,\langle [\frac{1}{4},\frac{1}{4}],[\frac{1}{104},\frac{1}{103}]\rangle ),\hfill \\ (x_2,\langle [\frac{1}{4},\frac{1}{3}],[\frac{1}{103},\frac{1}{101}]\rangle ,\langle [\frac{1}{4},\frac{1}{3}],[\frac{1}{104},\frac{1}{103}]\rangle )\hfill \end{array}\right\}.\end{array}$$

iii.The intersection of IVLDFSs ${\breve{⋏}}_k$ for all $k\in \{1,2,\cdots ,100\}$ is

$$\begin{array}{c}\hfill {\displaystyle \bigcap _{k=1}^{100}}{\breve{⋏}}_k=\left\{\begin{array}{c}(x_1,\langle [\frac{1}{102},\frac{1}{101}],[\frac{1}{5},1]\rangle ,\langle [\frac{1}{103},\frac{1}{103}],[\frac{1}{5},\frac{1}{4}]\rangle ),\hfill \\ (x_2,\langle [\frac{1}{103},\frac{1}{102}],[\frac{1}{4},\frac{1}{2}]\rangle ,\langle [\frac{1}{103},\frac{1}{102}],[\frac{1}{5},\frac{1}{4}]\rangle )\hfill \end{array}\right\}.\end{array}$$

Proposition 1.

Let ${\breve{⋏}}_k\in IVLDFS\left(\mathfrak{X}\right)$$(k=1,2,\cdots ,s)$, then ${\breve{⋏}}_k^c$, ${\displaystyle \bigcup _{k=1}^s}{\breve{⋏}}_k$ and ${\displaystyle \bigcap _{k=1}^s}{\breve{⋏}}_k$ are also IVLDFSs in the universal discourse set $\mathfrak{X}$.

Proof. 

Let ${\breve{⋏}}_k\in IVLDFS\left(\mathfrak{X}\right)$$(k=1,2,\cdots ,s)$. Let us prove that ${\displaystyle \bigcap _{k=1}^s}{\mathfrak{L}}_j$ is also an IVLDFS. The others can be similarly proved. To verify that ${\displaystyle \bigcup _{k=1}^s}{\breve{⋏}}_k$ is an IVLDFS, it should be demonstrated that the following conditions are valid: $0\le {}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U+{}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U\le 1$ and $0\le {}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U{\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)+{}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U{\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)\le 1$ for each ${\breve{\xi }}_i\in \mathfrak{X}$. Suppose that ${}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U={}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_{k^{☆}}}^U=\xi$ for any ${\breve{\xi }}_i\in \mathfrak{X}$. By Definition 6, we write $0\le {}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_{k^{☆}}}^U+{}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_{k^{☆}}}^U\le 1$ and so ${}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_{k^{☆}}}^U\le 1-\xi$. Since ${}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U=\underset{k\in \{1,2,\cdots ,s\}}{min}{}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_k}^U$, from Definition 9 (c), we have ${}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U\le {}^{\left(i\right)}{\aleph }_{{\breve{⋏}}_{k^{☆}}}^U$ and so ${}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U\le 1-\xi$. We compute that $0\le (1-\xi ){\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)+\xi {\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)\le 1$ since ${\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)\in [0,1]$ for each $x_i\in \mathfrak{X}$. Since ${}^{\left(i\right)}{\breve{\wp }}_{{\breve{⋏}}_{k^{☆}}}^U\in [0,1]$ (i.e., $\xi \in [0,1]$), by Definition 6 we obtain $0\le {}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U+{}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U\le 1$ and $0\le {}^{\left(i\right)}{\aleph }_{\tilde{\cap }{\breve{⋏}}_k}^U{\mathcal{M}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)+{}^{\left(i\right)}{\breve{\wp }}_{\tilde{\cap }{\breve{⋏}}_k}^U{\mathcal{N}}_{\tilde{\cap }{\breve{⋏}}_k}^U\left({\breve{\xi }}_i\right)\le 1$ for each ${\breve{\xi }}_i\in \mathfrak{X}$. Hence, the proof is completed. □

Proposition 2.

Let ${\breve{⋏}}_1,{\breve{⋏}}_2,{\breve{⋏}}_3\in IVLDFS\left(\mathfrak{X}\right)$. Then, the following commutative, associative, distributive, and De Morgan rules are valid for $\ast ,\diamond \in \{\cup ,\cap \}$:

(i)${\breve{⋏}}_1\ast {\breve{⋏}}_2={\breve{⋏}}_2\ast {\breve{⋏}}_1$ (commutative);

(ii)${\breve{⋏}}_1\ast ({\breve{⋏}}_2\ast {\breve{⋏}}_3)=({\breve{⋏}}_1\ast {\breve{⋏}}_2)\ast {\breve{⋏}}_3$ (associative);

(iii)${\breve{⋏}}_1\ast ({\breve{⋏}}_2\diamond {\breve{⋏}}_3)=({\breve{⋏}}_1\ast {\breve{⋏}}_2)\diamond ({\breve{⋏}}_1\ast {\breve{⋏}}_3)$ (distributive);

(iv)${({\breve{⋏}}_1\ast {\breve{⋏}}_2)}^c={\breve{⋏}}_1^c\diamond {\breve{⋏}}_2^c$. (De Morgan laws).

Proof. 

They can be demonstrated by using the concepts of the complement, union, and intersection in Definition 9, and hence the proof is omitted. □

3.3. Operational Laws of Interval-Valued Linear Diophantine Fuzzy Numbers

Definition 10.

Some operational laws such as sum, product, scalar multiplication, and scalar power for two IVLDFNs

$${{\overline{⨿}}^{\gimel }}_1=(\langle {[}^1{\mu }^{\gimel },{}^1{\nu }^{\gimel }],{[}^1{\alpha }^{\gimel },{}^1{\beta }^{\gimel }]\rangle ,\langle {[}^1{\gamma }^{\gimel },{}^1{\delta }^{\gimel }],{[}^1{\psi }^{\gimel },{}^1{\eta }^{\gimel }]\rangle )$$

and

$${{\overline{⨿}}^{\gimel }}_2=(\langle {[}^2{\mu }^{\gimel },{}^2{\nu }^{\gimel }],{[}^2{\alpha }^{\gimel },{}^2{\beta }^{\gimel }]\rangle ,\langle {[}^2{\gamma }^{\gimel },{}^2{\delta }^{\gimel }],{[}^2{\psi }^{\gimel },{}^2{\eta }^{\gimel }]\rangle )$$

are described below:

(a)The sum of two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ is

$$\begin{array}{c}\hfill {{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2=\left(\begin{array}{c}〈\left[{(}^1{\mu }^{\gimel }+{}^2{\mu }^{\gimel }-{}^1{\mu }^{\gimel }{}^2{\mu }^{\gimel }),{(}^1{\nu }^{\gimel }+{}^2{\nu }^{\gimel }-{}^1{\nu }^{\gimel }{}^2{\nu }^{\gimel })\right],\left[{(}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel }),{(}^1{\beta }^{\gimel }{}^2{\beta }^{\gimel })\right]〉,\hfill \\ 〈\left[{(}^1{\gamma }^{\gimel }+{}^2{\gamma }^{\gimel }-{}^1{\gamma }^{\gimel }{}^2{\gamma }^{\gimel }),{(}^1{\delta }^{\gimel }+{}^2{\delta }^{\gimel }-{}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel })\right],\left[{(}^1{\psi }^{\gimel }{}^2{\psi }^{\gimel }),{(}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel })\right]〉\hfill \end{array}\right).\end{array}$$

(b)The product of two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ is

$$\begin{array}{c}\hfill {{\overline{⨿}}^{\gimel }}_1\otimes {{\overline{⨿}}^{\gimel }}_2=\left(\begin{array}{c}〈\left[{(}^1{\mu }^{\gimel }{}^2{\mu }^{\gimel }),{(}^1{\nu }^{\gimel }{}^2{\nu }^{\gimel })\right],\left[{(}^1{\alpha }^{\gimel }+{}^2{\alpha }^{\gimel }-{}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel }),{(}^1{\alpha }^{\gimel }+{}^2{\alpha }^{\gimel }-{}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel })\right]〉,\hfill \\ 〈\left[{(}^1{\gamma }^{\gimel }{}^2{\gamma }^{\gimel }),{(}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel })\right],\left[{(}^1{\psi }^{\gimel }+{}^2{\psi }^{\gimel }-{}^1{\psi }^{\gimel }{}^2{\psi }^{\gimel }),{(}^1{\eta }^{\gimel }+{}^2{\eta }^{\gimel }-{}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel })\right]〉\hfill \end{array}\right).\end{array}$$

(c)The scalar multiplication of IVLDFN ${{\overline{⨿}}^{\gimel }}_1$ is

$$\begin{array}{c}\hfill {\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_1=\left(\begin{array}{c}〈\left[1-{(1-{}^1{\mu }^{\gimel })}^{{\beta }^{\gimel }},1-{(1-{}^1{\nu }^{\gimel })}^{{\beta }^{\gimel }}\right],\left[{{(}^1{\alpha }^{\gimel })}^{{\beta }^{\gimel }},{{(}^1{\beta }^{\gimel })}^{{\beta }^{\gimel }}\right]〉,\hfill \\ 〈\left[1-{(1-{}^1{\gamma }^{\gimel })}^{{\beta }^{\gimel }},1-{(1-{}^1{\delta }^{\gimel })}^{{\beta }^{\gimel }}\right],\left[{{(}^1{\eta }^{\gimel })}^{{\beta }^{\gimel }},{{(}^1{\eta }^{\gimel })}^{{\beta }^{\gimel }}\right]〉\hfill \end{array}\right),\end{array}$$

where ${\beta }^{\gimel }$ is a positive real number.

(d)The scalar power of IVLDFN ${{\overline{⨿}}^{\gimel }}_1$ is

$$\begin{array}{c}\hfill {{\overline{⨿}}^{\gimel }}_1^{{\beta }^{\gimel }}=\left(\begin{array}{c}〈\left[{{(}^1{\mu }^{\gimel })}^{{\beta }^{\gimel }},{{(}^1{\nu }^{\gimel })}^{{\beta }^{\gimel }}\right],\left[1-{(1-{}^1{\alpha }^{\gimel })}^{{\beta }^{\gimel }},1-{(1-{}^1{\beta }^{\gimel })}^{{\beta }^{\gimel }}\right]〉,\hfill \\ 〈\left[{{(}^1{\gamma }^{\gimel })}^{{\beta }^{\gimel }},{{(}^1{\delta }^{\gimel })}^{{\beta }^{\gimel }}\right],\left[1-{(1-{}^1{\psi }^{\gimel })}^{{\beta }^{\gimel }},1-{(1-{}^1{\eta }^{\gimel })}^{{\beta }^{\gimel }}\right]〉\hfill \end{array}\right),\end{array}$$

where ${\beta }^{\gimel }$ is a positive real number.

Example 3.

Let ${{\overline{⨿}}^{\gimel }}_1=\left(\langle [0.4,0.8],[0,0.7]\rangle ,\langle [0.1,0.6],[0.2,0.3]\rangle \right)$ and

${{\overline{⨿}}^{\gimel }}_2=\left(\langle [0.5,1],[0.1,0.6]\rangle ,\langle [0.3,0.3],[0.2,0.4]\rangle \right)$ be two IVLDFSs, with ${\beta }^{\gimel }=2$. Then, we have:

i.${{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2=\left(\langle [0.7,1],[0,0.42]\rangle ,\langle [0.37,0.72],[0.04,0.12]\rangle \right)$;

ii.${{\overline{⨿}}^{\gimel }}_1\otimes {{\overline{⨿}}^{\gimel }}_2=\left(\langle (0.2,0.8),(0.1,0.88)\rangle ,\langle (0.03,0.18),(0.36,0.58)\rangle \right)$;

iii.$4{{\overline{⨿}}^{\gimel }}_1=\left(\langle [0.8704,0.9984],[0,2401]\rangle ,\langle [0.3439,0.9744],[0.0016,0.0081]\rangle \right)$;

iv.${{\overline{⨿}}^{\gimel }}_2^4=\left(\langle [0.0625,0.1],[0.3439,0.9744]\rangle ,\langle [0.0081,0.0081],[0.5904,0.8704]\rangle \right)$.

Proposition 3.

Let ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ be two IVLDFNs, with ${\beta }^{\gimel }>0$, then ${{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2$, ${{\overline{⨿}}^{\gimel }}_1\otimes {{\overline{⨿}}^{\gimel }}_2$, ${\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_1^{{\beta }^{\gimel }}$ are also IVLDFNs.

Proof. 

Let ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ be two IVLDFNs. Then, by Definition 6, we have $0\le {}^j{\delta }^{\gimel }+{}^j{\eta }^{\gimel }\le 1$ and $0\le {}^j{\delta }^{\gimel }{}^j{\nu }^{\gimel }+{}^j{\eta }^{\gimel }{}^j{\beta }^{\gimel }\le 1$ for $j=1,2$. Assume that ${{\overline{⨿}}^{\gimel }}_3={{\overline{⨿}}^{\gimel }}_1\otimes {{\overline{⨿}}^{\gimel }}_2$, where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^3{\mu }^{\gimel }={}^1{\mu }^{\gimel }{}^2{\mu }^{\gimel },{}^3{\nu }^{\gimel }={}^1{\nu }^{\gimel }{}^2{\nu }^{\gimel }{,}^3{\alpha }^{\gimel }={}^1{\alpha }^{\gimel }{+}^2{\alpha }^{\gimel }{-}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel },{}^3{\beta }^{\gimel }={}^1{\beta }^{\gimel }{+}^2{\beta }^{\gimel }{-}^1{\mathcal{N}}_{{\breve{⋏}}_1}^U{}^2{\beta }^{\gimel },\hfill \\ {}^3{\gamma }^{\gimel }={}^1{\gamma }^{\gimel }{}^2{\gamma }^{\gimel },{}^3{\delta }^{\gimel }={}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel },{}^3{\psi }^{\gimel }={}^1{\psi }^{\gimel }+{}^2{\psi }^{\gimel }-{}^1{\psi }^{\gimel }{}^2{\psi }^{\gimel },{}^3{\eta }^{\gimel }={}^1{\eta }^{\gimel }+{}^2{\eta }^{\gimel }-{}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel }\hfill \end{array}\right\}\end{array}$$

It is obvious that ${\gamma }^{\gimel }+{\delta }^{\gimel }\ge 0$, ${\gamma }^{\gimel }{\delta }^{\gimel }\ge 0$ and ${\gamma }^{\gimel }+{\delta }^{\gimel }\ge {\gamma }^{\gimel }{\delta }^{\gimel }$ for ${\gamma }^{\gimel },{\delta }^{\gimel }\in [0,1]$, and so we have ${}^3{\mu }^{\gimel }$, ${}^3{\nu }^{\gimel }$, ${}^3{\alpha }^{\gimel }$, ${}^3{\beta }^{\gimel }$, ${}^3{\gamma }^{\gimel }$, ${}^3{\delta }^{\gimel }$, ${}^3{\psi }^{\gimel }$, ${}^3{\psi }^{\gimel }$$\ge 0$. In addition, if ${{\gamma }^{\gimel }}^L\le {{\gamma }^{\gimel }}^U$ and ${{\delta }^{\gimel }}^L\le {{\delta }^{\gimel }}^U$ for ${{\gamma }^{\gimel }}^L,{{\gamma }^{\gimel }}^U,{{\delta }^{\gimel }}^L,{{\delta }^{\gimel }}^U\in [0,1]$, then ${{\gamma }^{\gimel }}^L{{\delta }^{\gimel }}^L\le {{\gamma }^{\gimel }}^U{{\delta }^{\gimel }}^U$ and ${{\gamma }^{\gimel }}^L+{{\delta }^{\gimel }}^L-{{\gamma }^{\gimel }}^L{{\delta }^{\gimel }}^L\le {{\gamma }^{\gimel }}^U+{{\delta }^{\gimel }}^U-{{\gamma }^{\gimel }}^U{{\delta }^{\gimel }}^U$. Since $0\le {}^j{\delta }^{\gimel }+{}^j{\eta }^{\gimel }\le 1$$(j=1,2)$, we can write $0\le {}^j{\eta }^{\gimel }\le 1-{}^j{\delta }^{\gimel }$ for $j=1,2$. Then, we obtain

$$\begin{array}{ccc}\hfill {}^3{\delta }^{\gimel }+{}^3{\eta }^{\gimel }& =& {}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel }+{}^1{\eta }^{\gimel }+{}^2{\eta }^{\gimel }-{}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel }\hfill \\ & \le & {}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel }+1-{}^1{\delta }^{\gimel }+1-{}^2{\delta }^{\gimel }-(1-{}^1{\delta }^{\gimel })(1-{}^2{\delta }^{\gimel })\hfill \\ & =& {}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel }+1-{}^1{\delta }^{\gimel }+1-{}^2{\delta }^{\gimel }-1+{}^1{\delta }^{\gimel }+{}^2{\delta }^{\gimel }-{}^1{\delta }^{\gimel }{}^1{\delta }^{\gimel }\hfill \\ & =& 1\hfill \end{array}$$

On the other hand, we obtain ${}^3{\delta }^{\gimel }+{}^3{\eta }^{\gimel }\ge 0$, since ${}^1{\eta }^{\gimel }+{}^2{\eta }^{\gimel }\ge {}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel }$ for ${}^1{\eta }^{\gimel },{}^2{\eta }^{\gimel }\in [0,1]$. Thus, we obtain that $0\le {}^3{\delta }^{\gimel }{}^3{\nu }^{\gimel }+{}^3{\eta }^{\gimel }{}^3{\beta }^{\gimel }\le 1$ since $0\le {}^3{\delta }^{\gimel }+{}^3{\eta }^{\gimel }\le 1$ and ${}^3{\nu }^{\gimel },{}^3{\beta }^{\gimel }\in [0,1]$. Hence, we have the result that ${{\overline{⨿}}^{\gimel }}_1\otimes {{\overline{⨿}}^{\gimel }}_2$ is an IVLDFN. Likewise, it is proved that ${{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2$, ${\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_1$, and ${{\overline{⨿}}^{\gimel }}_1^{{\beta }^{\gimel }}$ are also IVLDFNs. □

Proposition 4.

Let ${{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2$, and ${{\overline{⨿}}^{\gimel }}_3$ be the IVLDFNs and ${\beta }^{\gimel },{{\beta }^{\gimel }}_1,{{\beta }^{\gimel }}_2>0$. Then, we have:

(i)${{\overline{⨿}}^{\gimel }}_1\circ {{\overline{⨿}}^{\gimel }}_2={{\overline{⨿}}^{\gimel }}_2\circ {{\overline{⨿}}^{\gimel }}_1$ for $\circ \in \{\oplus ,\otimes \}$;

(ii)$({{\overline{⨿}}^{\gimel }}_1\circ {{\overline{⨿}}^{\gimel }}_2)\circ {{\overline{⨿}}^{\gimel }}_3={{\overline{⨿}}^{\gimel }}_1\circ ({{\overline{⨿}}^{\gimel }}_2\circ {{\overline{⨿}}^{\gimel }}_3)$ for $\circ \in \{\oplus ,\otimes \}$;

(iii)${\beta }^{\gimel }({{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2)={\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_1\oplus {\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_2$;

(iv)${({{\overline{⨿}}^{\gimel }}_1\otimes {{\overline{⨿}}^{\gimel }}_2)}^{{\beta }^{\gimel }}={{\overline{⨿}}^{\gimel }}_1^{{\beta }^{\gimel }}\otimes {{\overline{⨿}}^{\gimel }}_2^{{\beta }^{\gimel }}$;

(v)$({{\beta }^{\gimel }}_1+{{\beta }^{\gimel }}_2){{\overline{⨿}}^{\gimel }}_1={{\beta }^{\gimel }}_1{{\overline{⨿}}^{\gimel }}_1\oplus {{\beta }^{\gimel }}_2{{\overline{⨿}}^{\gimel }}_1$;

(vi)${{\overline{⨿}}^{\gimel }}_1^{{{\beta }^{\gimel }}_1+{{\beta }^{\gimel }}_2}={{\overline{⨿}}^{\gimel }}_1^{{{\beta }^{\gimel }}_1}\otimes {{\overline{⨿}}^{\gimel }}_1^{{{\beta }^{\gimel }}_2}$.

Proof. 

Let us prove part (i) for ⊗ and part (iii). The others can be demonstrated similarly.

(i) Let ${\ell }_1$ and ${\ell }_2$ be two CLDFNs. Then, we have

$$\begin{array}{ccc}{{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2\hfill & =& \left(\begin{array}{c}〈\left[{(}^1{\mu }^{\gimel }+{}^2{\mu }^{\gimel }-{}^1{\mu }^{\gimel }{}^2{\mu }^{\gimel }),{(}^1{\nu }^{\gimel }+{}^2{\nu }^{\gimel }-{}^1{\nu }^{\gimel }{}^2{\nu }^{\gimel })\right],\left[{(}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel }),{(}^1{\beta }^{\gimel }{}^2{\beta }^{\gimel })\right]〉,\hfill \\ 〈\left[{(}^1{\gamma }^{\gimel }+{}^2{\gamma }^{\gimel }-{}^1{\gamma }^{\gimel }{}^2{\gamma }^{\gimel }),{(}^1{\delta }^{\gimel }+{}^2{\delta }^{\gimel }-{}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel })\right],\left[{(}^1{\psi }^{\gimel }{}^2{\psi }^{\gimel }),{(}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel })\right]〉\hfill \end{array}\right)\hfill \\ & =& \left(\begin{array}{c}〈\left[{(}^2{\mu }^{\gimel }+{}^1{\mu }^{\gimel }-{}^2{\mu }^{\gimel }{}^1{\mu }^{\gimel }),{(}^2{\nu }^{\gimel }+{}^1{\nu }^{\gimel }-{}^2{\nu }^{\gimel }{}^1{\nu }^{\gimel })\right],\left[{(}^2{\alpha }^{\gimel }{}^1{\alpha }^{\gimel }),{(}^2{\beta }^{\gimel }{}^1{\beta }^{\gimel })\right]〉,\hfill \\ 〈\left[{(}^2{\gamma }^{\gimel }+{}^1{\gamma }^{\gimel }-{}^2{\gamma }^{\gimel }{}^1{\gamma }^{\gimel }),{(}^2{\delta }^{\gimel }+{}^1{\delta }^{\gimel }-{}^2{\delta }^{\gimel }{}^1{\delta }^{\gimel })\right],\left[{(}^2{\psi }^{\gimel }{}^1{\psi }^{\gimel }),{(}^2{\eta }^{\gimel }{}^1{\eta }^{\gimel })\right]〉\hfill \end{array}\right)\hfill \\ & =& {{\overline{⨿}}^{\gimel }}_2\oplus {{\overline{⨿}}^{\gimel }}_1.\hfill \end{array}$$

(iii) Let ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ be two IVLDFNs and ${\beta }^{\gimel }>0$, then

$$\begin{array}{c}\hfill {\beta }^{\gimel }({{\overline{⨿}}^{\gimel }}_1\oplus {{\overline{⨿}}^{\gimel }}_2)={\beta }^{\gimel }\left(\begin{array}{c}〈\left[{(}^1{\mu }^{\gimel }+{}^2{\mu }^{\gimel }-{}^1{\mu }^{\gimel }{}^2{\mu }^{\gimel }),{(}^1{\nu }^{\gimel }+{}^2{\nu }^{\gimel }-{}^1{\nu }^{\gimel }{}^2{\nu }^{\gimel })\right],\left[{(}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel }),{(}^1{\beta }^{\gimel }{}^2{\beta }^{\gimel })\right]〉,\hfill \\ 〈\left[{(}^1{\gamma }^{\gimel }+{}^2{\gamma }^{\gimel }-{}^1{\gamma }^{\gimel }{}^2{\gamma }^{\gimel }),{(}^1{\delta }^{\gimel }+{}^2{\delta }^{\gimel }-{}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel })\right],\left[{(}^1{\psi }^{\gimel }{}^2{\psi }^{\gimel }),{(}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel })\right]〉\hfill \end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill =\left(\begin{array}{c}〈\left[1-{\left(1-{(}^1{\mu }^{\gimel }+{}^2{\mu }^{\gimel }-{}^1{\mu }^{\gimel }{}^2{\mu }^{\gimel })\right)}^{{\beta }^{\gimel }},1-{\left(1-{(}^1{\nu }^{\gimel }+{}^2{\nu }^{\gimel }-{}^1{\nu }^{\gimel }{}^2{\nu }^{\gimel })\right)}^{{\beta }^{\gimel }}\right],\left[{{(}^1{\alpha }^{\gimel }{}^2{\alpha }^{\gimel })}^{{\beta }^{\gimel }},{{(}^1{\beta }^{\gimel }{}^2{\beta }^{\gimel })}^{{\beta }^{\gimel }}\right]〉,\hfill \\ 〈\left[1-{\left(1-{(}^1{\gamma }^{\gimel }+{}^2{\gamma }^{\gimel }-{}^1{\gamma }^{\gimel }{}^2{\gamma }^{\gimel })\right)}^{{\beta }^{\gimel }},1-{\left(1-{(}^1{\delta }^{\gimel }+{}^2{\delta }^{\gimel }-{}^1{\delta }^{\gimel }{}^2{\delta }^{\gimel })\right)}^{{\beta }^{\gimel }}\right],\left[{{(}^1{\psi }^{\gimel }{}^2{\psi }^{\gimel })}^{{\beta }^{\gimel }},{{(}^1{\eta }^{\gimel }{}^2{\eta }^{\gimel })}^{{\beta }^{\gimel }}\right]〉\hfill \end{array}\right)\\ \hfill \end{array}$$

$$\begin{array}{c}\hfill =\left(\begin{array}{c}〈\begin{array}{c}\left[\begin{array}{c}\left(1-{(1-{}^1{\mu }^{\gimel })}^{{\beta }^{\gimel }}+1-{(1-{}^2{\mu }^{\gimel })}^{{\beta }^{\gimel }}-(1-{(1-{}^1{\mu }^{\gimel })}^{{\beta }^{\gimel }})(1-{(1-{}^2{\mu }^{\gimel })}^{{\beta }^{\gimel }})\right),\hfill \\ \left(1-{(1-{}^1{\nu }^{\gimel })}^{{\beta }^{\gimel }}+1-{(1-{}^2{\nu }^{\gimel })}^{{\beta }^{\gimel }}-(1-{(1-{}^1{\nu }^{\gimel })}^{{\beta }^{\gimel }})(1-{(1-{}^2{\nu }^{\gimel })}^{{\beta }^{\gimel }})\right)\hfill \end{array}\right],\hfill \\ \left[\begin{array}{c}\left({{(}^1{\alpha }^{\gimel })}^{{\beta }^{\gimel }}{{(}^2{\alpha }^{\gimel })}^{{\beta }^{\gimel }}\right),\left({{(}^1{\beta }^{\gimel })}^{{\beta }^{\gimel }}{{(}^2{\beta }^{\gimel })}^{{\beta }^{\gimel }}\right)\hfill \end{array}\right]\hfill \end{array}〉,\hfill \\ 〈\begin{array}{c}\left[\begin{array}{c}\left(1-{(1-{}^1{\gamma }^{\gimel })}^{{\beta }^{\gimel }}+1-{(1-{}^2{\gamma }^{\gimel })}^{{\beta }^{\gimel }}-(1-{(1-{}^1{\gamma }^{\gimel })}^{{\beta }^{\gimel }})(1-{(1-{}^2{\gamma }^{\gimel })}^{{\beta }^{\gimel }})\right),\hfill \\ \left(1-{(1-{}^1{\delta }^{\gimel })}^{{\beta }^{\gimel }}+1-{(1-{}^2{\delta }^{\gimel })}^{{\beta }^{\gimel }}-(1-{(1-{}^1{\delta }^{\gimel })}^{{\beta }^{\gimel }})(1-{(1-{}^2{\delta }^{\gimel })}^{{\beta }^{\gimel }})\right)\hfill \end{array}\right],\hfill \\ \left[\begin{array}{c}\left({{(}^1{\psi }^{\gimel })}^{{\beta }^{\gimel }}{{(}^2{\psi }^{\gimel })}^{{\beta }^{\gimel }}\right),\left({{(}^1{\eta }^{\gimel })}^{{\beta }^{\gimel }}{{(}^2{\eta }^{\gimel })}^{{\beta }^{\gimel }}\right)\hfill \end{array}\right]\hfill \end{array}〉\hfill \end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill ={\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_1\oplus {\beta }^{\gimel }{{\overline{⨿}}^{\gimel }}_2.\end{array}$$

□

To compare the two IVLDFNs, we define the concepts of the score function (SF) and accuracy function (AF) for an IVLDFN as follows.

Definition 11.

Let ${\overline{⨿}}^{\gimel }=(\langle [{\mu }^{\gimel },{\nu }^{\gimel }],[{\alpha }^{\gimel },{\beta }^{\gimel }]\rangle ,\langle [{\gamma }^{\gimel },{\delta }^{\gimel }],[{\psi }^{\gimel },{\eta }^{\gimel }]\rangle )$ be an IVLDFN.

(i)The n-SF for ${\overline{⨿}}^{\gimel }$ is denoted and defined by

$$\begin{array}{cc}\hfill S^n\left({\overline{⨿}}^{\gimel }\right)=& \frac{1}{4}\left(({\left({\mu }^{\gimel }\right)}^n-{\left({\alpha }^{\gimel }\right)}^n)+({\left({\nu }^{\gimel }\right)}^n-{\left({\beta }^{\gimel }\right)}^n)+({\left({\gamma }^{\gimel }\right)}^n-{\left({\psi }^{\gimel }\right)}^n)+({\left({\delta }^{\gimel }\right)}^n-{\left({\eta }^{\gimel }\right)}^n)\right),\\ & S^n\left({\overline{⨿}}^{\gimel }\right)\in [-1,1]\end{array}$$

where n is a positive integer.

(ii)The n-AF for ${\overline{⨿}}^{\gimel }$ is denoted and defined by

$$\begin{array}{cc}\hfill A^n\left({\overline{⨿}}^{\gimel }\right)=& \frac{1}{4}\left((\frac{{\left({\mu }^{\gimel }\right)}^n+{\left({\alpha }^{\gimel }\right)}^n)}{2})+(\frac{{\left({\nu }^{\gimel }\right)}^n+{\left({\beta }^{\gimel }\right)}^n}{2})+({\left({\gamma }^{\gimel }\right)}^n+{\left({\psi }^{\gimel }\right)}^n)+({\left({\delta }^{\gimel }\right)}^n+{\left({\eta }^{\gimel }\right)}^n)\right),\\ & A^n\left({\overline{⨿}}^{\gimel }\right)\in [-1,1]\end{array}$$

where n is a positive integer.

Note 2. Some special cases of n-SF and n-AF for an IVLDFN are stated as follows:

• If $n=1$, then the n-SF and n-AF are termed SF and AF for an IVLDFN, respectively. In addition, they are symbolized as $S\left({\overline{⨿}}^{\gimel }\right)$ and $A\left({\overline{⨿}}^{\gimel }\right)$ instead of $S^n\left({\overline{⨿}}^{\gimel }\right)$ and $A^n\left({\overline{⨿}}^{\gimel }\right)$, respectively.
• If $n=2$, then the n-SF and n-AF are termed the quadratic SF and quadratic AF for an IVLDFN, respectively.
• If $n=3$, then the n-SF and n-AF are termed the cubic SF and cubic AF for an IVLDFN, respectively.
• If $n=4$, then the n-SF and n-AF are termed the quartic SF and quartic AF for an IVLDFN, respectively.

In this paper, especially, we assume that $n=1$.

To rank two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$, the following comparison strategy is proposed:

1. If $S^n\left({{\overline{⨿}}^{\gimel }}_1\right)>S^n\left({{\overline{⨿}}^{\gimel }}_2\right)$, then ${{\overline{⨿}}^{\gimel }}_1>{{\overline{⨿}}^{\gimel }}_2$.

2. If $S^n\left({{\overline{⨿}}^{\gimel }}_1\right)<S^n\left({{\overline{⨿}}^{\gimel }}_2\right)$, then ${{\overline{⨿}}^{\gimel }}_1<{{\overline{⨿}}^{\gimel }}_2$.

3. If $S^n\left({{\overline{⨿}}^{\gimel }}_1\right)=S^n\left({{\overline{⨿}}^{\gimel }}_2\right)$, then

  i. if $A^n\left({{\overline{⨿}}^{\gimel }}_1\right)>A^n\left({{\overline{⨿}}^{\gimel }}_2\right)$ then ${{\overline{⨿}}^{\gimel }}_1>{{\overline{⨿}}^{\gimel }}_2$,

  ii. if $A^n\left({{\overline{⨿}}^{\gimel }}_1\right)<A^n\left({{\overline{⨿}}^{\gimel }}_2\right)$ then ${{\overline{⨿}}^{\gimel }}_1<{{\overline{⨿}}^{\gimel }}_2$,

  iii. if $A^n\left({{\overline{⨿}}^{\gimel }}_1\right)=A^n\left({{\overline{⨿}}^{\gimel }}_2\right)$ then ${{\overline{⨿}}^{\gimel }}_1={{\overline{⨿}}^{\gimel }}_2$.

Example 4.

We consider the IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ in Example 3. We have ${{\overline{⨿}}^{\gimel }}_1\prec {{\overline{⨿}}^{\gimel }}_2$, since $S\left({{\overline{⨿}}^{\gimel }}_1\right)=0.175<0.2=S\left({{\overline{⨿}}^{\gimel }}_2\right)$ for $n=1$. If $n=5$, then we calculate the 5-SFs as $S^5\left({{\overline{⨿}}^{\gimel }}_1\right)=0.0612$ and $S^5\left({{\overline{⨿}}^{\gimel }}_1\right)=0.2369$.

3.4. Frank T${}_c$T${}_n$

Theoretical set operators have played an important part in FS theory since its inception. Following the presentation of Zadeh’s min-max operators, many more AOs have been introduced in the FS literature [1]. All categories of specific AOs were incorporated with the concepts of T${}_c$T${}_n$, which meet the constraints of the disjunction and conjunction operators, respectively [50]. T${}_c$T${}_n$ are the most extensive binary function categories that translate the unit square to the unit interval, and they are related by the De Morgan duality, i.e., the t-conorm T${}_c$ can be defined as $T_c(x,y)=1-T_n(1-x,1-y)$$\forall x,y\in [0,1]$. Based on T${}_c$T${}_n$, Deschrijver and Kerre [51] presented the idea of the generalized intersection (GI) and generalized union (GU) of IFSs.

Definition 12

([51]). Let $\mathcal{I}=\{({\breve{\xi }}_i,\langle {\mathcal{M}}_{\mathcal{I}}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\mathcal{I}}\left({\breve{\xi }}_i\right)\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ and $\mathcal{Q}=\left\{\right({\breve{\xi }}_i,$$\langle {\mathcal{M}}_{\mathcal{Q}}\left({\breve{\xi }}_i\right),{\mathcal{N}}_{\mathcal{Q}}\left({\breve{\xi }}_i\right)\rangle ):{\breve{\xi }}_i\in \mathfrak{X}\}$ be two IFSs, then GI and GU between $\mathcal{I}$ and $\mathcal{Q}$ is defined as follows,

$$\mathcal{I}{\cap }_{T_n,T_c}\mathcal{Q}=\left\{〈\breve{\xi },T_n\left({\mathcal{M}}_{\mathcal{I}}\left(\breve{\xi }\right),{\mathcal{M}}_{\mathcal{Q}}\left(\breve{\xi }\right)\right),T_c\left({\mathcal{N}}_{\mathcal{I}}\left(\breve{\xi }\right),{\mathcal{N}}_{\mathcal{Q}}\left(\breve{\xi }\right)\right)〉\mid \breve{\xi }\in \mathfrak{X}\right\}$$

(1)

$$\mathcal{I}{\cup }_{T_n,T_c}\mathcal{Q}=\left\{〈\breve{\xi },T_n\left({\mathcal{M}}_{\mathcal{I}}\left(\breve{\xi }\right),{\mathcal{M}}_{\mathcal{Q}}\left(\breve{\xi }\right)\right),T_c\left({\mathcal{N}}_{\mathcal{I}}\left(\breve{\xi }\right),{\mathcal{N}}_{\mathcal{Q}}\left(\breve{\xi }\right)\right)〉\mid \breve{\xi }\in \mathfrak{X}\right\}$$

(2)

where any dual T${}_n$ and T${}_c$ pair can be used.

Definition 13

([12]). Let ${\beth }_i=\left(\left[a_i,b_i\right],\left[c_i,d_i\right]\right)(i=1,2)$ be any two IVIFNs, then the GI and GU between ${\beth }_1$ and ${\beth }_2$ are defined as follows:

$${\beth }_1{\otimes }_{T_n,T_c}{\beth }_2=\left(\left[T_n\left(a_1,a_2\right),T_n\left(b_1,b_2\right)\right],\left[T_c\left(c_1,c_2\right),T_c\left(d_1,d_2\right)\right]\right)$$

(3)

$${\beth }_1{\oplus }_{T_n,T_c}{\beth }_2=\left(\left[T_c\left(a_1,a_2\right),T_c\left(b_1,b_2\right)\right],\left[T_n\left(c_1,c_2\right),T_n\left(d_1,d_2\right)\right]\right)$$

(4)

where any dual T${}_n$ and T${}_c$ pair can be used.

Now, we extend Definition 13 to IVLDF contexts and define a GI and a GU of IVLDFNs on the basis of a T${}_n$ and T${}_c$.

Definition 14.

Let ${{\overline{⨿}}^{\gimel }}_1=(\langle {[}^1{\mu }^{\gimel },{}^1{\nu }^{\gimel }],{[}^1{\alpha }^{\gimel },{}^1{\beta }^{\gimel }]\rangle ,\langle {[}^1{\gamma }^{\gimel },{}^1{\delta }^{\gimel }],{[}^1{\psi }^{\gimel },{}^1{\eta }^{\gimel }]\rangle )$ and ${{\overline{⨿}}^{\gimel }}_2$ = $(\langle {[}^2{\mu }^{\gimel },{}^2{\nu }^{\gimel }],{[}^2{\alpha }^{\gimel },{}^2{\beta }^{\gimel }]\rangle ,\langle {[}^2{\gamma }^{\gimel },{}^2{\delta }^{\gimel }],{[}^2{\psi }^{\gimel },{}^2{\eta }^{\gimel }]\rangle )$ be any two IVLDFNs, then the GI and a GU between ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ are defined as follows:

$$\begin{array}{ccc}\hfill {{\overline{⨿}}^{\gimel }}_1{\otimes }_{T_n,T_s}{{\overline{⨿}}^{\gimel }}_2& =& \left(\langle \left[T_n\left({}^1{\mu }^{\gimel }{,}^2{\mu }^{\gimel }\right),T_n\left({}^1{\nu }^{\gimel }{,}^2{\nu }^{\gimel }\right)\right],\left[T_s\left({}^1{\alpha }^{\gimel }{,}^2{\alpha }^{\gimel }\right),T_s\left({}^1{\beta }^{\gimel }{,}^2{\beta }^{\gimel }\right)\right]\rangle ,\right.\hfill \\ & & \left.\langle \left[T_n\left({}^1{\gamma }^{\gimel }{,}^2{\gamma }^{\gimel }\right),T_n\left({}^1{\delta }^{\gimel }{,}^2{\delta }^{\gimel }\right)\right],\left[T_s\left({}^1{\psi }^{\gimel }{,}^2{\psi }^{\gimel }\right),T_s\left({}^1{\eta }^{\gimel }{,}^2{\eta }^{\gimel }\right)\right]\rangle \right)\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {{\overline{⨿}}^{\gimel }}_1{\oplus }_{T_n,T_s}{{\overline{⨿}}^{\gimel }}_2& =& \left(\langle \left[T_s\left({}^1{\mu }^{\gimel }{,}^2{\mu }^{\gimel }\right),T_s\left({}^1{\nu }^{\gimel }{,}^2{\nu }^{\gimel }\right)\right],\left[T_n\left({}^1{\alpha }^{\gimel }{,}^2{\alpha }^{\gimel }\right),T_n\left({}^1{\beta }^{\gimel }{,}^2{\beta }^{\gimel }\right)\right]\rangle ,\right.\hfill \\ & & \left.\langle \left[T_s\left({}^1{\gamma }^{\gimel }{,}^2{\gamma }^{\gimel }\right),T_s\left({}^1{\delta }^{\gimel }{,}^2{\delta }^{\gimel }\right)\right],\left[T_n\left({}^1{\psi }^{\gimel }{,}^2{\psi }^{\gimel }\right),T_n\left({}^1{\eta }^{\gimel }{,}^2{\eta }^{\gimel }\right)\right]\rangle \right)\hfill \end{array}$$

(6)

where any dual $\mathrm{t}$-norm $T_n$ and $\mathrm{t}$-conorm S pair can be used.

If the T${}_n$ and T${}_c$ are the products of $T_P(x,y)=x·y$ and the probabilistic sum $S_P(x,y)=x+y-x·y$, respectively, then the GI and the GI of two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$ become the algebraic product and the algebraic sum of two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$, respectively, as follows:

$$\begin{array}{cc}\hfill {{\overline{⨿}}^{\gimel }}_1{\otimes }_A{{\overline{⨿}}^{\gimel }}_2=& (〈\left[{}^1{{\mu }^{\gimel }}^2{\mu }^{\gimel }{,}^1{{\nu }^{\gimel }}^2{\nu }^{\gimel }\right],\left[{}^1{\alpha }^{\gimel }{+}^2{\alpha }^{\gimel }{-}^1{{\alpha }^{\gimel }}^2{\alpha }^{\gimel }{,}^1{\beta }^{\gimel }{+}^2{\beta }^{\gimel }{-}^1{{\beta }^{\gimel }}^2{\beta }^{\gimel }\right]〉,\hfill \\ & 〈\left[{}^1{{\gamma }^{\gimel }}^2{\gamma }^{\gimel }{,}^1{{\delta }^{\gimel }}^2{\delta }^{\gimel }\right],\left[{}^1{\psi }^{\gimel }{+}^2{\psi }^{\gimel }{-}^1{{\psi }^{\gimel }}^2{\psi }^{\gimel }{,}^1{\eta }^{\gimel }{+}^2{\eta }^{\gimel }{-}^1{{\eta }^{\gimel }}^2{\eta }^{\gimel }\right]〉)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {{\overline{⨿}}^{\gimel }}_1{\oplus }_A{{\overline{⨿}}^{\gimel }}_2=& (〈\left[{}^1{\mu }^{\gimel }{+}^2{\mu }^{\gimel }{-}^1{{\mu }^{\gimel }}^2{\mu }^{\gimel }{,}^1{\nu }^{\gimel }{+}^2{\nu }^{\gimel }{-}^1{{\nu }^{\gimel }}^2{\nu }^{\gimel }\right],\left[{}^1{{\alpha }^{\gimel }}^2{\alpha }^{\gimel }{,}^1{{\beta }^{\gimel }}^2{\beta }^{\gimel }\right]〉,\hfill \\ & 〈\left[{}^1{\gamma }^{\gimel }{+}^2{\gamma }^{\gimel }{-}^1{{\gamma }^{\gimel }}^2{\gamma }^{\gimel }{,}^1{\delta }^{\gimel }{+}^2{\delta }^{\gimel }{-}^1{{\delta }^{\gimel }}^2{\delta }^{\gimel }\right],\left[{}^1{{\psi }^{\gimel }}^2{\psi }^{\gimel }{,}^1{{\eta }^{\gimel }}^2{\eta }^{\gimel }\right]〉)\hfill \end{array}$$

The Frank T${}_n$ and T${}_c$ are given as [49]:

$$\begin{array}{c}{\left(T_n\right)}_F(x,y)={log}_{\zeta }\left(1+\frac{\left({\zeta }^x-1\right)\left({\zeta }^y-1\right)}{\zeta -1}\right),\forall x,y\in [0,1],\zeta \in (1,+\infty )\\ \\ {\left(T_c\right)}_F(x,y)=1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1-x}-1\right)\left({\zeta }^{1-y}-1\right)}{\zeta -1}\right),\forall x,y\in [0,1],\zeta \in (1,+\infty )\end{array}$$

4. The Frank Aggregation Operators for Interval-Valued Linear Diophantine Fuzzy Numbers

In this section, we propose Frank operations on IVLDFNs and assess certain desirable aspects of these operations, on which we will base the development of several Frank AOs for IVLDFNs.

4.1. The Frank Operations on IVLDFNs Based on Frank t-Norm and t-Conorm

Let the GI and GU on two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1=(\langle {[}^1{\mu }^{\gimel },{}^1{\nu }^{\gimel }],{[}^1{\beta }^{\gimel },{}^1{\alpha }^{\gimel }]\rangle ,\langle {[}^1{\gamma }^{\gimel },{}^1{\delta }^{\gimel }],{[}^1{\psi }^{\gimel },{}^1{\eta }^{\gimel }]\rangle )$ and ${{\overline{⨿}}^{\gimel }}_2=(\langle {[}^2{\mu }^{\gimel },{}^2{\nu }^{\gimel }],{[}^2{\alpha }^{\gimel },{}^2{\beta }^{\gimel }]\rangle ,\langle {[}^2{\gamma }^{\gimel },{}^2{\delta }^{\gimel }],{[}^2{\psi }^{\gimel },{}^2{\eta }^{\gimel }]\rangle )$ become the Frank product and Frank sum of two IVLDFNs ${{\overline{⨿}}^{\gimel }}_1$ and ${{\overline{⨿}}^{\gimel }}_2$, respectively, as follows:

${{\overline{⨿}}^{\gimel }}_1{\otimes }_F{{\overline{⨿}}^{\gimel }}_2$

$$=\left(\begin{array}{c}\langle \left[{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\mu }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\mu }^{\gimel }}-1\right)}{\zeta -1}\right),{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\nu }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\nu }^{\gimel }}-1\right)}{\zeta -1}\right)\right],\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\alpha }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\beta }^{\gimel }}-1\right)}{\zeta -1}\right),1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\beta }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\alpha }^{\gimel }}-1\right)}{\zeta -1}\right)\right]\rangle ,\hfill \\ \\ \langle \left[{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\gamma }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\gamma }^{\gimel }}-1\right)}{\zeta -1}\right),{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\delta }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\delta }^{\gimel }}-1\right)}{\zeta -1}\right)\right],\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\psi }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\psi }^{\gimel }}-1\right)}{\zeta -1}\right),1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\eta }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\eta }^{\gimel }}-1\right)}{\zeta -1}\right)\right]\rangle \hfill \end{array}\right)\zeta >1$$

${{\overline{⨿}}^{\gimel }}_1{\oplus }_F{{\overline{⨿}}^{\gimel }}_2$

$$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1-{}^1{\mu }^{\gimel }}-1\right)\left({\zeta }^{1-{}^2{\mu }^{\gimel }}-1\right)}{\zeta -1}\right),1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\nu }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\nu }^{\gimel }}-1\right)}{\zeta -1}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\alpha }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\beta }^{\gimel }}-1\right)}{\zeta -1}\right),{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\beta }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\alpha }^{\gimel }}-1\right)}{\zeta -1}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\gamma }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\gamma }^{\gimel }}-1\right)}{\zeta -1}\right),1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{1{-}^1{\delta }^{\gimel }}-1\right)\left({\zeta }^{1{-}^2{\delta }^{\gimel }}-1\right)}{\zeta -1}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\psi }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\psi }^{\gimel }}-1\right)}{\zeta -1}\right),{log}_{\zeta }\left(1+\frac{\left({\zeta }^{{}^1{\eta }^{\gimel }}-1\right)\left({\zeta }^{{}^2{\eta }^{\gimel }}-1\right)}{\zeta -1}\right)\right]\rangle \hfill \end{array}\right),\zeta >1$$

Theorem 1.

Let ${{\overline{⨿}}^{\gimel }}_1=\left(〈\left[{}^1{\mu }^{\gimel }{,}^1{\nu }^{\gimel }\right],\left[{}^1{\alpha }^{\gimel }{,}^1{\beta }^{\gimel }\right]〉,〈\left[{}^1{\gamma }^{\gimel }{,}^1{\delta }^{\gimel }\right],\left[{}^1{\psi }^{\gimel }{,}^1{\eta }^{\gimel }\right]〉\right)$ and ${{\overline{⨿}}^{\gimel }}_2=\left(〈\left[{}^2{\mu }^{\gimel }{,}^2{\nu }^{\gimel }\right],\left[{}^2{\beta }^{\gimel }{,}^2{\alpha }^{\gimel }\right]〉,〈\left[{}^2{\gamma }^{\gimel }{,}^2{\delta }^{\gimel }\right],\left[{}^2{\psi }^{\gimel }{,}^2{\eta }^{\gimel }\right]〉\right)$ be two IVLDFNs, and let ${{\overline{⨿}}^{\gimel }}_3={{\overline{⨿}}^{\gimel }}_1{\otimes }_F{{\overline{⨿}}^{\gimel }}_2$ and ${{\overline{⨿}}^{\gimel }}_4={{\overline{⨿}}^{\gimel }}_1{\oplus }_F{{\overline{⨿}}^{\gimel }}_2$. Then, both ${{\overline{⨿}}^{\gimel }}_3$ and ${{\overline{⨿}}^{\gimel }}_4$ are also IVLDFNs.

Proof. 

Here, we omit the proof. □

Theorem 2.

Let ${\overline{⨿}}^{\gimel }=\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$ be the IVLDFN. Then, the scalar multiplication $n{·}_F{\overline{⨿}}^{\gimel }$ is a mapping from $Z^{+}\times \Omega$ to Ω:

$$n{·}_F{\overline{⨿}}^{\gimel }=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{\mu }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{\nu }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{\alpha }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{\beta }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{\gamma }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{\delta }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{\psi }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{\eta }^{\gimel }}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right]\rangle \hfill \end{array}\right)$$

(7)

where $Z^{+}$ denotes the set of all the positive integers. Furthermore, $n{·}_F{\overline{⨿}}^{\gimel }$ is an IVLDFN if n is any +ve real number.

Proof. 

Here, we omit the proof. □

Theorem 3.

Let ${\overline{⨿}}^{\gimel }=$$\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$ be the IVLDFN. Then, the exponentiation operation ${{\overline{⨿}}^{\gimel }}^{{\wedge }_{F}n}$ is defined as follows:

$${{\overline{⨿}}^{\gimel }}^{{\wedge }_{F}n}=\left(\begin{array}{c}\langle \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^a-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^b-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right],\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-c}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-d}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^t-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^x-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right],\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-y}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-z}-1\right)}^n}{{(\zeta -1)}^{n-1}}\right)\right]\rangle \hfill \end{array}\right)$$

where ${{\overline{⨿}}^{\gimel }}^{{\wedge }_{F}n}={\overbrace{{\overline{⨿}}^{\gimel }{\otimes }_F{\overline{⨿}}^{\gimel }{\otimes }_F\cdots {\otimes }_F{\overline{⨿}}^{\gimel }}}^n.$ Moreover,${{\overline{⨿}}^{\gimel }}^{{\wedge }_{F}n}$ is an IVIFN if n is any +ve real number.

The following are some desired characteristics of the operational laws to consider.

Theorem 4.

For three IVLDFNs ${\overline{⨿}}^{\gimel },{{\overline{⨿}}^{\gimel }}_1$, and ${{\overline{⨿}}^{\gimel }}_2$, we have:

1.${{\overline{⨿}}^{\gimel }}_1{\oplus }_F{{\overline{⨿}}^{\gimel }}_2={{\overline{⨿}}^{\gimel }}_2{\oplus }_F{{\overline{⨿}}^{\gimel }}_1;$

2.${{\overline{⨿}}^{\gimel }}_1{\otimes }_F{{\overline{⨿}}^{\gimel }}_2={{\overline{⨿}}^{\gimel }}_2{\otimes }_F{{\overline{⨿}}^{\gimel }}_1;$

3.${\lambda }_F\left({{\overline{⨿}}^{\gimel }}_1{\oplus }_F{{\overline{⨿}}^{\gimel }}_2\right)={\lambda }_F{{\overline{⨿}}^{\gimel }}_1{\oplus }_F\lambda ·{{\overline{⨿}}^{\gimel }}_2,\lambda >0;$

4.${\left({{\overline{⨿}}^{\gimel }}_1{\otimes }_F{{\overline{⨿}}^{\gimel }}_2\right)}^{{\wedge }_F\lambda }={{\overline{⨿}}^{\gimel }}_1^{{\wedge }_F\lambda }{\otimes }_F{{\overline{⨿}}^{\gimel }}_2^{{\wedge }_F\lambda },\lambda >0$;

5.$\left(\lambda +\lambda \right)·{\overline{⨿}}^{\gimel }=\lambda {·}_F{\overline{⨿}}^{\gimel }{\oplus }_F\lambda {·}_F{\overline{⨿}}^{\gimel },\lambda ,\lambda >0$;

6.${{\overline{⨿}}^{\gimel }}^{{\wedge }_F\left(\lambda +\lambda \right)}={{\overline{⨿}}^{\gimel }}^{{\wedge }_F\lambda }{\otimes }_F{{\overline{⨿}}^{\gimel }}^{{\wedge }_F\lambda },\lambda ,\lambda >0$;

7.$\left(\lambda \lambda \right)·{\overline{⨿}}^{\gimel }{\overline{⨿}}^{\gimel }=\lambda {·}_F\left(\lambda ·{\overline{⨿}}^{\gimel }\right);$

8.${{\overline{⨿}}^{\gimel }}^{{\wedge }_F\left(\lambda \lambda \right)}={\left({{\overline{⨿}}^{\gimel }}^{{\wedge }_F\lambda }\right)}^{{\wedge }_F\lambda }$.

Proof. 

Due to space constraints, the proof is not included here. □

4.2. Interval-Valued Linear Diophantine Fuzzy Frank Aggregation Operators

We extend the Frank operations to incorporate IVLDFNs in this subsection, and then propose the interval-valued linear Diophantine fuzzy Frank weighted averaging (IVLDFFWA) operator and the interval-valued linear Diophantine fuzzy Frank weighted geometric (IVLDFFWG) operator.

Definition 15.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and let ${\omega }^{\gimel }={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^{\mathrm{T}}$ be the weight vector (WV) of ${{\overline{⨿}}^{\gimel }}_i$ with ${{\omega }^{\gimel }}_i\in [0,1]$ and ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$. Then, the IVLDFFWA operator is a mapping ${\mathsf{\Omega }}^n\to \mathsf{\Omega }$, such that

$$\mathrm{IVLDFFWA}\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)=\left({{\omega }^{\gimel }}_1{·}_F{{\overline{⨿}}^{\gimel }}_1\right){\oplus }_F\left({{\omega }^{\gimel }}_2{·}_F{{\overline{⨿}}^{\gimel }}_2\right){\oplus }_F\cdots {\oplus }_F\left({{\omega }^{\gimel }}_n{·}_F{{\overline{⨿}}^{\gimel }}_n\right)$$

Theorem 5.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and let $w={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^T$ be the WV of ${{\overline{⨿}}^{\gimel }}_i$, with ${{\omega }^{\gimel }}_i\in [0,1]$ and ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$. Then, the aggregated value using the IVLDFFWA operator is also found by

IVLDFFWA$\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)$$

(8)

Proof. 

First, we show that the Equation (9) holds true when ${\omega }^{\gimel }$ is any vector, i.e., when ${\omega }^{\gimel }$ is not constrained in any way.

IVLDFFWA$\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right),{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right),{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i-1}}\right)\right]\rangle \hfill \end{array}\right)$$

(9)

By using mathematical induction on $n:$ for $n=2$, since

${{\omega }^{\gimel }}_1{·}_F{{\overline{⨿}}^{\gimel }}_1=$

$$\begin{array}{c}\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{}^1{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^1{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^1{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)\right]\rangle \hfill \end{array}\right)\hfill \end{array}$$

${{\omega }^{\gimel }}_2{·}_F{{\overline{⨿}}^{\gimel }}_2=$

$$\begin{array}{c}\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{}^2{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^2{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^2{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^2{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)\right]\rangle \hfill \end{array}\right)\hfill \end{array}$$

we have

IVIFFWA$\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2\right)=\left({{\omega }^{\gimel }}_1{·}_F{{\overline{⨿}}^{\gimel }}_1\right){\oplus }_F\left({{\omega }^{\gimel }}_2{·}_F{{\overline{⨿}}^{\gimel }}_2\right)$

$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{}^1{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{}^2{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right),\right.\hfill \\ \\ \left.1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^2{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right),\right.\hfill \\ \\ \left.{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^1{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^2{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right)\right.,\hfill \\ \left.1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^1{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1{-}^2{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right),\right.\hfill \\ \\ \left.{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^1{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_1}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^2{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_2}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_2-1}}\right)}-1\right)}{\zeta -1}\right)\right]\rangle \hfill \end{array}\right)$

$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right),{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right),{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^2{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_1+{{\omega }^{\gimel }}_2-1}}\right)\right]\rangle \hfill \end{array}\right)$

That is, Equation (9) holds for $n=2$. Suppose that Equation (9) holds for $n=k$, i.e.,

IVLDFFWA$\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_k\right)$

$$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right),{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right),{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)\right]\rangle \hfill \end{array}\right)$$

then, when $n=k+1$, based on the operational laws for the IVIFNs, we have

$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_{k+1}\right)=IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_k\right){\oplus }_F{{\omega }^{\gimel }}_{k+1}{·}_F{{\overline{⨿}}^{\gimel }}_{k+1}$

$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\mu }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right),\right.\hfill \\ \left.1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\nu }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{c_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right)\right.,\hfill \\ \left.{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{d_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\gamma }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right),\right.\hfill \\ \left.1+\frac{\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\delta }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{y_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right)\right.,\hfill \\ \left.{log}_{\zeta }\left(1+\frac{{\prod }_{i=1}^k\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}}{{(\zeta -1)}^{{\sum }_{i=1}^k{{\omega }^{\gimel }}_i-1}}\right)}-1\right)\left({\zeta }^{lo{g}_{\zeta }\left(1+\frac{{\left({\zeta }^{z_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)}-1\right)}{\zeta -1}\right)\right]\rangle \hfill \end{array}\right)$

$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\mu }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\nu }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{c_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{d_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\mu }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right),1-{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{1-{{\nu }^{\gimel }}_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{c_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right),{log}_{\zeta }\left(1+\frac{{\left({\zeta }^{d_{k+1}}-1\right)}^{{{\omega }^{\gimel }}_{k+1}}}{{(\zeta -1)}^{{{\omega }^{\gimel }}_{k+1}-1}}\right)\right]\rangle ,\hfill \\ \end{array}\right)$

i.e., Equation (9) holds for $n=k+1$. Thus, Equation (9) holds for all n. Hence, Equation (9) is correct without any constraints for w. Therefore, when ${{\omega }^{\gimel }}_i\in [0,1]$ and ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$, Equation (9) reduces to Equation (8). □

Theorem 6.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ and ${{\overline{⨿}}^{\gimel }}_i^{\prime }$ = $\left(〈\left[{{}^i{\mu }^{\gimel }}^{\prime },{{}^i{\nu }^{\gimel }}^{\prime }\right],\left[{{}^i{\alpha }^{\gimel }}^{\prime },{{}^i{\beta }^{\gimel }}^{\prime }\right]〉,〈\left[{{}^i{\gamma }^{\gimel }}^{\prime },{{}^i{\delta }^{\gimel }}^{\prime }\right],\left[{{}^i{\psi }^{\gimel }}^{\prime },{{}^i{\eta }^{\gimel }}^{\prime }\right]〉\right)$ be two collections of IVLDFNs. If ${}^i{\mu }^{\gimel }\le {{}^i{\mu }^{\gimel }}^{\prime },{}^1{\nu }^{\gimel }\le {{}^1{\nu }^{\gimel }}^{\prime },{}^i{\alpha }^{\gimel }\ge {{}^i{\alpha }^{\gimel }}^{\prime }$, and ${}^i{\beta }^{\gimel }\ge {{}^i{\beta }^{\gimel }}^{\prime }$, ${}^i{\gamma }^{\gimel }\le {{}^i{\gamma }^{\gimel }}^{\prime },{}^i{\delta }^{\gimel }\le {{}^i{\delta }^{\gimel }}^{\prime },{}^i{\psi }^{\gimel }\ge {{}^i{\psi }^{\gimel }}^{\prime }$, and ${}^i{\eta }^{\gimel }\ge {{}^i{\eta }^{\gimel }}^{\prime }$ for all i, then

$$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\ge IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right).$$

Proof. 

For two collections of IVLDFNs ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ and ${{\overline{⨿}}^{\gimel }}_i^{\prime }=\left(〈\left[{{}^i{\mu }^{\gimel }}^{\prime },{{}^i{\nu }^{\gimel }}^{\prime }\right],\left[{{}^i{\alpha }^{\gimel }}^{\prime },{{}^i{\beta }^{\gimel }}^{\prime }\right]〉,〈\left[{{}^i{\gamma }^{\gimel }}^{\prime },{{}^i{\delta }^{\gimel }}^{\prime }\right],\left[{{}^i{\psi }^{\gimel }}^{\prime },{{}^i{\eta }^{\gimel }}^{\prime }\right]〉\right)$, if ${}^i{\mu }^{\gimel }\le {{}^i{\mu }^{\gimel }}^{\prime },{}^1{\nu }^{\gimel }\le {{}^1{\nu }^{\gimel }}^{\prime },{}^i{\alpha }^{\gimel }\ge {{}^i{\alpha }^{\gimel }}^{\prime }$, and ${}^i{\beta }^{\gimel }\ge {{}^i{\beta }^{\gimel }}^{\prime }$, ${}^i{\gamma }^{\gimel }\le {{}^i{\gamma }^{\gimel }}^{\prime },{}^i{\delta }^{\gimel }\le {{}^i{\delta }^{\gimel }}^{\prime },{}^i{\psi }^{\gimel }\ge {{}^i{\psi }^{\gimel }}^{\prime }$, and ${}^i{\eta }^{\gimel }\ge {{}^i{\eta }^{\gimel }}^{\prime }$ for all i, then

$$\begin{array}{ccc}\hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\mu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^1{\nu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\alpha }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\beta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\gamma }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\delta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\psi }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\eta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

(17)

By Definition 11:

$S\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)$

$$\begin{array}{cc}& =\frac{1}{4}(1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right))\hfill \\ & \le \frac{1}{4}(1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\mu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\alpha }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^1{\nu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\gamma }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\psi }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\delta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right))\hfill \\ & =S\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\right)\hfill \end{array}$$

If $S\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)<S\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\right)$, then

$$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)<IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right).$$

$$\mathrm{If}S\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)=S\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\right),\mathrm{i}.\mathrm{e}.,$$

$$\begin{array}{cc}& \frac{1}{4}(1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right))\hfill \\ & =\frac{1}{4}(1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\mu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\alpha }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^1{\nu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\gamma }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\psi }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\delta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right))\hfill \end{array}$$

then, by Equations (10)–(17)

$$\begin{array}{ccc}\hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\mu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^1{\nu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\alpha }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\beta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\gamma }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \le & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\delta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\psi }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ \hfill {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)& \ge & {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\eta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \end{array}$$

Thus, $H\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)$

$$\begin{array}{cc}& =\frac{1}{4}(1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right))\hfill \\ & =\frac{1}{4}(1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\mu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\alpha }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^1{\nu }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\gamma }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{{}^i{\psi }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)+\hfill \\ & 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{{}^i{\delta }^{\gimel }}^{\prime }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right))\hfill \\ & =H\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\right)\hfill \end{array}$$

which implies that

$$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)=IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right).$$

□

Theorem 7.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs. If all ${{\overline{⨿}}^{\gimel }}_i$ are equal, i.e., ${{\overline{⨿}}^{\gimel }}_i={\overline{⨿}}^{\gimel }=\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$ for all i, then

$$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)=IVLDFFWA({\overline{⨿}}^{\gimel },{\overline{⨿}}^{\gimel },\dots ,{\overline{⨿}}^{\gimel })={\overline{⨿}}^{\gimel }.$$

Proof. 

$IVIFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$$\begin{array}{cc}& =\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \langle \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)\hfill \end{array}$$

$$\begin{array}{cc}& =\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \langle \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \end{array}\right)\hfill \\ & =\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\left({\zeta }^{1-{\mu }^{\gimel }}-1\right)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\left({\zeta }^{1-{\nu }^{\gimel }}-1\right)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+{\left({\zeta }^{{\alpha }^{\gimel }}-1\right)}^{{\sum }_{i=1}{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\left({\zeta }^{{\beta }^{\gimel }}-1\right)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \langle \left[1-{log}_{\zeta }\left(1+{\left({\zeta }^{1-{\gamma }^{\gimel }}-1\right)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\left({\zeta }^{1-{\delta }^{\gimel }}-1\right)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \left[{log}_{\zeta }\left(1+{\left({\zeta }^{{\psi }^{\gimel }}-1\right)}^{{\sum }_{i=1}{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\left({\zeta }^{{\eta }^{\gimel }}-1\right)}^{{\sum }_{i=1}^n{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)\hfill \\ & =\left(\begin{array}{c}〈\left[1-{log}_{\zeta }\left({\zeta }^{1-{\mu }^{\gimel }}\right),1-{log}_{\zeta }\left({\zeta }^{1-{\nu }^{\gimel }}\right)\right],\left[{log}_{\zeta }\left({\zeta }^{{\alpha }^{\gimel }}\right),{log}_{\zeta }\left({\zeta }^{{\beta }^{\gimel }})\right)\right]〉,\hfill \\ 〈\left[1-{log}_{\zeta }\left({\zeta }^{1-{\gamma }^{\gimel }}\right),1-{log}_{\zeta }\left({\zeta }^{1-{\delta }^{\gimel }}\right)\right],\left[{log}_{\zeta }\left({\zeta }^{{\psi }^{\gimel }}\right),{log}_{\zeta }\left({\zeta }^{{\eta }^{\gimel }})\right)\right]〉,\hfill \end{array}\right)\hfill \\ & =\begin{array}{c}\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)\hfill \end{array}\hfill \end{array}$$

□

Theorem 8.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and

$$\begin{array}{c}{{\overline{⨿}}^{\gimel }}^{-}=\left(\left[\underset{i}{min}\left\{{}^i{\mu }^{\gimel }\right\},\underset{i}{min}\left\{{}^1{\nu }^{\gimel }\right\}\right],\left[\underset{i}{max}\left\{{}^i{\alpha }^{\gimel }\right\},\underset{i}{max}\left\{{}^i{\beta }^{\gimel }\right\}\right]\right)\hfill \\ {{\overline{⨿}}^{\gimel }}^{+}=\left(\left[\underset{i}{max}\left\{{}^i{\mu }^{\gimel }\right\},\underset{i}{max}\left\{{}^1{\nu }^{\gimel }\right\}\right],\left[\underset{i}{min}\left\{{}^i{\alpha }^{\gimel }\right\},\underset{i}{min}\left\{{}^i{\beta }^{\gimel }\right\}\right]\right)\hfill \end{array}$$

$${{\overline{⨿}}^{\gimel }}^{-}\le IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\le {{\overline{⨿}}^{\gimel }}^{+}$$

Proof. 

Consider

$IVIFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\prod }_{i=1}^n{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)$

$={\overline{⨿}}^{\gimel }=\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$

$\begin{array}{c}\mathrm{Because}\hfill \\ {min}_i\left\{{}^i{\mu }^{\gimel }\right\}\le {}^i{\mu }^{\gimel }\le {max}_i\left\{{}^i{\mu }^{\gimel }\right\},{min}_i\left\{{}^1{\nu }^{\gimel }\right\}\le {}^1{\nu }^{\gimel }\le {max}_i\left\{{}^1{\nu }^{\gimel }\right\},\hfill \\ {min}_i\left\{{}^i{\alpha }^{\gimel }\right\}\le {}^i{\alpha }^{\gimel }\le {max}_i\left\{{}^i{\alpha }^{\gimel }\right\},{min}_i\left\{{}^i{\beta }^{\gimel }\right\}\le {}^i{\beta }^{\gimel }\le {max}_i\left\{{}^i{\beta }^{\gimel }\right\},\hfill \\ {min}_i\left\{{}^i{\gamma }^{\gimel }\right\}\le {}^i{\gamma }^{\gimel }\le {max}_i\left\{{}^i{\gamma }^{\gimel }\right\},{min}_i\left\{{}^i{\delta }^{\gimel }\right\}\le {}^i{\delta }^{\gimel }\le {max}_i\left\{{}^i{\delta }^{\gimel }\right\},\hfill \\ {min}_i\left\{{}^i{\psi }^{\gimel }\right\}\le {}^i{\psi }^{\gimel }\le {max}_i\left\{{}^i{\psi }^{\gimel }\right\},\mathrm{and}{min}_i\left\{{}^i{\eta }^{\gimel }\right\}\le {}^i{\eta }^{\gimel }\le {max}_i\left\{{}^i{\eta }^{\gimel }\right\},\hfill \\ \mathrm{we}\mathrm{have}\hfill \end{array}$

$$\begin{array}{cc}\hfill \underset{i}{min}\left\{{}^i{\mu }^{\gimel }\right\}& =1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-min\left\{{}^i{\mu }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=a\hfill \\ & \le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-max\left\{{}^i{\mu }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\mu }^{\gimel }\right\}\hfill \\ \hfill \underset{i}{min}\left\{{}^1{\nu }^{\gimel }\right\}& =1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-min\left\{{}^1{\nu }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=b\hfill \\ & \le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-max\left\{{}^1{\nu }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^1{\nu }^{\gimel }\right\}\hfill \\ \hfill \underset{i}{min}\left\{{}^i{\alpha }^{\gimel }\right\}& ={log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{min\left\{{}^i{\alpha }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=c\hfill \\ & \le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{max\left\{{}^i{\alpha }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\alpha }^{\gimel }\right\}\hfill \\ \hfill \underset{i}{min}\left\{{}^i{\beta }^{\gimel }\right\}& ={log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }_i^{min\left\{{}^i{\beta }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=d\hfill \\ & \le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{max\left\{{}^i{\beta }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\beta }^{\gimel }\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \underset{i}{min}\left\{{}^i{\gamma }^{\gimel }\right\}& =1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-min\left\{{}^i{\gamma }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=t\hfill \\ & \le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-max\left\{{}^i{\gamma }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\gamma }^{\gimel }\right\}\hfill \\ \hfill \underset{i}{min}\left\{{}^i{\delta }^{\gimel }\right\}& =1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-min\left\{{}^i{\delta }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=x\hfill \\ & \le 1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-max\left\{{}^i{\delta }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\delta }^{\gimel }\right\}\hfill \\ \hfill \underset{i}{min}\left\{{}^i{\psi }^{\gimel }\right\}& ={log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{min\left\{{}^i{\psi }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=y\hfill \\ & \le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{max\left\{{}^i{\psi }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\psi }^{\gimel }\right\}\hfill \\ \hfill \underset{i}{min}\left\{{}^i{\eta }^{\gimel }\right\}& ={log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }_i^{min\left\{{}^i{\eta }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=z\hfill \\ & \le {log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{max\left\{{}^i{\eta }^{\gimel }\right\}}-1\right)}^{{{\omega }^{\gimel }}_i}\right)=\underset{i}{max}\left\{{}^i{\eta }^{\gimel }\right\}\hfill \end{array}$$

Then, we can obtain

$$\begin{array}{cc}\hfill S\left({\overline{⨿}}^{\gimel }\right)& =\frac{a-c+b-d+t-y+x-z}{2}\le \hfill \\ & \frac{1}{4}max\left\{{}^i{\mu }^{\gimel }\right\}-\underset{i}{min}\left\{{}^i{\alpha }^{\gimel }\right\}+\underset{i}{max}\left\{{}^1{\nu }^{\gimel }\right\}-\underset{i}{min}\left\{{}^i{\beta }^{\gimel }\right\}+max\left\{{}^i{\gamma }^{\gimel }\right\}-\underset{i}{min}\left\{{}^i{\psi }^{\gimel }\right\}+\hfill \\ & \underset{i}{max}\left\{{}^i{\delta }^{\gimel }\right\}-\underset{i}{min}\left\{{}^i{\eta }^{\gimel }\right\}\hfill \\ & =S\left({{\overline{⨿}}^{\gimel }}^{+}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill S\left({\overline{⨿}}^{\gimel }\right)& =\frac{a-c+b-d+t-y+x-z}{2}\ge \hfill \\ & \frac{1}{4}\underset{i}{min}\left\{{}^i{\mu }^{\gimel }\right\}-\underset{i}{max}\left\{{}^i{\alpha }^{\gimel }\right\}+\underset{i}{min}\left\{{}^1{\nu }^{\gimel }\right\}-\underset{i}{max}\left\{{}^i{\beta }^{\gimel }\right\}+\underset{i}{min}\left\{{}^i{\gamma }^{\gimel }\right\}-\hfill \\ & \underset{i}{max}\left\{{}^i{\psi }^{\gimel }\right\}+\underset{i}{min}\left\{{}^i{\delta }^{\gimel }\right\}-\underset{i}{max}\left\{{}^i{\eta }^{\gimel }\right\}\hfill \\ & =S\left({{\overline{⨿}}^{\gimel }}^{-}\right)\hfill \\ & \mathrm{If}S\left({\overline{⨿}}^{\gimel }\right)\le S\left({{\overline{⨿}}^{\gimel }}^{+}\right)\mathrm{and}S\left({\overline{⨿}}^{\gimel }\right)\ge S\left({{\overline{⨿}}^{\gimel }}^{-}\right),\mathrm{then},\mathrm{by}\mathrm{definition}\mathrm{of}\mathrm{SF}\hfill \end{array}$$

$${{\overline{⨿}}^{\gimel }}^{-}\le IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\le {{\overline{⨿}}^{\gimel }}^{+}.$$

(18)

□

Theorem 9.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$, and let ${{\overline{⨿}}^{\gimel }}_i^{\prime }$ = $\left(〈\left[{{}^i{\mu }^{\gimel }}^{\prime },{{}^i{\nu }^{\gimel }}^{\prime }\right],\left[{{}^i{\alpha }^{\gimel }}^{\prime },{{}^i{\beta }^{\gimel }}^{\prime }\right]〉,〈\left[{{}^i{\gamma }^{\gimel }}^{\prime },{{}^i{\delta }^{\gimel }}^{\prime }\right],\left[{{}^i{\psi }^{\gimel }}^{\prime },{{}^i{\eta }^{\gimel }}^{\prime }\right]〉\right)(i=1,2,\dots ,n)$ be the two collections of IVLDFNs, and $w={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^T$ be the WV of ${{\overline{⨿}}^{\gimel }}_i(i=1,2,\dots ,n)$, satisfying ${{\omega }^{\gimel }}_i\in [0,1]$, ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$. If $r>0$ is a real number and ${\overline{⨿}}^{\gimel }=\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$ is an IVLDFN, then:

1.$IVLDFFWA\left(r{·}_F{{\overline{⨿}}^{\gimel }}_1,r{·}_F{{\overline{⨿}}^{\gimel }}_2,\dots ,r_F{{\overline{⨿}}^{\gimel }}_n\right)$$=r{·}_{F}IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$;

2.$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1{\oplus }_F{\overline{⨿}}^{\gimel },{{\overline{⨿}}^{\gimel }}_2{\oplus }_F{\overline{⨿}}^{\gimel },\dots ,{{\overline{⨿}}^{\gimel }}_n{\oplus }_F{\overline{⨿}}^{\gimel }\right)=IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right){\oplus }_F{\overline{⨿}}^{\gimel }$;

3.$IVLDFFWA\left(r_F{{\overline{⨿}}^{\gimel }}_1{\oplus }_F{\overline{⨿}}^{\gimel },r_F{{\overline{⨿}}^{\gimel }}_2{\oplus }_F{\overline{⨿}}^{\gimel },\dots ,r_F{{\overline{⨿}}^{\gimel }}_n{\oplus }_F{\overline{⨿}}^{\gimel }\right)=r{·}_{F}IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right){\oplus }_F{\overline{⨿}}^{\gimel }$;

4.$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1{\oplus }_F{{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2{\oplus }_F{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n{\oplus }_F{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)=IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right){\oplus }_{F}IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)$.

Proof. 

Here, we omit the proof. □

We next define an interval-valued linear Diophantine fuzzy Frank weighted geometric (IVIFFWG) operator as follows.

Definition 16.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and let $w={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^{\mathrm{T}}$ be the WV of ${{\overline{⨿}}^{\gimel }}_i(i=1,2,\dots ,n)$ with ${{\omega }^{\gimel }}_i\in [0,1]$ and ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$. Then, the IVLDFFWG operator is a mapping ${\Omega }^n\to \Omega$, such that

$$\mathrm{IVLDFFWG}\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)=\left({{\omega }^{\gimel }}_1{·}_F{{\overline{⨿}}^{\gimel }}_1\right){\otimes }_F\left({{\omega }^{\gimel }}_2{·}_F{{\overline{⨿}}^{\gimel }}_2\right){\otimes }_F\cdots {\otimes }_F\left({{\omega }^{\gimel }}_n{·}_F{{\overline{⨿}}^{\gimel }}_n\right)$$

Theorem 10.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and $w={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^T$ be the WV of ${{\overline{⨿}}^{\gimel }}_i(i=1,2,\dots ,n)$, satisfying ${{\omega }^{\gimel }}_i\in [0,1]$ and ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$, then the aggregated value by using the IVLDFFWG operator is also an IVLDFN, and

$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$$=\left(\begin{array}{c}\langle \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \langle ,\left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{1-{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)$$

(19)

The IVLDFFWG operator has several of the same desirable features as the IVLDFFWA operator. It is worth noting that the IVLDFFWG operator’s proof is very comparable to these characteristics. As a result, we only enumerate these qualities without providing any proofs.

Theorem 11.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs. If all ${{\overline{⨿}}^{\gimel }}_i(i=1,2,\dots ,n)$ are equal, i.e., ${{\overline{⨿}}^{\gimel }}_i={\overline{⨿}}^{\gimel }=\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉\right.$, $\left.〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$ for all i, then

$$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)=IVLDFFWG({\overline{⨿}}^{\gimel },{\overline{⨿}}^{\gimel },\dots ,{\overline{⨿}}^{\gimel })={\overline{⨿}}^{\gimel }$$

Theorem 12.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ and ${{\overline{⨿}}^{\gimel }}_i^{\prime }$ = $\left(〈\left[{{}^i{\mu }^{\gimel }}^{\prime },{{}^1{\nu }^{\gimel }}^{\prime }\right],\left[{{}^i{\alpha }^{\gimel }}^{\prime },{{}^i{\beta }^{\gimel }}^{\prime }\right]〉,〈\left[{{}^i{\gamma }^{\gimel }}^{\prime },{{}^i{\delta }^{\gimel }}^{\prime }\right],\left[{{}^i{\psi }^{\gimel }}^{\prime },{{}^i{\eta }^{\gimel }}^{\prime }\right]〉\right)$ be two collections of IVLDFNs. If ${}^i{\mu }^{\gimel }\le {{}^i{\mu }^{\gimel }}^{\prime },{}^1{\nu }^{\gimel }\le {{}^1{\nu }^{\gimel }}^{\prime },{}^i{\alpha }^{\gimel }\ge {{}^i{\alpha }^{\gimel }}^{\prime }$, and ${}^i{\beta }^{\gimel }\ge {{}^i{\beta }^{\gimel }}^{\prime }$, ${}^i{\gamma }^{\gimel }\le {{}^i{\gamma }^{\gimel }}^{\prime },{}^i{\delta }^{\gimel }\le {{}^i{\delta }^{\gimel }}^{\prime },{}^i{\psi }^{\gimel }\ge {{}^i{\psi }^{\gimel }}^{\prime }$, and ${}^i{\eta }^{\gimel }\ge {{}^i{\eta }^{\gimel }}^{\prime }$ for all i, then

$$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\ge IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$$

Theorem 13.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and

$$\begin{array}{c}{{\overline{⨿}}^{\gimel }}^{-}=\left(\left[\underset{i}{min}\left\{{}^i{\mu }^{\gimel }\right\},\underset{i}{min}\left\{{}^1{\nu }^{\gimel }\right\}\right],\left[\underset{i}{max}\left\{{}^i{\alpha }^{\gimel }\right\},\underset{i}{max}\left\{{}^i{\beta }^{\gimel }\right\}\right]\right)\hfill \\ {{\overline{⨿}}^{\gimel }}^{+}=\left(\left[\underset{i}{max}\left\{{}^i{\mu }^{\gimel }\right\},\underset{i}{max}\left\{{}^1{\nu }^{\gimel }\right\}\right],\left[\underset{i}{min}\left\{{}^i{\alpha }^{\gimel }\right\},\underset{i}{min}\left\{{}^i{\beta }^{\gimel }\right\}\right]\right)\hfill \end{array}$$

$${{\overline{⨿}}^{\gimel }}^{-}\le IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)\le {{\overline{⨿}}^{\gimel }}^{+}$$

Theorem 14.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ and ${{\overline{⨿}}^{\gimel }}_i^{\prime }$ = $\left(〈\left[{{}^i{\mu }^{\gimel }}^{\prime },{{}^1{\nu }^{\gimel }}^{\prime }\right],\left[{{}^i{\alpha }^{\gimel }}^{\prime },{{}^i{\beta }^{\gimel }}^{\prime }\right]〉,〈\left[{{}^i{\gamma }^{\gimel }}^{\prime },{{}^i{\delta }^{\gimel }}^{\prime }\right],\left[{{}^i{\psi }^{\gimel }}^{\prime },{{}^i{\eta }^{\gimel }}^{\prime }\right]〉\right)(i=1,2,\dots ,n)$ be the two collections of IVLDFNs, and let $w={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^T$ be the WV of ${{\overline{⨿}}^{\gimel }}_i(i=1,2,\dots ,n)$, satisfying ${{\omega }^{\gimel }}_i\in [0,1]$, ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$. If $r>0$ is a real number and ${\overline{⨿}}^{\gimel }=\left(〈\left[{\mu }^{\gimel },{\nu }^{\gimel }\right],\left[{\alpha }^{\gimel },{\beta }^{\gimel }\right]〉,〈\left[{\gamma }^{\gimel },{\delta }^{\gimel }\right],\left[{\psi }^{\gimel },{\eta }^{\gimel }\right]〉\right)$ is an IVLDFN, then:

1.$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1^{{\wedge }_{F}r},{{\overline{⨿}}^{\gimel }}_2^{{\wedge }_{F}r},\dots ,{{\overline{⨿}}^{\gimel }}_n^{{\wedge }_{F}r}\right)$$={(IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right))}^{{\wedge }_{F}r}$;

2.$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1{\otimes }_F{\overline{⨿}}^{\gimel },{{\overline{⨿}}^{\gimel }}_2{\otimes }_F{\overline{⨿}}^{\gimel },\dots ,{{\overline{⨿}}^{\gimel }}_n{\otimes }_F{\overline{⨿}}^{\gimel }\right)=IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right){\otimes }_F{\overline{⨿}}^{\gimel }$;

3.$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1^{{\wedge }_{F}r}{\otimes }_F{\overline{⨿}}^{\gimel },{{\overline{⨿}}^{\gimel }}_2^{{\wedge }_{F}r}{\otimes }_F{\overline{⨿}}^{\gimel },\dots ,{{\overline{⨿}}^{\gimel }}_n^{{\wedge }_{F}r}{\otimes }_F{\overline{⨿}}^{\gimel }\right)={(IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right))}^{{\wedge }_{F}r}{\otimes }_F{\overline{⨿}}^{\gimel }$;

4.$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1{\otimes }_F{{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2{\otimes }_F{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n{\otimes }_F{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)=IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right){\otimes }_{F}IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1^{\prime },{{\overline{⨿}}^{\gimel }}_2^{\prime },\dots ,{{\overline{⨿}}^{\gimel }}_n^{\prime }\right)$.

The relationships between the IVLDFFWA and IVLDFFWG operators are discussed further in the following.

Theorem 15.

Let ${{\overline{⨿}}^{\gimel }}_i=\left(〈\left[{}^i{\mu }^{\gimel },{}^i{\nu }^{\gimel }\right],\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right]〉,〈\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right],\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right]〉\right)$ be the collection of IVLDFNs, and let $w={\left({{\omega }^{\gimel }}_1,{{\omega }^{\gimel }}_2,\dots ,{{\omega }^{\gimel }}_n\right)}^T$ be the WV of ${{\overline{⨿}}^{\gimel }}_i$, satisfying ${{\omega }^{\gimel }}_i\in [0,1]$ and ${\sum }_{i=1}^n{{\omega }^{\gimel }}_i=1$. Then, we have:

1. IVLDFFWA $\left({{\overline{⨿}}^{\gimel }}_1^c,{{\overline{⨿}}^{\gimel }}_2^c,\dots ,{{\overline{⨿}}^{\gimel }}_n^c\right)$ $={\left(IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)}^c$;

2. IVLDFFWG $\left({{\overline{⨿}}^{\gimel }}_1^c,{{\overline{⨿}}^{\gimel }}_2^c,\dots ,{{\overline{⨿}}^{\gimel }}_n^c\right)$ $={\left(IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)}^c$

Proof. 

1. For the left-hand side of the first equation, we have ${{\overline{⨿}}^{\gimel }}_i^c=$$\left(〈\left[{}^i{\alpha }^{\gimel },{}^i{\beta }^{\gimel }\right],\left[{}^i{\mu }^{\gimel },{}^1{\nu }^{\gimel }\right]〉\right.$, $\left.〈\left[{}^i{\psi }^{\gimel },{}^i{\eta }^{\gimel }\right],\left[{}^i{\gamma }^{\gimel },{}^i{\delta }^{\gimel }\right]〉\right)$ and

$IVIFFWA\left({{\overline{⨿}}^{\gimel }}_1^c,{{\overline{⨿}}^{\gimel }}_2^c,\dots ,{\tilde{x}}_n^c\right)$

$$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle \prod _{i=1}^n}{\left({\zeta }^{{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right).$$

For the right-hand side of the first equation,

${\left(IVIFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)}^c$

$$\begin{array}{c}={\left(\begin{array}{c}\langle \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)}^c\hfill \\ \\ =\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^1{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)\hfill \end{array}$$

Hence, we obtain

$$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1^c,{{\overline{⨿}}^{\gimel }}_2^c,\dots ,{{\overline{⨿}}^{\gimel }}_n^c\right)={\left(IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)\right)}^c.$$

Similarly, we obtain the second equation in the proof. □

5. Proposed Methodology

Let ${\overline{⨿}}^{\gimel }=\{{{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_m\}$ be the collection of alternatives, and let ${\wp }^{\partial }=\{{{\wp }^{\partial }}_1,{{\wp }^{\partial }}_2,\dots ,$${{\wp }^{\partial }}_n\}$ be the collection of criteria. DMs give a matrix according to their own standpoints $D={\left({\mathfrak{Y}}_{ij}\right)}_{m\times n}$, where ${\mathfrak{Y}}_{ij}$ is given for the alternatives ${{\overline{⨿}}^{\gimel }}_i\in {\overline{⨿}}^{\gimel }$ with respect to the attribute ${{\wp }^{\partial }}_j\in {\wp }^{\partial }$ by the DM. Normalization is not necessary if all performance indicators are of the same type; however, if MCDM has two types of performance indicators (beneficial types of indicators ${\tau }_b$ and cost types of indicators ${\tau }_c$), normalization is essential, and the matrix $D_{\left(p\right)}$ is transformed into a normalized matrix using the normalization formula $Y_{\left(p\right)}={\left({\aleph }_{ij}\right)}_{m\times n}$,

$${\left({\aleph }_{ij}\right)}_{m\times n}=\left\{\begin{array}{cc}{\left({\mathfrak{Y}}_{ij}\right)}^c;\hfill & j\in {\tau }_c\hfill \\ {\mathfrak{Y}}_{ij};\hfill & j\in {\tau }_b.\hfill \end{array}\right.$$

(20)

where ${\left({\mathfrak{Y}}_{ij}\right)}^c$ shows the complement of ${\mathfrak{Y}}_{ij}$.

The proposed AOs are incorporated into the MCDM, which necessitates the preceding procedures.

Decision-making algorithm

Step 1:

Acquire a decision matrix $D={\left({\mathfrak{Y}}_{ij}\right)}_{m\times n}$ in the form of IVLDFNs from the DM.

Step 2:

There is no need for normalization if all performance indicators are of an identical type; nevertheless, in MCDM, in many cases, there are two types of performance indicators. The matrix is turned into the transforming response matrix, in this case $Y={\left({\aleph }_{ij}\right)}_{m\times n}$, using the normalization formula in Equation (20).

Step 3:

This step deals with the application of AOs to the decision matrix to aggregate the IVLDF information for an alternative.

(a) Utilization of the IVLDFFWA operator to accumulate IVLDF information.

$IVLDFFWA\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$$=\left(\begin{array}{c}\langle \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right],\hfill \\ \\ \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)$$

(21)

or

(b) utilization of the IVLDFFWG operator to accumulate IVLDF information.

$IVLDFFWG\left({{\overline{⨿}}^{\gimel }}_1,{{\overline{⨿}}^{\gimel }}_2,\dots ,{{\overline{⨿}}^{\gimel }}_n\right)$

$$=\left(\begin{array}{c}\langle \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\mu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\nu }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\alpha }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\beta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle ,\hfill \\ \\ \langle \left[{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\gamma }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{{}^i{\delta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\hfill \\ \\ \left[1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\psi }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right),1-{log}_{\zeta }\left(1+{\displaystyle {\prod }_{i=1}^n}{\left({\zeta }^{1-{}^i{\eta }^{\gimel }}-1\right)}^{{{\omega }^{\gimel }}_i}\right)\right]\rangle \hfill \end{array}\right)$$

(22)

Step 4:

All cumulative alternative assessment scores are computed.

Step 5:

The alternatives are rated using the score feature, and the best option is chosen.

6. Application to Emergency Decision-Making

To demonstrate the potential application and significance of the presented approach, we present a real case study of an emergency caused by the pandemic of nCOVID-19 that occurred in Pakistan.

The virus was found in December 2019 in China, and the World Health Organization (WHO) declared it an epidemic disease on 11 March 2020. The WHO gave the virus the name “Novel Coronavirus (nCOVID-19)”. On 30 January 2020, the Director-General of the WHO declared the pandemic to be an international public health emergency. More than 217 million confirmed cases and 4.51 million confirmed deaths have been reported worldwide, as of 1 September 2020 [52].

The virus has a high potential for further spread. The WHO has declared the epidemic a public health emergency of global significance. There is no doubt that this disease has caused significant economic and environmental damage, as well as issues resulting from a lack of personal protective equipment (PPE). The ability to increase the PPE supply is limited, and current demand for masks and respirators will not be met, especially if widespread, improper PPE use continues. In the areas of clinical management, disease prevention and monitoring, and mathematical modeling, the WHO collaborates with public health experts and laboratory partnerships. In such situations, it is critical to provide an effective emergency response in order to avoid further losses and safeguard people’s health. In both healthcare and community settings, preventive and mitigation measures are critical. As a result of emergency decisions, health professionals must act quickly in order to save lives and effectively control the situation in order to prevent further deaths. In order to reduce the overall risk of this disease, six basic public health emergency factors must be considered:

• First aid training $\left({{\wp }^{\partial }}_1\right)$: Because this disease spreads quickly, it is necessary to train people or to avoid people who have symptoms of this disease, in order to control it. Individuals are strongly advised to take a fully supervised practical or online first aid course to learn how to respond to medical emergencies.
• Increased PPE $\left({{\wp }^{\partial }}_2\right)$: Another cause of problems is a scarcity of testing kits. The scenario will be ameliorated by manufacturing more testing kits, the elimination of confirmation requirements, and local governments’ decisions to eventually isolate all suspected cases. Masks, respirators, gloves, and gowns are being sent to places all over the world. Face masks offer only a limited level of protection in terms of keeping the virus from spreading. As a result, the simplest strategy to prevent spread is to practice proper personal hygiene. However, the global market for PPE is experiencing severe instability.
• Inter-city transportation banned $\left({{\wp }^{\partial }}_3\right)$: For the protection of residents, local governments should take action or issue an announcement prohibiting intra-city mobility, forcing patients to visit local clinics. In addition, all aircraft and subway services have been suspended, and all types of celebrations have been cancelled. Everyone must keep a gap of at least 3 feet between themselves and anyone who coughs or sneezes.
• Coordination and planning at the government level $\left({{\wp }^{\partial }}_4\right)$: To prepare for the heightened uncertainty caused by nCOVID-19, each government requires the highest level of collaboration from its provinces.
• Monitoring $\left({{\wp }^{\partial }}_5\right)$: Every country should appoint health and emergency decision makers to study and monitor the country’s current situation and provide recommendations on how to improve it.
• Clinical management $\left({{\wp }^{\partial }}_6\right)$: After the virus has spread, vaccination is a highly effective method of reducing the spread of lethal diseases. Vaccines are quite effective, with only a few severe side effects. Furthermore, no effective therapeutics exist for nCOVID-19. Clinical management demands the prompt implementation of permitted disease prevention and control methods, as well as assistance with complication management and strategic organ care as needed.

Assume there are four emergency alternatives: risk communication (RC), border and city lockdowns (BCL), healthcare systems (HS), and consulted specialists (CS). Health experts evaluated the information using IVLDFNs.

Emergency Decision-Making Using Frank Aggregation Operators

We have six criteria and four alternatives, as discussed above.

The algorithmic steps are calculated as follows.

Step 1:

Acquire a decision matrix $D={\left({\mathfrak{Y}}_{ij}\right)}_{m\times n}$ in the form of IVLDFNs, from the DM that is given in Figure 1.

Step 2:

Here, all benefit-type criteria are met; there is no requirement for normalization.

Step 3:

In this step, we use the IVLDFFWG operator to aggregate the IVLDF information for each alternative given in

Table 4. Aggregated IVLDF information.

Alternative	Aggregated Value
BCL	$\left(\langle [0.48201,0.58578],[0.47642,0.67946]\rangle ,\langle [0.17731,0.30362],[0.18192,0.25407]\rangle \right)$
RC	$\left(\langle [0.37256,0.54711],[0.38902,0.61873]\rangle ,\langle [0.19040,0.33623],[0.21347,0.30173]\rangle \right)$
HS	$\left(\langle [0.44127,0.52939],[0.41386,0.66965]\rangle ,\langle [0.30338,0.28840],[0.19082,0.20009\rangle \right)$
CS	$\left(\langle [0.29367,0.57377],[0.46052,0.57202]\rangle ,\langle [0.16136,0.29385],[0.20285,0.29702]\rangle \right)$

.

Step 4:

We compute all cumulative alternative assessment scores given in

Table 5. Score values of aggregated IVLDF information.

Alternative	Score Value
BCL	−0.0107843
RC	−0.0191628
HS	−0.2220026
CS	−0.0524384

.

Step 5:

The alternatives are rated using the score feature. Hence, BCL is the best choice according to the attributes.

7. Sensitivity Analysis Regarding the Parameter $\mathbf{\zeta }$

Several values may be assigned to the parameter $\zeta$ based on the DM’s preferences. We assigned $\zeta$ values ranging from 4 to 80 and computed the scores of these four alternatives to investigate the variance in the rankings of the four alternatives based on the value of the parameter $\zeta$.

Table 6. Sensitivity analysis regarding the parameter $\zeta$.

$\mathit{\zeta }$	Score Value				Ranking	Optimal Alternative
	BCL	RC	HS	CS
4	−0.01078	−0.01916	−0.22200	−0.052438	BCL ≻ HS ≻ RC ≻ CS	BCL
8	−0.00519	−0.01303	−0.12367	−0.04573	BCL ≻ HS ≻ RC≻ CS	BCL
12	−0.00229	−0.00978	−0.10328	−0.04227	BCL ≻ HS ≻ RC≻ CS	BCL
20	0.00096	0.00040	−0.096714	−0.03826	BCL ≻ HS ≻ RC≻ CS	BCL
40	0.00478	0.00516	−0.07539	−0.03356	HS ≻ BCL ≻ RC≻ CS	HS
80	0.00535	0.00629	−0.04405	−0.02956	HS ≻ BCL ≻ RC≻ CS	HS

summarizes the scores of the alternatives determined by the IVLDFFWG operator, and we can see that as the value of $\zeta$ increases from 4 to 80, the score of each alternative increases. When $\zeta \in [4,80]$, we can see that the ordering of the four choices is always the same, and BCL is always the best option. When $\zeta \in [4,80]$, we can observe that the ordering of the four alternatives is not always the same, and BCL is not always the best option. We can see that the IVLDFFWG operator increases monotonically with respect to the parameter $\zeta$ when $\zeta \in (1,+\infty )$. If we change the parameter $\zeta$, then the optimum alternative may be changed. In our example, when we put $\zeta =40$, the optimum solution changed. In Table 6, the fluctuations in the scores are readily seen in relation to the values of the parameter $\zeta$.

We set the parameter according to the requirements, for example, the risk factor in our case. If the decision maker determines that there is a high risk factor, the value of the parameter is large; if the decision maker determines that there is a low risk factor, the value of the parameter is small. Therefore, the decision maker determines the value of this parameter based on the nature of the situation. A doctor decides the value of this parameter in a medical diagnosis; if a medicine has many side effects, the value of this parameter will be larger, and if a medicine has few side effects, the value of this parameter will be smaller.

8. Comparison Analysis

This section contrasts the proposed AOs with the current AOs. Solving the data using pre-existing AOs, we obtain a similar optimal solution. This illustrates the long-term durability and usefulness of the AOs. The method described here is more practical and superior to many other existing AOs. A comparative analysis is given in

Table 7. Comparison of proposed operators with some existing operators.

Authors	AO	Ranking	Optimal Alternative
Zhang [12]	IVIFFWA	BCL ≻ RC ≻ HS ≻ CS	BCL
	IVIFFWG	BCL ≻ CS ≻ RC ≻ HS	BCL
Wu and Su [16]	IVIF-PHWA	BCL ≻ HS ≻ RC ≻ CS	BCL
Zhou and He [13]	IIFOPWA	BCL ≻ HS ≻ CS ≻ RC	BCL
	IIFOPWG	BCL ≻ CS ≻ RC ≻ HS	BCL
Meng at el. [10]	GBIVIFGC	BCL ≻ HS ≻ RC ≻ CS	BCL
Meng at el. [14]	IG-IVIFHSA	BCL ≻ CS ≻ HS ≻ RC	BCL
Proposed	GBIVIFGC	BCL ≻ HS ≻ RC ≻ CS	BCL

.

9. Conclusions

The notion of interval-valued linear Diophantine fuzzy sets provides more accurate information in emergency decision-making. We extended the concepts of FT${}_c$T${}_n$ to interval-valued linear Diophantine fuzzy numbers and developed operational rules for IVLDFNs. Based on these new operating principles, some novel AOs such as the IVLDFFWA and IVLDFFWG operators were developed, to handle scenarios where the given inputs are IVLDFNs. Various desirable properties of these AOs, as well as their interconnections, were thoroughly investigated. Finally, the developed AOs were used in an MCDM problem related to emergency decision-making during the nCOVID-19 pandemic, using interval-valued linear Diophantine fuzzy information. The sensitivity analysis showed that the parameter $\zeta$ reflects the DMs’ preferences and that the DMs can choose the appropriate values of $\zeta$ depending on their preferences. By varying the value of the parameter $\zeta$, we can obtain different SFs, diverse rankings of the alternatives, and numerous optimal alternatives. That is, the eventual optimal decisions may differ depending on the value of the parameter $\zeta$. As a result, combining the parameters with the developed AOs can give us more possibilities and flexibility than using the pre-existing AOs, because they allow us to select numerous values for the parameter. The current work does not address how to discover the parameter $\zeta$ for the suggested operators in practical circumstances, which is an exciting topic that should be investigated further in the future. Furthermore, in the future, we intend to use the suggested AOs and the technique in a wide range of practical applications, including image processing, game theory, pattern recognition, cluster analysis, and uncertain programming.