Linguistic interval-valued q-rung orthopair fuzzy TOPSIS method for decision making problem with incomplete weight

Abstract

The aim of this paper is to introduce the notion of linguistic interval-valued q-rung orthopair fuzzy set (LIVq-ROFS) as a generalization of linguistic q-rung orthopair fuzzy set. We develop some basic operations, score and accuracy functions to compare the LIVq-ROF values (LIVq-ROFVs). Based on the proposed operations a series of aggregation techniques to aggregate the LIVq-ROFVs and some of their desirable properties are discussed in detail. Moreover, a TOPSIS method is developed to solve a multi-criteria decision making (MCDM) problem under LIVq-ROFS setting. Furthermore, a MCDM approach is proposed based on the developed operators and TOPSIS method, then a practical decision making example is given in order to explain the proposed method. To illustrate to effectiveness and application of the proposed method a comparative study is also conducted.

Keywords

Linguistic interval-valued q-rung orthopair fuzzy set (LIVq-ROFS)LIVq-ROF aggregation operators TOPSIS method MCDM problem

1 Introduction

Fuzzy set (FS) presented by Zadeh [32] consists of membership degree and mathematically represented by a membership function assigning each membership degree (MD) belong to the interval [0, 1]. Due to the outstanding ability to model vagueness FS has gain attention. Several extensions of FS were made to deal with uncertainty, for instance, intuitionistic fuzzy set (IFS) [2], Pythagorean fuzzy set (PFS) [28], hesitant fuzzy set [28], Pythagorean hesitant fuzzy set [12], cubic Pythagorean fuzzy set [1], Pythagorean cubic fuzzy set [13], extended Pythagorean fuzzy set [3], q-rung orthopair fuzzy set (q-ROFS) [32, 33] etc. Since FS only consider the membership degree therefore, Atanassov generalized the concept of FS and presented the IFS, which is consisting by a MD and a non-membership degree (NMD) that fulfill the condition that MD + NMD ⩽ 1. Due to the effectiveness of the IFS numerous researchers have focused on IFS and applied the concept to many real-world multi-criteria decisions making (MCDM) problems [4, 19]. However, in many practical situation decision makers (DMs) may deal with the MD and NMD such that MD + NMD1 and thus cannot utilize the intuitionistic fuzzy information. To do this Yager [31] proposed the theory of PFSs and fulfill the condition that (MD) ² + (NMD) ² ⩽ 1. After the appearance of PFS, the concept achieved researcher’s attention and applied the concept to a verity of decision making problems. In Khan et al. [14] presented prioritized aggregation operators for Pythagorean fuzzy information, also Khan et al. [15] developed Einstein T-norm and T-conorm based operators to deal with MCDM problems under PF environment. Khan et al. [16, 17] developed interval-valued Pythagorean fuzzy Choquet integral operators. However, the PFN has also some restrictions on the scope of information. To beat this imperfection, Yager [9 10] initiated the notion of q-rung orthopair FSs (q-ROFSs) which fulfill the condition: a^q + b^q≤1, where a is the membership degree b is the non-membership degree and q ⩾ 1. In [11], Joshi et al. presented the idea of interval-valued q-ROFS (IVq-ROFS) as a generalization of q-ROFS and developed a MCDM approach in IVq-ROFS environment.

All the above concepts express only quantitative information; however, in many practical situations such as evaluation of the performance various kind of goods and stocks, and evaluating Universities faculty for tenure and promotion, in which the given information cannot be evaluated accurately by means of numerical values but may be in the form of words or sentences or in other word called linguistic variables (LVs). For instance, assessing the design or comfort of a car, the LVs, poor, fair and good are commonly be used. To overcome this limitation, the idea of linguistic term set (LTS) was first introduced by Zadeh [36, 37]. In recent years it is an important and interesting research topic that has been attracting more and more attention. Xu [29] extended the discrete LTS and introduced the notion of continuous LTS. Zhang et al. [39] developed an algorithm to characterize a linguistic distribution assessment (LDA) by means of a hesitant linguistic distribution (HLD). In dealing with larg-scale group decision making (LGDM) problem the authors developed an algorithm with multigranular unbalanced hesitant fuzzy linguistic information based on these algorithms. Yu et al. [34] proposed a consensus which considers the weighted vector of DMs and criteria on the basis of fuzzy envelope decision matrices. They devised an algorithm to help the DMs reach consensus in MCDM under the multi-granular hesitant fuzzy linguistic terms sets environment. Based on the logarithmic least square method Zhang et al. [40] developed two-sided matching MCDM with self-confidence and fuzzy preference. Extended the notion of IFS with linguistic term set Chen at al. [5] proposed the notion of linguistic IFS (LIFS), which is characterized by the linguistic membership degree (LMD) and linguistic non-membership degree (LND) such that for any linguistic intuitionistic fuzzy value s = (s_x, s_y), where s_p represents the linguistic variable p ∈ [0, l], x + y ⩽ l here l represent the cardinality of the LTS. After the appearance of LIFS, many researchers [20 , 38] have investigated researches to deal with MCDM problems under LIFS environment. However in the real decision making problem there may be situation in which the DMs gives their preference towards any object in the form linguistic intuitionistic fuzzy value (LIFV) as s = (s₅, s₆), where s_p represents the linguistic variable p ∈ [0, 8], then clearly this preference value cannot be able to solve with LIFS, as 5 + 6 > 8. Thus, under LIFS environment it is unable to make a decision during the MCDM process. Therefore, to overcome this situation Garg [6] introduced the concept of linguistic Pythagorean fuzzy set (LPFS) and is characterized by LMD an LNMD such that their square sum is less than or equal to the cardinality of the linguistic set. But the LPFS failed to apply if the DM gives their preference in the form of linguistic Pythagorean fuzzy value (LPFV) as s = (s₅, s₇), where s_p represents the linguistic variable p ∈ [0, 8], then clearly this preference value cannot be able to solve with LIFS, as 5² + 7² = 7464 = 8². Therefore Lui and Lui [23, 24] introduced the concept of linguistic q-rung orthopair fuzzy set (Lq-ROFS) as a generalization of LIFS and LPFS, which is characterized by the LMD and LNMD such that 5^q + 7^q ⩽ 8^q where q ⩾ 3 (in this case). The authors developed some aggregation operators to deal with decision making problem with linguistic q-rung orthopair fuzzy setting. Lin et al. [26] developed interactional partitioned Heronian mean operator to deal with multi-criteria decision-making problems. Based on linguistic scale function Liu et al. [25] proposed TOPSIS method under linguistic q-rung orthopair fuzzy set.

Moreover, in [8] Garg and Kumar introduced the concept of linguistic interval-valued intuitionistic fuzzy set (LIV-IFS) as a generalization of LIFS and developed aggregation technique to deal with MCDM problems. Further Garg [7] introduced the concept of linguistic interval-valued Pythagorean fuzzy set (LIVPFS). The author introduced some operational laws and developed aggregation operators to fuse the linguistic interval-valued Pythagorean fuzzy information. However in the real decision making problem there may be situation in which the DM gives their preference towards any object in the form linguistic interval-valued Pythagorean fuzzy value (LIVPFV) as ([s₅, s₆] , [s₄, s₇]), where s_p represents the linguistic variable p ∈ [0, 8], then clearly this preference value cannot be able to solve with LIVPFS, as 6² + 7² > 8². Therefore, in this situation the above mention procedures are unable to make a decision during the process of decision making. Therefore, there is a need to modify these theories to resolve these issues in a more effective way. Hence, to overcome this limitation in this paper by keeping the advantage of IVq-ROFS and LFS, we introduced the concept of linguistic interval-valued q-rung orthopair fuzzy set (LIVq-ROFS). In a MCDM problem when the membership degrees and non-membership degrees needs to be characterized by interval-valued linguistic term rather than linguistic term, LIVq-ROFS is a preferable choice because it has a great ability to handle ambiguous and imprecise information. However, the linguistic interval-valued q-rung orthopair fuzzy value (LIVq-ROFV) can keep the entire interval valued proposed by the DMs, that is it keeps more information about the DMs opinions, the information that is normally dismissed. We first define some basic operation such as sum, product, score function, accuracy functions and discuss some of their properties. Moreover, based on the operational laws we develop a series of aggregation operators such as linguistic interval-valued q-rung orthopair fuzzy weighted averaging (LIVq-ROFWA) operator, linguistic interval-valued q-rung orthopair fuzzy ordered weighted averaging (LIVq-ROFOWA), linguistic interval-valued q-rung orthopair fuzzy hybrid weighted averaging (LIVq-ROFHWA), linguistic interval-valued q-rung orthopair fuzzy weighted geometric (LIVq-ROFWG), linguistic interval-valued q-rung orthopair fuzzy ordered weighted geometric (LIVq-ROFOWG) and linguistic interval-valued q-rung orthopair fuzzy hybrid weighted geometric (LIVq-ROFHWG) to aggregate to linguistic interval-valued q-rung orthopair fuzzy information. Also, some desirable propertied of the proposed operators are investigated. We have also developed a MCDM approach based on TOPSIS method under LIV-q-ROF environment. The main objectives of the paper under LIV-ROF setting are:

To propose LIVq-ROFS and represent the preference of the DMs in terms of linguistic interval-valued numbers.

To propose basic operational laws of LIVq-ROFS, score and accuracy functions.

To develop some aggregation operators under LIVq-ROFS environment to aggregate the DMs opinions.

To develop TOPSIS method with generalized distance under LIVq-ROFS environment.

To develop a MCDM problem based on TOPSIS method with LIVq-ROF setting.

The remaining paper has organized in following:

In section 2 the concept of LIV-q-ROFS is introduced and some operational laws are initiated. Further to camper the LIVq-ROF values (LIVq-ROFVs) the concept of score function and accuracy function are introduced. Moreover, based on the proposed operational laws a series of LIVq-ROF aggregation operators are developed. In section 3 a MCDM approach with generalized distance measure and TOPSIS is proposed under LIVq-ROFS environment. In section 4 a numerical example is given in order to illustrate the proposed method. In section 5 the sensitivity analysis given. Concluding remark is in section 6.

2 Linguistic interval-valued q-rung orthopair fuzzy aggregation operators

In this section we present some basic definition and elementary operations, score and accuracy function to compare the LIVq-ROFVs.

2.1 Linguistic interval-valued q-rung orthopair fuzzy sets

Definition 1. Suppose $ϒ = ({\dot{a}}_{1}, {\dot{a}}_{2}, \dots, {\dot{a}}_{n})$ is any set and ${\bar{S}}_{[0, p]} = {{\tilde{s}}_{υ} | {\tilde{s}}_{0} ⩽ {\tilde{s}}_{υ} ⩽ {\tilde{s}}_{p}, υ \in [0, p]}$ is a LTS which is continuous. An LIVq-ROFS is defined in the finite universe of discourse A can be defined as follows: $L = {({\dot{a}}_{i}, {\tilde{s}}_{M} ({\dot{a}}_{i}), {\tilde{s}}_{N} ({\dot{a}}_{i})) | {\dot{a}}_{i} \in A},$ (1) where ${\tilde{s}}_{M} ({\dot{a}}_{i}) = [{\tilde{s}}_{m_{1}} ({\dot{a}}_{i}), {\tilde{s}}_{m_{2}} ({\dot{a}}_{i})],$ ${\tilde{s}}_{N} ({\dot{a}}_{i}) = [{\tilde{s}}_{n_{1}} ({\dot{a}}_{i}), {\tilde{s}}_{n_{2}} ({\dot{a}}_{i})]$ are all subset of [s₀, s_l] and are said to be the LMD and LNMD of the element ${\dot{a}}_{i} \in A$ to L. For any ${\dot{a}}_{i} \in A$ $({\tilde{s}}_{m_{1}} ({\dot{a}}_{i}))^{q} + ({\tilde{s}}_{n_{1}} ({\dot{a}}_{i}))^{q} ⩽ {\tilde{s}}_{p}^{q}$ (i.e. $m_{1}^{q} + n_{1}^{q} ⩽ p^{q}$ ) always satisfies for ${\dot{a}}_{i} \in A$ . We denote the pair $([{\tilde{s}}_{m_{1}} ({\dot{a}}_{i}), {\tilde{s}}_{m_{2}} ({\dot{a}}_{i})], [{\tilde{s}}_{n_{1}} ({\dot{a}}_{i}), {\tilde{s}}_{n_{2}} ({\dot{a}}_{i})])$ by $\tilde{ς} = ([{\tilde{s}}_{\hat{ϑ}}, {\tilde{s}}_{\hat{θ}}], [{\tilde{s}}_{\hat{ϕ}}, {\tilde{s}}_{\hat{φ}}])$ and called the linguistic interval-valued q-rung orthopair fuzzy value (LIVq-ROFV). The set of all LIVq-ROFVs is denoted by A_[0,p]. The linguistic indeterminacy degree of ${\dot{a}}_{i} \in A$ to L is denoted by $[π ({\dot{a}}_{i}), ψ ({\dot{a}}_{i})]$ and is defined by: $π ({\dot{a}}_{i}) = {\tilde{s}}_{(p^{q} - (\hat{θ} ({\dot{a}}_{i}))^{q} - (\hat{φ} ({\dot{a}}_{i}))^{q})^{\frac{1}{q}}}$ and $ψ ({\dot{a}}_{i}) = {\tilde{s}}_{(p^{q} - (\hat{ϑ} ({\dot{a}}_{i}))^{q} - (\hat{ϕ} ({\dot{a}}_{i}))^{q})^{\frac{1}{q}}}$

Let A_[0,p] be the set of all LIVq-ROFVs based on ${\bar{S}}_{[0, p]}$ and ${\tilde{ς}}_{j} = ([{\tilde{s}}_{{\hat{ϑ}}_{j}}, {\tilde{s}}_{{\hat{θ}}_{j}}], [{\tilde{s}}_{{\hat{ϕ}}_{j}}, {\tilde{s}}_{{\hat{φ}}_{j}}]) (j = 1, 2, 3)$ ∈A_[0,p], then the complement, union and intersection operations for LIVq-ROFVs are defined as follows:

1. ${\tilde{ς}}_{1}^{c} = ([{\tilde{s}}_{{\hat{ϕ}}_{1}}, {\tilde{s}}_{{\hat{φ}}_{1}}], [{\tilde{s}}_{{\hat{ϑ}}_{1}}, {\tilde{s}}_{{\hat{θ}}_{1}}])$

2. $\begin{matrix} {\tilde{ς}}_{1} \cup {\tilde{ς}}_{2} = ([{\tilde{s}}_{{\hat{ϑ}}_{1}}, {\tilde{s}}_{{\hat{θ}}_{1}}], [{\tilde{s}}_{{\hat{ϕ}}_{1}}, {\tilde{s}}_{{\hat{φ}}_{1}}]) \cup ([{\tilde{s}}_{{\hat{ϑ}}_{2}}, {\tilde{s}}_{{\hat{θ}}_{2}}], [{\tilde{s}}_{{\hat{ϕ}}_{2}}, {\tilde{s}}_{{\hat{φ}}_{2}}]) \\ = ([{\tilde{s}}_{{\hat{ϑ}}_{1}} \lor {\tilde{s}}_{{\hat{ϑ}}_{2}}, {\tilde{s}}_{{\hat{θ}}_{1}} \lor {\tilde{s}}_{{\hat{θ}}_{2}}], [{\tilde{s}}_{{\hat{ϕ}}_{1}} \land {\tilde{s}}_{{\hat{ϕ}}_{2}}, {\tilde{s}}_{{\hat{φ}}_{1}} \land {\tilde{s}}_{{\hat{φ}}_{2}}]) \end{matrix}$

3. $\begin{matrix} {\tilde{ς}}_{1} \cap {\tilde{ς}}_{2} = ([{\tilde{s}}_{{\hat{ϑ}}_{1}}, {\tilde{s}}_{{\hat{θ}}_{1}}], [{\tilde{s}}_{{\hat{ϕ}}_{1}}, {\tilde{s}}_{{\hat{φ}}_{1}}]) \cup ([{\tilde{s}}_{{\hat{ϑ}}_{2}}, {\tilde{s}}_{{\hat{θ}}_{2}}], [{\tilde{s}}_{{\hat{ϕ}}_{2}}, {\tilde{s}}_{{\hat{φ}}_{2}}]) \\ = ([{\tilde{s}}_{{\hat{ϑ}}_{1}} \land {\tilde{s}}_{{\hat{ϑ}}_{2}}, {\tilde{s}}_{{\hat{θ}}_{1}} \land {\tilde{s}}_{{\hat{θ}}_{2}}], [{\tilde{s}}_{{\hat{ϕ}}_{1}} \lor {\tilde{s}}_{{\hat{ϕ}}_{2}}, {\tilde{s}}_{{\hat{φ}}_{1}} \lor {\tilde{s}}_{{\hat{φ}}_{2}}]) \end{matrix}$

Further we develop the following operational laws to fuse the LIVq-ROFVs. For ${\tilde{ς}}_{j} = ([{\tilde{s}}_{{\hat{ϑ}}_{j}}, {\tilde{s}}_{{\hat{θ}}_{j}}], [{\tilde{s}}_{{\hat{ϕ}}_{j}}, {\tilde{s}}_{{\hat{φ}}_{j}}]) (j = 1, 2)$ ∈A_[0,l] we have,

1. ${\tilde{ς}}_{1} \oplus {\tilde{ς}}_{2} =$ $({\tilde{s}}_{[\sqrt[q]{{\hat{ϑ}}_{1}^{q} + {\hat{ϑ}}_{2}^{q} - \frac{{\hat{ϑ}}_{1}^{q} {\hat{ϑ}}_{2}^{q}}{p^{q}}}, \sqrt[q]{{\hat{θ}}_{1}^{q} + {\hat{θ}}_{2}^{q} - \frac{{\hat{θ}}_{1}^{q} {\hat{θ}}_{2}^{q}}{p^{q}}}]}, {\tilde{s}}_{[\frac{{\hat{ϕ}}_{1} {\hat{ϕ}}_{2}}{p}, \frac{{\hat{φ}}_{1} {\hat{φ}}_{2}}{p}]})$ ;

2. ${\tilde{ς}}_{1} \otimes {\tilde{ς}}_{2} =$ $({\tilde{s}}_{[\frac{{\hat{ϑ}}_{1} {\hat{ϑ}}_{2}}{p}, \frac{{\hat{θ}}_{1} {\hat{θ}}_{2}}{p}]}, {\tilde{s}}_{[\sqrt[q]{{\hat{ϕ}}_{1}^{q} + {\hat{ϕ}}_{2}^{q} - \frac{{\hat{ϕ}}_{1}^{q} {\hat{ϕ}}_{2}^{q}}{p^{q}}}, \sqrt[q]{{\hat{φ}}_{1}^{q} + {\hat{φ}}_{2}^{q} - \frac{{\hat{φ}}_{1}^{q} {\hat{φ}}_{2}^{q}}{p^{q}}}]})$

3. $λ {\tilde{ς}}_{1} =$ $({\tilde{s}}_{[p \sqrt[q]{1 - (1 - \frac{{\hat{ϑ}}_{1}^{q}}{p^{q}})^{λ}}, p \sqrt[q]{1 - (1 - \frac{{\hat{θ}}_{1}^{q}}{p^{q}})^{λ}}]}, {\tilde{s}}_{[p (\frac{{\hat{ϕ}}_{1}}{p})^{λ}, p (\frac{{\hat{φ}}_{1}}{p})^{λ}]}), λ ≻ 0,$

4. ${\tilde{ς}}_{1}^{λ} =$ $({\tilde{s}}_{[p (\frac{{\hat{ϑ}}_{1}}{p})^{λ}, p (\frac{{\hat{θ}}_{1}}{p})^{λ}]}, {\tilde{s}}_{[p \sqrt[q]{1 - (1 - \frac{{\hat{ϕ}}_{1}^{q}}{p^{q}})^{λ}}, p \sqrt[q]{1 - (1 - \frac{{\hat{φ}}_{1}^{q}}{p^{q}})^{λ}}]}), λ ≻ 0 .$

For the comparison between LIVq-ROFVs, the score and accuracy values can be defined by:

Definition 2. Let $\tilde{ς} = ([{\tilde{s}}_{\hat{ϑ}}, {\tilde{s}}_{θ}], [{\tilde{s}}_{\hat{ϕ}}, {\tilde{s}}_{\hat{φ}}])$ be a LIVq-ROFVs with ${\tilde{s}}_{\hat{ϑ}}, {\tilde{s}}_{\hat{φ}} \in \bar{S}$ . The score and accuracy values of $\tilde{ς}$ can be defined by: $Ω (\tilde{ς}) = {\tilde{s}}_{\sqrt[q]{\frac{p^{q} + {\hat{ϑ}}^{q} + {\hat{θ}}^{q} - {\hat{ϕ}}^{q} - {\hat{φ}}^{q}}{4}}}$

and $Ψ (\tilde{ς}) = {\tilde{s}}_{\sqrt[q]{\frac{{\hat{ϑ}}^{q} + {\hat{θ}}^{q} + {\hat{ϕ}}^{q} + {\hat{φ}}^{q}}{2}}}$

It is easily verified that $0 ⩽ \frac{p^{q} + {\hat{ϑ}}^{q} + {\hat{θ}}^{q} - {\hat{ϕ}}^{q} - {\hat{φ}}^{q}}{4} ⩽ p^{q}$ and $\frac{{\hat{ϑ}}^{q} + {\hat{θ}}^{q} + {\hat{ϕ}}^{q} + {\hat{φ}}^{q}}{2} ⩽ p^{q}$ , this means that ${\tilde{s}}_{\sqrt[q]{\frac{p^{q} + {\hat{ϑ}}^{q} + {\hat{θ}}^{q} - {\hat{ϕ}}^{q} - {\hat{φ}}^{q}}{4}}}, {\tilde{s}}_{\sqrt[q]{\frac{{\hat{ϑ}}^{q} + {\hat{θ}}^{q} + {\hat{ϕ}}^{q} + {\hat{φ}}^{q}}{2}}} \in \bar{S}$ .

Definition 3. Let ${\tilde{ς}}_{j} = ([{\tilde{s}}_{{\hat{ϑ}}_{1}}, {\tilde{s}}_{{\hat{θ}}_{j}}], [{\tilde{s}}_{{\hat{ϕ}}_{j}}, {\tilde{s}}_{{\hat{φ}}_{j}}]) (j = 1, 2)$ ∈A_[0,p],

If $Ω ({\tilde{ς}}_{1}) ≺ Ω ({\tilde{ς}}_{2})$ , then ${\tilde{ς}}_{1} ≺ {\tilde{ς}}_{2}$ ;

If $Ω ({\tilde{ς}}_{1}) ≻ Ω ({\tilde{ς}}_{2})$ , then ${\tilde{ς}}_{1} ≻ {\tilde{ς}}_{2}$ ;

If $Ω ({\tilde{ς}}_{1}) = Ω ({\tilde{ς}}_{2})$ , then

If $Ψ ({\tilde{ς}}_{1}) = Ψ ({\tilde{ς}}_{2})$ , then ${\tilde{ς}}_{1} = {\tilde{ς}}_{2}$ .

If $Ψ ({\tilde{ς}}_{1}) ≺ Ψ ({\tilde{ς}}_{2})$ , then ${\tilde{ς}}_{1} ≺ {\tilde{ς}}_{2}$ ;

If $Ψ ({\tilde{ς}}_{1}) ≻ Ψ ({\tilde{ς}}_{2})$ , then ${\tilde{ς}}_{1} ≻ {\tilde{ς}}_{2}$ .

In the following subsection based on operational laws we define aggregation operators to aggregate to LIVq-ROFVs and discussed some desirable properties.

2.2 Aggregation operators for LIVq-ROFS

Subsequently to fuse all the DM information and achieve the optimal alternative, the aggregation operators played a vital role. Thus, authors have developed some aggregation operators based on linguistic interval-valued q-rung orthopair fuzzy information.

Definition 4. Let ${\tilde{ς}}_{j} \in A_{[0, p]}$ (j = 1, 2, . . . , m) be the collection of LIVq-ROFVs, and ɛ = (ɛ₁, ɛ₂, . . . , ɛ_m) ^T is a weighted vector of ${\tilde{ς}}_{j}$ (j = 1, 2, . . . , m) with ɛ_j ∈ [0, 1] and $Σ_{j = 1}^{m} ɛ_{j}$ . Then,

1). The LIVq-ROFWA_ɛ operator is a mapping $LIVq - {ROFWA}_{ɛ} : {\tilde{ς}}_{j}^{m} \to {\tilde{ς}}_{j}$ of dimension m, and $\begin{matrix} LIVq - {ROFWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \oplus_{j = 1}^{m} ɛ_{j} {\tilde{ς}}_{j} \\ = (\begin{matrix} {\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}]}, \\ {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϕ}}_{j}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{φ}}_{j}}{p})^{ɛ_{j}}]} \end{matrix}) \end{matrix}$ (2)

2). The LIVq-ROFOWA_ɛ operator is a mapping $LIVq - {ROFOWA}_{ɛ} : {\tilde{ς}}_{j}^{m} \to {\tilde{ς}}_{j}$ of dimension m, and

$\begin{matrix} LIVq - {ROFOWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \oplus_{j = 1}^{m} ɛ_{j} {\tilde{ς}}_{\dot{σ} (j)} \\ = (\begin{matrix} {\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{\dot{σ} (j)}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{\dot{σ} (j)}^{q}}{p^{q}})^{ɛ_{j}}}]}, \\ {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϕ}}_{\dot{σ} (j)}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{φ}}_{\dot{σ} (j)}}{p})^{ɛ_{j}}]} \end{matrix}) \end{matrix}$ (3)

Where $\dot{σ} (j)$ (j = 1,2, ... ,m) represents an arrangement of (1,2,..., m), such that ${\tilde{ς}}_{\dot{σ} (j - 1)} ⩾ {\tilde{ς}}_{\dot{σ} (j)}$ ∀j.

3). The LIVq-ROFHWA operator is a mapping $LIVq - {ROFHWA}_{ɛ, χ} : {\tilde{ς}}_{j}^{m} \to {\tilde{ς}}_{j}$ of dimension m, which has associated weighting vector χ = (χ₁, χ₂, . . . , χ_m) ^T such that χ_j ∈ [0, 1], $Σ_{j = 1}^{m} χ_{j} = 1$ such that

$\begin{matrix} LIVq - {ROFOHWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \oplus_{j = 1}^{m} ɛ_{j} {\tilde{ς}}_{\dot{σ} (j)}^{'} \\ = (\begin{matrix} {\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{({\hat{ϑ}}_{\dot{σ} (j)}^{'})^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{({\hat{θ}}_{\dot{σ} (j)}^{'})^{q}}{p^{q}})^{ɛ_{j}}}]}, \\ {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϕ}}_{\dot{σ} (j)}^{'}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{φ}}_{\dot{σ} (j)}^{'}}{p})^{ɛ_{j}}]} \end{matrix}) \end{matrix}$ (4)

Where ${\tilde{ς}}_{\dot{σ} (j)}^{'} = m ɛ_{j} {\tilde{ς}}_{j}$ , m = balancing coefficient and $\dot{σ} (j)$ (j = 1,2, ... ,m) represents the arrangement of (1,2,...,m), such that ${\tilde{ς}}_{\dot{σ} (j - 1)}^{'} ⩾ {\tilde{ς}}_{\dot{σ} (j)}^{'}$ ∀j.

Definition 5. Let ${\tilde{ς}}_{j} \in A_{[0, p]}$ (j = 1, 2, . . . , m) be the collection of LIVq-ROFVs and ɛ = (ɛ₁, ɛ₂, . . . , ɛ_m) ^T is a weighted vector of ${\tilde{ς}}_{j}$ (j = 1, 2, . . . , m) with ɛ_j ∈ [0, 1] and $Σ_{j = 1}^{m} ɛ_{j}$ . Then,

1). The LIVq-ROFWG_ɛ operator is a mapping $LIVq - {ROFWG}_{ɛ} : {\tilde{ς}}_{j}^{m} \to {\tilde{ς}}_{j}$ of dimension m, and $\begin{matrix} LIVq - {ROFWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \otimes_{j = 1}^{m} {\tilde{ς}}_{j}^{ɛ_{j}} \\ = (\begin{matrix} {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϑ}}_{j}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{θ}}_{j}}{p})^{ɛ_{j}}]}, \\ {\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϕ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{φ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}]} \end{matrix}) \end{matrix}$ (5)

2). The LIVq-ROFOWG_ɛ operator is a mapping $LIVq - {ROFOWG}_{ɛ} : {\tilde{ς}}_{j}^{m} \to {\tilde{ς}}_{j}$ of dimension m, and $\begin{matrix} LIVq - {ROFOWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \otimes_{j = 1}^{m} {\tilde{ς}}_{\dot{σ} (j)}^{ɛ_{j}} \\ = (\begin{matrix} {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϑ}}_{\dot{σ} (j)}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{θ}}_{\dot{σ} (j)}}{p})^{ɛ_{j}}]}, \\ {\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϕ}}_{\dot{σ} (j)}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{φ}}_{\dot{σ} (j)}^{q}}{p^{q}})^{ɛ_{j}}}]} \end{matrix}) \end{matrix}$ (6)

Where $\dot{σ} (j)$ (j = 1,2, ... ,m) represents the arrangement of (1,2,..., m), such that ${\tilde{ς}}_{\dot{σ} (j - 1)} ⩾ {\tilde{ς}}_{\dot{σ} (j)}$ ∀j.

3). The LIVq-ROFHWA operator is a mapping $LIVq - {ROFHWG}_{ɛ, χ} : {\tilde{ς}}_{j}^{m} \to {\tilde{ς}}_{j}$ of dimension m, which has associated weighting vector χ = (χ₁, χ₂, . . . , χ_m) ^T such that χ_j ∈ [0, 1], $Σ_{j = 1}^{m} χ_{j} = 1$ such that

$\begin{matrix} LIVq - {ROFOHWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \otimes_{j = 1}^{m} ({\tilde{ς}}_{\dot{σ} (j)}^{'})^{ɛ_{j}} \\ = (\begin{matrix} {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϑ}}_{\dot{σ} (j)}^{'}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{θ}}_{\dot{σ} (j)}^{'}}{p})^{ɛ_{j}}]}, \\ {\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{({\hat{ϕ}}_{\dot{σ} (j)}^{'})^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{φ}}_{\dot{σ} (j)}^{'})^{q}}{p^{q}})^{ɛ_{j}}}]} \end{matrix}) \end{matrix}$ (7)

Where ${\tilde{ς}}_{\dot{σ} (j)}^{'} = ({\tilde{ς}}_{j})^{m ɛ_{j}}$ , m = balancing coefficient and $\dot{σ} (j)$ (j = 1,2, ... ,m) represents an arrangement of (1,2,...,m), such that ${\tilde{ς}}_{\dot{σ} (j - 1)}^{'} ⩾ {\tilde{ς}}_{\dot{σ} (j)}^{'}$ ∀j.

In the following we present some properties of the developed operators.

Theorem 1. Suppose ${\tilde{ς}}_{j} \in A_{[0, p]}$ (1,2,..., m) are the LIVq-ROFVs and ɛ = (ɛ₁, ɛ₂, . . . , ɛ_m) ^T a weight vector of ${\tilde{ς}}_{j}$ (j = 1, 2, . . . , m) with ɛ_j ∈ [0, 1] and $Σ_{j = 1}^{m} ɛ_{j}$ .

1) (Idempotency) If all ${\tilde{ς}}_{j}$ (j = 1, 2, . . . , m) are equal, i.e. ${\tilde{ς}}_{j} = \tilde{ς}$ for all j. Then

$\begin{matrix} {LIVq - ROFWA}_{ɛ} / {LIVq - ROFWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \tilde{ς} \\ {LIVq - ROFOWA}_{ɛ} / {LIVq - ROFOWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \tilde{ς} \\ {LIVq - ROFHWA}_{ɛ} / {LIVq - ROFHWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) = \tilde{ς} \end{matrix}$

2) (Monotonicity) Let ${\tilde{ς}}_{j}^{*} \in A_{[0, p]}$ (j = 1, 2, . . . , m) be collections of LIVq-ROFVs. If ${\tilde{s}}_{{\hat{ϑ}}_{j}} ⩽ {\tilde{s}}_{{\hat{ϑ}}_{j}^{*}}$ and ${\tilde{s}}_{{\hat{φ}}_{j}} ⩾ {\tilde{s}}_{{\hat{φ}}_{j}^{*}}$ for all j, then

$\begin{matrix} \begin{matrix} {LIVq - ROFWA}_{ɛ} / {LIVq - ROFWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) \\ ⩽ {LIVq - ROFWA}_{ɛ} / {LIVq - ROFWG}_{ɛ} ({\tilde{ς}}_{1}^{*}, {\tilde{ς}}_{2}^{*}, . . ., {\tilde{ς}}_{m}^{*}) \end{matrix} \\ \begin{matrix} {LIVq - ROFOWA}_{ɛ} / {LIVq - ROFOWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) \\ ⩽ {LIVq - ROFOWA}_{ɛ} / {LIVq - ROFOWG}_{ɛ} ({\tilde{ς}}_{1}^{*}, {\tilde{ς}}_{2}^{*}, . . ., {\tilde{ς}}_{m}^{*}) \end{matrix} \\ \begin{matrix} {LIVq - ROFHWA}_{ɛ} / {LIVq - ROFHWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) \\ ⩽ {LIVq - ROFHWA}_{ɛ} / {LIVq - ROFHWG}_{ɛ} ({\tilde{ς}}_{1}^{*}, {\tilde{ς}}_{2}^{*}, . . ., {\tilde{ς}}_{m}^{*}) \end{matrix} \end{matrix}$

3) (Boundary) Let ${\tilde{ς}}^{-} = (\land_{j} ({\tilde{s}}_{{\hat{ϑ}}_{j}}), \lor_{j}^{(} {\tilde{s}}_{{\hat{φ}}_{j}}))$ , ${\tilde{ς}}^{+} = (\lor_{j} ({\tilde{s}}_{{\hat{ϑ}}_{j}}), \land_{j}^{(} {\tilde{s}}_{{\hat{φ}}_{j}}))$ . Then

$\begin{matrix} {\tilde{ς}}^{-} ⩽ {LIVq - ROFWA}_{ɛ} / {LIVq - ROFWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩽ {\tilde{ς}}^{+} \\ {\tilde{ς}}^{-} ⩽ {LIVq - ROFOWA}_{ɛ} / {LIVq - ROFOWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩽ {\tilde{ς}}^{+} \\ {\tilde{ς}}^{-} ⩽ {LIVq - ROFHWA}_{ɛ} / {LIVq - ROFHWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩽ {\tilde{ς}}^{+} \end{matrix}$

Proof. Since ${\tilde{ς}}_{j} \in A_{[0, p]}$ (j = 1, 2, ⋯ , m) is a collection of LIVq-ROFVs which implies that ${\tilde{s}}_{{\hat{ϑ}}_{j}}, {\tilde{s}}_{{\hat{φ}}_{j}} \in {\bar{S}}_{[0, p]}$ and ${\hat{θ}}_{j}^{q} + {\hat{φ}}_{j}^{q} ⩽ p^{q}$ . Then

1) If ${\tilde{ς}}_{j}$ , (j = 1, 2, ⋯ , m) are equal, i.e. ${\tilde{ς}}_{j} = \tilde{ς}$ for all j, then

$\begin{matrix} {LIVq - ROFWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) \\ = ({\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{j}^{q}}{l^{q}})^{ɛ_{j}}}]}, {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϕ}}_{j}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{φ}}_{j}}{p})^{ɛ_{j}}]}) \end{matrix}$ $= ({\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}^{q}}{p^{q}})^{ɛ_{j}}}]}, {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{\hat{ϕ}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{\hat{φ}}{p})^{ɛ_{j}}]})$ $= (\begin{matrix} {\tilde{s}}_{[p \sqrt[p]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}^{q}}{p^{q}})^{Σ_{j = 1}^{m} ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}^{q}}{p^{q}})^{Σ_{j = 1}^{m} ɛ_{j}}})^{\frac{1}{q}}]}, \\ {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{\hat{ϕ}}{p})^{Σ_{j = 1}^{m} ɛ_{j}}, Π_{j = 1}^{m} p (\frac{\hat{φ}}{p})^{Σ_{j = 1}^{m} ɛ_{j}}]} \end{matrix})$ $= ({\tilde{s}}_{[p \sqrt[q]{1 - (1 - \frac{{\hat{ϑ}}^{q}}{p^{q}})}, p \sqrt[q]{1 - (1 - \frac{{\hat{θ}}^{q}}{p^{q}})}]}, {\tilde{s}}_{[p (\frac{\hat{ϕ}}{p}), p (\frac{\hat{φ}}{p})]}) = ({\tilde{s}}_{[\hat{ϑ}, \hat{θ}]}, {\tilde{s}}_{[\hat{ϕ}, \hat{φ}]})$

2) If ${\tilde{s}}_{{\hat{ϑ}}_{j}} ⩽ {\tilde{s}}_{{\hat{ϑ}}_{j}^{*}}, {\tilde{s}}_{{\hat{θ}}_{j}} ⩽ {\tilde{s}}_{{\hat{θ}}_{j}^{*}}$ for all j it implies that ${\hat{ϑ}}_{j} ⩽ {\hat{ϑ}}_{j}^{*}, {\hat{θ}}_{j} ⩽ {\hat{θ}}_{j}^{*},$ then we have

$\begin{matrix} \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}} ⩾ \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}}, \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}} ⩾ \frac{{\hat{θ}}_{j}^{q}}{p^{q}} \\ \Rightarrow 1 - \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}} ⩽ 1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}}, 1 - \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}} ⩽ 1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}} \\ \Rightarrow (1 - \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}} ⩽ (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}, 1 - \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}} ⩽ (1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}})^{ɛ_{j}} \\ \begin{matrix} \Rightarrow Π_{j = 1}^{m} (1 - \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}} ⩽ Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}, \\ Π_{j = 1}^{m} (1 - \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}} ⩽ Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}})^{ɛ_{j}} \end{matrix} \\ \begin{matrix} \Rightarrow 1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}} ⩽ 1 - Π_{j = 1}^{m} (1 - \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}}, \\ 1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}})^{ɛ_{j}} ⩽ 1 - Π_{j = 1}^{m} (1 - \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}} \end{matrix} \\ \begin{matrix} \Rightarrow p^{q} (1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}) ⩽ p^{q} (1 - Π_{j = 1}^{m} (1 - \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}}), \\ p^{q} (1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}) ⩽ p^{q} (1 - Π_{j = 1}^{m} (1 - \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}}) \end{matrix} \end{matrix}$

Also ${\tilde{s}}_{{\hat{ϕ}}_{j}} ⩾ {\tilde{s}}_{{\hat{ϕ}}_{j}^{*}}, {\tilde{s}}_{{\hat{φ}}_{j}} ⩾ {\tilde{s}}_{{\hat{φ}}_{j}^{*}}$ for all j, this implies that ${\hat{ϕ}}_{j} ⩾ {\hat{ϕ}}_{j}^{*}, {\hat{φ}}_{j} ⩾ {\hat{φ}}_{j}^{*}$ , then we have, $\frac{{\hat{ϕ}}_{j}}{p} ⩾ \frac{{\hat{ϕ}}_{j}^{*}}{p}, \frac{{\hat{φ}}_{j}}{p} ⩾ \frac{{\hat{φ}}_{j}^{*}}{p}$

$\Rightarrow Π_{j = 1}^{m} (\frac{{\hat{ϕ}}_{j}}{p})^{ɛ_{j}} ⩾ Π_{j = 1}^{m} (\frac{{\hat{ϕ}}_{j}^{*}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} (\frac{{\hat{φ}}_{j}}{p})^{ɛ_{j}} ⩾ Π_{j = 1}^{m} (\frac{{\hat{φ}}_{j}^{*}}{p})^{ɛ_{j}}$

Therefore by Definition (5) we have $\begin{matrix} ({\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{ϑ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{{\hat{θ}}_{j}^{q}}{p^{q}})^{ɛ_{j}}}]}, {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϕ}}_{j}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{ϑ}}_{j}}{p})^{ɛ_{j}}]}) \\ ⩽ ({\tilde{s}}_{[p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{({\hat{ϑ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}}}, p \sqrt[q]{1 - Π_{j = 1}^{m} (1 - \frac{({\hat{θ}}_{j}^{*})^{q}}{p^{q}})^{ɛ_{j}}}]}, {\tilde{s}}_{[Π_{j = 1}^{m} p (\frac{{\hat{ϕ}}_{j}^{*}}{p})^{ɛ_{j}}, Π_{j = 1}^{m} p (\frac{{\hat{φ}}_{j}^{*}}{p})^{ɛ_{j}}]}) \end{matrix}$

Hence ${LIVq - ROFWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩽ LIVq - {ROFWA}_{ɛ} ({\tilde{ς}}_{1}^{*}, {\tilde{ς}}_{2}^{*}, . . ., {\tilde{ς}}_{m}^{*})$

3) As $\land_{j} ({\tilde{s}}_{{\hat{ϑ}}_{j}}) ⩽ {\tilde{s}}_{{\hat{ϑ}}_{j}} ⩽ \lor_{j}^{(} {\tilde{s}}_{{\hat{ϑ}}_{j}})$ , $\land_{j} ({\tilde{s}}_{{\hat{θ}}_{j}}) ⩽ {\tilde{s}}_{{\hat{θ}}_{j}} ⩽ \lor_{j} ({\tilde{s}}_{{\hat{θ}}_{j}})$ , $\lor_{j} ({\tilde{s}}_{{\hat{ϕ}}_{j}}) ⩽ {\tilde{s}}_{{\hat{ϕ}}_{j}} ⩽ \land_{j} ({\tilde{s}}_{{\hat{ϕ}}_{j}})$ , $\lor_{j} ({\tilde{s}}_{{\hat{φ}}_{j}}) ⩽ {\tilde{s}}_{{\hat{φ}}_{j}} ⩽ \land_{j} ({\tilde{s}}_{{\hat{φ}}_{j}})$ for all j. Then by Part 1 and Part 2 we have ${\tilde{ς}}^{-} ⩽ LIV {q - ROFWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩽ {\tilde{ς}}^{+}$

This completes the proof.

Lemma 1. [30] Let $a_{j} ⩾ 0, {\tilde{b}}_{j} > 0 (j = 1, 2, . . ., m)$ , $\sum_{j = 1}^{m} {\tilde{b}}_{j} = 1, then \prod_{j = 1}^{m} {({\tilde{a}}_{j})}^{{\tilde{b}}_{j}} ⩽ \sum_{j = 1}^{m} {\tilde{b}}_{j} {\tilde{a}}_{j}$ where equality holds only if ${\tilde{ς}}_{1} = {\tilde{ς}}_{2} = . . . = {\tilde{ς}}_{m}$ .

Based on Lemma 1, we have derived the following theorem:

Theorem 2. Let ${\tilde{ς}}_{j} \in A_{[0, p]}$ (j = 1, 2, ⋯ , m) be the collection of L-q-ROFVs. Then

1) ${LIVq - ROFWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩾ LIVq - {ROFWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m})$

2) ${LIVq - ROFOWA}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩾ LIVq - {ROFOWG}_{ɛ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m})$

3) ${LIVq - ROFHWA}_{ɛ, χ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m}) ⩾ LIVq - {ROFHWG}_{ɛ, χ} ({\tilde{ς}}_{1}, {\tilde{ς}}_{2}, . . ., {\tilde{ς}}_{m})$

where equality holds only if ${\tilde{ς}}_{1} = {\tilde{ς}}_{2} = . . . = {\tilde{ς}}_{m}$ .

3 TOPSIS method for Linguistic interval-valued q-rung Orthopair fuzzy

In this section we describe a MCDM model based on TOPSIS to solve a faculty selection decision making problem with LIVq-ROFSs setting. The MCDM problem may be expressed in the form of a decision matrix (DM), where the columns represent the set of criteria and the rows represent alternatives. Thus, for DM A_m×n, consider a set of m alternatives and n criteria. The unknown weight vector of k DMsis denoted by χ = (χ₁, χ₂, . . . , χ_k) ^T such that χ_j ∈ [0, 1], $Σ_{t = 1}^{k} χ_{t} = 1$ and unknown weight vector of n criteria is denoted by ɛ = (ɛ₁, ɛ₂, ⋯ , ɛ_n) ^T with ɛ_j ∈ [0, 1], $Σ_{j = 1}^{n} ɛ_{j} = 1$ . The linguistic interval-valued q-rung orthopair decision matrices (LIVq-ROF-DMs) provided by the DMs is denoted by ${\bar{A}}^{(k)} = {([{\tilde{s}}_{{\hat{ϑ}}_{ij}}, {\tilde{s}}_{{\hat{θ}}_{ij}}]^{(k)}, [{\tilde{s}}_{{\hat{ϕ}}_{ij}}, {\tilde{s}}_{{\hat{φ}}_{ij}}]^{(k)})}_{m \times n}$ . Since criteria are of two type (i) cost and (ii) benefit. In the decision making process if criteria is of cost type than we will change the criteria into benefit type by normalizing the LIVq-ROF-DMs ${\bar{A}}^{* (k)} = ([{\tilde{s}}_{{\hat{ϑ}}_{ij}}^{*}, {\tilde{s}}_{{\hat{θ}}_{ij}}^{*}]^{(k)}, {[{\tilde{s}}_{{\hat{ϕ}}_{ij}}^{*}, {\tilde{s}}_{{\hat{φ}}_{ij}}^{*}]^{(k)})}_{m \times n}$ as follows:

$\begin{matrix} ([{\tilde{s}}_{{\hat{ϑ}}_{ij}}^{*}, {\tilde{s}}_{{\hat{θ}}_{ij}}^{*}]^{(k)}, [{\tilde{s}}_{{\hat{ϕ}}_{ij}}^{*}, {\tilde{s}}_{{\hat{φ}}_{ij}}^{*}]^{(k)}) \\ = {\begin{matrix} \begin{matrix} [{\tilde{s}}_{{\hat{ϑ}}_{ij}}, {\tilde{s}}_{{\hat{θ}}_{ij}}]^{(k)}, [{\tilde{s}}_{{\hat{ϕ}}_{ij}}, {\tilde{s}}_{{\hat{φ}}_{ij}}]^{(k)} \\ for benefit criteria, i = 1, 2, . . ., m; j = 1, 2, . . ., n . \end{matrix} \\ \begin{matrix} ([{\tilde{s}}_{{\hat{ϑ}}_{ij}}, {\tilde{s}}_{{\hat{θ}}_{ij}}]^{(k)}, [{\tilde{s}}_{{\hat{ϕ}}_{ij}}, {\tilde{s}}_{{\hat{φ}}_{ij}}]^{(k)})^{c} \\ for cost criteria, i = 1, 2, . . ., m; j = 1, 2, . . ., n . \end{matrix} \end{matrix} \end{matrix}$

where $([{\tilde{s}}_{{\hat{ϑ}}_{ij}}, {\tilde{s}}_{{\hat{θ}}_{ij}}]^{(k)}, [{\tilde{s}}_{{\hat{ϕ}}_{ij}}, {\tilde{s}}_{{\hat{φ}}_{ij}}]^{(k)})^{c}$ is the complement of $([{\tilde{s}}_{{\hat{ϑ}}_{ij}}, {\tilde{s}}_{{\hat{θ}}_{ij}}]^{(k)}, [{\tilde{s}}_{{\hat{ϕ}}_{ij}}, {\tilde{s}}_{{\hat{φ}}_{ij}}]^{(k)})$ .

Steps of the developed approach are as under:

Step 1. Construct LIVq-ROF decision matrices.

Step 2. When the weight of the DMs is given then utilize them. If not, then using the method of computing the DMs as discussed in [27].

Step 3. Apply the developed LIVq-ROF operators to aggregate the LIVq-ROFVs.

Step 4. When the weights of the criteria are given than utilize. If not, then use the method of optimization problem to compute the weight [10]: ${\begin{matrix} max (T_{i} (ɛ)) = Σ_{j = 1}^{n} ɛ T_{ij} \\ s . t ɛ = Δ \\ Σ_{j = 1}^{n} ɛ_{j} = 1 \end{matrix}$

Where T_ij = S (K_ij) is the score values K_ij. Δ represents all possible weight sets that can be determined by the known weight information. In general, there are several kinds of relationships among the weights of attributes as follows [18]: ${\begin{matrix} {ɛ_{i} ⩾ ɛ_{j}} \\ {ɛ_{i} - ɛ_{j} ⩾ γ_{i} (> 0)} \\ {ɛ_{i} ⩾ γ_{i} ɛ_{j}}, 0 ⩽ γ_{i} ⩽ 1 \\ {δ_{i} ⩽ ɛ_{i} ⩽ δ_{i} + γ_{i}}, 0 ⩽ δ_{i} ⩽ δ_{i} + γ_{i} ⩽ 1 \\ {ɛ_{i} - ɛ_{j} ⩾ ɛ_{k} - ɛ_{l}}, for j \neq k \neq l \end{matrix}$

By solving the above model, the optimal weight solution corresponding to the alternative K_i is obtained: ɛ = (ɛ₁, ɛ₂, ⋯ , ɛ_n) ^T.

Step 5. Calculated the linguistic interval-valued q-rung Orthopair fuzzy positive ideal solution (LIVq-ROF-PIS), denoted ${\tilde{ς}}^{+} = ({\tilde{ς}}_{1}^{+}, {\tilde{ς}}_{2}^{+}, . . ., {\tilde{ς}}_{n}^{+})$ , and the linguistic interval-valued q-rung Orthopair fuzzy negative ideal solution (LIVq-ROF-NIS), denoted ${\tilde{ς}}^{-} = ({\tilde{ς}}_{1}^{-}, {\tilde{ς}}_{2}^{-}, . . ., {\tilde{ς}}_{n}^{-})$ by utilizing the following equations are defined as follows: $\begin{matrix} {\tilde{ς}}^{+} = ([{\tilde{s}}_{{\hat{ϑ}}_{max_{j}}}, {\tilde{s}}_{{\hat{θ}}_{max_{j}}}], [{\tilde{s}}_{{\hat{ϕ}}_{min_{j}}}, {\tilde{s}}_{{\hat{φ}}_{min_{j}}}]) \\ = (\begin{matrix} [\lor_{1 ⩽ i ⩽ m} {\tilde{s}}_{{\hat{ϑ}}_{ij}}, \lor_{1 ⩽ i ⩽ m} {\tilde{s}}_{{\hat{θ}}_{ij}}], \\ [\land_{1 ⩽ i ⩽ m} {\tilde{s}}_{{\hat{ϕ}}_{ij}}, \land_{1 ⩽ i ⩽ m} {\tilde{s}}_{{\hat{φ}}_{ij}}] \end{matrix}) \end{matrix}$ (8) and $\begin{matrix} {\tilde{ς}}^{-} = ([{\tilde{s}}_{{\hat{ϑ}}_{min_{j}}}, {\tilde{s}}_{{\hat{θ}}_{min_{j}}}], [{\tilde{s}}_{{\hat{ϕ}}_{max_{j}}}, {\tilde{s}}_{{\hat{φ}}_{max_{j}}}]) \\ = (\begin{matrix} [\land_{1 ⩽ i ⩽ m} s_{ϑ_{ij}}, \land_{1 ⩽ i ⩽ m} s_{θ_{ij}}], \\ [\lor_{1 ⩽ i ⩽ m} s_{ϕ_{ij}}, \lor_{1 ⩽ i ⩽ m} s_{φ_{ij}}] \end{matrix}) \end{matrix}$ (9)

Step 6. Let ${\tilde{ς}}_{j} = ([{\tilde{s}}_{{\hat{ϑ}}_{j}}, {\tilde{s}}_{{\hat{θ}}_{j}}], [{\tilde{s}}_{{\hat{ϕ}}_{j}}, {\tilde{s}}_{{\hat{φ}}_{j}}]) \in A_{[0, p]}$ (j = 1, 2, ⋯ , m) and ${\tilde{ξ}}_{j} = ([{\tilde{s}}_{{\hat{ϑ}}_{j}^{'}}, {\tilde{s}}_{{\hat{θ}}_{j}^{'}}], [{\tilde{s}}_{{\hat{ϕ}}_{j}^{'}}, {\tilde{s}}_{{\hat{φ}}_{j}^{'}}]) \in A_{[0, p]}$ (j = 1, 2, ⋯ , m) be two collections of LIVq-ROFSs. Then linguistic interval-valued q-rung Orthopair fuzzy generalized distance between ${\tilde{ς}}_{j}, {\tilde{ξ}}_{j} (j = 1, 2, . . ., m)$ is denoted by $d ({\tilde{ς}}_{j}, {\tilde{ξ}}_{j})$ and is defined by

$\begin{matrix} d ({\tilde{ς}}_{j}, {\tilde{ξ}}_{j}) \\ = {(\begin{matrix} \frac{1}{4 (p^{q})^{δ}} [({\hat{ϑ}}_{j}^{q} - {\hat{ϑ}}^{'}_{j}^{q})^{δ} + ({\hat{θ}}_{j}^{q} - {\hat{θ}}^{'}_{j}^{q})^{δ} + ({\hat{π}}_{j}^{q} - {\hat{π}}^{'}_{j}^{q})^{δ} \\ + ({\hat{ϕ}}_{j}^{q} - {\hat{ϕ}}^{'}_{j}^{q})^{δ} + ({\hat{φ}}_{j}^{q} - {\hat{φ}}^{'}_{j}^{q})^{δ} + ({\hat{ψ}}_{j}^{q} - {\hat{ψ}}^{'}_{j}^{q})^{δ}] \end{matrix})}^{\frac{1}{δ}} \end{matrix}$ (10)

If we put δ = 2, then the above distance become Euclidean distance between LIVq-ROFSs. If we put δ = 1, then the above distance become Hamming distance between LIVq-ROFSs.

According to generalized linguistic interval-valued q-rung orthopair fuzzy distance, calculate the distance between the alternative K_i and the LIVq-ROF-PIS ${\tilde{ς}}^{+}$ : $\begin{matrix} d_{j}^{+} ({\tilde{ς}}_{ij}, {\tilde{ς}}^{+}) \\ = {(\sum_{i = 1}^{m} (\begin{matrix} \frac{1}{2 (p^{q})^{δ}} [({\hat{ϑ}}_{ij}^{q} - ({\hat{ϑ}}_{i}^{+})^{q})^{δ} \\ + ({\hat{φ}}_{ij}^{q} - ({\hat{φ}}_{i}^{+})^{q})^{δ} + ({\hat{π}}_{1}^{q} - ({\hat{π}}_{i}^{+})^{q})^{δ}] \end{matrix}))}^{\frac{1}{δ}} \end{matrix}$ and the distance between the alternative K_i and the LIVq-ROF-NIS α^-: $\begin{matrix} d_{j}^{-} ({\tilde{ς}}_{ij}, {\tilde{ς}}^{-}) \\ = {(\sum_{i = 1}^{m} (\begin{matrix} ɛ_{i} \frac{1}{2 (p^{q})^{δ}} [({\hat{ϑ}}_{ij}^{q} - ({\hat{ϑ}}_{i}^{-})^{q})^{δ} \\ + ({\hat{φ}}_{ij}^{q} - ({\hat{φ}}_{i}^{-})^{q})^{δ} + ({\hat{π}}_{1}^{q} - ({\hat{π}}_{i}^{-})^{q})^{δ}] \end{matrix}))}^{\frac{1}{δ}} \end{matrix}$

Step 7. Calculate the closeness coefficient of each alternative: $Λ ({Cd}_{i}) = \frac{d_{j}^{+} ({\tilde{ς}}_{j}, {\tilde{ς}}^{+})}{d_{j}^{+} ({\tilde{ς}}_{j}, {\tilde{ς}}^{+}) + d_{j}^{-} ({\tilde{ς}}_{j}, {\tilde{ς}}^{-})}$ (11)

Step 8. Rank all the alternatives Cd_i (i = 1, 2, . . . , m) according to the closeness coefficients Λ (Cd_i), the greater the value Λ (Cd_i), the better the alternative Cd_i.

4 Illustrative example

In this section the proposed TOPSIS method for the selection process of Assistant Professor (AP) in Kohat University of Science and Technology (KUST), a leading university in Khyber Pakhtonkhwa, Pakistan. Suppose that KUST need to choose a best alternative for a teaching position of AP. The information was collected by conducting semi-structured discussion with KUST’s selection board committee, department head and office of Human resources. After the initial scrutiny four candidates (alternatives) Cd = {Cd₁, Cd₂, Cd₃, Cd₄} were selected for further process. A commission of three DMs, DM = {DM₁, DM₂, DM₃} requested distinctly proceeds to their own evaluation for the significance weights of selection criteria and the ratings of four potential alternatives. Four selection criteria C = {C₁, C₂, C₃, C₄} are considered on the basis of fair discussion with the members of commission, containing professional experience (C₁); publications (C₂); fluency in English language (C₃); and personality (C₄). The evaluation procedure in as follows:

Three DMs decided the suitability ranking of the four potential alternatives versus the decision criteria by utilizing the LIVq-ROFS. The weights of these attributes are ɛ = {ɛ₁, ɛ₂, ɛ₃, ɛ₄} such that ɛ_j ⩾ 0 and $\sum_{j = 1}^{4} ɛ_{j} = 1$ . To evaluate the four alternatives with respect to their criteria, the commission chooses a pair of linguistic terms from the following LTS:

{s₀ = EP = extremely poor, s₁ = VP = very poor, s₂ P = poor, s₃ = SP = slightly poor, s₄ = F = fair, s₅ = SG = slightly good, s₆ = G = good, s₇ VG = very good, s₈ = EG = extremely good. Three DMs DM = {DM₁, DM₂, DM₃} evaluated the alternatives Cd = {Cd₁, Cd₂, Cd₃, Cd₄} with respect to the criteria C = {C₁, C₂, C₃, C₄} and construct the following three LIVq-ROF-DMs $A^{(k)} = [{([s_{ϑ_{ij}}, s_{θ_{ij}}], [s_{ϕ_{ij}}, s_{φ_{ij}}])}_{m \times n}^{(k)}$ (see Table 1).

Table 1
Linguistic interval-valued q-rung Orthopair fuzzy assessment Information by expert

DMs Alter/Criteria C ₁ C ₂ C ₃ C ₄

DM ₁ Cd ₁ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$

Cd ₂ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$

Cd ₃ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{2}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$

Cd ₄ $([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{5}, {\tilde{s}}_{1}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$

DM ₂ Cd ₁ $([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{1}, {\tilde{s}}_{3}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$

Cd ₂ $([{\tilde{s}}_{1}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$

Cd ₃ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{2}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$

Cd ₄ $([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$

DM ₃ Cd ₁ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$

Cd ₂ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{4}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$

Cd ₃ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{5}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{1}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$

Cd ₄ $([{\tilde{s}}_{2}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$ $([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{1}, {\tilde{s}}_{2}])$

DMs	Alter/Criteria	C ₁	C ₂	C ₃	C ₄
DM ₁	Cd ₁	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$
	Cd ₂	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$
	Cd ₃	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{2}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$
	Cd ₄	$([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{5}, {\tilde{s}}_{1}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$
DM ₂	Cd ₁	$([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{1}, {\tilde{s}}_{3}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$
	Cd ₂	$([{\tilde{s}}_{1}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$
	Cd ₃	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{2}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$
	Cd ₄	$([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{2}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$
DM ₃	Cd ₁	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$
	Cd ₂	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{1}, {\tilde{s}}_{2}], [{\tilde{s}}_{3}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{4}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{4}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$
	Cd ₃	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{5}], [{\tilde{s}}_{1}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{1}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{2}, {\tilde{s}}_{3}])$
	Cd ₄	$([{\tilde{s}}_{2}, {\tilde{s}}_{4}], [{\tilde{s}}_{2}, {\tilde{s}}_{6}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{3}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{2}, {\tilde{s}}_{6}], [{\tilde{s}}_{3}, {\tilde{s}}_{5}])$	$([{\tilde{s}}_{3}, {\tilde{s}}_{5}], [{\tilde{s}}_{1}, {\tilde{s}}_{2}])$

Step 2. Compute the DMs weights by using the method discussed in [30].

Step 3. By utilizing the LIVq-ROFWA operator the collective linguistic q-rung Orthopair fuzzy information are presented in Table 2.

Step 4. Utilizing Step 4 above we get the criteria weight as: $ɛ = {ɛ_{1} = 0.20, ɛ_{2} = 0.18, ɛ_{3} = 0.27, ɛ_{4} = 0.35}$

Step 5. According to Equation and Equation we get the LIVq-ROF-PIS and LIVq-ROF-NIS respectively as: $\begin{matrix} LIVq - ROF - PIS \\ = {\begin{matrix} ([{\tilde{s}}_{2 . 8211}, s_{5 . 1460}], [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{3 . 9378}]), \\ ([{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{5 . 5951}], [{\tilde{s}}_{1 . 5453}, {\tilde{s}}_{3 . 0155}]), \\ ([{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{5 . 0946}], [{\tilde{s}}_{1 . 4004}, {\tilde{s}}_{2 . 8820}]), \\ ([{\tilde{s}}_{3 . 0000}, {\tilde{s}}_{5 . 5640}], [{\tilde{s}}_{1 . 3059}, {\tilde{s}}_{4 . 1161}]) \end{matrix}} \\ LIVq - ROF - NIS \\ = {\begin{matrix} ([{\tilde{s}}_{1 . 6402}, {\tilde{s}}_{2 . 9913}], [{\tilde{s}}_{2 . 7186}, {\tilde{s}}_{5 . 4372}]), \\ ([{\tilde{s}}_{1 . 8478}, {\tilde{s}}_{4 . 4546}], [{\tilde{s}}_{2 . 7572}, {\tilde{s}}_{5 . 7401}]), \\ ([{\tilde{s}}_{1 . 8478}, {\tilde{s}}_{4 . 3376}], [{\tilde{s}}_{2 . 6117}, {\tilde{s}}_{4 . 6599}]), \\ ([{\tilde{s}}_{2 . 3320}, {\tilde{s}}_{4 . 0636}], [{\tilde{s}}_{2 . 2974}, {\tilde{s}}_{5 . 2264}]) \end{matrix}} \end{matrix}$

Step 6. According to Equation and Equation we get distance between the alternative Cd_i and the LIVq-ROF-PIS ${\tilde{ς}}^{+}$ , and distance between the alternative Cd_i and the LIVq-ROF-NIS ${\tilde{ς}}^{-}$ respectively as: $d_{1}^{+} ({Cd}_{1}, {\tilde{ς}}_{1}^{+}) = 0 . 0729, d_{1}^{-} ({Cd}_{1}, {\tilde{ς}}_{1}^{-}) = 0 . 0730$ $d_{2}^{+} ({Cd}_{2}, {\tilde{ς}}_{2}^{+}) = 0 . 0748, d_{2}^{-} ({Cd}_{2}, {\tilde{ς}}_{2}^{-}) = 0 . 0734$ $d_{3}^{+} ({Cd}_{3}, {\tilde{ς}}_{3}^{+}) = 0 . 0568, d_{3}^{-} ({Cd}_{3}, {\tilde{ς}}_{3}^{-}) = 0 . 1043$ $d_{4}^{+} ({Cd}_{4}, {\tilde{ς}}_{4}^{+}) = 0 . 1056, d_{4}^{-} ({Cd}_{4}, {\tilde{ς}}_{4}^{-}) = 0 . 0711$

Step 7. According to Equation (11) we calculate the relative closeness coefficient of each alternatives as follows: $Λ ({Cd}_{1}) = 0 . 5003, Λ ({Cd}_{2}) = 0 . 4954,$ $Λ ({Cd}_{3}) = 0 . 6472, Λ ({Cd}_{4}) = 0 . 4025$

Step 8. Rank all the alternatives Cd_j (j = 1, 2, 3, 4) according to the relative closeness coefficients $Λ ({Cd}_{j}) : {Cd}_{3} > {Cd}_{1} > {Cd}_{2} > {Cd}_{4}$

Based on the closeness coefficient greater the value of closeness coefficient better the alternative is. So, the ranking of alternatives is Cd₃ > Cd₄ > Cd₂ > Cd₁ and the most desirable alternative is Cd₃.

5 Comparison analysis and test validity

5.1 Comparison with existing methods

The developed approach has been contrasted with existing approaches such as linguistic intuitionistic fuzzy sets [5] and linguistic Pythagorean fuzzy sets [6] and interval valued linguistic intuitionistic fuzzy sets [8]. The ranking results obtained by existing methods and the proposed TOPSIS method have depicted in Table 3.

Table 2
Collective linguistic interval-valued q-rung Orthopair fuzzy assessment information

C ₁ C ₂ C ₃ C ₄

Cd ₁ $(\begin{matrix} [{\tilde{s}}_{1 . 6402}, {\tilde{s}}_{4 . 9709}], \\ [{\tilde{s}}_{2 . 7186}, {\tilde{s}}_{3 . 9378}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 613 0}, {\tilde{s}}_{4 . 7460}], \\ [{\tilde{s}}_{1 . 5453}, {\tilde{s}}_{5 . 7401}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{1 . 8478}, {\tilde{s}}_{5 . 0946}], \\ [{\tilde{s}}_{1 . 7052}, {\tilde{s}}_{3 . 6421}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{3 . 0000}, {\tilde{s}}_{4 . 5865}], \\ [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{4 . 1161}] \end{matrix})$

Cd ₂ $(\begin{matrix} [{\tilde{s}}_{2 . 3986}, {\tilde{s}}_{4 . 3025}], \\ [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{5 . 4372}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{1 . 8478}, {\tilde{s}}_{5 . 5951}], \\ [{\tilde{s}}_{2 . 4354}, {\tilde{s}}_{3 . 6421}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{4 . 5865}], \\ [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{4 . 4140}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 3320}, {\tilde{s}}_{5 . 2286}], \\ [{\tilde{s}}_{2 . 2974}, {\tilde{s}}_{5 . 2264}] \end{matrix})$

Cd ₃ $(\begin{matrix} [{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{5 . 1460}], \\ [{\tilde{s}}_{1 . 8650}, {\tilde{s}}_{4 . 3957}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 3320}, {\tilde{s}}_{5 . 5916}], \\ [{\tilde{s}}_{2 . 0819}, {\tilde{s}}_{5 . 4631}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 5304}, {\tilde{s}}_{4 . 7460}], \\ [{\tilde{s}}_{1 . 4004}, {\tilde{s}}_{2 . 8820}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 6140}, {\tilde{s}}_{5 . 5640}], \\ [{\tilde{s}}_{2 . 2070}, {\tilde{s}}_{4 . 8506}] \end{matrix})$

Cd ₄ $(\begin{matrix} [{\tilde{s}}_{1 . 6402}, {\tilde{s}}_{2 . 9913}], \\ [{\tilde{s}}_{2 . 7186}, {\tilde{s}}_{4 . 6599}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{4 . 4546}], \\ [{\tilde{s}}_{2 . 7572}, {\tilde{s}}_{3 . 0155}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 0899}, {\tilde{s}}_{4 . 3376}], \\ [{\tilde{s}}_{2 . 6117}, {\tilde{s}}_{4 . 6599}] \end{matrix})$ $(\begin{matrix} [{\tilde{s}}_{2 . 7505}, {\tilde{s}}_{4 . 0636}], \\ [{\tilde{s}}_{1 . 3059}, {\tilde{s}}_{4 . 1638}] \end{matrix})$

	C ₁	C ₂	C ₃	C ₄
Cd ₁	$(\begin{matrix} [{\tilde{s}}_{1 . 6402}, {\tilde{s}}_{4 . 9709}], \\ [{\tilde{s}}_{2 . 7186}, {\tilde{s}}_{3 . 9378}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 613 0}, {\tilde{s}}_{4 . 7460}], \\ [{\tilde{s}}_{1 . 5453}, {\tilde{s}}_{5 . 7401}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{1 . 8478}, {\tilde{s}}_{5 . 0946}], \\ [{\tilde{s}}_{1 . 7052}, {\tilde{s}}_{3 . 6421}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{3 . 0000}, {\tilde{s}}_{4 . 5865}], \\ [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{4 . 1161}] \end{matrix})$
Cd ₂	$(\begin{matrix} [{\tilde{s}}_{2 . 3986}, {\tilde{s}}_{4 . 3025}], \\ [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{5 . 4372}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{1 . 8478}, {\tilde{s}}_{5 . 5951}], \\ [{\tilde{s}}_{2 . 4354}, {\tilde{s}}_{3 . 6421}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{4 . 5865}], \\ [{\tilde{s}}_{1 . 6901}, {\tilde{s}}_{4 . 4140}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 3320}, {\tilde{s}}_{5 . 2286}], \\ [{\tilde{s}}_{2 . 2974}, {\tilde{s}}_{5 . 2264}] \end{matrix})$
Cd ₃	$(\begin{matrix} [{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{5 . 1460}], \\ [{\tilde{s}}_{1 . 8650}, {\tilde{s}}_{4 . 3957}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 3320}, {\tilde{s}}_{5 . 5916}], \\ [{\tilde{s}}_{2 . 0819}, {\tilde{s}}_{5 . 4631}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 5304}, {\tilde{s}}_{4 . 7460}], \\ [{\tilde{s}}_{1 . 4004}, {\tilde{s}}_{2 . 8820}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 6140}, {\tilde{s}}_{5 . 5640}], \\ [{\tilde{s}}_{2 . 2070}, {\tilde{s}}_{4 . 8506}] \end{matrix})$
Cd ₄	$(\begin{matrix} [{\tilde{s}}_{1 . 6402}, {\tilde{s}}_{2 . 9913}], \\ [{\tilde{s}}_{2 . 7186}, {\tilde{s}}_{4 . 6599}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 8211}, {\tilde{s}}_{4 . 4546}], \\ [{\tilde{s}}_{2 . 7572}, {\tilde{s}}_{3 . 0155}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 0899}, {\tilde{s}}_{4 . 3376}], \\ [{\tilde{s}}_{2 . 6117}, {\tilde{s}}_{4 . 6599}] \end{matrix})$	$(\begin{matrix} [{\tilde{s}}_{2 . 7505}, {\tilde{s}}_{4 . 0636}], \\ [{\tilde{s}}_{1 . 3059}, {\tilde{s}}_{4 . 1638}] \end{matrix})$

Table 3

Comparison analysis with existing methods

Methods	Λ (Cd₁)	Λ (Cd₂)	Λ (Cd₃)	Λ (Cd₄)	Ranking of assessment information
Garg and Kumar [9]	0.5091	0.4590	0.6383	0.3901	Cd₃ > Cd₁ > Cd₂ > Cd₄
Ou et al. [27]	0.6010	0.4472	0.6944	0.3342	Cd₃ > Cd₁ > Cd₂ > Cd₄
Han et al. [10]	0.5928	0.4548	0.6944	0.3377	Cd₃ > Cd₁ > Cd₂ > Cd₄
Proposed method	0.5003	0.4954	0.6471	0.4025	Cd₃ > Cd₁ > Cd₂ > Cd₄

It has observed that the result obtained through the proposed methodology is similar to the existing methodologies developed in [9], [10] and [27]. Here the advantages of the proposed concept are listed below:

LIVq-ROFS is more precise than LIV-IFS, LIV-PFS, Lq-ROFS, LIFS and LPFS. In other word the linguistic interval-valued q-rung Orthopair fuzzy assessment information is the generalization of the LIV-PFS, LIV-IFS and Lq-ROFS. So, authors can realize that the LIVq-ROFS has more broad application prospect than the LIV-PFS, LIV-IFS and Lq-ROFS.

The method proposed in this study is the optimization of the earlier contemporaries developed in [28, 29]. The Novel TOPSIS method under the LIVq-ROFS environment can deal with more MCDM problems than the TOPSIS method with LIV-PFS, LIV-IFS and Lq-ROFS. Because the qth power sum of membership degree and nonmembership degree is less than or equal 1 in LIVq-ROFS information, if we take q = 1, then it became the LIVIFS, the proposed TOPSIS method is better than the existing method proposed in [9 , 24].

5.2 Sensitivity analysis

This section focuses on the analysis of the proposed approach with respect to ranking of alternatives for different values of δ parameter. The effects of the decision parameter δ on the closeness coefficient are examined and their comparing results are outlined in Table 4.

Table 4
Ranking of the assessment information for different valued of δ

Λ (Cd₁) Λ (Cd₂) Λ (Cd₃) Λ (Cd₄) Ranking of assessment information

δ = 1 0.5003 0.4954 0.6471 0.4025 Cd₃ > Cd₁ > Cd₂ > Cd₄

δ = 2 0.4597 0.5422 0.5672 0.4814 Cd₃ > Cd₂ > Cd₄ > Cd₁

δ = 3 0.4411 0.5569 0.5216 0.5159 Cd₂ > Cd₃ > Cd₄ > Cd₁

δ = 4 0.4290 0.5644 0.4951 0.5373 Cd₂ > Cd₄ > Cd₃ > Cd₁

δ = 5 0.4198 0.5699 0.4793 0.5519 Cd₂ > Cd₄ > Cd₃ > Cd₁

δ = 6 0.4127 0.5734 0.4696 0.5620 Cd₂ > Cd₄ > Cd₃ > Cd₁

δ = 7 0.4072 0.5763 0.4632 0.5692 Cd₂ > Cd₄ > Cd₃ > Cd₁

δ = 8 0.4030 0.5785 0.4590 0.5744 Cd₂ > Cd₄ > Cd₃ > Cd₁

δ = 9 0.3998 0.5801 0.4560 0.5784 Cd₂ > Cd₄ > Cd₃ > Cd₁

δ = 10 0.3973 0.5812 0.4538 0.5815 Cd₄ > Cd₂ > Cd₃ > Cd₁

δ = 11 0.3953 0.5820 0.4522 0.5839 Cd₄ > Cd₂ > Cd₃ > Cd₁

δ = 12 0.3937 0.5826 0.4510 0.5858 Cd₄ > Cd₂ > Cd₃ > Cd₁

δ = 13 0.3924 0.5830 0.4501 0.5874 Cd₄ > Cd₂ > Cd₃ > Cd₁

δ = 14 0.3913 0.5833 0.4493 0.5887 Cd₄ > Cd₂ > Cd₃ > Cd₁

δ = 15 0.3904 0.5836 0.4487 0.5898 Cd₄ > Cd₂ > Cd₃ > Cd₁

	Λ (Cd₁)	Λ (Cd₂)	Λ (Cd₃)	Λ (Cd₄)	Ranking of assessment information
δ = 1	0.5003	0.4954	0.6471	0.4025	Cd₃ > Cd₁ > Cd₂ > Cd₄
δ = 2	0.4597	0.5422	0.5672	0.4814	Cd₃ > Cd₂ > Cd₄ > Cd₁
δ = 3	0.4411	0.5569	0.5216	0.5159	Cd₂ > Cd₃ > Cd₄ > Cd₁
δ = 4	0.4290	0.5644	0.4951	0.5373	Cd₂ > Cd₄ > Cd₃ > Cd₁
δ = 5	0.4198	0.5699	0.4793	0.5519	Cd₂ > Cd₄ > Cd₃ > Cd₁
δ = 6	0.4127	0.5734	0.4696	0.5620	Cd₂ > Cd₄ > Cd₃ > Cd₁
δ = 7	0.4072	0.5763	0.4632	0.5692	Cd₂ > Cd₄ > Cd₃ > Cd₁
δ = 8	0.4030	0.5785	0.4590	0.5744	Cd₂ > Cd₄ > Cd₃ > Cd₁
δ = 9	0.3998	0.5801	0.4560	0.5784	Cd₂ > Cd₄ > Cd₃ > Cd₁
δ = 10	0.3973	0.5812	0.4538	0.5815	Cd₄ > Cd₂ > Cd₃ > Cd₁
δ = 11	0.3953	0.5820	0.4522	0.5839	Cd₄ > Cd₂ > Cd₃ > Cd₁
δ = 12	0.3937	0.5826	0.4510	0.5858	Cd₄ > Cd₂ > Cd₃ > Cd₁
δ = 13	0.3924	0.5830	0.4501	0.5874	Cd₄ > Cd₂ > Cd₃ > Cd₁
δ = 14	0.3913	0.5833	0.4493	0.5887	Cd₄ > Cd₂ > Cd₃ > Cd₁
δ = 15	0.3904	0.5836	0.4487	0.5898	Cd₄ > Cd₂ > Cd₃ > Cd₁

On the basis of change in parameter we can get the ordering of the LIVq-Rung Orthopair fuzzy assessment information which is listed in Table 6. According to Table 6, we can find that the ranking result have changed for different valued of δ. When the parameter lies between 1 and 2 i.e., 1 ⩽ δ ⩽ 2, the ranking result is Cd₃ > Cd₁ > Cd₂ > Cd₄ and Cd₃ > Cd₂ > Cd₄ > Cd₁; when the parameter 3 ⩽ δ ⩽ 9, the ranking result is Cd₂ > Cd₄ > Cd₃ > Cd₁; when the parameter 10 ⩽ δ ⩽ 15, the ranking result is Cd₄ > Cd₂ > Cd₃ > Cd₁ and so on. With the change of parameter, the ranking result for the best alternative also changes. Also by increasing the value of parameter δ the values of the closeness coefficient, Λ (Cd₁) and Λ (Cd₂) increase as the value of Validity Test for Proposed Approach.

To assess to the legitimacy of DM strategies, [34] built up the following testing criteria.

Criterion 1: “A powerful DM technique doesn’t change the list of the best option by supplanting a non-optimal alternative with a more terrible one without moving the relating significance of each decision criteria.”

Criterion 2: “To a viable DM technique must be fulfilled transitive property.”

Criterion 3: “By decomposing a DM problem into the sub-DM problems and applying the same method to this sub-DM problem to rank the alternatives, the collective ranking of alternatives must be identical to the ranking of un-decomposed DM problem.

5.2.1 Validity test by criterion 1

For test criterion 1, we change the non-optimal alternative Cd₁ by self-assertive more appalling alternative $C {\tilde{d}}_{1}$ of Table 4, and summarized for δ = 1 as $C {\tilde{d}}_{1} = {\begin{matrix} ([s_{1 . 0000}, s_{2 . 9709}], [s_{1 . 7186}, s_{2 . 9378}]), \\ ([s_{1 . 613 0}, s_{3 . 7460}], [s_{1 . 5453}, s_{3 . 7401}]), \\ ([s_{1 . 8478}, s_{3 . 0946}], [s_{1 . 7052}, s_{2 . 6421}]), \\ ([s_{2 . 0000}, s_{3 . 5865}], [s_{1 . 6901}, s_{2 . 1161}]) \end{matrix}}$

At that point, by applying the proposed TOPSIS Method to convert information, we get the values of closeness coefficients for the alternative $C {\tilde{d}}_{i}$ i = 1, 2, 3, 4 as $Λ (C {\tilde{d}}_{1}) = 0 . 5278$ , Λ (Cd₂) =0 . 4455, Λ (Cd₃) =0 . 5369 and Λ (Cd₄) =0 . 4614. Hence, ranking of the alternatives is Cd₃ > Cd₁ > Cd₄ > Cd₂. Hence, the most desirable alternative remains Cd₃ which proved the test criterion 1.

5.2.2 Test of validity by criterion 2 and criterion 3

Under this test, the given MCDM problem is decomposed into three smaller MCDM problems with alternatives {Cd₁, Cd₂, Cd₃}, {Cd₁, Cd₃, Cd₄} and {Cd₂, Cd₃, Cd₄}.

By applying the proposed technique to each smaller problem, we get the order and positions of the alternatives and are presented in Table 5. This order of alternatives is the same as that of given problem and consequently show the transitive property. Thus, the proposed MCDM approach satisfies the states of the test criteria 2 and 3.

Table 5
Ranking and closeness coefficients of assessment information for test criteria 2 and 3

Closeness coefficients and ranking of assessment for {Cd₁, Cd₂, Cd₃}

Λ (Cd₁) Λ (Cd₂) Λ (Cd₃) Ranking of assessment information

0.5003 0.4965 0.6459 Cd₂ < Cd₁ < Cd₃

Closeness coefficients and ranking of assessment for {Cd₁, Cd₃, Cd₄}

Λ (Cd₁) Λ (Cd₃) Λ (Cd₄) Ranking of assessment information

0.5003 0.6459 0.4025 Cd₄ < Cd₁ < Cd₃

Closeness coefficients and ranking of assessment for {Cd₁, Cd₃, Cd₄}

Λ (Cd₂) Λ (Cd₃) Λ (Cd₄) Ranking of assessment information

0.4965 0.6459 0.4025 Cd₄ < Cd₂ < Cd₃

	Closeness coefficients and ranking of assessment for {Cd₁, Cd₂, Cd₃}
Λ (Cd₁)	Λ (Cd₂)	Λ (Cd₃)		Ranking of assessment information
0.5003	0.4965	0.6459	Cd₂ < Cd₁ < Cd₃
	Closeness coefficients and ranking of assessment for {Cd₁, Cd₃, Cd₄}
Λ (Cd₁)	Λ (Cd₃)	Λ (Cd₄)		Ranking of assessment information
0.5003	0.6459	0.4025	Cd₄ < Cd₁ < Cd₃
	Closeness coefficients and ranking of assessment for {Cd₁, Cd₃, Cd₄}
Λ (Cd₂)	Λ (Cd₃)	Λ (Cd₄)		Ranking of assessment information
0.4965	0.6459	0.4025	Cd₄ < Cd₂ < Cd₃

6 Conclusion

In this article we introduced a novel TOPSIS method for solving a MCDM problem under LIVq-ROFS environments. First, we established some aggregation operations namely, the LIVq-ROFWA operator, the LIVq-ROFOWA operator, the LIVq-ROFHWA operator, the LVq-ROFWG operator, the LIVq-ROFOWG operator, LIVq-ROFHWG operator. We discussed some properties of the developed operators. Further a MCDM approach is proposed by using TOPSIS and new generalized distance measure. To show the applicability and effectiveness we given a numerical example and compare the proposed approach with existing methods. We further analyzed the proposed approach and show the behaver of distance measure by giving different values of the parameter δ.

In future work, the result of the paper can be extended to another uncertain fuzzy environment [1 , 13].

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