The operational properties of linguistic interval valued q-Rung orthopair fuzzy information and its VIKOR model for multi-attribute group decision making

Abstract

As a generalization of linguistic q-rung orthopair fuzzy set (Lq-ROFS), linguistic interval valued q-Rung orthopair fuzzy set (LIVq-ROFS) is a new concept to deal with complex and uncertain decision making problems which Lq-ROFS cannot handle. Due to the lack of information in decision making process, decision makers mostly prefer to give their preferences in interval form rather than a crisp number. In this situations, LIVq-ROFS appears up as a useful tool. In this work, we define operational laws of LIVq-ROFS and prove some properties. Furthermore, we propose the conception of the LIVq-ROF weighted averaging operator and give its formula by mathematical induction. To compare two or more linguistic interval valued q-Rung orthopair fuzzy numbers (LIVq-ROFNs), the improved form of score function is also given. Considering the powerfulness of LIVq-ROFSs handling ambiguity and complex uncertainty in practical problems, the key innovation of this paper is to develop the linguistic interval-valued q-rung orthopair fuzzy VIKOR model that is significantly different from the existing VIKOR methodology. The computing steps of this newly created model are briefly presented. Finally, the effectiveness of model is verified by an example and through comparative analysis, the superiority of VIKOR method is further illustrated.

Keywords

Linguistic interval-valued q-rung orthopair fuzzy sets multiple attribute group decision making aggregation operators VIKOR model linguistic variable

1 Introduction

Multi-attribute group decision-making (MAGDM) is a significant tool in decision-making method and has been widely used in various evaluation processes since its emergence. The main purpose is to obtain the best choice from a limited set of alternatives based on the ranking or cumulative attribute value of all alternatives, where the value of each attribute is provided by a group of experts. Due to the increasing convolution of decision-making, the ambiguity of decision-making and the ambiguity of human subjective priorities, it is becoming increasingly challenge for decision-makers to provide accurate and numerical decisions. In order to solve this shortcoming, fuzzy set (FS) theory is introduced for the MAGDM problems, and it has been fully considered by researchers. Zadeh [1] in 1965 presented FS, originally used membership function to describe the judgment outcomes, instead of exact number. Later, Atanassov [2, 3] put forward the intuitionist fuzzy set (IFS) which deals with both membership degrees (MDs) and non-membership degrees (NMDs). Hereafter, extensions of IFS open a wide area of research for the researchers [4 –10]. When the assessment set by decision makers (DM) surpasses the described condition of intuitionist fuzzy numbers (IFNs), then IFS fails to solve such problems. For instance, if any expert gives evaluation values 0.6 and 0.8, MD and NMD, respectively. The sum of MD and NMD is greater than 1. Then a new set with name of Pythagorean fuzzy set (PFS) [11 –13] has developed as a useful tool for defining uncertainty of MAGDM problems when IFS failed. The defined condition of the proposed concept is that square sum of MD and NBD is below and equal to one. So PFS has more capability than IFSs to express fuzzy information.

Furthermore, some extensions of the PFSs have been established in recent times, for example hesitant PFS (HPFS), interval-valued PFS and their aggregation operators [14, 15]. Xu and Zhang [16] generalized the TOPSIS model to deal efficiently with the decision-making problems under the PFS situation. Later, Abbasov and Yager [17] studied the relationship between the Pythagorean fuzzy number (PFN) and complex numbers. On the basis of interactive Hamacher operations, Wang et al. [18] proposed some Pythagorean fuzzy interactive Hamacher aggregation operators. Garg [19, 20] defined neutral addition, scalar multiplication and power operational laws, based on these laws, he proposed some aggregation operators under Pythagorean fuzzy environment. To integrate PFNs, Wang and Li [21] extended the power Bonferroni mean (PBM) operator using interaction operational laws of PFNs. Peng and Ma [22] presented new score functions to compare two PFNs. Although, PFSs are extensively studied and used to decision-making problems. But they still have some limitations. When the psychological diagnosis of decision making (DM) is too much complex and inconsistent, then corresponding decision making information is challenging to demonstrate by the help of PFS.

To deal with such a situation, Yager [23, 24] gave a new idea of q-rung orthopair FSs (q-ROFSs). The defined condition for this concept is, the sum of q^th power of MD, NMD and their sum must be less than or equal to 1, i.e, m^q + n^q ⩽ 1. for example, during the evaluation if, an expert gives the degree of satisfaction and dissatisfaction as 0.8 and 0.7, respectively. The final evaluation result given by the decision making will be (0.8, 0.7), then we can see 0.8 + 0.7 > 1 and also (0.8) ² + (0.7) ² > 1. IFS and PFS both are failed to evaluate this attribute value. In this case, if we take the value of q = 3, we can see (0.8) ³ + (0.7) ³ < 1. It is clear that q-ROFSs has more space than IFS and PFS. IFSs and PFSs are the special forms of q-ROFSs. If we take q = 1 and q = 2, respectively. q-ROFSs got a huge intention from the researchers since its development. Liu and Wang [25] proposed the q-ROF accumulation operators. Maclaurin symmetric mean (MSM) operator and power average (PA) operator were introduced based on q-ROFNs and applied to MAGDM problems by Liu and Ming [26]. Peng and Dai [27] defined new operations and developed a technique for MCDM problems under the q-ROF weighted exponential aggregation operator. Some q-ROF Heronian mean operators were established by Wei, et al. [28] and used them to solve multi attribute decision making (MADM) problems. Under q-ROFS, Yang and Pang [29] came up with some new definitions of partitioned Bonferroni mean operators and established a MADM model. Joshi et al. [30] derived the generalized form of q-ROFS as interval valued q-ROFS. Liu and Liu [31] defined some power Bonferroni mean operators using linguistic q-ROF information. Xu, et al. [32] extended hesitant q-ROFS to q-rung dual hesitant orthopair fuzzy set (q-RDHOFS) and accessible some q-RDHOF operators related to heronian mean.

The study introduced above only deals with the quantitative features and failed when any decision-maker wants to give his qualitative decision making which is the method of manipulating qualitative information. When making quality assessments for replacement items in decision-making processes, experts use linguistic variables to evaluate the objects first presented by Zadeh [33]. The linguistic variable technique gives more flexible ways to deal with this situation in which IFS, PFS, and q-ROFS failed. DM can give the values in words and sentences rather than using any numbers, such as “very low”, “low”, “fair”, “good”, “excellent” etc. Researchers have come up with lots of ideas when dealing with group decision-making problems using linguistic techniques [34]. However, this approach gained lots of interest and used in many fields. But it may lose some information. To surpass this deficiency, extensions and improvements have been developed, such as Herrera et al. [35 –37] established the model of group decision making (GDM) by using linguistic term sets (LTS). Martinez and Herrera [38] derived 2-tuple linguistic representation model. Zadeh [39] gave the model of computing with words (CWW). Rodríguez et al. [40] presented the idea of a hesitant fuzzy LTS (HFLTS). Moreover, many extensions of HFLTSs were also provided by the researchers, such as hesitant fuzzy uncertain linguistic sets (HFULS) [41], HFL aggregation operators [42], interval-valued HFLSs [43], dual HFLSs [44], interval-valued dual HFLSs [45] and dual HFULSs [46]. Wei [47] proposed a hybrid geometric mean operator based on LTS.

Later, Chen et al. [48] offered the idea of a linguistic IFS (LIFS), which takes into account both linguistic MDs and linguistic NMDs. Zhang [49] used t-norm and t-conorm to put forward a series of aggregation operators. Wang and Liu [50] improved operational laws for LIFSs and based on these laws presented aggregation operators. A lot of extensions were provided by the researchers with time such as HFL information, interval-valued intuitionistic linguistic fuzzy sets, [51 –53]. Amin et al. [54 –57] introduced the new idea of trapezoidal linguistic cubic hesitant fuzzy number, also defined some operations and Einstein hybrid aggregation operators for triangular cubic linguistic hesitant fuzzy sets. Based on triangular neutrosophic cubic hesitant fuzzy environment, they established HIV infection model. Fahmi et al. [58 –64] came up with some new concepts such as, triangular cubic linguistic hesitant fuzzy sets, cubic uncertain linguistic sets, triangular cubic linguistic hesitant fuzzy aggregation and cubic uncertain linguistic powered Einstein aggregation operators.

Garg [65] brought up the idea of a linguistic PFS (LPFS) composing the concept of PFS and LTS. The difference between LPFS and LIFS is that LPFS can be LIFS, but LIFS cannot be LPFS. This means LPFS has an extensive range than LIFS to demonstrate the assessment information given by the experts. Lin et al. [66, 67] presented the aggregation operators and also come up with the TOPSIS approach for LPFS to resolve the decision problems. Harsh [68] stretched out the idea of LPFS to linguistic interval valued Pythagorean fuzzy sets (LIVPFS). He also defined some aggregation operators based on operation laws of LIVPFNs.

Moreover, the conception of a linguistic q-ROFS (Lq-ROFS) was presented by Lin et al. [69]. The idea of a linguistic q-ROFS is presented by combing the two concepts q-ROFS set and linguistic variable (LV). Lq-ROFS has space more than LIFS and LIPFS.

When dealing with decision-making problems in real life, such as choosing alternative schemes for developing the brownfield, there are many attributes to evaluate the brownfield, including land utilization rate, soil recovery methods from environmental engineering, financing channels and public acceptance etc. Due to lack of information, DM’s cannot accurately prove the precision of their judgments with clear numbers. In this situation to make experts more free to provide their evaluations more conveniently, they mostly prefer to give their preferences in linguistic variables rather than a crisp number. Using the extensions of FSs and linguistic variables such as linguistic interval valued intuitionistic fuzzy set (LIVIFS) and linguistic interval valued pythagorean fuzzy set (LIVPFS), there were still some flaws in this theory. For example, if a DM provides his assessment in terms of LIVIFS and LIVPFS with cardinality 9 as ([s₃, s₆] , [s₂, s₈]), then it is clear that, 6 + 8 > 9 and 6² + 8² > 9², both ideas are failed to solve this information provided by the decision-maker. Thus, LIVIFS and LIVPFS have some limitations. To meet this gap recently, Ali et al. [70] came up with a new idea of LIVq-ROFS.

In addition to those mentioned above, MAGDM methods have a key role in the decision making processes. Due to the complexity and ambiguity of decision problems, many techniques have been developed to solve decision making problems. One of them well known technique is VIKOR model. VIKOR has an advantage in presenting a ranking method for positive attributes and negative attributes when it is used and studied in decision support. For this cause, the VIKOR should be considered for use as a decision support tool for future study. Many extension of VIKOR model have been given by the researchers. For example, Nasim and Ebrahim [71] used VIKOR model for supplier selection under fuzzy environment. Chang [72] used VIKOR model with fuzzy information for the evaluation of hospitals service quality. Bausys and Zavadskas [73] established MCDM approach by extending VIKOR model using interval neutrosophic set environment. Using Pythagorean fuzzy information Chen [74] developed VIKOR model to solve MCDM problem. Gao et al. [75] used extended VIKOR model under interval valued q-rung orthopair fuzzy information. Kutlu and Kahraman [76] came up with the extension of VIKOR model using spherical fuzzy environment and etc.

However, to date, it is clear that the researchers didn’t discuss the properties of operational laws and LIVq-ROF aggregation operator. Furthermore, VIKOR model with LIVq-ROF information has not been studied yet. Therefore, motivation of our work is to discuss these flaws and to take LIVq-ROF-VIKOR model into consideration, in which the individual’s evaluation of the alternatives and the importance of the criteria are expressed by LIVq-ROFSs. The aim of our manuscript is to create an enlarged VIKOR model with the traditional VIKOR method and LIVq-ROF information to settle MAGDM problems more effectively.

The contribution of this paper is summarized as follows:

We constructed new score function to compare two or more LIVq-ROFNs.

We gave the definition of operational laws of LIVq-ROFS and proved some properties. Furthermore, we proposed the conception of the LIVq-ROF weighted averaging operator and obtained its formula by mathematical induction.

We extended VIKOR model under linguistic interval valued q-rung orthopair fuzzy information to settle MAGDM problems more effectively.

Effectiveness of the proposed model was also shown by an illustrated example, which is the development and utilization of brownfield in Hefei, China.

For this reason, the rest of this paper is arranged as follows: In Section 2, some basic preliminary concepts associated to LIVq-ROFS are discussed. In Section 3, the score function and accuracy function of LIVq-ROFN are defined, the working rules are determined, and some of their characteristics are discussed. In Section 4, some aggregation operators related to LIVq-ROF are also proposed. In Section 5, we developed the VIKOR model based on the LIVq-ROF structure to solve the MAGDM problem. In Section 6, we provide an example. Finally, Section 7 provides conclusions.

2 Preliminaries

In this section, we introduce some basic concepts related to linguistic interval valued q-rung orthopair fuzzy sets.

Definition 2.1 [23] Let X be a non-empty fixed set. Then q-rung orthopair fuzzy set A is presented by, ∀x ∈ X $A = {〈 x, (μ_{A} (x), υ_{A} (x)) 〉},$ (2.1) where (μ_A (x) : X → [0, 1] is called the MD and υ_A (x) : X → [0, 1] is called NMD of x to A, satisfying the condition

${(μ_{A} (x))}^{q} + {(υ_{A} (x))}^{q} ⩽ 1, (q ⩾ 1) .$ (2.2)

The degree of indeterminacy of elements x ∈ X is given by $π_{A} = {(1 - (μ_{A} (x)^{q} - υ_{A} (x)^{q})}^{1 / q} .$ (2.3)

A pair A = (μ_A, υ_A) is called q-rung orthopair fuzzy number (q-ROFN).

Definition 2.2 [30] Let X be a non-empty fixed set. Then interval valued q-rung orthopair fuzzy set $\tilde{A}$ is defined as follows, ∀x ∈ X $\tilde{A} = {〈 x, (μ_{\tilde{A}} (x), υ_{\tilde{A}} (x)) 〉 | x \in X},$ (2.4) where $μ_{\tilde{A}} (x) = [α (x), β (x)] \subseteq [0, 1]$ is called the membership degree and $υ_{\tilde{A}} (x) = [ξ (x), ς (x)] \subseteq [0, 1]$ is called non-membership degree of x to $\tilde{A}$ , satisfying the condition (β (x)) ^q + (ς (x)) ^q ⩽ 1, (q ⩾ 1), ∀x ∈ X. A pair $\tilde{A} = (μ_{\tilde{A}}, υ_{\tilde{A}})$ is called interval valued q-rung orthopair fuzzy number (IVq-ROFN).

Decision makers may give their perception in linguistic number rather than a numerical number. In this situation, linguistic variable is considered and its concept is given as follows.

Definition 2.3 [34] Let S ={ s_t|t = 0, 1, . . . , τ } be a finite linguistic term set with odd cardinality. Here s_t stand for the possible value for the linguistic variable expressed by the expert. In general, s₀ lower and s_τ denotes the upper limits of LT.

Additionally, s_t has the following feature.

Ordered set: s_v > s_t, if v> t ;

negation operator: neg (s_v) = s_t such that v + t = τ .

max { s_v, s_t } = s_v if v ⩾ t

min { s_v, s_t } = s_t if v ⩾ t

3 Linguistic interval valued q-rung orthopair fuzzy set

The concept of linguistic interval valued q-rung orthopair fuzzy set (LIVq-ROFS) is given as follows.

Definition 3.1 [70] Let X be a universe of discourse and S ={ s_α|α ∈ [0, t] } be a continuous linguistic set. Then a linguistic interval-valued q-rung orthopair fuzzy set (LIVq-ROFS) $L$ associated with X can be written as $L = {(x, s_{μ (x)}, s_{υ (x)}) | x \in X},$ (3.1) where $s_{μ (x)} = [s_{μ L (x)}, s_{μ U (x)}] \subseteq [s_{0}, s_{t}], s_{ν (x)} = [s_{ν L (x)}, s_{ν U (x)}] \subseteq [s_{0}, s_{t}]$ are the interval MD and interval NMD, respectively, which holds that

${(μ U (x))}^{q} + {(ν U (x))}^{q} ⩽ t^{q} with q ⩾ 1 .$ (3.2)

The interval indeterminacy degree of LIVq-ROFS is given as follows,

$\begin{matrix} s_{π (x)} = [s_{π L (x)}, s_{π U (x)}] \\ = [s_{\sqrt[q]{t^{q} - {(μ U (x))}^{q} - {(ν U (x))}^{q}}}, s_{\sqrt[q]{t^{q} - {(μ L (x))}^{q} - {(ν L (x))}^{q}}}] \end{matrix}$ (3.3)

For convenience, α = ([s_a, s_b] , [s_c, s_d]) is called linguistic interval-valued q-rung orthopair fuzzy number (LIVq-ROFN), where $s_{a}, s_{b}, s_{c}, s_{d} \in [s_{0}, s_{t}], and b^{q} + d^{q} ⩽ t^{q} .$ (3.4)

Corollary 3.1 If $s_{μ_{L} (x)} = s_{μ_{U} (x)}$ and $s_{ν_{L} (x)} = s_{ν_{U} (x)}$ , then LIVq-ROF reduces to Lq-ROF.

Definition 3.2 Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2 be two LIVq-ROFN. Then,

$\begin{matrix} (1) If a_{1} = a_{2}, b_{1} = b_{2}, c_{1} = c_{2}, d_{1} = d_{2}, \\ then α_{1} = α_{2} \end{matrix}$ (3.5)

$\begin{matrix} (2) a_{1} ⩽ a_{2}, b_{1} ⩽ b_{2}, c_{1} ⩾ c_{2}, d_{1} ⩾ d_{2} \\ then α_{1} ⩽ α_{2} \end{matrix}$ (3.6)

$\begin{matrix} (3) Negation of α_{1} is defined as α_{1}^{c} \\ = ([s_{c_{1}}, s_{d_{1}}], [s_{a_{1}}, s_{b_{1}}]) \end{matrix}$ (3.7)

$\begin{matrix} (4) α_{1} \cup α_{2} = ([max {s_{a_{1}}, s_{a_{2}}}, max {s_{b_{1}}, s_{b_{2}}}], \\ [min {s_{c_{1}}, s_{c_{2}}}, min {s_{d_{1}}, s_{d_{2}}}]) \end{matrix}$ (3.8)

$\begin{matrix} (5) α_{1} \cap α_{2} = ([min {s_{a_{1}}, s_{a_{2}}}, min {s_{b_{1}}, s_{b_{2}}}], \\ [max {s_{c_{1}}, s_{c_{2}}}, max {s_{d_{1}}, s_{d_{2}}}]) \end{matrix}$ (3.9)

Definition 3.3 Let α = ([s_a, s_b] , [s_c, s_d]) be a LIVq-ROFN, then the score fusnction S (α) and accuracy function H (α) of α are defined as,

$S (α) = s_{{((2 t^{q} + a^{q} - c^{q} + b^{q} - d^{q}) / 4)}^{1 / q}}$ (3.10)

$H (α) = s_{{((a^{q} + b^{q} + c^{q} + d^{q}) / 2)}^{1 / q}}$ (3.11)

Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2 be two LIVq-ROFN. Based on score function and accuracy function, we define the following comparison laws for ranking two LIVq-ROFNs:

If S (α₁) < S (α₂), then α₁ < α₂.

If S (α₁) = S (α₂), then we have two ways to elaborate this situation,

If H (α₁) < H (α₂), then α₁ < α₂.

If H (α₁) = H (α₂), then α₁ = α₂.

Definition 3.4 [70] If α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2 are two LIVq-ROFNs, and parameter λ > 0, then some operational laws are defined as follows. $(1) α_{1} \oplus α_{2} = {[s_{t {(1 - \prod_{i = 1}^{2} (1 - a_{i}^{q} / t^{q}))}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}))}^{1 / q}}], [s_{t \prod_{i = 1}^{2} (c_{i} / t)}, s_{t \prod_{i = 1}^{2} (d_{i} / t)}]}$ (3.12)

$(2) α_{1} \otimes α_{2} = {[s_{t \prod_{i = 1}^{2} (a_{i} / t)}, s_{t \prod_{i = 1}^{2} (b_{i} / t)}], [s_{t {(1 - \prod_{i = 1}^{2} (1 - c_{i}^{q} / t^{q}))}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} (1 - d_{i}^{q} / t^{q}))}^{1 / q}}]}$ (3.13)

$(3) λ α_{1} = {[s_{t {(1 - {(1 - a_{1}^{q} / t^{q})}^{λ})}^{1 / q}}, s_{t {(1 - {(1 - b_{1}^{q} / t^{q})}^{λ})}^{1 / q}}], [s_{t {(c_{1} / t)}^{λ}}, s_{t {(d_{1} / t)}^{λ}}]}$ (3.14)

$\begin{matrix} (4) α_{1}^{λ} = {[s_{t {(a_{1} / t)}^{λ}}, s_{t {(b_{1} / t)}^{λ}}], \\ [s_{t {(1 - {(1 - c_{1}^{q} / t^{q})}^{λ})}^{1 / q}}, s_{t {(1 - {(1 - d_{1}^{q} / t^{q})}^{λ})}^{1 / q}}]} \end{matrix}$ (3.15)

$(5) α^{c} = ([s_{c}, s_{d}], [s_{a}, s_{b}])$ (3.16)

Theorem 3.1 Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2 be two LIVq-ROFNs, then α₁ ⊕ α₂, α₁ ⊗ α₂, λα₁, $α_{1}^{λ}$ and α^c defined in definition 3.4 are all LIVq-ROFNs.

Proof. We shall only prove α₁ ⊕ α₂ is LIVq-ROFN.

Since α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2 are two LIVq-ROFNs, according to Equation (3.4), we have a_i, b_i, c_i, d_i ∈ [0, t], a_i < b_i, c_i < d_i, and $b_{i}^{q} + d_{i}^{q} ⩽ t^{q}, i = 1, 2$ .

Therefore, it is obtained that $0 ⩽ a_{i}^{q} / t^{q} < b_{i}^{q} / t^{q} ⩽ 1, 0 ⩽ c_{i} / t < d_{i} / t ⩽ 1,$ (3.17)

which imply that

$\begin{array}{l} 0 ⩽ 1 - b_{i}^{q} / t^{q} < 1 - a_{i}^{q} / t^{q} ⩽ 1, \\ and 0 ⩽ \prod_{i = 1}^{2} (c_{i} / t) < \prod_{i = 1}^{2} (d_{i} / t) ⩽ 1 \end{array}$ (3.18)

So we have $0 ⩽ \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}) < \prod_{i = 1}^{2} (1 - a_{i}^{q} / t^{q}) ⩽ 1,$ (3.19) It is clear that

$\begin{matrix} 0 ⩽ t {(1 - \prod_{i = 1}^{2} (1 - a_{i}^{q} / t^{q}))}^{1 / q} \\ < t {(1 - \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}))}^{1 / q} ⩽ t, \end{matrix}$ (3.20)

$0 ⩽ t \prod_{i = 1}^{2} (c_{i} / t) < t \prod_{i = 1}^{2} (d_{i} / t) ⩽ t .$ (3.21)

According to Equations (3.12), (3.20) and (3.21), ${\begin{matrix} [s_{t {[1 - (1 - a_{1}^{q} / t^{q}) (1 - a_{2}^{q} / t^{q})]}^{1 / q}}, s_{t {[1 - (1 - b_{1}^{q} / t^{q}) (1 - b_{2}^{q} / t^{q})]}^{1 / q}}] \subseteq [s_{0}, s_{t}] \\ [s_{c_{1} c_{2} / t}, s_{d_{1} d_{2} / t}] \subseteq [s_{0}, s_{t}] \end{matrix},$ (3.22)

Since $b_{i}^{q} + d_{i}^{q} ⩽ t^{q}, i = 1, 2$ ,

we have $d_{i}^{q} ⩽ t^{q} - b_{i}^{q}, i = 1, 2$ (3.23)

Accordingly, we can get $\begin{matrix} 0 ⩽ {[t {(1 - \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}))}^{1 / q}]}^{q} + {(d_{1} d_{2} / t)}^{q} \\ = t^{q} (1 - \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q})) + {d_{1}^{q} d_{2}^{q} / t}^{q} \\ ⩽ t^{q} - t^{q} \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}) + {(t^{q} - b_{1}^{q}) (t^{q} - b_{2}^{q}) / t}^{q} \\ = t^{q} - {(t^{q} - b_{1}^{q}) (t^{q} - b_{2}^{q}) / t}^{q} + {(t^{q} - b_{1}^{q}) (t^{q} - b_{2}^{q}) / t}^{q} = t^{q} \end{matrix}$ (3.24)

According to Equations (3.22), (3.24) and definition 3.1, we have shown that α₁ ⊕ α₂ is a LIVq-ROFN.

Similarly, we can easily show that α₁ ⊗ α₂, λα₁, $α_{1}^{λ}$ and α^c are also LIVq-ROFNs.

Theorem 3.2. Let α, α₁ and α₂ be three LIVq-ROFNs and λ, λ₁, λ₂ be three positive real numbers. Then,

α₁ ⊕ α₂ = α₂ ⊕ α₁

λα₁ ⊕ λα₂ = λ (α₁ ⊕ α₂)

α₁ ⊗ α₂ = α₂ ⊗ α₁

${(α_{1} \otimes α_{2})}^{λ} = α_{1}^{λ} \otimes α_{2}^{λ}$

λ₁α ⊕ λ₂α = (λ₁ + λ₂) α

$α_{1}^{c} \oplus α_{2}^{c} = {(α_{1} \otimes α_{2})}^{c}$

Proof. We shall prove only two parts 2) and 5) in the following.

2) For any two LIVq-ROFNs α₁ and α₂, we can get by Equation (3.15). $λ α_{i} = {[s_{t {(1 - {(1 - a_{i}^{q} / t^{q})}^{λ})}^{1 / q}}, s_{t {(1 - {(1 - b_{i}^{q} / t^{q})}^{λ})}^{1 / q}}], [s_{t {(c_{i} / t)}^{λ}}, s_{t {(d_{i} / t)}^{λ}}]}, i = 1, 2$ (3.25)

According to Equations (3.25), (3.12) and (3.14), it is obtained that

$\begin{matrix} λ α_{1} \oplus λ α_{2} = {[s_{t {(1 - \prod_{i = 1}^{2} {(1 - a_{i}^{q} / t^{q})}^{λ})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} {(1 - b_{i}^{q} / t^{q})}^{λ})}^{1 / q}}], [s_{t \prod_{i = 1}^{2} {(c_{i} / t)}^{λ}}, s_{t \prod_{i = 1}^{2} {(d_{i} / t)}^{λ}}]} \\ = {[s_{t {(1 - {(\prod_{i = 1}^{2} (1 - a_{i}^{q} / t^{q}))}^{λ})}^{1 / q}}, s_{t {(1 - {(\prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}))}^{λ})}^{1 / q}}], [s_{t {(\prod_{i = 1}^{2} (c_{i} / t))}^{λ}}, s_{t {(\prod_{i = 1}^{2} (d_{i} / t))}^{λ}}]} \\ = λ (α_{1} \oplus α_{2}) \end{matrix}$ (3.26)

Hence we get λα₁ ⊕ λα₂ = λ (α₁ ⊕ α₂), which completes the proof.

5) For any LIVq-ROFN α = ([s_a, s_b] , [s_c, s_d]), by Equation (3.14), we can get,

$λ_{i} α = {[s_{t {(1 - {(1 - a^{q} / t^{q})}^{λ_{i}})}^{1 / q}}, s_{t {(1 - {(1 - b^{q} / t^{q})}^{λ_{i}})}^{1 / q}}], [s_{t {(c / t)}^{λ_{i}}}, s_{t {(d / t)}^{λ_{i}}}]}, i = 1, 2$ (3.27)

According to Equations (3.27), (3.12) and (3.14)

$\begin{matrix} λ_{1} α \oplus λ_{2} α = {[s_{t {(1 - \prod_{i = 1}^{2} {(1 - a^{q} / t^{q})}^{λ_{i}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} {(1 - b^{q} / t^{q})}^{λ_{i}})}^{1 / q}}], [s_{t \prod_{i = 1}^{2} {(c / t)}^{λ_{i}}}, s_{t \prod_{i = 1}^{2} {(d / t)}^{λ_{i}}}]} \\ = {[s_{t {(1 - {(1 - a^{q} / t^{q})}^{λ_{1} + λ_{2}})}^{1 / q}}, s_{t {(1 - {(1 - b^{q} / t^{q})}^{λ_{1} + λ_{2}})}^{1 / q}}], [s_{t {(c / t)}^{λ_{1} + λ_{2}}}, s_{t {(d / t)}^{λ_{1} + λ_{2}}}]} \\ = (λ_{1} + λ_{2}) α \end{matrix}$ (3.28)

Which completes the proof.

Rest proofs of 1), 3), 4) and 6) are similar.

Theorem 3.3 Let α₁, α₂ be two LIVq-ROFNs, then

(α₁ ∪ α₂) ⊕ (α₁ ∩ α₂) = α₁ ⊕ α₂

(α₁ ∪ α₂) ⊗ (α₁ ∩ α₂) = α₁ ⊗ α₂

Proof. Here only part (1) we shall prove,

Suppose that α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2 are two LIVq-ROFNs, $\begin{matrix} (α_{1} \cup α_{2}) \oplus (α_{1} \cap α_{2}) \\ = ([max {s_{a_{1}}, s_{a_{2}}}, max {s_{b_{1}}, s_{b_{2}}}], [min {s_{c_{1}}, s_{c_{2}}}, \\ min {s_{d_{1}}, s_{d_{2}}}]) \oplus \\ ([min {s_{a_{1}}, s_{a_{2}}}, min {s_{b_{1}}, s_{b_{2}}}], [max {s_{c_{1}}, s_{c_{2}}}, \\ max {s_{d_{1}}, s_{d_{2}}}]) \end{matrix}$ (3.29)

Let’s discuss it in 16 different cases

Case1, If

$\begin{matrix} max {s_{a_{1}}, s_{a_{2}}} = s_{a_{1}}, max {s_{b_{1}}, s_{b_{2}}} \\ = s_{b_{1}}, max {s_{c_{1}}, s_{c_{2}}} \\ = s_{c_{1}}, max {s_{d_{1}}, s_{d_{2}}} = s_{d_{1}}, \end{matrix}$ (3.30)

then

$\begin{matrix} min {s_{a_{1}}, s_{a_{2}}} = s_{a_{2}}, min {s_{b_{1}}, s_{b_{2}}} \\ = s_{b_{2}}, min {s_{c_{1}}, s_{c_{2}}} \\ = s_{c_{2}}, min {s_{d_{1}}, s_{d_{2}}} = s_{d_{2}}, \end{matrix}$ (3.31)

According to Equations (3.12), (3.29)–(3.31), we have $\begin{matrix} (α_{1} \cup α_{2}) \oplus (α_{1} \cap α_{2}) \\ = ([s_{a_{1}}, s_{b_{1}}], [s_{c_{2}}, s_{d_{2}}]) \oplus ([s_{a_{2}}, s_{b_{2}}], [s_{c_{1}}, s_{d_{1}}]) \\ = [s_{t {(1 - \prod_{i = 1}^{2} (1 - a_{i}^{q} / t^{q}))}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}))}^{1 / q}}], \\ [s_{t \prod_{i = 1}^{2} (c_{i} / t)}, s_{t \prod_{i = 1}^{2} (d_{i} / t)}] \\ = α_{1} \oplus α_{2} \end{matrix}$

Similarly, we can discuss other 15 different cases, which completes the proof.

The proof of part (2) is similar to part (1), which is omitted here.

4 Linguistic interval valued q-Rung orthopair fuzzy aggregation operators

In this section, we have developed some series of linguistic interval valued q-Rung orthopair aggregation operators for different LIVq-ROFNs. Let Ω be the set of all LIVq-ROFNs.

4.1 LIVq-ROF weighted averaging (LIVq-ROFWA) operator

Definition 4.1 [70] Let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2, …, n be the collection of LIVq-ROFNs. Then, the LIVq-ROFWA operator of dimension n is a mapping LIVqROFWA : rmOmegaⁿ → rmOmega and

$\begin{matrix} LIVqROFWA (α_{1}, α_{2}, \dots, α_{n}) \\ = \oplus_{i = 1}^{n} (ω_{i} α_{i}), \end{matrix}$ (4.1) where ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector of the α_i (i = 1, 2, …, n) with ω_i > 0 and $\sum_{i = 1}^{n} ω_{i} = 1$ .

Theorem 4.1 For a collection of LIVq-ROFNs α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]), i = 1, 2, …, n. The aggregated value of LIVq-ROFN’s using LIVq-ROFWA is still a LIVq-ROFN and is given by $\begin{matrix} LIVqROFWA (α_{1}, α_{2}, \dots, α_{n}) \\ = ([s_{t {(1 - \prod_{i = 1}^{n} {(1 - a_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{n} {(1 - b_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}], \\ [s_{t \prod_{i = 1}^{n} {(c_{i} / t)}^{ω_{i}}}, s_{t \prod_{i = 1}^{n} {(d_{i} / t)}^{ω_{i}}}]) \end{matrix}$ (4.2)

Where ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector of the α_i (i = 1, 2, …, n) with ω_i > 0 and $\sum_{i = 1}^{n} ω_{i} = 1$ .

Proof. The obtained aggregated value of LIVq-ROFN is still LIVq-ROFN by using LIVq-ROFWA operator. The result given in Equation (4.2) can be proved by using mathematical induction on n. For each i = 1, 2, …, n, α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) is a LIVq-ROFNs, which implies that a_i, b_i, c_i, d_i ∈ [0, t] and b^q + d^q ⩽ t^q. Then for mathematical induction following steps have been taken.

Step 1: For n = 2, let α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]), i = 1, 2 be two LIVq-ROFNs. Then, by Equation (3.14), we get $\begin{matrix} ω_{i} α_{i} = {[s_{t {(1 - {(1 - a_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}, s_{t {(1 - {(1 - b_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}], \\ [s_{t {(c_{i} / t)}^{ω_{i}}}, s_{t {(d_{i} / t)}^{ω_{i}}}]} \end{matrix}$ (4.3)

By Equations (3.12) and (4.2), it follows that $\begin{matrix} α_{1} \oplus α_{2} = {[s_{t {(1 - \prod_{i = 1}^{2} (1 - a_{i}^{q} / t^{q}))}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} (1 - b_{i}^{q} / t^{q}))}^{1 / q}}], \\ [s_{t \prod_{i = 1}^{2} (c_{i} / t)}, s_{t \prod_{i = 1}^{2} (d_{i} / t)}]} \end{matrix}$ $\begin{matrix} LIVqROFWA (α_{1}, α_{2}) = ω_{1} α_{1} \oplus ω_{2} α_{2} \\ = (\begin{matrix} [s_{t {(1 - {(1 - a_{1}^{q} / t^{q})}^{ω_{1}})}^{1 / q}}, s_{t {(1 - {(1 - b_{1}^{q} / t^{q})}^{ω_{1}})}^{1 / q}}], \\ [s_{t {(c_{1} / t)}^{ω_{1}}}, s_{t {(d_{1} / t)}^{ω_{1}}}] \end{matrix}) \oplus (\begin{matrix} [s_{t {(1 - {(1 - a_{2}^{q} / t^{q})}^{ω_{2}})}^{1 / q}}, s_{t {(1 - {(1 - b_{2}^{q} / t^{q})}^{ω_{2}})}^{1 / q}}], \\ [s_{t {(c_{2} / t)}^{ω_{2}}}, s_{t {(d_{2} / t)}^{ω_{2}}}] \end{matrix}) \\ = (\begin{matrix} [s_{t {(1 - {(1 - a_{1}^{q} / t^{q})}^{ω_{1}} * {(1 - a_{2}^{q} / t^{q})}^{ω_{2}})}^{1 / q}}, s_{t {(1 - {(1 - b_{1}^{q} / t^{q})}^{ω_{1}} * {(1 - b_{2}^{q} / t^{q})}^{ω_{2}})}^{1 / q}}], \\ [s_{t {(c_{1} / t)}^{ω_{1}} * {(c_{2} / t)}^{ω_{2}}}, s_{t {(d_{1} / t)}^{ω_{1}} * {(d_{2} / t)}^{ω_{2}}}] \end{matrix}) \\ = ([s_{t {(1 - \prod_{i = 1}^{2} {(1 - a_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{2} {(1 - b_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}], [s_{t \prod_{i = 1}^{2} {(c_{i} / t)}^{ω_{i}}}, s_{t \prod_{i = 1}^{2} {(d_{i} / t)}^{ω_{i}}}]), \end{matrix}$

which shows that the result holds for n = 2.

Step 2: if Equation (4.2) is true for n = k, i.e.

$\begin{matrix} LIVqROFWA (α_{1}, α_{2}, \dots, α_{k}) \\ = ([s_{t {(1 - \prod_{i = 1}^{k} {(1 - a_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{k} {(1 - b_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}], [s_{t \prod_{i = 1}^{k} {(c_{i} / t)}^{ω_{i}}}, s_{t \prod_{i = 1}^{k} {(d_{i} / t)}^{ω_{i}}}]) \end{matrix}$ (4.4)

when n = k + 1, according to Equation (4.4), we get $\begin{matrix} LIVqROFWA (α_{1}, α_{2}, \dots, α_{k}, α_{k + 1}) = \oplus_{i = 1}^{k + 1} ω_{i} α_{i} = (\oplus_{i = 1}^{k} ω_{i} α_{i}) \oplus ω_{k + 1} α_{k + 1} \\ = ([s_{t {(1 - \prod_{i = 1}^{k} {(1 - a_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{k} {(1 - b_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}], [s_{t \prod_{i = 1}^{k} {(c_{i} / t)}^{ω_{i}}}, s_{t \prod_{i = 1}^{k} {(d_{i} / t)}^{ω_{i}}}]) \\ \oplus {[s_{t {(1 - {(1 - a_{k + 1}^{q} / t^{q})}^{ω_{k + 1}})}^{1 / q}}, s_{t {(1 - {(1 - b_{k + 1}^{q} / t^{q})}^{ω_{k + 1}})}^{1 / q}}], [s_{t {(c_{k + 1} / t)}^{ω_{k + 1}}}, s_{t {(d_{k + 1} / t)}^{ω_{k + 1}}}]} \\ = {[s_{t {(1 - \prod_{i = 1}^{k} {(1 - a_{i}^{q} / t^{q})}^{ω_{i}} * {(1 - a_{k + 1}^{q} / t^{q})}^{ω_{k + 1}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{k} {(1 - b_{i}^{q} / t^{q})}^{ω_{i}} * {(1 - b_{k + 1}^{q} / t^{q})}^{ω_{k + 1}})}^{1 / q}}], [s_{t \prod_{i = 1}^{k} {(c_{i} / t)}^{ω_{i}} * {(c_{k + 1} / t)}^{ω_{k + 1}}}, s_{t \prod_{i = 1}^{k} {(d_{i} / t)}^{ω_{i}} * {(d_{k + 1} / t)}^{ω_{k + 1}}}]} \\ = ([s_{t {(1 - \prod_{i = 1}^{k + 1} {(1 - a_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}, s_{t {(1 - \prod_{i = 1}^{k + 1} {(1 - b_{i}^{q} / t^{q})}^{ω_{i}})}^{1 / q}}], [s_{t \prod_{i = 1}^{k + 1} {(c_{i} / t)}^{ω_{i}}}, s_{t \prod_{i = 1}^{k + 1} {(d_{i} / t)}^{ω_{i}}}]) \end{matrix}$

Hence, the result holds for n = k + 1 and thus, the Equation (4.2) holds for all positive integer n.

Corollary 4.1. If q = 1 LIVq-ROFWA operator reduces to linguistic interval valued intuitionistic fuzzy weighted averaging operators.

Corollary 4.2. If q = 2 LIVq-ROFWA operator reduces to linguistic interval valued pythagorean fuzzy weighted averaging operators.

Similarly, we can get the formula of weighted averaging operator of LIVPFNs as follows.

Theorem 4.2. For a collection of LIVPFNs α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]), i = 1, 2, …, n. The aggregated value of LIVPFNs using LIVPFN weighted averaging operator is still a LIVPFN, and we have $\begin{matrix} LIVPFWA (α_{1}, α_{2}, \dots, α_{n}) \\ = ([s_{t {(1 - \prod_{i = 1}^{n} {(1 - a_{i}^{2} / t^{2})}^{ω_{i}})}^{1 / 2}}, s_{t {(1 - \prod_{i = 1}^{n} {(1 - b_{i}^{2} / t^{2})}^{ω_{i}})}^{1 / 2}}], \\ [s_{t \prod_{i = 1}^{n} {(c_{i} / t)}^{ω_{i}}}, s_{t \prod_{i = 1}^{n} {(d_{i} / t)}^{ω_{i}}}]) \end{matrix}$ (4.5)

5 MAGDM with linguistic interval valued q-Rung orthopair fuzzy information

In this section, two practical methods, i.e., VIKOR model and aggregation based method, for MAGDM problems are demonstrated, where the assessment of decision maker takes the mode of LIVq-ROFs.

5.1 The VIKOR model for LIVq-ROF MAGDM problems

Assume that there is a set of m alternatives ${A_{1}, A_{2}, \dots, A_{m}}$ , evaluated by finite n attributes {G₁, G₂, … G_n } with weight vectors (ω₁, ω₂, …, ω_n), which meets the condition that $0 ⩽ ω_{j} ⩽ 1, \sum_{i = 1}^{n} ω_{j} = 1$ . Let {D⁽¹⁾, D⁽²⁾, … , D^(L) } be the L decision makers with weight vectors (ρ₁, ρ₂, …, ρ_L) such that $0 ⩽ ρ_{l} ⩽ 1, \sum_{l = 1}^{L} ρ_{l} = 1$ . Assume that the l-th decision maker has evaluated the alternative $A_{i}$ under G_j and provide the decision matrix information in terms of LIVq-ROFNs, $R^{(l)} = {(r_{ij}^{(l)})}_{m \times n}$ , l = 1, 2, …, L, where $r_{ij}^{(l)} = ([s_{A_{ij}}^{(l)}, s_{B_{ij}}^{(l)}], [s_{C_{ij}}^{(l)}, s_{D_{ij}}^{(l)}])$ , $[s_{A_{ij}}^{(l)}, s_{B_{ij}}^{(l)}] \subseteq [s_{0}, s_{t}]; i = 1, 2, \dots, m, j = 1, 2, \dots, n$ indicates the MD of alternatives $A_{i} (i = 1, 2, \dots, m)$ satisfying the attributes G_j (j = 1, 2, …, n) and $[s_{C_{ij}}^{(l)}, s_{D_{ij}}^{(l)}] \subseteq [s_{0}, s_{t}]; i = 1, 2, \dots, m, j = 1, 2, \dots, n$ is the NMD of the alternatives $A_{i} (i = 1, 2, \dots, m)$ not satisfying the attributes G_j (j = 1, 2, …, n) given by the decision maker, also (B_ij) ^q + (D_ij) ^q ⩽ t^q ; i = 1, 2, …, m, j = 1, 2, …, n .

To get the best alternatives in group decision making, the LIVqROF-VIKOR model is established. The computing steps are simply demonstrated as follows.

Step 1. Standardize each decision matrix $R^{(l)} = (r_{ij}^{(l)})_{m \times n}, l = 1, 2, \dots, L$ (5.1)

where $r_{ij}^{(l)} = ([s_{A_{ij}}^{(l)}, s_{B_{ij}}^{(l)}], [s_{C_{ij}}^{(l)}, s_{D_{ij}}^{(l)}])$ , i = 1, 2, …, m, j = 1, 2, …, n

Step 2. Based on experts decision matrices, utilize the LIVq-ROFWA to aggregate all the decision matrices R^(l) of all decision makers into a collective decision matrix R = (r_ij) _m×n according to Equation (4.2), where ∀i = 1, 2, …, m, j = 1, 2, …, n, $r_{ij} = LIVqROFWA (r_{ij}^{(1)}, r_{ij}^{(2)}, \dots, r_{ij}^{(L)}) = \oplus_{l = 1}^{L} ρ_{l} \cdot r_{ij}^{(l)}$ (5.2)

Suppose that we denote $r_{ij} = ([s_{A}^{ij}, s_{B}^{ij}], [s_{C}^{ij}, s_{D}^{ij}])$ by Equation (5.2), i = 1, 2, …, m, j = 1, 2, …, n.

Step 3. Calculate linguistic interval valued q-rung orthopair fuzzy positive ideal solution (PIS) $r_{j}^{+}$ and linguistic interval valued q-rung orthopair fuzzy negative ideal solution (NIS) $r_{j}^{-}$ by the following equations: $r_{j}^{+} = ([s_{A_{ij}}^{+}, s_{B_{ij}}^{+}], [s_{C_{ij}}^{+}, s_{D_{ij}}^{+}]); i = 1, 2, \dots, m, j = 1, 2, \dots, n$ (5.3)

$r_{j}^{-} = ([s_{A_{ij}}^{-}, s_{B_{ij}}^{-}], [s_{C_{ij}}^{-}, s_{D_{ij}}^{-}]); i = 1, 2, \dots, m, j = 1, 2, \dots, n$ (5.4)

For benefit attribute: $r_{j}^{+} = ([max_{i} s_{A_{ij}}, max_{i} s_{B_{ij}}], [min_{i} s_{C_{ij}}, min_{i} s_{C_{ij}}]); j = 1, 2, \dots, n$ (5.5)

$r_{j}^{-} = ([min_{i} s_{A_{ij}}, min_{i} s_{B_{ij}}], [max_{i} s_{C_{ij}}, max_{i} s_{D_{ij}}]); j = 1, 2, \dots, n$ (5.6)

For cost attribute: $r_{j}^{+} = ([min_{i} s_{A_{ij}}, min_{i} s_{B_{ij}}], [max_{i} s_{C_{ij}}, max_{i} s_{D_{ij}}]); j = 1, 2, \dots, n$ (5.7)

$r_{j}^{-} = ([max_{i} s_{A_{ij}}, max_{i} s_{B_{ij}}], [min_{i} s_{C_{ij}}, min_{i} s_{C_{ij}}]); j = 1, 2, \dots, n$ (5.8)

Step 4. Compute the hesitant group utility measure.

According to the formulas (5.6) and (5.7), compute the linguistic interval valued q-rung orthopair fuzzy group utility measure Φ_i (i = 1, 2, …, m) and linguistic interval valued q-rung orthopair fuzzy individual regret measure of the alternatives $A_{i} (i = 1, 2, \dots, m)$ as follows. $Φ_{i} = \sum_{j = 1}^{n} ω_{j} \frac{d (r_{j}^{+}, r_{ij})}{d (r_{j}^{+}, r_{j}^{-})}, i = 1, 2, \dots, m$ (5.9)

$y_{i} = max_{j} (ω_{j} \frac{d (r_{j}^{+}, r_{ij})}{d (r_{j}^{+}, r_{j}^{-})}), i = 1, 2, \dots, m$ (5.10)

Where 0 ⩽ ω_j ⩽ 1 is the weighting vector of attributes satisfying $0 ⩽ ω_{j} ⩽ 1, \sum_{j = 1}^{n} ω_{j} = 1$ and d denotes the linguistic interval valued q-rung orthopair fuzzy distance measures. d is hamming distance measures given as follows

If α_i = ([s_{a
_i}, s_{b
_i}] , [s_{c
_i}, s_{d
_i}]) ; i = 1, 2, then

$d_{H} (α_{1}, α_{2}) = \frac{1}{4 t^{q}} [| a_{1}^{q} - a_{2}^{q} | + | b_{1}^{q} - b_{2}^{q} | + | c_{1}^{q} - c_{2}^{q} | + | d_{1}^{q} - d_{2}^{q} |]$ (5.11)

Step 4. Calculate the compromise measure $Q_{i}$ of the alternative $A_{i} (i = 1, 2, \dots, m)$ as follows. $Q_{i} = β \frac{(Φ_{i} - Φ^{+})}{(Φ^{-} - Φ^{+})} + (1 - β) \frac{(y_{i} - y^{+})}{(y^{-} - y^{+})}$ (5.12)

where $Φ^{+} = min_{i} (Φ_{i}), Φ^{-} = max_{i} (Φ_{i}), y^{+} = min_{i} (y_{i}), y^{-} = max_{i} (y_{i}),$ (5.13)

and β is the coefficient of the strategy of the decision making. If β > 0.5 denotes “the maximum group utility, if β < 0.5 then denotes minimum regret. Usually taking β = 0.5, which is called equality.

Step 5. Rank $Q_{i}$ in descending order to select the best one alternative. Of course, smaller the $Q_{i}$ , the best alternative $A_{i}$ will be.

6 An illustrative example

6.1 Numerical example for LIVq-ROFNs MAGDM problem

Brownfield and green field are a pair of opposite concepts, some heavy pollution enterprises have adjusted the production location, the original factory sites become the brownfield. In addition, abandoned gas stations, garbage disposal stations, oil storage tanks, cargo stacks and warehouses, railway yards and other places may become the sources of brownfield. After the toxic substances in the brownfield have seeped into the ground, they can slowly evaporate and release the toxic substances through the soil and pipes, and the toxicity can last for decades.

There is a brownfield in the city of Hefei, China. The environmental protection department invited three experts {D¹, D², D³} with weight vector ρ = (0.2, 0.4, 0.4) ^T in the field of environmental protection and economy to evaluate the use of the brownfield. There are four alternative schemes for developing the brownfield, namely agricultural land, residential land, park parking lot and commercial land, which are denoted by ${A_{1}, A_{2}, A_{3}, A_{4}}$ . There are four attributes to evaluate the brownfield, which are land utilization rate, soil recovery methods from environmental engineering, financing channels and public acceptance, which are represented by {G₁, G₂, G₃, G₄}, with the corresponding weight vector being ω = (0.2, 0.3, 0.2, 0.3) ^T.

Then the evaluation matrices provided by three experts are R⁽¹⁾, R⁽²⁾ and R⁽³⁾, respectively, which are expressed by LIVq-ROF information with t = 8 and q = 3. The proposed MAGDM method in Section 5 is applied to fuse different expert’s opinions to get the best alternative.

Step 1. Construct the decision matrix $R^{(l)} = {(r_{ij}^{(l)})}_{4 \times 4}$ under the given information by experts are shown in Tables 1 –3.

Table 1
Evaluation matrix $R^{(1)} = {(r_{ij}^{(1)})}_{4 \times 4}$ provided by expert D¹

G ₁ G ₂ G ₃ G ₄

$A_{1}$ ([s₅, s₆] , [s₁, s₂]) ([s₄, s₆] , [s₁, s₁]) ([s₄, s₅] , [s₂, s₃]) ([s₆, s₇] , [s₂, s₃])

$A_{2}$ ([s₅, s₇] , [s₂, s₃]) ([s₃, s₅] , [s₂, s₄]) ([s₁, s₄] , [s₁, s₂]) ([s₂, s₃] , [s₁, s₂])

$A_{3}$ ([s₃, s₄] , [s₁, s₂]) ([s₃, s₆] , [s₂, s₄]) ([s₅, s₆] , [s₃, s₄]) ([s₄, s₆] , [s₁, s₁])

$A_{4}$ ([s₃, s₇] , [s₂, s₄]) ([s₃, s₄] , [s₁, s₄]) ([s₁, s₅] , [s₂, s₄]) ([s₄, s₆] , [s₃, s₄])

	G ₁	G ₂	G ₃	G ₄
$A_{1}$	([s₅, s₆] , [s₁, s₂])	([s₄, s₆] , [s₁, s₁])	([s₄, s₅] , [s₂, s₃])	([s₆, s₇] , [s₂, s₃])
$A_{2}$	([s₅, s₇] , [s₂, s₃])	([s₃, s₅] , [s₂, s₄])	([s₁, s₄] , [s₁, s₂])	([s₂, s₃] , [s₁, s₂])
$A_{3}$	([s₃, s₄] , [s₁, s₂])	([s₃, s₆] , [s₂, s₄])	([s₅, s₆] , [s₃, s₄])	([s₄, s₆] , [s₁, s₁])
$A_{4}$	([s₃, s₇] , [s₂, s₄])	([s₃, s₄] , [s₁, s₄])	([s₁, s₅] , [s₂, s₄])	([s₄, s₆] , [s₃, s₄])

Table 2

Evaluation matrix $R^{(2)} = {(r_{ij}^{(2)})}_{4 \times 4}$ provided by expert D²

	G ₁	G ₂	G ₃	G ₄
$A_{1}$	([s₄, s₆] , [s₃, s₄])	([s₃, s₆] , [s₂, s₃])	([s₃, s₄] , [s₁, s₃])	([s₂, s₃] , [s₁, s₂])
$A_{2}$	([s₂, s₃] , [s₁, s₂])	([s₆, s₇] , [s₂, s₃])	([s₁, s₅] , [s₂, s₄])	([s₄, s₅] , [s₂, s₃])
$A_{3}$	([s₁, s₄] , [s₁, s₂])	([s₅, s₆] , [s₃, s₄])	([s₃, s₄] , [s₁, s₄])	([s₃, s₆] , [s₂, s₄])
$A_{4}$	([s₃, s₄] , [s₁, s₂])	([s₃, s₇] , [s₂, s₄])	([s₅, s₇] , [s₂, s₃])	([s₅, s₆] , [s₁, s₂])

Table 3

Evaluation matrix $R^{(3)} = {(r_{ij}^{(3)})}_{4 \times 4}$ provided by expert D³

	G ₁	G ₂	G ₃	G ₄
$A_{1}$	([s₁, s₆] , [s₃, s₄])	([s₂, s₆] , [s₁, s₄])	([s₃, s₄] , [s₁, s₃])	([s₃, s₄] , [s₁, s₂])
$A_{2}$	([s₁, s₄] , [s₁, s₂])	([s₂, s₃] , [s₁, s₂])	([s₄, s₆] , [s₃, s₄])	([s₃, s₆] , [s₂, s₃])
$A_{3}$	([s₃, s₄] , [s₁, s₄])	([s₃, s₅] , [s₂, s₄])	([s₁, s₅] , [s₂, s₄])	([s₃, s₆] , [s₂, s₄])
$A_{4}$	([s₆, s₇] , [s₂, s₃])	([s₃, s₄] , [s₁, s₄])	([s₅, s₇] , [s₂, s₃])	([s₅, s₆] , [s₁, s₂])

Step 2. We compute the collective decision matrix R = (r_ij) _4×4 obtained by aggregating three experts decision matrices R⁽¹⁾, R⁽²⁾ and R⁽³⁾ using LIVq-ROFWA according to Equation 4.2. For example, $\begin{matrix} r_{11} = LIVqROFWA (r_{11}^{(1)}, r_{11}^{(2)}, r_{11}^{(3)}) = 0.2 r_{11}^{(1)} \oplus 0.4 r_{11}^{(2)} \oplus 0.4 r_{11}^{(3)} \\ = 0.2 ([s_{5}, s_{6}], [s_{1}, s_{2}]) \oplus 0.4 ([s_{4}, s_{6}], [s_{3}, s_{4}]) \oplus 0.4 ([s_{1}, s_{6}], [s_{3}, s_{4}]) \\ = ([s_{3.7660}, s_{6.0000}], [s_{2.4082}, s_{3.4822}]) \end{matrix}$

Similarly, we get other r_ij, i = 1, 2, 3, 4j = 1, 2, 3, 4. The collective decision matrix R = (r_ij) _4×4 is shown in Table 4.

Table 4

The collective decision matrix $R = {(r_{ij})}_{4 \times 4}$ by LIVq-ROFWA

	G ₁	G ₂	G ₃	G ₄
$A_{1}$	([s_3.7600, s_6.0000] , [s_2.4082, s_3.4822])	([s_3.0090, s_6.0000] , [s_1.3195, s_2.7019])	([s_3.2597, s_4.2529] , [s_1.1487, s_3.0000])	([s_4.0372, s_5.0849] , [s_1.1487, s_2.1689])
$A_{2}$	([s_3.1512, s_5.0849] , [s_1.1487, s_2.1689])	([s_4.7587, s_5.9242] , [s_1.5157, s_2.7019])	([s_3.0073, s_5.3604] , [s_2.0477, s_3.4822])	([s_3.3775, s_5.2931] , [s_1.7411, s_2.7663])
$A_{3}$	([s_2.5593, s_4.0000] , [s_1.0000, s_2.6390])	([s_4.0990, s_5.6722] , [s_2.3522, s_4.0000])	([s_3.3802, s_4.9743] , [s_1.6438, s_4.0000])	([s_3.2597, s_6.0000] , [s_1.7411, s_3.0314])
$A_{4}$	([s_4.8478, s_6.4021] , [s_1.5157, s_2.7019])	([s_3.0000, s_5.9313] , [s_1.3195, s_4.0000])	([s_4.6857, s_6.7863] , [s_2.0000, s_3.1777])	([s_4.8418, s_6.0000] , [s_1.2457, s_2.2974])

Step 3. According to Table 4 and Equations (5.5) and (5.6), we can get LIVq-ROF positive ideal solution $r^{+} = (r_{1}^{+}, r_{2}^{+}, r_{3}^{+}, r_{4}^{+})$ and negative ideal solution $r^{-} = (r_{1}^{-}, r_{2}^{-}, r_{3}^{-}, r_{4}^{-})$ as follows. $\begin{matrix} r^{+} = (([s_{4.8748}, s_{6.4021}], [s_{1.0000}, s_{2.1689}]), \\ ([s_{4.7587}, s_{6.0000}], [s_{1.3195}, s_{2.7019}]), \\ ([s_{4.6857}, s_{6.7863}], [s_{1.1487}, s_{3.0000}]), \\ ([s_{4.8418}, s_{6.0000}], [s_{1.1487}, s_{2.1689}])) \end{matrix}$ $\begin{matrix} r^{-} = (([s_{2.5593}, s_{4.0000}], [s_{2.4082}, s_{3.4822}]), \\ ([s_{3.0000}, s_{5.6722}], [s_{2.3522}, s_{4.0000}]), \\ ([s_{3.0073}, s_{4.2529}], [s_{2.0477}, s_{4.0000}]), \\ ([s_{3.2597}, s_{5.0849}], [s_{1.7411}, s_{3.0314}])) \end{matrix}$

Step 4. Compute the group utility measure Φ_i (i = 1, 2, 3, 4) and individual regret measure $y_{i} = max_{j} (ω_{j} \frac{d (r_{j}^{+}, r_{ij})}{d (r_{j}^{+}, r_{j}^{-})}), i = 1, 2, 3, 4$ of alternatives $A_{i} (i = 1, 2, 3, 4)$ by using Equations (5.9) and (5.10).

For example, since ω = (0.2, 0.3, 0.2, 0.3) ^T, we have $\begin{matrix} Φ_{1} = 0.2 \frac{d (r_{1}^{+}, r_{11})}{d (r_{1}^{+}, r_{1}^{-})} + 0.3 \frac{d (r_{2}^{+}, r_{12})}{d (r_{2}^{+}, r_{2}^{-})} \\ + 0.2 \frac{d (r_{3}^{+}, r_{13})}{d (r_{3}^{+}, r_{3}^{-})} + 0.3 \frac{d (r_{4}^{+}, r_{14})}{d (r_{4}^{+}, r_{4}^{-})} = 0.6176 \\ y_{1} = max (0.2 \frac{d (r_{1}^{+}, r_{11})}{d (r_{1}^{+}, r_{1}^{-})}, 0.3 \frac{d (r_{2}^{+}, r_{12})}{d (r_{2}^{+}, r_{2}^{-})}, \\ 0.2 \frac{d (r_{3}^{+}, r_{13})}{d (r_{3}^{+}, r_{3}^{-})}, 0.3 \frac{d (r_{4}^{+}, r_{14})}{d (r_{4}^{+}, r_{4}^{-})}) = 0.2146, \end{matrix}$

Similarly, we get $\begin{matrix} Φ_{1} = 0.6176, Φ_{2} = 0.5420, Φ_{3} = 0.7322, Φ_{4} = 0.2520 \\ y_{1} = 0.2146, y_{2} = 0.2555, y_{3} = 0.2258, y_{4} = 0.2346 \end{matrix}$

By Equation (5.13), we have $\begin{matrix} Φ^{+} = min_{1 ⩽ i ⩽ 4} (Φ_{i}) = 0.2520, Φ^{-} = max_{1 ⩽ i ⩽ 4} (Φ_{i}) = 0.7322, \\ y^{+} = min_{1 ⩽ i ⩽ 4} (y_{i}) = 0.2146, y^{-} = max_{1 ⩽ i ⩽ 4} (y_{i}) = 0.2555, \end{matrix},$

Step 5. Calculate the compromise measure $Q_{i} (i = 1, 2, 3, 4)$ of alternatives $A_{i} (i = 1, 2, 3, 4)$ by using Equation (5.12). Here we take β = 0.5, therefore, $Q_{1} = 0.5 \frac{Φ_{1} - Φ^{+}}{Φ^{-} - Φ^{+}} + 0.5 \frac{y_{1} - y^{+}}{y^{-} - y^{+}} = 0.3812$

Similarly, we get $Q_{2} = 0.8022, Q_{3} = 0.6367, Q_{4} = 0.2448$

Step 6. Rank all the alternatives according to $Q_{i}$ in descending order. $Q^{*} = min_{1 ⩽ i ⩽ 4} (Q_{i}) = Q_{4} = 0.2448$

Therefore, the best alternative is A₄.

6.2 Comparative analyses for LIVq-ROFNs MAGDM problems

In this subsection, we shall compare our presented VIKOR model for LIVq-ROFNs with other existing for LIVq-ROF decision making tool TOPSIS method for LIVq-ROFNs proposed by Ali et al. in [70] to indicate that the model we developed is scientifically effective. Data taken from [70] are shown in Table 5.

Table 5
Linguistic interval-valued q-rung orthopair fuzzy assessment information by three experts

C ₁ C ₂ C ₃ C ₄

DM ₁ cd ₁ ([s₂, s₆] , [s₃, s₅]) ([s₂, s₄] , [s₂, s₆]) ([s₁, s₂] , [s₃, s₄]) ([s₃, s₄] , [s₁, s₆])

cd ₂ ([s₃, s₅] , [s₂, s₄]) ([s₂, s₆] , [s₃, s₅]) ([s₂, s₄] , [s₂, s₆]) ([s₂, s₃] , [s₃, s₅])

cd ₃ ([s₃, s₄] , [s₁, s₂]) ([s₃, s₅] , [s₂, s₆]) ([s₂, s₆] , [s₂, s₃]) ([s₂, s₆] , [s₃, s₅])

cd ₄ ([s₂, s₃] , [s₃, s₅]) ([s₃, s₄] , [s₅, s₁]) ([s₃, s₄] , [s₁, s₆]) ([s₁, s₂] , [s₃, s₄])

DM ₂ cd ₁ ([s₁, s₂] , [s₃, s₄]) ([s₃, s₄] , [s₁, s₆]) ([s₂, s₆] , [s₁, s₃]) ([s₃, s₅] , [s₂, s₄])

cd ₂ ([s₁, s₄] , [s₂, s₆]) ([s₂, s₆] , [s₂, s₃]) ([s₃, s₅] , [s₂, s₄]) ([s₂, s₆] , [s₃, s₅])

cd ₃ ([s₃, s₅] , [s₂, s₆]) ([s₂, s₆] , [s₃, s₅]) ([s₃, s₄] , [s₁, s₂]) ([s₃, s₅] , [s₂, s₆])

cd ₄ ([s₁, s₂] , [s₃, s₄]) ([s₃, s₅] , [s₂, s₄]) ([s₁, s₂] , [s₃, s₄]) ([s₃, s₄] , [s₁, s₆])

DM ₃ cd ₁ ([s₂, s₆] , [s₂, s₃]) ([s₂, s₆] , [s₃, s₅]) ([s₂, s₃] , [s₃, s₅]) ([s₃, s₄] , [s₂, s₃])

cd ₂ ([s₃, s₄] , [s₁, s₆]) ([s₁, s₂] , [s₃, s₄]) ([s₃, s₄] , [s₁, s₄]) ([s₃, s₄] , [s₁, s₆])

cd ₃ ([s₂, s₆] , [s₃, s₅]) ([s₂, s₅] , [s₁, s₆]) ([s₁, s₄] , [s₂, s₆]) ([s₂, s₆] , [s₂, s₃])

cd ₄ ([s₂, s₄] , [s₂, s₆]) ([s₂, s₃] , [s₃, s₅]) ([s₂, s₆] , [s₃, s₅]) ([s₃, s₅] , [s₁, s₂])

	C ₁	C ₂	C ₃	C ₄
DM ₁ cd ₁	([s₂, s₆] , [s₃, s₅])	([s₂, s₄] , [s₂, s₆])	([s₁, s₂] , [s₃, s₄])	([s₃, s₄] , [s₁, s₆])
cd ₂	([s₃, s₅] , [s₂, s₄])	([s₂, s₆] , [s₃, s₅])	([s₂, s₄] , [s₂, s₆])	([s₂, s₃] , [s₃, s₅])
cd ₃	([s₃, s₄] , [s₁, s₂])	([s₃, s₅] , [s₂, s₆])	([s₂, s₆] , [s₂, s₃])	([s₂, s₆] , [s₃, s₅])
cd ₄	([s₂, s₃] , [s₃, s₅])	([s₃, s₄] , [s₅, s₁])	([s₃, s₄] , [s₁, s₆])	([s₁, s₂] , [s₃, s₄])
DM ₂ cd ₁	([s₁, s₂] , [s₃, s₄])	([s₃, s₄] , [s₁, s₆])	([s₂, s₆] , [s₁, s₃])	([s₃, s₅] , [s₂, s₄])
cd ₂	([s₁, s₄] , [s₂, s₆])	([s₂, s₆] , [s₂, s₃])	([s₃, s₅] , [s₂, s₄])	([s₂, s₆] , [s₃, s₅])
cd ₃	([s₃, s₅] , [s₂, s₆])	([s₂, s₆] , [s₃, s₅])	([s₃, s₄] , [s₁, s₂])	([s₃, s₅] , [s₂, s₆])
cd ₄	([s₁, s₂] , [s₃, s₄])	([s₃, s₅] , [s₂, s₄])	([s₁, s₂] , [s₃, s₄])	([s₃, s₄] , [s₁, s₆])
DM ₃ cd ₁	([s₂, s₆] , [s₂, s₃])	([s₂, s₆] , [s₃, s₅])	([s₂, s₃] , [s₃, s₅])	([s₃, s₄] , [s₂, s₃])
cd ₂	([s₃, s₄] , [s₁, s₆])	([s₁, s₂] , [s₃, s₄])	([s₃, s₄] , [s₁, s₄])	([s₃, s₄] , [s₁, s₆])
cd ₃	([s₂, s₆] , [s₃, s₅])	([s₂, s₅] , [s₁, s₆])	([s₁, s₄] , [s₂, s₆])	([s₂, s₆] , [s₂, s₃])
cd ₄	([s₂, s₄] , [s₂, s₆])	([s₂, s₃] , [s₃, s₅])	([s₂, s₆] , [s₃, s₅])	([s₃, s₅] , [s₁, s₂])

Step 1. Using equal weights of experts the collective matrix is given in Table 6.

Table 6

The collective decision matrix $R = {(r_{ij})}_{4 \times 4}$ by LIVq-ROFWA

	C ₁	C ₂	C ₃	C ₄
cd ₁	([s_1.7840, s_5.4123] , [s_2.6207, s_3.9149])	([s_2.4337, s_4.9570] , [s_1.8171, s_5.6462])	([s_1.7840, s_4.5677] , [s_2.0801, s_3.9149])	([s_3.0000, s_4.4026] , [s_1.5874, s_4.1602])
cd ₂	([s_2.6440, s_4.4026] , [s_1.5874, s_5.2415])	([s_1.7840, s_5.4123] , [s_2.6207, s_3.9149])	([s_2.7478, s_4.4026] , [s_1.5874, s_4.5789])	([s_2.4337, s_4.8109] , [s_2.0801, s_5.3133])
cd ₃	([s_2.7478, s_5.1972] , [s_1.8171, s_3.9149])	([s_2.4337, s_5.4070] , [s_1.8171, s_5.6462])	([s_2.2971, s_4.9570] , [s_1.5874, s_3.3019])	([s_2.4337, s_5.7320] , [s_2.2894, s_4.4814])
cd ₄	([s_1.7840, s_3.2260] , [s_2.6207, s_4.9324])	([s_2.7478, s_4.1964] , [s_1.8171, s_4.6416])	([s_2.2971, s_4.7353] , [s_2.0801, s_4.9324])	([s_2.6440, s_4.0866] , [s_1.4422, s_3.6342])

Step 2. According to Table 6. We can get LIVq-ROF positive ideal solution $r^{+} = (r_{1}^{+}, r_{2}^{+}, r_{3}^{+}, r_{4}^{+})$ and negative ideal solution $r^{-} = (r_{1}^{-}, r_{2}^{-}, r_{3}^{-}, r_{4}^{-})$ as follows. $\begin{matrix} r^{+} = (([s_{2.7478}, s_{5.4123}], [s_{1.5874}, s_{3.9149}]), \\ ([s_{2.7478}, s_{5.4123}], [s_{1.8171}, s_{3.9149}]), \\ ([s_{2.7478}, s_{4.9570}], [s_{1.5874}, s_{3.3019}]), \\ ([s_{3.0000}, s_{5.7320}], [s_{1.4422}, s_{3.6342}])) \end{matrix}$ $\begin{matrix} r^{-} = (([s_{1.7840}, s_{3.2260}], [s_{2.6207}, s_{5.2415}]), \\ ([s_{1.7840}, s_{4.1964}], [s_{2.6207}, s_{5.6462}]), \\ ([s_{1.7840}, s_{4.4026}], [s_{2.0801}, s_{4.9324}]), \\ ([s_{2.4337}, s_{4.0866}], [s_{2.2894}, s_{5.3133}])) \end{matrix}$

Step 3. Group utility measure Φ_i (i = 1, 2, 3, 4) and individual regret measure $y_{i} = max_{j} (ω_{j} \frac{d (r_{j}^{+}, r_{ij})}{d (r_{j}^{+}, r_{j}^{-})}), i = 1, 2, 3, 4$ of alternatives cd_i (i = 1, 2, 3, 4) are obtained as follows.

Since ω = (0.20, 0.18, 0.27, 0.35) ^T, so we can get $\begin{matrix} Φ_{1} = 0.4705, Φ_{2} = 0.6241, Φ_{3} = 0.2233, Φ_{4} = 0.6789 \\ y_{1} = 0.1839, y_{2} = 0.2838, y_{3} = 0.0985, y_{4} = 0.2176 \end{matrix}$

By Equation (5.13), we have $\begin{matrix} Φ^{+} = min_{1 ⩽ i ⩽ 4} (Φ_{i}) = 0.2233, Φ^{-} = max_{1 ⩽ i ⩽ 4} (Φ_{i}) = 0.6789, \\ y^{+} = min_{1 ⩽ i ⩽ 4} (y_{i}) = 0.0985, y^{-} = max_{1 ⩽ i ⩽ 4} (y_{i}) = 0.2838 . \end{matrix},$

Step 4. Calculate the compromise measure $Q_{i} (i = 1, 2, 3, 4)$ of alternatives cd_i (i = 1, 2, 3, 4) by using Equation (5.12). Here we take β = 0.5, therefore, $Q_{1} = 0.5 \frac{Φ_{1} - Φ^{+}}{Φ^{-} - Φ^{+}} + 0.5 \frac{y_{1} - y^{+}}{y^{-} - y^{+}} = 0.5016$

Similarly, we get $Q_{2} = 0.9399, Q_{3} = 0.0000, Q_{4} = 0.8213$

Step 5. Rank all the alternatives according to $Q_{i}$ in descending order. $Q^{* *} = min_{1 ⩽ i ⩽ 4} (Q_{i}) = Q_{3} = 0.0000$

Apparently, the ordering of $Q_{i}$ is $Q_{3} > Q_{1} > Q_{4} > Q_{2}$ and the best choice is cd₃.

From the above example, we can see that the best alternative obtained by proposed method is also same as in [70] using TOPSIS method, which shows the practicality and effectiveness of our proposed approach.

7 Conclusions

In this article, we presented the extension of the VIKOR model under LIVq-ROF environment to express the uncertainty of data. Some operational laws, Score functions, accuracy function, and aggregation operator associated with their proofs are also defined to aggregate the information. Finally, the VIKOR model is established based on the LIVq-ROF structure to solve the MAGDM problem and illustrated with a numerical example. It has been seen that the technique used in this article has a wide range to express uncertain information.

In terms of LIVq-ROF information, this paper discusses the weighted average operator. In the future, we can define the weighted geometric average operator, ordered weighted average operator and other information fusion methods, and discuss their corresponding properties. The established technique can be extended to another uncertain fuzzy environment [77 –79] and more decision making methods, such as the multi-attributive border approximation area comparison (MABAC) method, ELECTRE I method, can be used to deal with LIVq-ROF environment.

Footnotes

Acknowledgments

The work was supported by National Natural Science Foundation of China (Nos. 71871001, 71771001, 71701001, 72001001); Natural Science Foundation for Distinguished Young Scholars of Anhui Province (No. 1908085J03); Research Funding Project of Academic and technical leaders and reserve candidates in Anhui Province (No.2018H179).

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