A new approach of interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator based on decision making problem

Abstract

In this paper, we define the new concept of linguistic interval-valued intuitionistic neutrosophic fuzzy numbers and operational laws. We discuss the two aggregation operators, which are defined as linguistic interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator and linguistic interval-valued intuitionistic neutrosophic fuzzy hybrid weighted averaging operator, for collection of data. We define the MAGDM problem under the LIVINF environment. Finally, we define a numerical example.

Keywords

Linguistic neutrosophic set linguistic intuitionistic fuzzy number linguistic interval-valued intuitionistic neutrosophic fuzzy numbers interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator linguistic interval-valued intuitionistic neutrosophic fuzzy hybrid weighted averaging operator MCDM Numerical application

1 Introduction

The fuzzy set was first introduced by Zadeh in 1965 which is the extension of classical set theory [1]. Since fuzzy set theory has been successfully applied to the various field of multi-attribute decision making. After that this concept was extended and developed, such that as the interval Valued Fuzzy Set (IVFSs) [2], Interval Valued Intuitionistic Fuzzy Set (IVIFSs) [3], Hesitant Fuzzy Set (HFSs) [4], etc.

Fuzzy set theory was developed but it cannot give the proper information of physical problem. For instance, when someone asked a question, he may think that the possibility of the correct answer is 0.7, and the possibility of a false answer is 0.3, and the degree of uncertainty 0.4. This issue is beyond the scope of a fuzzy set and intuitionistic fuzzy set. Hence, Smarandache gives the concept of neutrosophic logic and neutrosophic set in 1999 [5]. This concept is the extension of the standard interval [0,1] used for the intuitionistic fuzzy set.

After that many scholars have begun to study the practical application of neutrosophic set in multi-attributes decision-making. Wang et al. [6] and Ye [7] define the aggregation operator for single-valued neutrosophic set and proposed a corresponding multi-attributes decision-making method. Majumdar et al. [8] defined the distance, similarity, and entropy of two single-valued neutrosophic sets based on cross-entropy, and proposed a corresponding multi-attributes decision-making method. Wang et al. [9] define Interval-valued neutrosophic set and their logic operation rules. Zhang et al. [10] defined the operation rules for interval-valued neutrosophic set based on T-norm and S-norm and proposed a corresponding multi-attributes decision-making method.

Zhao et al. [13] utilized to manage the time-varying weightings corresponding to the upper membership functions. Aliya et al. [14] introduced the triangular neutrosophic cubic fuzzy Einstein weighted averaging (TNCFEWA), triangular neutrosophic cubic fuzzy Einstein ordered weighted averaging (TNCFEOWA) and triangular neutrosophic cubic fuzzy Einstein hybrid weighted averaging (TNCFEHWA) operators. Fei [15] introduced as preliminaries of our research, flexible aggregation approach of probabilistic evidence in the context of forensic crime investigations. Yang et al. [16] proposed under single-valued neutrosophic environment, novel SVNNWBM aggregation operator and operational laws based on Einstein. Li et al. [17] introduced the distance measurement between two LANs and the power-weighted averaging operator and the power-weighted geometric operator. Aiwu et al. [18] proved to be the maximum approximation to the original data and maintains most of the information during data processing. Wu et al. [19] proposed the three-fold unit prioritized hybrid weighted aggregation (UPHWA) operator, the importance and the ordered position of attributes and into IVIF-PHWA operator. Wang et al. [20] proposed the interval-valued intuitionistic fuzzy Einstein weighted averaging operator, interval-valued intuitionistic fuzzy Einstein ordered weighted averaging operator, and interval-valued intuitionistic fuzzy Einstein hybrid weighted averaging operator. Yu [21] proposed the generalized interval-valued intuitionistic fuzzy weighted geometric (GIIFWG) and generalized interval-valued intuitionistic fuzzy ordered weighted geometric (GIIFOWG) operators.

To deal with some decision problems, we must first select a suitable information expression tool to describe the evaluation information. In order to fully express people’s decision-making consciousness and complex conditions, according to the above discussion about fuzzy information expression, we know the LIVINFNs can fully express people’s decision-making consciousness and describe the decision information precisely by a parameter which can flexibly adjust the scope of information expression. At the same time, the attributes in decision-making problems are not completely independent, and there are more or less relationships between different attributes. Therefore, to deal with such complex decision problems, we need to adopt some effective information LIVINFN to deal with realistic MADM problems. Based on above analysis, we know the LIVINFWA operator is a very good tool, it can not only deal with MADM problems with independent attributes, but also with correlations among different attributes. In addition, it is more general that some existing LIVINFN, such as LIVINFWA operator and so on because they are the special case of LIVINFHWA operator because they can be obtained by adjusting some variable parameters. However, up to now, no scholars have studied the traveling projects selection problems using LIVINFN, nor have they studied the LIVINFWA operator under LN fuzzy environment to deal with complex decision information.

In a multiple attribute decision making problem with linguistic interval-valued intuitionistic neutrosophic fuzzy information, there are generally a finite set of alternatives and a collection of attributes. It collects the information about attribute values and attribute weights, needs weighted aggregation of the attribute values across all attributes for each alternative to get an overall attribute value.

Similarly, in a group decision making problem with linguistic interval-valued intuitionistic neutrosophic fuzzy information, for each alternative, the decision makers generally need to provide their evaluations by means of linguistic variables. Then, all the individual evaluation information is fused to become a group opinion, which can sufficiently reflects the opinion of every member of the group. As a result, the group evaluation of the alternatives should be as close as possible to all the decision makers’ individual opinion. This means that the computation of the distance between all individual evaluations and the aggregated group evaluation on a given alternative is an integral part of the solution to a linguistic decision making problem.

Since neutrosophic fuzzy number handles the situation when indeterminacy arises and thus generalize the concept of intuitionistic fuzzy set (IFS). Keeping this in view the idea of intuitionistic neutrosophic set (INS) was proposed. Thus we have generalized the idea of intuitionistic neutrosophic set (INS) to interval-valued intuitionistic neutrosophic set (IVINS) which has a wider range of values to handle imprecise and vague information.

This paper are organized as follows. Section 2, we give some fundamental thought and properties of basic concepts. Section 3, we exhibit of linguistic interval-valued intuitionistic neutrosophic fuzzy numbers and operational laws. Section 4, we exhibit a series of aggregation operators based on the LIVINFNs. Section 5, develop an approach to MCDM. Section 6, the application of the developed approach in group decision-making problem is shown by an illustrative example. Section 7, we discuss in comparative analysis. Finally, we give the conclusions in Section 8.

2 Basic concept

Definition 2.1. [1] Let Φ ≠ X and by a fuzzy set is defined as γ ={ 〈 x, μ_γ(x) 〉 |x ∈ X } in which μ_γ(x) is a mapping from X to [0, 1] represent membership function of an element x in X·

Definition 2.2. [3] Let Φ ≠ X . By an intuitionistic fuzzy set, we mean a form γ ={ 〈x, μ_γ(x), η_γ(x) 〉 |x ∈ X } simply denoted by γ =〈μ_γ,η_γ 〉, where μ : = X → [0, 1] and η : = X → [0, 1] in which μ_γ is the membership and η_γ is the non-membership of γ respectively such that 0 ⩽ μ_γ(X) + η_γ (x) ⩽ 1 .

Definition 2.3. [5] Let X be a fixed set, A neutrosophic set T in X is define as $T = {〈 u, X_{T (u)}, Y_{T (u)}, Z_{T (u)}; u \in X 〉}$ where $X_{T (u)}, Y_{T (u)}, Z_{T (u)}$ are the truth membership function, the indertemency function and the falcity membership function respectively, such that X, Y, Z : X →]0^-, 1⁺ [ and

Definition 2.4. [9] Let X be a fixed set, and let an element u ∈ X. An interval-valued neutrosophic set Z in X is Z = { u, (A_A (u) , B_A (u) , C_A (u)) ; u ∈ X } , where A_A (u) , B_A (u) , C_A (u) are the truth-membership, indeterminacy-membership and falsity membership function separately, for each point u in X we have that 0 ⩽ sup A_A (u) + sup B_A (u) + sup C_A (u) ⩽ 3.

Definition 2.5. [11] Let X be a fixed set, for a continuous linguistic term set S_[0, h], a LIFS Z is defined as $Z = {\begin{matrix} (\begin{matrix} u, s_{α (u)}, \\ s_{β (u)} \end{matrix}) \\ | u \in X \end{matrix}},$ where s_α, s_β ∈ S_[0, h] represent the linguistic membership and non-membership degree, respectively, and 0 ⩽ α + β ⩽ h . The linguestic indeterminacy is defined as s_ϖ = s_h-α-β . Usually, the pair (s_α, s_β) is called the linguistic intuitionistic fuzzy numbers (LIFN), and it is often written as γ^. = (s_α, s_β) where s_α, s_β ∈ S_[0, h] and α + β ⩽ h . if s_α, s_β ∈ S, then the LIFN is called original, otherwise actual.

3 Linguistic interval-valued intuitionistic neutrosophic fuzzy numbers and operational laws

Definition 3.1. Let $A = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], \\ [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], \\ [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}}$ and $B = {\begin{matrix} [s_{w_{2}^{l}}, s_{w_{2}^{u}}], \\ [s_{Γ_{2}^{l}}, s_{Γ_{2}^{u}}], \\ [s_{α_{2}^{l}}, s_{α_{2}^{u}}], \\ [s_{β_{2}^{l}}, s_{β_{2}^{u}}] \\ [s_{c_{2}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{2}^{l}}, s_{d_{2}^{u}}] \end{matrix}}$ be two LIVINFNs and λ be any real number. Then we have

$A \oplus B = {\begin{matrix} [s_{w_{1}^{l}} + s_{w_{2}^{l}} - s_{w_{1}^{l}} s_{w_{2}^{l}}, s_{w_{1}^{u}} + s_{w_{2}^{u}} - s_{w_{1}^{u}} s_{w_{2}^{u}}], \\ [s_{Γ_{1}^{l}} + s_{Γ_{2}^{l}} - s_{Γ_{1}^{l}} s_{Γ_{2}^{l}}, s_{Γ_{1}^{u}} + s_{Γ_{2}^{u}} - s_{Γ_{1}^{u}} s_{Γ_{2}^{u}}], \\ [s_{α_{1}^{l}} s_{α_{2}^{l}}, s_{α_{1}^{u}} s_{α_{2}^{u}}], [s_{β_{1}^{l}} s_{β_{2}^{l}}, s_{β_{1}^{u}} s_{β_{2}^{u}}], \\ [s_{c_{1}^{l}} s_{c_{2}^{l}}, s_{c_{1}^{u}} s_{c_{2}^{u}}], [s_{d_{1}^{l}} s_{d_{2}^{l}}, s_{d_{1}^{u}} s_{d_{2}^{u}}] \end{matrix}},$ $A \otimes B = {\begin{matrix} [s_{w_{1}^{l}} s_{w_{2}^{l}}, s_{w_{1}^{u}} s_{w_{2}^{u}}], [s_{Γ_{1}^{l}} s_{Γ_{2}^{l}}, s_{Γ_{1}^{u}} s_{Γ_{2}^{u}}], \\ [s_{α_{1}^{l}} + s_{α_{2}^{l}} - s_{α_{1}^{u}} s_{α_{2}^{u}}], [s_{β_{1}^{l}} + s_{β_{2}^{l}} - s_{β_{1}^{u}} s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}} + s_{c_{2}^{l}} - s_{c_{1}^{u}} s_{c_{2}^{u}}], [s_{d_{1}^{l}} + s_{d_{2}^{l}} - s_{d_{1}^{u}} s_{d_{2}^{u}}] \end{matrix}},$

$λ A = {\begin{matrix} [s_{1 - (1 - w_{1}^{l})^{λ}}, s_{1 - (1 - w_{1}^{u})^{λ}}], \\ [s_{1 - (1 - Γ_{1}^{l})^{λ}}, s_{1 - (1 - Γ_{1}^{u})^{λ}}], \\ [(s_{α_{1}^{l}})^{λ}, (s_{α_{1}^{u}})^{λ}], [(s_{β_{1}^{l}})^{λ}, (s_{β_{1}^{u}})^{λ}], \\ [(s_{c_{1}^{l}})^{λ}, (s_{c_{1}^{u}})^{λ}], [(s_{d_{1}^{l}})^{λ}, (s_{d_{1}^{u}})^{λ}] \end{matrix}},$ $A^{λ} = {\begin{matrix} [(s_{w_{1}^{l}})^{λ}, (s_{w_{1}^{u}})^{λ}], [(s_{Γ_{1}^{l}})^{λ}, (s_{Γ_{1}^{u}})^{λ}], \\ [s_{1 - (1 - α_{1}^{l})^{λ}}, s_{1 - (1 - α_{1}^{u})^{λ}}], \\ [1 - (1 - s_{β_{1}^{l}})^{λ}, 1 - (1 - s_{β_{1}^{u}})^{λ}], \\ [s_{1 - (1 - c_{1}^{l})^{λ}}, s_{1 - (1 - c_{1}^{u})^{λ}}], \\ [s_{1 - (1 - d_{1}^{l})^{λ}}, s_{1 - (1 - d_{1}^{u})^{λ}}] \end{matrix}} .$

Example 3.2. Let $A = {\begin{matrix} [s_{1}, s_{3}], \\ [s_{2}, s_{4}], \\ [s_{3}, s_{4}], \\ [s_{1}, s_{2}], \\ [s_{2}, s_{3}], \\ [s_{3}, s_{4}] \end{matrix}}$ and $B = {\begin{matrix} [s_{1}, s_{2}], \\ [s_{2}, s_{2}], \\ [s_{3}, s_{5}], \\ [s_{2}, s_{3}], \\ [s_{3}, s_{5}], \\ [s_{3}, s_{4}] \end{matrix}}$ be two LIVINFNs and λ be any real number. Then we have $A + B = {\begin{cases} [s_{1} + s_{1} - s_{1} s_{1}, s_{3} + s_{2} - s_{3} s_{2}], \\ [s_{2} + s_{2} - s_{2} s_{2}, s_{4} + s_{2} - s_{4} s_{2}], \\ [s_{3} s_{3}, s_{4} s_{5}], [s_{1} s_{2}, s_{2} s_{3}], \\ [s_{2} s_{3}, s_{3} s_{5}], [s_{3} s_{3}, s_{4} s_{4}], \end{cases}}$ $A \otimes B = {\begin{matrix} [s_{1} s_{1}, s_{3} s_{2}], [s_{2} s_{2}, s_{4} s_{2}], \\ [s_{3} + s_{3} - s_{4} s_{5}], [s_{1} + s_{2} - s_{2} s_{3}], \\ [s_{2} + s_{3} - s_{3} s_{5}], [s_{3} + s_{3} - s_{4} s_{4}] \end{matrix}},$ $λ = (0.25, 0.25, 0.50)^{T}$ $λ A = {\begin{matrix} [1 - (1 - s_{1})^{λ}, 1 - (1 - s_{3})^{λ}], \\ [1 - (1 - s_{2})^{λ}, 1 - (1 - s_{4})^{λ}], \\ [(s_{3})^{λ}, (s_{4})^{λ}], [(s_{1})^{λ}, (s_{2})^{λ}], \\ [(s_{2})^{λ}, (s_{3})^{λ}], [(s_{3})^{λ}, (s_{4})^{λ}] \end{matrix}},$ $A^{λ} = {\begin{matrix} [(s_{1})^{λ}, (s_{3})^{λ}], [(s_{2})^{λ}, (s_{4})^{λ}], \\ [1 - (1 - s_{3})^{λ}, 1 - (1 - s_{4})^{λ}], \\ [1 - (1 - s_{1})^{λ}, 1 - (1 - s_{2})^{λ}], \\ [1 - (1 - s_{2})^{λ}, 1 - (1 - s_{3})^{λ}], \\ [1 - (1 - s_{3})^{λ}, 1 - (1 - s_{4})^{λ}] \end{matrix}} .$

Definition 3.3. Let $α_{j} = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], \\ [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], \\ [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}}$ be the collection of LIVINFNs. Then the score function is defined as $S (a) = \frac{{\begin{matrix} [s_{w_{1}^{l}} - s_{w_{1}^{u}}] + [s_{Γ_{1}^{l}} - s_{Γ_{1}^{u}}] + [s_{α_{1}^{l}} - s_{α_{1}^{u}}] + \\ [s_{β_{1}^{l}} - s_{β_{1}^{u}}] + [s_{c_{1}^{l}} - s_{c_{1}^{u}}] + [s_{d_{1}^{l}} - s_{d_{1}^{u}}] \end{matrix}}}{12} .$

Fig. 1

Different score function.

Example 3.4. Let $α_{1} = {\begin{matrix} [s_{2}, s_{3}], [s_{3}, s_{4}], \\ [s_{3}, s_{5}], [s_{1}, s_{6}], \\ [s_{4}, s_{7}], [s_{1}, s_{3}] \end{matrix}}, α_{2} = {\begin{matrix} [s_{5}, s_{6}], [s_{2}, s_{6}], \\ [s_{4}, s_{5}], [s_{3}, s_{6}], \\ [s_{3}, s_{8}], [s_{2}, s_{5}] \end{matrix}}$ and $α_{3} = {\begin{matrix} [s_{4}, s_{7}], [s_{5}, s_{8}], \\ [s_{7}, s_{9}], [s_{6}, s_{7}], \\ [s_{4}, s_{6}], [s_{3}, s_{6}] \end{matrix}}$ be the collection of LIVINFNs. Then the score function is defined $\begin{matrix} S_{1} (a) = \frac{{[s_{2 - 3 - 4 - 3 - 5 - 1 - 6 - 4 - 7 - 1 - 3}]}}{12}, \\ S_{2} (a) = \frac{{[s_{5 - 6 - 2 - 6 - 4 - 5 - 3 - 6 - 3 - 8 - 2 - 5}]}}{12}, \\ S_{3} (a) = \frac{{[s_{4 - 7 - 5 - 8 - 7 - 9 - 6 - 7 - 4 - 6 - 3 - 6}]}}{12} \\ S_{1} (a) = - s_{1.66}, S_{2} (a) = - s_{1.4166}, \\ S_{3} (a) = - s_{1.621} . \end{matrix}$

Definition 3.5. Let $α_{j} = {\begin{matrix} [s_{w^{l}}, s_{w^{u}}], \\ [s_{Γ^{l}}, s_{Γ^{u}}], \\ [s_{α^{l}}, s_{α^{u}}], \\ [s_{β^{l}}, s_{β^{u}}], \\ [s_{c^{l}}, s_{c^{u}}], \\ [s_{d^{l}}, s_{d^{u}}] \end{matrix}}$ be the collection of LIVINFNs. Then the accuracy function is defined as $H (a) = \frac{{\begin{matrix} [s_{w^{l}} + s_{w^{u}}] + [s_{Γ^{l}} + s_{Γ^{u}}] + [s_{α^{l}} + s_{α^{u}}] + \\ [s_{β^{l}} + s_{β^{u}}] + [s_{c^{l}} + s_{c^{u}}] + [s_{d^{l}} + s_{d^{u}}] \end{matrix}}}{12} .$

Fig. 2

Different accuracy function.

Example 3.6. Let $α_{1} = {\begin{matrix} [s_{2}, s_{3}], [s_{3}, s_{4}], \\ [s_{3}, s_{5}], [s_{1}, s_{6}], \\ [s_{4}, s_{7}], [s_{1}, s_{3}] \end{matrix}}$ , $α_{2} = {\begin{matrix} [s_{5}, s_{6}], [s_{2}, s_{6}], \\ [s_{4}, s_{5}], [s_{3}, s_{6}], \\ [s_{3}, s_{8}], [s_{2}, s_{5}] \end{matrix}}$ and $α_{3} = {\begin{matrix} [s_{4}, s_{7}], [s_{5}, s_{8}], \\ [s_{7}, s_{9}], [s_{6}, s_{7}], \\ [s_{4}, s_{6}], [s_{3}, s_{6}] \end{matrix}}$ be the collection of LIVINFNs. Then the accuracy function is defined as $\begin{matrix} H_{1} (a) = \frac{{[s_{2 + 3 + 3 + 4 + 1 + 6 + 4 + 7 + 1 + 3}]}}{12}, \\ H_{2} (a) = \frac{{[s_{5 + 6 + 2 + 6 + 4 + 5 + 3 + 6 + 3 + 8 + 2 + 5}]}}{12}, \\ H_{3} (a) = \frac{{[s_{4 + 7 + 5 + 8 + 7 + 9 + 6 + 7 + 4 + 6 + 3 + 6}]}}{12} \\ H_{1} (a) = s_{3.5}, H_{2} (a) = s_{4.58}, H_{3} (a) = s_{5.91} . \end{matrix}$

4 Aggregation operators based on the LIVINFNs

In this section, we define the linguistic interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator and linguistic interval-valued intuitionistic neutrosophic fuzzy hybrid weighted averaging operator.

4.1 Linguistic interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator

Definition 4.1.1. Let $α_{j} = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], \\ [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], \\ [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}}$ be the collection of LIVINFNs. The LIVINFWA operator is a map LIVINFWA: Ωⁿ - > Ω defined by ${LIVINFWA}_{W} (α_{1}, α_{2}, . . . α_{n}) = \oplus_{j = 1}^{n} w_{i} α_{j}$ , where w = (w₁, w₂, . . . . w_n) ^T be the weight vector of α_j and $w_{t} > 0, \sum_{t = 1}^{n} w_{t} = 1 .$

Theorem 4.1.2. The aggregated value of LIVINFNs $α_{j} = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], \\ [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], \\ [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}} (j = 1, 2, . . n)$ by using the LIVINFWA operator is also a LIVINFNs, and $\begin{matrix} {LIVINFWA}_{w} (α_{1,} α_{2, . . . . . .} α_{n}) = \\ {\begin{matrix} [s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (α_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (c_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (d_{1}^{u}))}^{w}}] \end{matrix}} \end{matrix}$

Proof We will prove the above theorem by mathematical induction

Let it is true for n = 2.

${\begin{matrix} [s_{(1 - {(1 - w_{1}^{l}) + (1 - w_{1}^{l})})^{w}}, s_{{(1 - {(1 - w_{1}^{u}) + (1 - w_{2}^{u}})}^{w}}], \\ [s_{{(1 - ({(1 - Γ_{1}^{l}) + (1 - Γ_{2}^{l})})}^{w}}, s_{{(1 - (1 - Γ_{1}^{u}) + (1 - Γ_{2}^{u}))}^{w}}], \\ [s_{{((α_{1}^{l} + α_{2}^{l}))}^{w}}, s_{{((α_{1}^{u} + α_{2}^{u}))}^{w}}], \\ [s_{{((β_{1}^{l} + β_{2}^{l}))}^{w}}, s_{{((β_{1}^{u} + β_{2}^{u}))}^{w}}], \\ [s_{{((c_{1}^{l} + c_{2}^{l}))}^{w}}, s_{{((c_{1}^{u} + c_{2}^{u}))}^{w}}], \\ [s_{{((d_{1}^{l} + d_{2}^{l}))}^{w}}, s_{{((d_{1}^{u} + d_{2}^{u}))}^{w}}] \end{matrix}}$

holds for n = 2

Let it is true for n = k ${\begin{matrix} [s_{{(1 - \prod_{j = 1}^{k} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{k} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (α_{1}^{l}))}^{w}}, s_{\prod_{j = 1}^{k} {((α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (d_{1}^{u}))}^{w}}] \end{matrix}}$

holds for n = k

Now for n = k + 1 ${\begin{matrix} [s_{{(1 - \prod_{j = 1}^{k} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{k} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (α_{1}^{l}))}^{w}}, s_{\prod_{j = 1}^{k} {((α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (d_{1}^{u}))}^{w}}] \end{matrix}} \oplus w_{t} α_{t}$ ${\begin{matrix} [s_{{(1 - \prod_{j = 1}^{k} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{k} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (α_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k} (d_{1}^{u}))}^{w}}] \end{matrix}} \oplus$ ${\begin{matrix} [s_{{(1 - \prod_{j = 1}^{k + 1} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k + 1} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{k + 1} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k + 1} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (α_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (d_{1}^{u}))}^{w}}] \end{matrix}}$ ${\begin{matrix} [s_{{(1 - \prod_{j = 1}^{k + 1} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{k + 1} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{k + 1} (1 - Γ_{1}^{l}))}^{w},} s_{{(1 - \prod_{j = 1}^{k + 1} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (α_{1}^{l}))}^{w}}, s {(\prod_{j = 1}^{k + 1} (α_{1}^{u}))}^{w}], \\ [s_{{(\prod_{j = 1}^{k + 1} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{k + 1} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{k + 1} (d_{1}^{u}))}^{w}}] \end{matrix}}$

Fig. 3

Proposed method.

It holds for n = k + 1. So by mathematical induction, it is true.

4.2 Linguistic interval-valued intuitionistic neutrosophic fuzzy hybrid weighted averaging operator

Definition 4.2.1. Let $α_{j} = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], \\ [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], \\ [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}} (j = 1, 2, . . n)$ be the collection of LIVINFNs, w = (w_1,w₂, . . . . w_n) ^t is the associated weight vector w_t > 0, $\sum_{t = 1}^{n} w_{t} = 1 .$ If a mapping Ωⁿ - > Ω of dimension n satisfies LIVINFHw,w $(α_{1}, α_{2}, . . . α_{n}) = \oplus_{j = 1}^{n} w_{j} α_{ϑ^{.} (j)}$ then LIVINFHw, w is called LIVINF hybrid averaging operator, where α_ϑ^.(j) is the j^th largest LIVINFNs of all LIVINFNs $α_{j}^{.} = {nw}_{t} α_{j} (j = 1, 2, . . . n) .$

Theorem 4.2.2. Let $α_{t} = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], \\ [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], \\ [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], \\ [s_{c_{1}^{l}}, s_{c_{1}^{u}}], \\ [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}}$ be the collection of LIVINFNs then the aggregated value by using the LIVINFHA operator is also a LIVINFNs and is given by LIVINFHA $\begin{matrix} {LIVINFHAw}_{W} (α_{1}, α_{2}, . . . α_{n}) = \\ {\begin{matrix} [s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (α_{1}^{l}))}^{w}}, s_{(\prod_{j = 1}^{n} (α_{1}^{u}))^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (c_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (d_{1}^{u}))}^{w}}] \end{matrix}} \end{matrix}$

5 MCDM with LIVINF information

In this sector, we present a decision-making approach based on the proposed operator for solving the MAGDM problem under the LIVINF environment.

Let we have a group decision making problem in which there are m alternatives A_1,A_2, . . . . A_m and n attribuates C_1,C₂, . . . C_n whose weight vector are w_t, t = 1, 2, . . n such that w_t > 0 and $\sum_{t = 1}^{n} w_{t} = 1$ . Let λ = (λ₁, λ₂, . . λ_t) be the set of decision-makers and w = (w₁, w₂, . . . , w_n) ^T be the weight vector of λ_t (t = 1, 2, . . , n) with w_t > 0 and $\sum_{t = 1}^{n} w_{t} = 1$ . Suppose that the characteristic information of the alternatives A_k (k = 1, 2, . . m) over the attributes C_t (t = 1, 2, . . n) is evaluated by decision-maker λ_t (t = 1, 2, . . , n) and gives the preference in the form of LIVINFNs $α_{kt} = {\begin{matrix} [s_{w_{1}^{l}}, s_{w_{1}^{u}}], [s_{Γ_{1}^{l}}, s_{Γ_{1}^{u}}], [s_{α_{1}^{l}}, s_{α_{1}^{u}}], \\ [s_{β_{1}^{l}}, s_{β_{1}^{u}}], [s_{c_{1}^{l}}, s_{c_{1}^{u}}], [s_{d_{1}^{l}}, s_{d_{1}^{u}}] \end{matrix}}$ , and hence formulated the LIVINF decision matrix.

Based on these the following steps have been summarized for describing the group decision-making approach based on the proposed operation as;

Step 1: Calculate the LIVINF decision matrix

Step 2: Calculate the LIVINFWA and w = (w₁, w₂, . . . , w_n) ^T $\begin{matrix} {LIVIFNWA}_{w} (α_{1}, α_{2}, . . ., α_{n}) = \\ {\begin{matrix} [s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{l}))}^{w}}, s_{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{u}))}], \\ [s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (α_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (c_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (d_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (d_{1}^{u}))}^{w}}] \end{matrix}} \end{matrix}$

Step 3: Calculate the LIVINFWA and w = (w₁, w₂, . . . , w_n) ^T.

LIVINFWA $_{W} (α_{1}, α_{2}, . . . α_{n}) = \oplus_{j = 1}^{n} w_{t} α_{ϑ^{.} (t)}^{.}$ $\begin{matrix} {LIVINFWA}_{W} (α_{1}, α_{2}, . . . α_{n}) = \\ {\begin{matrix} [s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - w_{1}^{u}))}^{w}}], \\ [s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{l}))}^{w}}, s_{{(1 - \prod_{j = 1}^{n} (1 - Γ_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (α_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (α_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (β_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (β_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (c_{1}^{l}))}^{w}}, s_{{(\prod_{j = 1}^{n} (c_{1}^{u}))}^{w}}], \\ [s_{{(\prod_{j = 1}^{n} (d_{1}^{l}))}^{w}}, d_{{(\prod_{j = 1}^{n} (d_{1}^{u}))}^{w}}] \end{matrix}} \end{matrix}$

Step 4: Calculate the score function $S (a) = \frac{{\begin{matrix} [s_{w_{1}^{l}} - s_{w_{1}^{u}}] + [s_{Γ_{1}^{l}} - s_{Γ_{1}^{u}}] + [s_{α_{1}^{l}} - s_{α_{1}^{u}}] \\ + [s_{β_{1}^{l}} - s_{β_{1}^{u}}] + [s_{c_{1}^{l}} - s_{c_{1}^{u}}] + [s_{d_{1}^{l}} - s_{d_{1}^{u}}] \end{matrix}}}{12}$

Step 5: Find the ranking

6 Numerical application

An investment company selects three mines, A₁, A₂ and A₃ as alternatives and considers three factors as the evaluation criteria: (i) C₁ is the geology factor; (ii) C₂ is the mineral reserve risk; (iii) C₃ is the development level of the market. The DMs, Dh. h = 1, 2, 3, gives the evaluation values of alternatives A_i . i = 1, 2, 3 on the criteria C_j (j = 1, 2, 3) in the form of linguistic interval-valued intuitionistic neutrosophic fuzzy numbers. The linguistic interval-valued intuitionistic neutrosophic fuzzy decision matrices are constructed and listed in Tables 1 –3.

Table 1
Interval-valued intuitionistic neutrosophic fuzzy decision

C ₁ C ₂ C ₃

A ₁ ${\begin{matrix} [s_{2}, s_{3}], [s_{3}, s_{4}], \\ [s_{3}, s_{5}], [s_{1}, s_{6}], \\ [s_{4}, s_{7}], [s_{1}, s_{3}] \end{matrix}}$ ${\begin{matrix} [s_{6}, s_{7}], [s_{2}, s_{4}], \\ [s_{3}, s_{5}], [s_{4}, s_{6}], \\ [s_{5}, s_{7}], [s_{3}, s_{6}] \end{matrix}}$ ${\begin{matrix} [s_{4}, s_{6}], [s_{3}, s_{5}], \\ [s_{3}, s_{6}], [s_{2}, s_{3}], \\ [s_{3}, s_{5}], [s_{3}, s_{4}] \end{matrix}}$

A ₂ ${\begin{matrix} [s_{2}, s_{3}], [s_{3}, s_{5}], \\ [s_{2}, s_{6}], [s_{1}, s_{6}], \\ [s_{5}, s_{7}], [s_{1}, s_{3}] \end{matrix}}$ ${\begin{matrix} [s_{5}, s_{6}], [s_{2}, s_{6}], \\ [s_{4}, s_{5}], [s_{3}, s_{6}], \\ [s_{3}, s_{8}], [s_{2}, s_{5}] \end{matrix}}$ ${\begin{matrix} [s_{4}, s_{6}], [s_{4}, s_{9}], \\ [s_{7}, s_{9}], [s_{5}, s_{7}], \\ [s_{3}, s_{7}], [s_{4}, s_{6}] \end{matrix}}$

A ₃ ${\begin{matrix} [s_{3}, s_{5}], [s_{4}, s_{7}], \\ [s_{1}, s_{4}], [s_{2}, s_{5}], \\ [s_{3}, s_{6}], [s_{2}, s_{3}] \end{matrix}}$ ${\begin{matrix} [s_{3}, s_{5}], [s_{5}, s_{8}], \\ [s_{6}, s_{7}], [s_{4}, s_{6}], \\ [s_{1}, s_{3}], [s_{2}, s_{4}] \end{matrix}}$ ${\begin{matrix} [s_{4}, s_{5}], [s_{2}, s_{3}], \\ [s_{7}, s_{8}], [s_{3}, s_{4}], \\ [s_{3}, s_{5}], [s_{3}, s_{8}] \end{matrix}}$

	C ₁	C ₂	C ₃
A ₁	${\begin{matrix} [s_{2}, s_{3}], [s_{3}, s_{4}], \\ [s_{3}, s_{5}], [s_{1}, s_{6}], \\ [s_{4}, s_{7}], [s_{1}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{6}, s_{7}], [s_{2}, s_{4}], \\ [s_{3}, s_{5}], [s_{4}, s_{6}], \\ [s_{5}, s_{7}], [s_{3}, s_{6}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{6}], [s_{3}, s_{5}], \\ [s_{3}, s_{6}], [s_{2}, s_{3}], \\ [s_{3}, s_{5}], [s_{3}, s_{4}] \end{matrix}}$
A ₂	${\begin{matrix} [s_{2}, s_{3}], [s_{3}, s_{5}], \\ [s_{2}, s_{6}], [s_{1}, s_{6}], \\ [s_{5}, s_{7}], [s_{1}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{5}, s_{6}], [s_{2}, s_{6}], \\ [s_{4}, s_{5}], [s_{3}, s_{6}], \\ [s_{3}, s_{8}], [s_{2}, s_{5}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{6}], [s_{4}, s_{9}], \\ [s_{7}, s_{9}], [s_{5}, s_{7}], \\ [s_{3}, s_{7}], [s_{4}, s_{6}] \end{matrix}}$
A ₃	${\begin{matrix} [s_{3}, s_{5}], [s_{4}, s_{7}], \\ [s_{1}, s_{4}], [s_{2}, s_{5}], \\ [s_{3}, s_{6}], [s_{2}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{3}, s_{5}], [s_{5}, s_{8}], \\ [s_{6}, s_{7}], [s_{4}, s_{6}], \\ [s_{1}, s_{3}], [s_{2}, s_{4}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{5}], [s_{2}, s_{3}], \\ [s_{7}, s_{8}], [s_{3}, s_{4}], \\ [s_{3}, s_{5}], [s_{3}, s_{8}] \end{matrix}}$

Table 2

Interval-valued intuitionistic neutrosophic fuzzy decision

	C ₁	C ₂	C ₃
A ₁	${\begin{matrix} [s_{2}, s_{4}], [s_{4}, s_{5}], \\ [s_{1}, s_{3}], [s_{4}, s_{6}], \\ [s_{1}, s_{2}], [s_{1}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{2}, s_{6}], [s_{2}, s_{4}], \\ [s_{5}, s_{7}], [s_{4}, s_{6}], \\ [s_{3}, s_{4}], [s_{2}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{6}], [s_{4}, s_{5}], \\ [s_{2}, s_{3}], [s_{2}, s_{7}], \\ [s_{4}, s_{5}], [s_{2}, s_{4}] \end{matrix}}$
A ₂	${\begin{matrix} [s_{2}, s_{3}], [s_{4}, s_{5}], \\ [s_{3}, s_{6}], [s_{1}, s_{6}], \\ [s_{5}, s_{7}], [s_{4,} s_{3}] \end{matrix}}$	${\begin{matrix} [s_{3}, s_{6}], [s_{2}, s_{5}], \\ [s_{2}, s_{4}], [s_{4}, s_{6}], \\ [s_{2}, s_{3}], [s_{3}, s_{5}] \end{matrix}}$	${\begin{matrix} [s_{3}, s_{5}], [s_{2}, s_{7}], \\ [s_{4}, s_{6}], [s_{5}, s_{9}], \\ [s_{3}, s_{2}], [s_{4}, s_{2}] \end{matrix}}$
A ₃	${\begin{matrix} [s_{3}, s_{7}], [s_{5}, s_{6}], \\ [s_{2}, s_{6}], [s_{3}, s_{5}], \\ [s_{4}, s_{5}], [s_{1}, s_{6}] \end{matrix}}$	${\begin{matrix} [s_{2}, s_{6}], [s_{5}, s_{7}], \\ [s_{6}, s_{9}], [s_{4}, s_{5}], \\ [s_{5}, s_{7}], [s_{1}, s_{4}] \end{matrix}}$	${\begin{matrix} [s_{2}, s_{3}], [s_{4}, s_{7}], \\ [s_{7}, s_{8}], [s_{6}, s_{9}], \\ [s_{4}, s_{6}], [s_{3}, s_{5}] \end{matrix}}$

Table 3

Interval-valued intuitionistic neutrosophic fuzzy decision

	C ₁	C ₂	C ₃
A ₁	${\begin{matrix} [s_{4}, s_{5}], [s_{4}, s_{6}], \\ [s_{2}, s_{3}], [s_{1}, s_{6}], \\ [s_{5}, s_{7}], [s_{2}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{3}, s_{4}], [s_{3}, s_{4}], \\ [s_{2}, s_{3}], [s_{4}, s_{6}], \\ [s_{6}, s_{7}], [s_{4}, s_{7}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{6}], [s_{4}, s_{7}], \\ [s_{5}, s_{8}], [s_{2}, s_{3}], \\ [s_{4}, s_{7}], [s_{6}, s_{9}] \end{matrix}}$
A ₂	${\begin{matrix} [s_{4}, s_{9}], [s_{4}, s_{7}], \\ [s_{2}, s_{4}], [s_{1}, s_{6}], \\ [s_{5}, s_{7}], [s_{1}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{9}], [s_{3}, s_{7}], \\ [s_{4}, s_{5}], [s_{7}, s_{9}], \\ [s_{8}, s_{9}], [s_{3}, s_{5}] \end{matrix}}$	${\begin{matrix} [s_{3}, s_{5}], [s_{4}, s_{6}], \\ [s_{3}, s_{6}], [s_{4}, s_{8}], \\ [s_{3}, s_{7}], [s_{4}, s_{9}] \end{matrix}}$
A ₃	${\begin{matrix} [s_{4}, s_{6}], [s_{5}, s_{9}], \\ [s_{1}, s_{4}], [s_{4}, s_{7}], \\ [s_{4}, s_{8}], [s_{2}, s_{3}] \end{matrix}}$	${\begin{matrix} [s_{3}, s_{6}], [s_{5}, s_{3}], \\ [s_{6}, s_{8}], [s_{4}, s_{9}], \\ [s_{1}, s_{3}], [s_{1}, s_{4}] \end{matrix}}$	${\begin{matrix} [s_{4}, s_{7}], [s_{5}, s_{8}], \\ [s_{7}, s_{9}], [s_{6}, s_{7}], \\ [s_{4}, s_{6}], [s_{3}, s_{6}] \end{matrix}}$

Table 4

LIVINFWA

	C ₁	C ₂	C ₃
A ₁	${\begin{matrix} [s_{2.30}, s_{2.53}], [s_{3.0001}, s_{2.67}], \\ , [s_{1.39}, s_{3.63}], \\ [s_{2.05}, s_{3.00}], [s_{1.18}, s_{2.20}] \end{matrix}}$	${\begin{matrix} [s_{2.73}, s_{3.94}], [s_{2.18}, s_{2.20}], \\ [s_{2.20}, s_{3.05}], [s_{2.71}, s_{3.63}], \\ [s_{2.94}, s_{3.54}], [s_{2.14}, s_{3.19}] \end{matrix}}$	${\begin{matrix} [s_{3.20}, s_{4.18}], [s_{3.00}, s_{3.99}], \\ [s_{2.26}, s_{3.29}], [s_{1.64}, s_{2.70}], \\ [s_{2.53}, s_{2.76}], [s_{2.36}, s_{3.29}] \end{matrix}}$
A ₂	${\begin{matrix} [s_{2.73}, s_{6.65}], [s_{4.24}, s_{10.79}], \\ [s_{3.46}, s_{12.00}], [s_{1}, s_{14.69}], \\ [s_{11.18}, s_{18.52}], [s_{2.00}, s_{5.19}] \end{matrix}}$	${\begin{matrix} [s_{5.89}, s_{15.14}], [s_{2.41}, s_{11.95}], \\ [s_{5.65}, s_{11.18}], [s_{9.16}, s_{14.69}], \\ [s_{6.92}, s_{14.69}], [s_{3.46}, s_{11.18}] \end{matrix}}$	${\begin{matrix} [s_{4.46}, s_{9.94}], [s_{7.00}, s_{16.79}], \\ [s_{9.16}, s_{18.00}], [s_{10.00}, s_{19.79}], \\ [s_{5.19}, s_{9.69}], [s_{8.00}, s_{10.39}] \end{matrix}}$
A ₃	${\begin{matrix} [s_{2.90}, s_{4.47}], [s_{2.73}, s_{4.15}], \\ [s_{1.19}, s_{3.27}], [s_{2.28}, s_{3.82}], \\ [s_{2.73}, s_{4.15}], [s_{1.43}, s_{2.82}] \end{matrix}}$	${\begin{matrix} [s_{2.43}, s_{4.31}], [s_{2.94}, s_{4.16}], \\ [s_{4.04}, s_{5.04}], [s_{2.94}, s_{4.28}], \\ [s_{1.51}, s_{2.93}], [s_{1.19}, s_{2.94}] \end{matrix}}$	${\begin{matrix} [s_{2.90}, s_{2.73}], [s_{3.05}, s_{4.16}], \\ [s_{4.56}, s_{5.22}], [s_{3.37}, s_{4.21}], \\ [s_{2.73}, s_{3.85}], [s_{2.35}, s_{4.15}] \end{matrix}}$

Step 1: Calculate the interval-valued intuitionistic neutrosophic fuzzy decision Table 1, 2 and 3.

Step 2: Calculate the LIVINFWA operator and w = (0.24, 0.50, 0.26) ^T.

Step 3: Calculate the LIVINFWA operator.

Step 4: Calculate the score function S₁ (a) = - s_0.1801, S₂ (a) = - s_16.5148, S₃ (a) = - s_0.4632.

Table 5

Table5

	C ₁
A ₁	${\begin{matrix} [s_{2.4677}, s_{2.8937}], [s_{2.4512}, s_{2.5367}], \\ , [s_{1.5483}, s_{2.3565}], \\ [s_{1.9558}, s_{2.2495}], [s_{1.5343}, s_{2.1243}] \end{matrix}}$
A ₂	${\begin{matrix} [s_{6.3314}, s_{27.5751}], [s_{6.2354}, s_{42.1423}], \\ [s_{13.3816}, s_{49.1414}], [s_{9.5077}, s_{65.3498}], \\ [s_{20.0318}, s_{51.3444}], [s_{7.4404}, s_{24.5534}] \end{matrix}}$
A ₃	${\begin{matrix} [s_{2.5322}, s_{3.1753}], [s_{2.2491}, s_{3.4513}], \\ [s_{2.2316}, s_{3.1842}], [s_{2.2491}, s_{3.0048}], \\ [s_{1.7877}, s_{2.7183}], [s_{1.4338}, s_{2.5091}] \end{matrix}}$

Step 5: Find the ranking S₂ (a) > S₃ (a) > S₁ (a) and S₂ (a) is the best one.

7 Comparison analysis

To check the legitimacy and viability of the proposed methodology, a near report is led utilizing the techniques linguistic interval-valued intuitionistic fuzzy number [12] and linguistic neutrosophic [17], which are unique instances of the linguistic interval-valued intuitionistic neutrosophic fuzzy number, to the equivalent illustrative model.

7.1 A comparison analysis with the existing MCDM method linguistic interval-valued intuitionistic fuzzy number

The linguistic interval-valued intuitionistic fuzzy number can be considered as a special case of linguistic interval-valued intuitionistic neutrosophic fuzzy numbers when there is only four element in membership and non-membership degree. For comparison, the linguistic interval-valued intuitionistic fuzzy number can be transformed to linguistic interval-valued intuitionistic fuzzy number by calculating the average value of the membership and nonmember ship degrees. After transformation, the linguistic interval-valued intuitionistic fuzzy number information is given in Table 6.

Fig. 4

Different score function.

Table 6

Linguistic Interval-valued Atanassov intuitionistic fuzzy

	C ₁	C ₂	C ₃
A ₁	{ [s₂, s₃] , [s₃, s₄]}	{ [s₆, s₇] , [s₂, s₄]}	{ [s₄, s₆] , [s₃, s₅]}
A ₂	{ [s₂, s₃] , [s₃, s₅]}	{ [s₅, s₆] , [s₂, s₆]}	{ [s₄, s₆] , [s₄, s₅}
A ₃	{ [s₃, s₅] , [s₄, s₇]}	{ [s₃, s₅] , [s₄, s₆]}	{ [s₄, s₅] , [s₂, s₃]}

Table 7

LIVAIFWA operator

A ₁	{ [s_0.1840, s_0.3086] , [s_0.2601, s_0.3722]}
A ₂	{, [s_0.0699, s_0.1746}
A ₃	{ [s_0.1570, s_0.2379] , [s_0.2701, s_0.3939]}

Calculate the LIVAIFWA operator

Calculate the score function S₁ (a) = s_8.4651, S₂ (a) = s_8.6331, S₃ (a) = - s_8.4327.

Fig. 5

Score function of linguistici nterval - valued intuitionistic fuzzy number.

Obviously, the ranking is derived from the method proposed by Garg et al. [12], is different from the result of the proposed method. The linguistic interval-valued intuitionistic neutrosophic fuzzy numbers are more flexible than linguistic interval-valued intuitionistic fuzzy number because they consider the situations where decision-makers would like to use several possible values to express the membership and non-membership degrees.

7.2 A Linguistic Neutrosophic Multi-criteria Group Decision-Making Approach with EDAS Method

The linguistic neutrosophic fuzzy number can be considered as a special case of linguistic interval-valued intuitionistic neutrosophic fuzzy numbers when there is only four element in membership and non-membership degree. For comparison, the linguistic neutrosophic fuzzy number can be transformed to linguistic neutrosophic fuzzy number by calculating the average value of the membership and nonmember ship degrees. After transformation, the linguistic neutrosophic fuzzy number information is given in Table 8.

Table 8
The linguistic neutrosophic decision matrix 8

C ₁ C ₂ C ₃

A ₁ [s₂, s₃, s₄] [s₃, s₅, s₆] [s₃, s₆, s₂]

A ₂ [s₂, s₃, s₅] [s₄, s₅, s₆] [s₄, s₇, s₉]

A ₃ [s₃, s₅, s₇] [s₁, s₂, s₄] [s₃, s₅, s₈]

	C ₁	C ₂	C ₃
A ₁	[s₂, s₃, s₄]	[s₃, s₅, s₆]	[s₃, s₆, s₂]
A ₂	[s₂, s₃, s₅]	[s₄, s₅, s₆]	[s₄, s₇, s₉]
A ₃	[s₃, s₅, s₇]	[s₁, s₂, s₄]	[s₃, s₅, s₈]

Table 9

LNFWA operator

A ₁	[s_2.001, s_2.945, s_2.532]
A ₂	[s_5.657, s_10.247, s_16.432]
A ₃	[s_1.771, s_2.765, s_4.084]

Calculate the LNFWA operator and w = (0.24, 0.50, 0.26) ^T

Calculate the score function S₁ (a) =0.087, S₂ (a) = -2.837, S₃ (a) =2.103 . Ranking S₃ (a) > S₁ (a) > S₂ (a) .

Obviously, the ranking is derived from the method proposed by Y. Y. Li [17], is different from the result of the proposed method. The linguistic interval-valued intuitionistic neutrosophic fuzzy numbers are more flexible than linguistic neutrosophic fuzzy number because they consider the situations where decision-makers would like to use several possible values to express the membership and non-membership degrees.

Comparison analysis in Table 10

Table 10

The ranking values of the above discussion is given in Table 10

Method	Ranking
Proposed method	S₂ (a) > S₁ (a) > S₃ (a)
LIVIFWA operator [12]	S₂ (a) > S₁ (a) > S₃ (a)
LNFWA operator [17]	S₃ (a) > S₁ (a) > S₂ (a)

The following advantages of our proposal can be summarized based on the above comparison analyses. The linguistic interval-valued intuitionistic neutrosophic fuzzy number is very suitable for illustrating uncertain or fuzzy information in MCDM problems because the membership and non-membership degrees can be two sets of several possible values, which cannot be achieved by linguistic interval-valued intuitionistic fuzzy number. On the bases of basic operations, aggregation operators and comparison method of linguistic interval-valued intuitionistic neutrosophic fuzzy number can be also used to process linguistic interval-valued intuitionistic fuzzy number, because the linguistic interval-valued intuitionistic neutrosophic fuzzy number can be considered as the generalized form of a linguistic interval-valued intuitionistic fuzzy number. The defined operations of linguistic interval-valued intuitionistic neutrosophic fuzzy number give us more accurate than the existing operators.

The results of the score value of the numerical example, are Tabulated below, Table 11

Table 11

Table11

Method	Score function
TNCFEHWA Aliya et al. [14]	S₂ (a) > S₁ (a) > S₃ (a)
IVFDM using soft L. Fei [15]	S₁ (a) > S₃ (a) > S₂ (a)
SVNNWBMEA operator L. Yang [16]	S₂ (a) > S₁ (a) > S₃ (a)
LN Li [17]	S₃ (a) > S₁ (a) > S₂ (a)
IVNS Z. Aiwu et al. [18]	S₂ (a) > S₁ (a) > S₃ (a)
IVIFPHWA operator H. Wu [19]	S₁ (a) > S₃ (a) > S₂ (a)
IVIFEHWA operator Wang [20]	S₂ (a) > S₃ (a) > S₁ (a)
GIVIFGA Operator Yu [21]	S₁ (a) > S₃ (a) > S₂ (a)

8 Conclusion

In this paper, we define a new concept of linguistic interval-valued intuitionistic neutrosophic fuzzy numbers and operational laws. We discuss the two aggregation operators, which are defined as linguistic interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator and linguistic interval-valued intuitionistic neutrosophic fuzzy hybrid weighted averaging operator, for collection of data. We define the MAGDM problem under the LIVINF environment. Finally, we define a numerical example. Moreover, we apply the developed aggregation operator to multiple attribute group decision-making with linguistic interval-valued intuitionistic neutrosophic fuzzy information. At last, a numerical example is used to illustrate the validity of the exhibit approach in group decision-making problems. In group decision-making problems, because the expert usually comes from different specialty field and have different backgrounds and levels of knowledge, they usually have diverging opinions.

Footnotes

Acknowledgments

The author 4 extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups Program under grant number R.G.P2/52/40

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A new approach of interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator based on decision making problem

Abstract

Keywords

1 Introduction

2 Basic concept

3 Linguistic interval-valued intuitionistic neutrosophic fuzzy numbers and operational laws

4.1 Linguistic interval-valued intuitionistic neutrosophic fuzzy weighted averaging operator

5 MCDM with LIVINF information

6 Numerical application

7.1 A comparison analysis with the existing MCDM method linguistic interval-valued intuitionistic fuzzy number

Table 8 The linguistic neutrosophic decision matrix 8 C 1 C 2 C 3 A 1 [s2, s3, s4] [s3, s5, s6] [s3, s6, s2] A 2 [s2, s3, s5] [s4, s5, s6] [s4, s7, s9] A 3 [s3, s5, s7] [s1, s2, s4] [s3, s5, s8]

Footnotes

Acknowledgments

References

Table 8
The linguistic neutrosophic decision matrix 8

C ₁ C ₂ C ₃

A ₁ [s₂, s₃, s₄] [s₃, s₅, s₆] [s₃, s₆, s₂]

A ₂ [s₂, s₃, s₅] [s₄, s₅, s₆] [s₄, s₇, s₉]

A ₃ [s₃, s₅, s₇] [s₁, s₂, s₄] [s₃, s₅, s₈]