Aggregation operators on triangular cubic fuzzy numbers and its application to multi-criteria decision making problems

Abstract

In this paper, we introduce the concept of triangular cubic fuzzy numbers. We discuss some basic operational laws of triangular cubic fuzzy numbers and then develop triangular cubic fuzzy weighted average (TCFWA) operator. We also define crisp weighted possibility means of TCFNs and hamming distance between TCFNs. Furthermore, we extend the classical VIKOR method to solve the MCDM method based on triangular cubic fuzzy numbers. The new ranking method for TCFNs is used to rank the alternatives. Finally, an illustrative example is given to verify and demonstrate the practicality and effectiveness of the proposed method.

Keywords

The crisp weighted possibility means of TCFNs weighted average operator of TCFNs hamming distance for TCFNs A MCDM method VIKOR method using TCFNS

1 Introduction

It is not easy for the decision makers to give the precise valuation on complex things in practical decision making problems because of the effects of the subjective and objective reasons. As there exists some hesitations for decision makers to access the fuzzy and uncertain quantities. Therefor in [8], Atanassov introduced the concept of intuitionistic fuzzy set (IFS), which is the generalization of fuzzy set [50] and characterized by membership and non-membership degrees. Intuitionistic fuzzy set is more suitable to deal with uncertainty and ambiguity. Intuitionistic fuzzy set is more suitable to deal with uncertainty and ambiguity. In the real world, due to the complexity, fuzziness and uncertainties of the objective things, the criteria involved in the decision making problem may not be appropriate to express them by exact numerical values. It is more suitable to describe them by means of linguistic variables. Hence, linguistic information has frequently been applied to MCDM problems. In [27], Lui and Teng proposed, the extended TODIM method to deal with MADM problems in which the attribute values are in the form of 2-dimension uncertain linguistic variable. In [25] Liu and Shi extended the HM operator to the neutrosophic uncertain linguistic set for multi-attribute group decision making and proposed some Heronian mean operators based on neutrosophic uncertain linguistic numbers, namely the neutrosophic uncertain linguistic number improved generalized weighted Heronian mean (NULNIGWHM) operator and the neutrosophic uncertain linguistic number improved generalized geometric weighted Heronian mean (NULNIGGWHM) operator. In [30], Liu et al. proposed some aggregation operators, including 2-dimension uncertain linguistic generalized weighted average operator, 2-dimension uncertain linguistic generalized ordered weighted average operator, and 2-dimension uncertain linguistic generalized hybrid weighted average operator for multi-attribute decision making problems. Liu and Tang [29], Proposed the interval neutrosophic uncertain linguistic Choquet averaging (INULCA) operator and the interval neutrosophic uncertain linguistic Choquet geometric (INULCG) operator, generalized Shapley INULCA (GS-INULCA) operator and the generalized Shapley INULCG (GS-INULCG) operator for mulit-attribute decision making. Liu [24] proposed the interval-valued intuitionistic fuzzy power Heronian aggregation (IVIFPHA) operator, interval-valued intuitionistic fuzzy power weight Heronian aggregation (IVIFPWHA) operator and at the same time, we presented a new similarity function of IVIFNs as support degree in power weighting, and it has a good reliability and accuracy. Liu et al. [28] extend the PBM operator and the PGBM operator based on the interaction operational laws of intuitionistic fuzzy sets (IFSs) to propose the interaction PBM (IFIPBM) operator for intuitionistic fuzzy numbers (IFNs), the weighted interaction PBM (IFWIPBM) operator for IFNs, the interaction PGBM (IFIPGBM) operator for IFNs and the weighted interaction PGBM (IFWIPGBM) operator for IFNs. Liu et al. [26] proposed the multi-valued neutrosophic weighted Bonferroni mean (MVNWBM) operator and the multi-valued neutrosophic weighted geometric Bonferroni mean (MVNWGBM) operator and some properties of them are also investigated. In [23] Lue and Wang developed some aggregation operators to fuse the decision information represented by LIFNs, including the improved linguistic intuitionistic fuzzy weighted averaging (ILIFWA) operator and the improved linguistic intuitionistic fuzzy weighted power average (ILIFWPA) operator and proposed some new methods to deal with the multi-attribute group decision making (MAGDM) problems under the linguistic intuitionistic fuzzy environment. Later, in [1], Atanassov and Gargov extended the concept of intuitionistic fuzzy set and introduced the concept of interval-valued intuitionistic fuzzy set (IVIFS). Many researchers [3 , 12] studied, intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set such as operators [8 , 49], operations [1 , 9] and distances [5, 34], and have been applied to many different fields, such as decision making [16 , 48], supplier selection [11, 43], investment option [43, 45] et al.

Since the domain of domain of intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set are discrete sets, therefore they are only used to specify the degree to which the criterion does or does not belong to some fuzzy concepts [36]. Thus to remove this defect in [33] Shu et al. gave the defined the concept of triangular intuitionistic fuzzy number and operational laws of triangular intuitionistic fuzzy number. An important characteristic of the triangular intuitionistic fuzzy set is that its domain is a consecutive set. Due to the flexibility of triangular intuitionistic fuzzy numbers many authors have paid attention to the research on triangular intuitionistic fuzzy numbers [17 , 52–57], these researches can be roughly classified into two types: decision making methods [1 , 57–59] and aggregation operators, which are respectively reviewed as follows:

In the feature of decision-making methods, Li [19] acute out and altered some errors in the definition of the operational laws of triangular intuitionistic fuzzy numbers presented by Shu et al. [33]. Li [20] discussed the idea of the triangular intuitionistic fuzzy number and the ranking method of the triangular intuitionistic fuzzy number on the foundation of the notion of a ratio of the value index to the equivocalness index as well as applications to MADM problems. Nan et al. [31] defined the ranking order relatives of triangular intuitionistic fuzzy number, which are practical to matrix games with payoffs of the triangular intuitionistic fuzzy number. Wan [37] introduced the notions of possibility mean and variance for triangular intuitionistic fuzzy numbers, acquired a new decision method established on possibility mean and variance of triangular intuitionistic fuzzy numbers. Li et al. [21] defined the values and obscurities of the membership degree and non-membership for a triangular intuitionistic fuzzy number as well as the value index ambiguity-index and developed a ranking method established on appraise and ambiguity. Wang [39] proposed new arithmetic operations and logic operators for triangular intuitionistic fuzzy numbers and applied them to fault analysis of a printed circuit board assembly system. Through the existing literature, we can found that the aggregation operators of triangular intuitionistic fuzzy numbers are still quite limited and the methods for ranking triangular intuitionistic fuzzy numbers are a bit complicated, which are inconvenient to compare triangular intuitionistic fuzzy numbers.

Cubic sets are the generalizations of fuzzy sets and intuitionistic fuzzy sets, in which there are two representations, one is used for the degree of membership and other is used for the degree of non-membership. The membership function is grip in the form of interval whereas non-membership is thought over the normal fuzzy set [18]. Due to the motivation and inspiration of the above discussion in this paper we generalized the concept of triangular fuzzy sets and triangular intuitionistics fuzzy sets and introduce the concept of triangular cubic fuzzy sets. If we take only one element in the membership degree of the triangular cubic fuzzy number, i.e. instead of interval we take a fuzzy number, than we get triangular intuitionistic fuzzy numbers, similarly if we take membership degree as fuzzy number and non-membership degree equal to zero, than we get triangular fuzzy numbers.

In Section 2, we discussed some basic concepts Fuzzy set theory, Interval-Valued Fuzzy Set, and Cubic Set Theory. In Section 3, we discussed some basic concepts and operation laws related to triangular cubic fuzzy numbers and crisp weighted possibility means are defined. In Section 4, we develop Weighted average operator of TCFNs and hamming distance of the TCFN are defined. In Section 5, we develop an MCDM method approach based on an extended VIKOR method using TCFNS; MCDM method using TCFN’s are developed. Finally, an illustrative example is given to verify the developed approach. In Section 6, we discuss in comparison analyses.

2 Preliminaries

In this section, we define some basic concepts of Fuzzy set theory and Cubic Set Theory.

Definition 2.1. [51] Let H be a universe of discourse. The idea of fuzzy set was presented by Zadeh and defined as following; $J = {h, Γ_{J} (h) | h \in H}$ (1)

A fuzzy set in a set H is defined Γ_J : H → I, is a membership function, Γ_J (h) denoted the degree of membership of the element h to the set H, where I = [0, 1]. The collection of all fuzzy subsets of H is denoted by I^H. Define a relation on I^H as follows:

$\begin{matrix} (\forall Γ, η \in I^{H}) (Γ \leq η \\ \Leftrightarrow (\forall h \in H) (Γ (h) \leq η (h))) . \end{matrix}$ (2)

Definition 2.2. [1, 4] An Atanassov intuitionistic fuzzy set on H is a set $J = {h, Γ (h), η (h) : h \in H}$ (3) where Γ_J and η_j are membership and non-membership function, respectively $Γ_{J} (h) : h \to [0, 1], h \in H \to Γ_{J} (h) \in [0, 1]$ (4) $η_{J} (h) : h \to [0, 1], h \in H \to η_{J} (h) \in [0, 1]$ (5) and $0 \leq Γ_{J} (h) + η_{J} (h) \leq 1 for all h \in H$ (6) $π_{J} (h) = 1 - Γ_{J} (h) - η_{J} (h)$ (7)

Definition 2.3. [18] Let H be a nonempty set. By a cubic set in H we mean a structure F = {h, α (h) , β (h) : h ∈ H} in which α is an IVF set in H and β is a fuzzy set in H . A cubic set F = {h, α (h) , β (h) : h ∈ H} is simply denoted by F = 〈α, β〉 . Denote by C^H the collection of all cubic sets in H . A cubic set F = 〈α, β〉 in which α (h) =0 and β (h) =1 (resp. α (h) =1 and β (h) =0 for all h ∈ H is denoted by 0 (resp. 1). A cubic set D = 〈λ, ξ〉, in which λ (h) =0 and ξ (h) =0(resp. λ (h) =1 and ξ (h) = (1) for all h ∈ H is denoted by 0 (resp. 1).

Definition 2.4. [18] Let H be a nonempty set. A cubic set F = (C, λ) in H is said to be an internal cubic set if $C^{-} (h) \leq λ (h) \leq C^{+} (h) for all h \in H .$

Definition 2.5. Let H be a non-empty set. A cubic set F = (C, λ) in H is said to be an external cubic set if λ (h) ∉ (C^- (h) , C⁺ (h)) for all h ∈ H .

3 Triangular cubic fuzzy numbers

In this section, we exhibit some basic concepts comprising the definitions, operations and crisp weighted possibility means of the TCFNs, whichever are used in the subsections.

3.1 The definition and arithmetical operations of TCFNs

In this subsection, we shall exist some fundaments ideas related to cubic triangular fuzzy numbers and some new operators with cubic triangular fuzzy numbers are also defined.

Definition 3.1.1. Let $\tilde{b}$ be the triangular cubic fuzzy number on the set of real numbers, its IVFS is defined as: $λ_{\tilde{b}} (h) = {\begin{matrix} [\frac{(h - r)}{(s - r)} ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], & (r \leq h < r) \\ [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], & h = s, s \\ [\frac{(t - h)}{(t - s)} ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], & (r \leq h < t) \\ 0 & otherwise \end{matrix}$ and its triangular fuzzy set is defined $Γ_{\tilde{b}} (h) = {\begin{matrix} \frac{(h - r) η_{\tilde{b}}}{(s - r)}, & r \leq h < s \\ η_{\tilde{b}} & h = s \\ \frac{(t - h) η_{\tilde{b}}}{(t - s)} & s < h \leq t \\ 0 & otherwise \end{matrix}$ where $0 \leq λ_{\tilde{b}} (h) \leq 1, 0 \leq Γ_{\tilde{b}} (h) \leq 1$ and r, s, t are real numbers, such that r ≤ s ≤ t . The values of ω_{b
^-}, ω_{b
⁺} consequently the maximum values of the degree of membership and $η_{\tilde{b}}$ minimum degree of a fuzzy set. Then the TCFN $\tilde{b}$ basically denoted by $\tilde{b} = 〈 (r, s, t); [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}} 〉 .$ Further, the TCFN reduced to a TCFN. Moreover, if $ω_{\tilde{b}}^{-} = 1, ω_{\tilde{b}}^{+} = 1$ and $η_{\tilde{b}} = 0$ , if the TCFN $\tilde{b}$ is called a normal TCFN denoted as $\tilde{b} = 〈 (r, s, t); [1, 1], (0) 〉 .$

Therefore, the TCFN considered now can be regarded as generalized TCFN. Such numbers remand the doubt information in a more flexible approach than normal fuzzy numbers as the values $ω_{b}^{-}, ω_{b}^{+}, η_{\tilde{b}} \in [0, 1]$ can be interpreted as the degree of confidence in the quantity characterized by r, s, t . Then $\tilde{b}$ is called triangular cubic fuzzy number (TCFN).

Definition 3.1.2. Let ${\tilde{b}}_{1} = 〈 (r_{1}, s_{1}, t_{1}); [ω_{{\tilde{b}}_{1}}^{-}, ω_{{\tilde{b}}_{1}}^{+}], η_{{\tilde{b}}_{1}} 〉$ and ${\tilde{b}}_{2} = 〈 (r_{2}, s_{2}, t_{2}); [ω_{{\tilde{b}}_{2}}^{-}, ω_{{\tilde{b}}_{2}}^{+}], η_{{\tilde{b}}_{2}} 〉$ be two TCFNs and ξ be any real number. The operational laws over TCFNs are solid as under: $\begin{matrix} 1) & {\tilde{b}}_{1} + {\tilde{b}}_{2} = 〈 (r_{1} + r_{2}, s_{1} + s_{2}, t_{1} + t_{2}), \\ [min (ω_{{\tilde{b}}_{1}}^{-}, ω_{{\tilde{b}}_{2}}^{-}), min (ω_{{\tilde{b}}_{1}}^{+}, ω_{{\tilde{b}}_{2}}^{+})], \\ max (η_{{\tilde{b}}_{1}}, η_{{\tilde{b}}_{2}}) 〉, \\ 2) & {\tilde{b}}_{1} - {\tilde{b}}_{2} = 〈 (r_{1} - r_{2}, s_{1} - s_{2}, t_{1} - t_{2}), \\ [min (ω_{{\tilde{b}}_{1}}^{-}, ω_{{\tilde{b}}_{2}}^{-}), min (ω_{{\tilde{b}}_{1}}^{+}, ω_{{\tilde{b}}_{2}}^{+})], \\ max (η_{{\tilde{b}}_{1}}, η_{{\tilde{b}}_{2}}) 〉, \\ 3) & {\tilde{b}}_{1} {\tilde{b}}_{2} = 〈 (r_{1} r_{2}, s_{1} s_{2}, t_{1} t_{2}), \\ [min (ω_{{\tilde{b}}_{1}}^{-}, ω_{{\tilde{b}}_{2}}^{-}), min (ω_{{\tilde{b}}_{1}}^{+}, ω_{{\tilde{b}}_{2}}^{+})], \\ max (η_{{\tilde{b}}_{1}}, η_{{\tilde{b}}_{2}})] \\ 4) & ξ {\tilde{b}}_{1} = 〈 (ξ r_{1}, ξ s_{1}, ξ t_{1}), [ω_{{\tilde{b}}_{1}}^{-}, ω_{{\tilde{b}}_{1}}^{+}], η_{{\tilde{b}}_{1}} 〉 \\ if ξ \geq 0 \\ 5) & ξ {\tilde{b}}_{1} = 〈 (r_{1}^{ξ}, s_{1}^{ξ}, t_{1}^{ξ}), [(ω_{{\tilde{b}}_{1}}^{-})^{ξ}, (ω_{{\tilde{b}}_{1}}^{+})^{ξ}], \\ 1 - (1 - η_{{\tilde{b}}_{1}})^{ξ} 〉 if ξ \geq 0 \end{matrix}$

Theorem 3.1.3.Let ${\tilde{b}}_{1} = 〈 (r_{1}, s_{1}, t_{1}); [ω_{{\tilde{b}}_{1}}^{-}, ω_{{\tilde{b}}_{1}}^{+}], η_{{\tilde{b}}_{1}} 〉$ and ${\tilde{b}}_{2} = 〈 (r_{2}, s_{2}, t_{2}); [ω_{{\tilde{b}}_{2}}^{-}, ω_{{\tilde{b}}_{2}}^{+}], η_{{\tilde{b}}_{2}} 〉$ be twoTCFNs, then the operational laws over ${\tilde{b}}_{1}$ and ${\tilde{b}}_{2}$ are shown as follows: $\begin{matrix} (1) & {\tilde{b}}_{1} + {\tilde{b}}_{2} = {\tilde{b}}_{2} + {\tilde{b}}_{1}, \\ (2) & {\tilde{b}}_{1} \otimes {\tilde{b}}_{2} = {\tilde{b}}_{2} \otimes {\tilde{b}}_{1}, \\ (3) & ξ ({\tilde{b}}_{1} + {\tilde{b}}_{2}) = ξ {\tilde{b}}_{1} + ξ {\tilde{b}}_{2}, ξ \geq 0 \\ (4) & ξ_{1} \tilde{b} + ξ_{2} \tilde{b} = (ξ_{1} + ξ_{2}) \tilde{b}, ξ_{1}, ξ_{2} \geq 0 \\ (5) & {\tilde{b}}^{ξ_{1}} \otimes {\tilde{b}}^{ξ_{2}} = {\tilde{b}}^{ξ_{1} + ξ_{2}}, ξ_{1}, ξ_{2} \geq 0 \\ (6) & {\tilde{b}}_{1}^{ξ} \otimes {\tilde{b}}_{2}^{ξ} = ({\tilde{b}}_{1} \otimes {\tilde{b}}_{2})^{ξ}, ξ \geq 0 . \end{matrix}$

3.2 The crisp weighted possibility means of TCFNs

Definition 3.2.1. For instance TCFNs $\tilde{b} = 〈 (r, s, t); [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}} 〉,$ the (α, β)-cut set, the α-cut set and the β-cut set are define as

$\begin{matrix} {\tilde{b}}_{α, β} & = & 〈 h | λ_{\tilde{b}} (h) \geq α, Γ_{\tilde{b}} (h) \leq β 〉, \\ {\tilde{b}}_{α} & = & 〈 {h | λ_{\tilde{b}} (h) \geq α 〉 . \end{matrix}$ (8) and $\begin{matrix} {\tilde{b}}_{β} = 〈 {h | Γ_{\tilde{b}} (h) \leq β} 〉 \\ {\tilde{b}}_{α} = 〈 L_{\tilde{b}} (α), R_{\tilde{b}} (α) 〉 \\ {\tilde{b}}_{α} = 〈 r + \frac{α (s - r)}{ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}}, - t + \frac{α (t - s)}{ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}} 〉 . \end{matrix}$ (9) $\begin{matrix} {\tilde{b}}_{β} = [L_{\tilde{a}} (β), R_{\tilde{a}} (β)] \\ {\tilde{b}}_{β} = 〈 s + \frac{β (s - r)}{η_{\tilde{b}}}, - t + \frac{β (t - s)}{η_{\tilde{b}}} 〉 . \end{matrix}$ (10)

Definition 3.2.2. Let ${\tilde{b}}_{α} = 〈 [{L_{b} (α), R_{b} (α)}] 〉$ be the a-cut set of a TCFN $\tilde{b} = 〈 (r, s, t); [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], (η_{\tilde{b}})] 〉 .$

The φ weighted lower and upper possibility means of the ivtfs λ for the TCFN $\tilde{b}$ are defined as $N_{λ} (\tilde{b}) = \int_{0}^{[ω_{b^{-}}, ω_{b^{+}}]} φ (pos [\tilde{b} \leq L_{b} (α)]) L_{b} (α) d α,$ (11) $Q_{λ} (\tilde{b}) = \int_{0}^{[ω_{b^{-}}, ω_{b^{+}}]} φ (pos [\tilde{b} \geq R_{b} (α)]) R_{b} (α) d α$ (12) respectively, where $φ : {0, [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} \to R$ are a non-negative, monotone increasing weighted function satisfying $\begin{matrix} \int_{0}^{[ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} φ (α) d α = (ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}) and φ (0) = 0, \\ \int_{0}^{[ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} φ (α) d α = (ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}) and φ (0) = 0 . \end{matrix}$

Here, Pos means possibility which is defined as $Pos [\tilde{b} \leq L_{a} (α)]) = \underset{h \leq L_{b} (α)}{\lor} λ_{a} (h) = α$ (13) $Pos [\tilde{b} \geq R_{a} (α)] = \underset{h \leq R_{b} (α)}{\lor} λ_{a} (h) = α$ (14) using Equations (13 and 14) we get $\begin{matrix} N_{λ} (\tilde{b}) & = & \int_{0}^{[ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} φ (pos [\tilde{b} \leq L_{\tilde{b}} (α)]) L_{\tilde{b}} (α) d α \\ = & \int_{0}^{[ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} φ (α) L_{\tilde{b}} (α) d α \end{matrix}$ (15) $\begin{matrix} Q_{λ} (\tilde{b}) & = & \int_{0}^{[ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} φ (pos [\tilde{b} \geq R_{\tilde{b}} (α)]) R_{\tilde{b}} (α) d α \\ = & \int_{0}^{[ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]} φ (α) R_{{\tilde{b}}^{-}} (α) d α^{-} \end{matrix}$ (16)

Clearly, $N_{λ} (\tilde{b})$ shows the φ weighted lower possibility weighted average of the minimum of the α-cut set and hence called the φ weighted lower possibility mean of interval values. $Q_{λ} (\tilde{b})$ shows the weighted upper possibility weighted average of the maximum of the α-cut set and thus called the φ weighted upper possibility medium of interval values.

Definition 3.2.3. Let ${\tilde{b}}_{β} = 〈 L_{\tilde{b}} (β), R_{\tilde{b}} (β) 〉$ be theβ-cut set of a TCFN $\tilde{b} = 〈 (r, s, t); [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}}] 〉 .$

The φ weighted lower and upper possibility means of the fuzzy set Γ for the TCFN $\tilde{b}$ are defined as $N_{Γ} (\tilde{b}) = \int_{0}^{η_{\tilde{b}}} ψ (pos [\tilde{b} \leq L_{\tilde{b}} (β)]) L_{\tilde{b}} (β) d β,$ (17) and $Q_{Γ} (\tilde{b}) = \int_{0}^{η_{\tilde{b}}} ψ (pos [\tilde{b} \geq R_{\tilde{b}} (β)]) R_{\tilde{b}} (β) d β$ (18) respectively, where $ψ : [η_{\tilde{b}}, 0] \to R$ is a non-negative, monotone decreasing weighting function satisfying $\int_{0}^{η_{\tilde{b}}} ψ (β) d β = η_{\tilde{b}} and ψ (1) = 0,$ and $Pos [\tilde{b} \leq L_{b} (β)] = \underset{h \geq L_{\tilde{b}} (β)}{\lor} Γ_{\tilde{b}} (h) = β$ (19) $Pos [\tilde{b} \geq R_{b} (β)] = \underset{x \geq R_{\tilde{b}} (β)}{\lor} Γ_{\tilde{b}} (h) = β$ (20)

It follows from Equations (19 and 20) $\begin{matrix} N_{Γ} (\tilde{b}) & = & \int_{0}^{η_{\tilde{b}}} ψ (pos [\tilde{b} \leq L_{\tilde{b}} (β)]) L_{\tilde{b}} (β) d β \\ = & \int_{0}^{η_{\tilde{b}}} ψ (β) L_{\tilde{b}} (β) d β \\ Q_{Γ} (\tilde{b}) & = & \int_{0}^{η_{\tilde{b}}} ψ Pos [\tilde{b} \geq R_{\tilde{b}} (β)] R_{\tilde{b}} (β) d β \\ = & \int_{0}^{η_{\tilde{b}}} ψ (β) R_{\tilde{b}} (β) d β \end{matrix}$

Most likely, $N_{Γ} (\tilde{b})$ seems the ψ weighted lower possibility weighted average of the minimum of the β-cut set, hence the testimony ψ weighted down possibility mean of a fuzzy set. Likewise, $Q_{Γ} (\tilde{b})$ return the ψ weighted upper possibility weighted average of the maximum of the β-cut set and is called ψ weighted upper possibility mean of a fuzzy set.

Definition 3.2.4. For a TCFN $\tilde{b} = 〈 (r, s, t); [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}} 〉$ the φ weighted possibility mean of ivfs λ is defined as $M_{λ} (\tilde{b}) = 〈 \frac{1}{2} [N_{λ} (\tilde{b}) + Q_{λ} (\tilde{b})] 〉$ (21) and the ψ weighted possibility mean of fuzzy set Γ is defined as $M_{Γ} (\tilde{b}) = 〈 \frac{1}{2} [N_{Γ} (\tilde{b}) + Q_{Γ} (\tilde{b})] 〉$ (22)

The significance of α-cut set of TCFN $\tilde{b}$ is regular by the weighing function and the significance of β-cut set is measured by the weighing function ψ. It can be assured that the selective information on every membership degree is synthetically lament by $M_{λ} (\tilde{b})$ , which can be detected as a central value presents the viewpoint of the membership function. Similarly, the information on every fuzzy set is formally imitated by $M_{Γ} (\tilde{b})$ , which can be beheld as a central value comprises the simulated of fuzzy set function.

Definition 3.2.5. The continuous ordered weighted average (C - OWA) operator can be used to any interval value fuzzy set $\tilde{c} = [c^{D}, c^{W}]$ into an accurate number $Z (\tilde{c})$ as follows:

$\begin{matrix} Z (\tilde{c}) = C - {OWA}_{p (t)} (\tilde{c}) \\ = \int_{0}^{1} \frac{da (t)}{dt} [c^{D} - t [c^{D} - c^{W}]] dt, \end{matrix}$ (27) where p (t) is a basic unit-interval monotonic (BUM) function, p : [0, 1] → [0, 1], which is fixed at the end points, i.e., p (0) =0, p (1) =1, and p (r) ≥ p (t) ≥ t. The choice of BUM function p (t) is an effect of the decision-makers fortune aptitude.

Definition 3.2.6. Let the weighted possibility mean interval for the TCFN $\tilde{b} = 〈 (r, s, t), [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}} 〉$ be $[M_{λ} (\tilde{b}), M_{Γ} (\tilde{b})]$ then the crisp weighted possibility mean $Z (\tilde{b})$ of TCFN $\tilde{b}$ by using Equation (28) is defined as follows:

$\begin{matrix} Z (\tilde{b}) & = & Z [M_{λ} (\tilde{b}), M_{Γ} (\tilde{b})] \\ = & C - {OWA}_{p (t)} [M_{λ} (\tilde{b}), M_{Γ} (\tilde{b})]] \\ = & \int_{0}^{1} \frac{da (t)}{dt} [M_{Γ} (\tilde{b}) - t [M_{Γ} (\tilde{b}) - M_{λ} (\tilde{b})] dt \end{matrix}$ (29)

3.3 Weighted average operator of TCFNs

Definition 3.3.1. For TCFNs $\tilde{b} = 〈 (r, s, t); [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}} 〉 (j = 1, 2, \dots, n)$ , the weighted average operator TCFN-WA : φⁿ → φ is defined as TCF-WA $TCF - WA ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, b_{n}^{\sim}) = 〈 \sum_{j = 1}^{n} w_{j} {\tilde{b}}_{j} 〉$ (30) where the set of all TCFNs is denoted by φ, W = (w₁, w₂, … w₃) ^T is the weight vector of $\tilde{b} (j = 1, 2, \dots, n)$ and w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1 .$ The TCFN-WA operator is called the triangular Cubic fuzzy weighted average operator.

Theorem 3.3.2. The aggregated value of TCFNs ${\tilde{b}}_{j} (j = 1, 2, \dots, n)$ under the TCFWA operator is also a TCFN given by

$\begin{matrix} TCFWA ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}) \\ = 〈 (\sum_{j = 1}^{n} w_{j} (r_{j}), \sum_{j = 1}^{n} w_{j} (s_{j}), \sum_{j = 1}^{n} w_{j} (t_{j})), \\ min_{j} [ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}], max_{j} (η_{{\tilde{b}}_{j}}) 〉 \end{matrix}$ (31)

Theorem 3.3.3. (Idempotency): Assume f is the TCFN-WA operator, if ${\tilde{b}}_{j} = b \forall j = 1, 2, \dots, n$ then $f ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots {\tilde{b}}_{n}) = \tilde{b} .$

Proof. Straight forward.

Theorem 3.3.4. (Monotonicity):Let $({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots {\tilde{b}}_{n}),$ and $({\tilde{e}}_{1}, {\tilde{e}}_{2}, \dots, {\tilde{e}}_{n})$ be two sets of TCFNs satisfying ${\tilde{b}}_{j} \geq {\tilde{e}}_{j}$ then $f [(b_{1}, b_{2}, \dots b_{n}} \geq f [{({\tilde{e}}_{1}, {\tilde{e}}_{2}, \dots, {\tilde{e}}_{n}}$ wherefis a TCFN-WA operator.

Proof.

Theorem 3.3.5.(Boundedness): The TCFN-WA operator lies between minimum and maximum operator, i.e. $\underset{1 \leq j \leq n}{min {} {\tilde{b}}_{j}} \leq f [({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}} \leq max_{1 \leq j \leq n} {{\tilde{b}}_{j}}$ .

Proof. Let $\underset{1 \leq j \leq n}{min {} {\tilde{b}}_{j}} = s$ and $max_{1 \leq j \leq n} {{\tilde{b}}_{j}} = t,$ then $\begin{matrix} f ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}) \\ = 〈 (\sum_{j = 1}^{n} w_{j} r_{j}, \sum_{j = 1}^{n} w_{j} s_{j}, \sum_{j = 1}^{n} w_{j} t_{j})); \\ min_{j} {ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}}], max_{j} {η_{{\tilde{b}}_{j}}} 〉 \\ \leq (\sum_{j = 1}^{n} w_{j} t, \sum_{j = 1}^{n} w_{j} t, \sum_{j = 1}^{n} w_{j} t); \\ min_{j} {ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}}], max_{j} {η_{{\tilde{b}}_{j}}} \\ = (t \sum_{j = 1}^{n} w_{j}, t \sum_{j = 1}^{n} w_{j}, t \sum_{j = 1}^{n} w_{j}); \\ min_{j} {ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}], max_{j} {η_{{\tilde{b}}_{j}}} \end{matrix}$

And $\begin{matrix} f [({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n})] \\ = 〈 (\sum_{j = 1}^{n} w_{j} r_{j}, \sum_{j = 1}^{n} w_{j} s_{j}, \sum_{j = 1}^{n} w_{j} t_{j}); min_{j} [w_{b_{j}}^{-}, w_{b_{j}}^{+}}]; max_{j} {η_{{\tilde{b}}_{j}}} 〉 \\ \geq [(\sum_{j = 1}^{n} w_{j} s, \sum_{j = 1}^{n} w_{j} s, \sum_{j = 1}^{n} w_{j} s); min_{j} [ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}],; max_{j} {η_{{\tilde{b}}_{j}}} 〉 \\ = [(s \sum_{j = 1}^{n} w_{j}, s \sum_{j = 1}^{n} w_{j}, s \sum_{j = 1}^{n} w_{j}); min_{j} [ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}], max_{j} {η_{{\tilde{b}}_{j}}} \end{matrix}$

Since $\sum_{j = 1}^{n} w_{j} = 1,$ we have $f ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}) \leq t$ and $f ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}) \geq s$ Therefore, $s \leq f [(b_{1}^{-}, b_{2}^{-}, \dots, {\tilde{b}}_{n}^{-}} \leq t$

This completes the proof.

3.4 Hamming distance for TCFNs

Definition 3.4.1. Let ${\tilde{b}}_{j} = 〈 (r_{j}, s_{j}, t_{j}), [ω_{b_{j}^{-}}, ω_{b_{j}^{+}}], (η_{{\tilde{b}}_{j}}) 〉 (j = 1, 2)$ , be two TCFNs. The hamming distance between ${\tilde{b}}_{1}$ and ${\tilde{b}}_{2}$ is defined as follows:

$\begin{matrix} d_{H} ({\tilde{b}}_{1}, {\tilde{b}}_{2}) \\ = \frac{1}{9} 〈 [| r_{1} - r_{2} | + | s_{1} - s_{2} | + | t_{1} - t_{2} |] \\ + max (| ω_{{\tilde{b}}_{1}}^{-} - ω_{{\tilde{b}}_{2}}^{-} |, | ω_{{\tilde{b}}_{1}}^{+} - ω_{{\tilde{b}}_{2}}^{+} |, \\ | η_{{\tilde{b}}_{1}} - η_{{\tilde{b}}_{2}} |) 〉 \end{matrix}$ (34)

The TCFN $\tilde{b} = 〈 (r, s, t), [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+})], (η_{\tilde{b}})] 〉$ reduce to a TFN $\tilde{b} = 〈 (r, s, t), [(ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+})], (η_{\tilde{b}})] 〉$ if $ω_{\tilde{b}}^{-} = 1, ω_{\tilde{b}}^{+} = 1$ and $η_{\tilde{b}} = 0 .$

Theorem 3.4.2. The hamming distance of TCFNs satisfy the following properties.

Non negativity: $d_{H} ({\tilde{b}}_{1}, {\tilde{b}}_{2}) \geq 0;$

Refeixivity: $d_{H} ({\tilde{b}}_{1}, {\tilde{b}}_{1}) \geq 0;$

Symmetry: $d_{H} ({\tilde{b}}_{1}, {\tilde{b}}_{2}) = d_{H} ({\tilde{b}}_{2}, {\tilde{b}}_{1});$

Triangular Inequalities: $d_{H} ({\tilde{b}}_{1}, {\tilde{b}}_{3}) = d_{H} ({\tilde{b}}_{1}, {\tilde{b}}_{2}) + d_{H} ({\tilde{b}}_{2}, {\tilde{b}}_{3}) .$

4 An MCDM method approach based on an extended VIKOR method using TCFNS

In this section, we will introduce a new method to solve multi-criteria decision-making problems with TCFNs and determine the standardize decision matrix as well as attributes. An MCDM access based on an extended VIKOR method for TCFNs is then proposed.

4.1 MCDM method using TCFNs

For some fuzzy multi-criteria decision-making problem, let B = {B₁, B₂, …, B_m} be a discrete set of alternatives, and C = {C₁, C₂, …, C_n} be the set of criteria, w = (w₁, w₂, …, w_n) ^T is the corresponding weight vector of the criteria, where $w_{j} \in [0, 1], \sum_{j = 1}^{n} w_{j} = 1,$ but the value w_j is unknown. The criteria value of alternative B_i on the criteria C_j is the triangular cubic fuzzy number $\tilde{b} = 〈 (r, s, t), [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}], η_{\tilde{b}} 〉 .$

To determine the fuzzy assessment data provided by the decision-maker for each alternative with respect to each attribute is fuzzified into the crisp weighted possibility mean values $G^{k} = (g_{ij}^{k})$ where

$\begin{matrix} g_{ij}^{k} & = & 〈 \frac{(r + 4 s - t), [ω_{\tilde{b}}^{-}, ω_{\tilde{b}}^{+}]}{6}, \\ (\frac{(r + 4 s - t) η_{b}}{6}) 〉 \end{matrix}$ (35)

4.2 Determine the criteria weight

There are a lots of methods to determine the criteria weight, such as maximizing deviation method [42], information entropy method [22] and other optimization method [41]. In this paper, we will assume maximizing deviation method to determine the criteria weight. The maximizing deviation method is proposed by Wang [42] to address multi-criteria decision-making problems with numerical information. For a multi-criteria decision-making problem, we need to compare the collective preference values to rank the alternatives, the larger ranking value r_i is, the better the comparable alternative B_i is. If the criteria values of all alternative have minor differences under criteria, it shows that such a criteria play a less important role in the priority procedure. Contrariwise, if some criteria make the performance values among all the alternatives have obvious differences, such an attribute plays an important role in selecting the best alternative. So to the view of sorting the alternatives, if one criterion has similar attribute values across alternatives, it should be assigned a small weight; otherwise, the criteria which make larger deviations should be evaluated a bigger weight, despite the degree of its own importance. Especially, if all usable alternatives score about equally with respect to a given attribute, then such a criteria will be judged unimportant by most decision makers. In another word, such a criteria should be assigned a very small weight [46].

Because of the conventional maximizing deviation method is generally suitable for criteria value taking the form of a crisp number and yet it fails in dealing with the triangular cubic fuzzy number. The steps of determining the attribute weights by the maximizing deviation method are shown as follows:

For the criteria C_j, the deviation value D_j (ω) of all alternative to all the other alternatives can be defined as follows: $D_{j} (w) = \sum_{i = 1}^{m} D_{ij} (w) = \sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj}) w_{j} .$

We can construct a non-linear programming model as follows: $M_{1} {\begin{matrix} max D_{j} (w) = \sum_{j = 1}^{m} \sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj}) w_{j} \\ such that \sum_{j = 1}^{n} w_{j}^{2} = 1, w_{j} \geq 0, \\ j = 1, 2, \dots, n \end{matrix}$ (36)

To solve the above model, let $L (w_{j}, ϑ) = \sum_{j = 1}^{m} \sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj}) w_{j} + \frac{1}{2} ϑ (\sum_{j = 1}^{n} w_{j}^{2} - 1)$ denote the Lagrange function of the constrained optimization problem (M₁), where ϑ is a real number. Then the partial derivatives of L are computed as ${\begin{matrix} \frac{\partial L}{\partial ω_{j}} = \sum_{j = 1}^{m} \sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj}) w_{j} \\ + ϑ (\sum_{j = 1}^{n} w_{j} = 0 \\ \frac{\partial L}{\partial ϑ} = \frac{1}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1) = 0 \end{matrix}$

It can be verified easily that $w_{j}^{*}$ are positive such that they do satisfy the constrained conditions in the model (M₁) and the solution is unique. $w_{j}^{*} = \frac{\sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj})}{\sqrt{\sum_{j = 1}^{m} {[\sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj})]}^{2}}}$

By normalizing $w_{j}^{*}$ to let the sum of w_j, we have $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj})}{\sum_{i = 1}^{n} \sum_{i = 1}^{m} \sum_{l = 1}^{m} d (r_{ij}, r_{lj})}$ (37)

4.3 The extended VIKOR method using TCFNS

Here, we extended the classical VIKOR method based on the above analysis. The steps are outlined as follows:

Step 1. Calculate the crisp weighted possibility mean values $G^{k} = (g_{ij}^{k})_{m \times n}$ of decision-makers by Equation (35) from the TCFN assessment date $b_{ij}^{k} (i = 1, 2, \dots, l; j = 1, 2, \dots, n; k = 1, 2, \dots, q)$ given by each DM_k based on attribute $b_{ij}^{k}$ with respect to alternative B_i .

Step 2. If the weight vector of criteria is completely unknown, utilize Equation (37) to find the optimal weight of criteria.

Step 3. Using the TCFN-WA operator (i.e., Equation (31)) compute the TCFN group decision values by aggregating the fuzzy assessment data given by each DM_kH = (h_ij) _m×n, where

$\begin{matrix} {\tilde{h}}_{ij} & = & 〈 [h_{1} (b_{j}), h_{2} (b_{j}), h_{3} (b_{j})], [ω_{{\tilde{h}}_{ij}}^{-}, ω_{{\tilde{h}}_{ij}}^{+}], (η_{{\tilde{h}}_{ij}}) 〉 \\ = & TCFN - WA ({\tilde{b}}_{ij}^{1}, {\tilde{b}}_{ij}^{2}, \dots \dots, {\tilde{b}}_{ij}^{n}) \\ = & \sum_{j = 1}^{n} ρ_{j} {\tilde{b}}_{ij}^{n} \\ = & 〈 (\sum_{j = 1}^{n} ρ_{k} r_{ij}^{n}, \sum_{j = 1}^{n} ρ_{k} s_{ij}^{n}, \sum_{j = 1}^{n} ρ_{k} t_{ij}^{n}), \\ min_{1 \leq j \leq n} [ω_{{\tilde{b}}_{j}}^{-}, ω_{{\tilde{b}}_{j}}^{+}], max_{1 \leq j \leq n} (η_{b_{j}}) 〉 \end{matrix}$ (38)

Step 4. With a view to except the effect of diverse physical dimensions on the final decision values, the group decision values $H = ({\tilde{h}}_{ij})_{m \times n}$ must be normalized into $\tilde{X} = ({\tilde{x}}_{ij})_{m \times n},$ where values, the group decision values, where

$\begin{matrix} ({\tilde{x}}_{ij})_{m \times n} = 〈 (x_{1 i} (b_{j}), x_{2 i} (b_{j}), x_{3 i} (b_{j})); [ω_{{\tilde{x}}_{ij}}^{-}, ω_{{\tilde{x}}_{ij}}^{+}], η_{{\tilde{x}}_{ij}} 〉 \\ x_{ti} (b_{j}) = 〈 \frac{max {- h_{3 - t, i} (b_{j})}}{max {h_{2 i} (b_{j}), h_{2 i}^{'} (b_{j})} - min {h_{1 i} (b_{j}), h_{1 i}^{'} (b_{j})}} 〉 \end{matrix}$ (39) and

$x_{ti}^{'} (b_{j}) = 〈 \frac{max {[h_{2 i} (b_{j}), h_{2 i}^{'} (b_{j})] - h_{3 - t, i} (b_{j})}}{max {[h_{2 i} (b_{j}), h_{2 i}^{'} (b_{j})]} - min [h_{1 i} (b_{j}), h_{1 i}^{'} (b_{j})}} 〉$ (40) and the subscript 3 - t of h_3-t,i (b_ij) begins the fact that normalized values satisfy $\begin{matrix} [x_{1} (b_{j}) \leq x_{2} (b_{j}) x_{3} (b_{j})], \\ [x_{1}^{'} (b_{j}) \leq x_{2}^{'} (b_{j}) \leq x_{3}^{'} (b_{j})] \end{matrix}$ is lifeless a TCFN. Also, for the benefit attributes, we have

$x_{ti} (b_{j}) = 〈 \frac{h_{ti} (b_{ij}) - min {[h_{1 i} (b_{j}), h_{1 i}^{'} (b_{j})]}}{max {[h_{2 i} (b_{j}), h_{2 i}^{'} (b_{j})]} - min [h_{1 i} (b_{j}), h_{1 i}^{'} (b_{j})}} 〉$ (41) and

$x_{ti} (b_{j}) = 〈 \frac{h_{ti}^{'} (b_{ij}) - min {[h_{1 i} (b_{j}), h_{1 i}^{'} (b_{j})}}{max {[h_{2 i} (b_{j}), h_{2 i}^{'} (b_{j})]} - min [h_{1 i} (b_{j}), h_{1 i}^{'} (b_{j})}} 〉$ (42)

Step 5. The positive ideal solution $B^{+} = {b_{1}^{+}, b_{2}^{+}, \dots, b_{n}^{+}}$ and $B^{-} = {b_{1}^{-}, b_{2}^{-}, \dots, b_{n}^{-}}$ negative ideal solution, are to be resolved, where $b_{j}^{+}$ indicates an eagerness level and $b_{j}^{-}$ represents least tolerable level and are given as follows: $\begin{matrix} b_{j}^{+} & = & 〈 [max_{1 \leq j \leq n} [{x_{1 i} (b_{j}), max_{1 \leq j \leq n} x_{2 i} (b_{j}), \\ max_{1 \leq j \leq n} x_{3 i} (b_{j})], max [ω_{x_{ij}^{-}}, ω_{x_{ij}^{+}}], min (η_{x_{ij}}) 〉 \end{matrix}$ (43) $\begin{matrix} b_{j}^{-} & = & 〈 [max_{1 \leq j \leq n} [{x_{1} (b_{j}) max_{1 \leq j \leq n} x_{2 i} (b_{j}), \\ max_{1 \leq j \leq n} x_{3 i} (b_{j})], min [ω_{x_{ij}^{-}}, ω_{x_{ij}^{+}}], max (η_{x_{ij}}) 〉 \end{matrix}$ (44)

Step 6. Using hamming distance, compute the group utility value U_G (B_i) and individual regret value R_I (B_i) with respect to alternative B_i, i = 1, 2, …, l as follows:

$U_{G} (B_{i}) = 〈 \sum_{j = 1}^{n} w_{j} \frac{d_{H} (b_{j}^{+}, {\tilde{x}}_{ij})}{d_{H} (b_{j}^{+}, b_{j}^{-})} 〉,$ (45)

$R_{I} (B_{i}) = 〈 max_{1 \leq j \leq n} {\sum_{j = 1}^{n} w_{j} \frac{d_{H} (b_{j}^{-}, {\tilde{x}}_{ij})}{d_{H} (b_{j}^{+}, b_{j}^{-})}} 〉$ (46)

where U_G represents the distance of the ith alternative to the positive ideal solution, which proves the best combination and R_I represents the distance of the ith alternative to the negative ideal solution, which designates the worst combination.

Step 7. For each alternative B_i compute the degree of closeness C (B_i) to the ideal solution asbelow: $C (B_{i}) = 〈 τ \frac{U_{G} (B_{i}) - U_{G}^{+}}{U_{G}^{-} - U_{G}^{+}} + (1 - τ) \frac{R_{I} (B_{i}) - R_{I}^{+}}{R_{I}^{-} - R_{I}^{+}} 〉$ (47) where $\begin{matrix} U_{G}^{+} & = & min_{1 \leq i \leq l} {U_{G} (B_{i})}, U_{G}^{-} = max_{1 \leq i \leq l} {U_{G} (B_{i})} \\ R_{I}^{+} & = & min_{1 \leq i \leq l} {R_{I} (B_{i})}, R_{I}^{-} = max_{1 \leq i \leq l} {R_{I} (B_{i})} \end{matrix}$

Step 8. Rank the alternative in increasing order C (B_i) , (i = 1, 2, …, l).

5 Numerical applications

A Bus company is interested in selecting the most appropriate green supplier for one of the key elements in its manufacturing process. After pre-evaluation, three suppliers B_i (i = 1, 2, 3) have remained as alternatives for further evaluation. Three criteria are considered as product quality C₁; technology capability C₂; environment management C₃; (whose weighting vector is completely unknown). They constructed the decision matrices $V^{k} = (v_{ij}^{k})_{m \times n}$ as follows:

Step 2. If the weight vector of criteria is completely unknown, utilize Equation (37) to obtain the optimal weight of criteria. ω = (0.1960, 0.3812, 0.4227) .

Step 3. Using the TCFN-WA operator (i.e., Equation (31).

Step 4. With a view to except the effect of diverse physical dimensions on the final decision values, the group decision values $H = ({\tilde{h}}_{ij})_{m \times n}$ must b normalized into $\tilde{X},$ using Equations (40–42) where values, the group decision values.

Step 5. We obtain the positive ideal solution and negative ideal solution of the normalized group decision values using Equations (43, 44), respectively. PIS $\begin{matrix} c_{1}^{+} & = & 〈 (0.9804, 0.9408, 0.9581), [0.9, 0.10], 0.5 〉 \\ c_{2}^{+} & = & 〈 (0.9905, 0.9467, 0.8824), [0.11, 0.13], 0.9 〉 \\ c_{3}^{+} & = & 〈 (0.9619, 0.9238, 0.9578), [0.13, 0.15], 0.7 〉 \\ NIS \\ c_{1}^{-} & = & 〈 (0.657, 0.619, 0.831); [0.5, 0.6], 0.12 〉 \\ c_{2}^{-} & = & 〈 (0.7042, 0.902, 0.429); [0.7, 0.8], 0.15 〉 \\ c_{3}^{-} & = & 〈 (0.6618, 0.902, 0.8432), [0.4, 0.6], 0.17 〉 \end{matrix}$

Step 6. Using hamming distance Equation (34), we compute the group utility value U_G (B_i) and individual regret value by using Equations (45, 46), i = 1, 2, …, l as follows: $\begin{matrix} U_{G} (B_{1}) = 0.4803, U_{G} (B_{2}) = 0.7000, \\ U_{G} (B_{3}) = 0.3731, \\ R_{I} (B_{1}) = 0.4235, R_{I} (B_{2}) = 0.4557, \\ R_{I} (B_{3}) = 0.3704 \end{matrix}$

Based on the above value, we have $\begin{matrix} U_{G}^{+} = 0.3731, U_{G}^{-} = 0.7000, \\ R_{I}^{+} = 0.3704, R_{I}^{-} = 0.4557 . \end{matrix}$

Step 7. Using Equation (47), we calculate the closeness of alternative B_i to the ideal solution provided in Table 5.

Step 8. The ranking order of alternatives with coefficient of decision mechanism τ = 0.5 in increasing order of C (B_i) , i = 1, 2, 3 is listed in Ranking of alternatives.

Observe that the counsel value of the coefficient of decision gears is τ = 0.5 since in this case the decision to get hold of through agreement among the different decision-makers. However, the final ranking of the alternatives gamble on densely on its value, i.e., different τ values may generate different ranking orders of the given alternatives. Here, we perform sensitivity analysis to get the impact of the change in τ value on the output or final ranking of the alternatives. The ranking orders of the alternatives with different values of coefficient of decision gears in increasing order of C (B_i) , i = 1, 2, 3.

6 Comparison analyses

In direction to verify the rationality and efficiency of the proposed approach, a comparative study is steered consuming the methods of Fuzzy-VIKOR Afful et al. [2], Intuitionistic fuzzy-VIKOR, Devi [15] and, which are special cases of TCFNs, to the similar expressive example.

6.1 A comparison analysis with the existing MCDM method with fuzzy-VIKOR

Let X = [x_ij] _m×n be a fuzzy decision matrix for a multi-criteria decision-making problem in which B₁, B₂, …, B_m are n possible alternatives and C₁, C₂, …, C_n are m criteria. So the performance of alternative B_j with respect to criterion C_i is denoted as x_ij. As illustrated, x_ij and w_i are expressed in FNs. x_ij = x₁, x₂ ; x₃ . The x can be also demonstrated as x_ij = [x₁, x₂ ; x₃] . It is worth noting that the use of interval value numbers gives an opportunity for experts to define lower and upper-bound values as fuzzy set for matrix’s elements and weights of criteria. Also, in a group decision environment with K persons, the importance of the criteria and the rating of alternatives with respect to each criterion can be calculated as: $x_{ij} = \frac{1}{k} [x_{ij}^{1} + x_{ij}^{2} + x_{ij}^{k}], w_{ij} = \frac{1}{k} [w_{ij}^{1} + w_{ij}^{2} + w_{ij}^{k}] .$ So the output is also an FN. Now, the proposed approach to develop the VIKOR for F data (F-VIKOR) can be defined as follows:

Step 1. Ascertain the best rating f⁺ and the worst rating f^- for all the criteria. If the criterion i symbolizes a benefit, then: $\begin{matrix} f^{+} = max f_{ij}, f^{-} = min f_{ij} . \\ f_{1}^{+} = 0.10, f_{2}^{+} = 0.15, f_{3}^{+} = 0.17, \\ f_{1}^{-} = 0.7, f_{2}^{-} = 0.9, f_{3}^{-} = 0.7 \end{matrix}$

Naturally, a candidate possessing scores $(f_{1}^{+}, f_{2}^{+}, \dots f_{n}^{+})$ would be ideal whereas a candidate having scores $(f_{1}^{-}, f_{2}^{-}, \dots f_{n}^{-})$ would be an anti-ideal candidate. It is assumed that such an ideal candidate does not exist; otherwise, the decision would be trivial.

Step 2. Calculate the S_j and R_j values for j = 1, 2, …, m, which symbolize the average and the worst group scores for the alternative B_j, respectively with the relations $\begin{matrix} S_{1} = 1.0334, S_{2} = 1.0664, S_{3} = 2.1886, \\ R_{1} = 0.3933, R_{2} = 0.4133, R_{3} = 0.7131 . \end{matrix}$

Based on the above value, $\begin{matrix} S^{+} = 1.0334, S^{-} = 2.1886, \\ R^{+} = 0.3933, R^{-} = 0.7131 . \end{matrix}$

So the relative importance weights of the criteria set by the DM. The smaller values of S_j and R_j correspond to the better average and the worse group scores for the alternative B_j respectively.

Step 3. Calculate the Q_j values for j = 1, …, m with the relation $\begin{matrix} S^{+} = 1.0334, S^{-} = 2.1886, \\ R^{+} = 0.3933, R^{-} = 0.7131 . \end{matrix}$

We have here, so the relative importance weights of the criteria set by the DM. The smaller values of respectively.

Step 4. Calculate the Q_j values for j = 1, …, m with the relation.

6.2 A comparison analysis with the existing MCDM method with Intuitionistic Fuzzy VIKOR [15]

In VIKOR method, numerical measure of the relative importance of attributes and the performance of each alternative on these attributes are very important. It is difficult to precisely determine the exact data as human judgements are often vague under many situations and conditions. Fuzzy sets and other non-standard fuzzy sets are efficient in tackling these uncertainties present in the provided data. Therefore, extension of VIKOR method to the non-standard fuzzy environment is natural. Out of these nonstandard fuzzy sets, IFSs are more efficient in dealing with uncertainty. As in many situations, available information is not sufficient for the exact definition of degree of membership for certain element. There may be some hesitation degree between membership and non-membership. Thus in many real life problems, due to insufficiency in information availability, IFSs with ill known membership grades are appropriate. IFSs have been found to be particularly useful to deal with uncertainty. In this paper, criteria values as well as criteria weights are considered as linguistic variables (Table 9).

Table 1
Decision matrix

C ₁ C ₂ C ₃

B ₁ $〈 \begin{matrix} (0.1, 0.2, 0.3), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$ $〈 \begin{matrix} (0.9, 0.10, 0.11), \\ [0.9, 0.10], 0.12 \end{matrix} 〉$ $〈 \begin{matrix} (0.4, 0.5, 0.6), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$

V = B ₂ $〈 \begin{matrix} (0.4, 0.5, 0.6), \\ [0.7, 0.8], 0.9 \end{matrix} 〉$ $〈 \begin{matrix} (0.13, 0.14, 0.15), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$ $〈 \begin{matrix} (0.7, 0.8, 0.9), \\ [0.11, 0.13], 0.15 \end{matrix} 〉$

B ₃ $〈 \begin{matrix} (0.4, 0.5, 0.8), \\ [0.4, 0.6], 0.8 \end{matrix} 〉$ $〈 \begin{matrix} (0.1, 0.2, 0.3), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$ $〈 \begin{matrix} (0.8, 0.9, 0.10), \\ [0.13, 0.15], 0.17 \end{matrix} 〉$

	C ₁	C ₂	C ₃
B ₁	$〈 \begin{matrix} (0.1, 0.2, 0.3), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$	$〈 \begin{matrix} (0.9, 0.10, 0.11), \\ [0.9, 0.10], 0.12 \end{matrix} 〉$	$〈 \begin{matrix} (0.4, 0.5, 0.6), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$
V =	B ₂ $〈 \begin{matrix} (0.4, 0.5, 0.6), \\ [0.7, 0.8], 0.9 \end{matrix} 〉$	$〈 \begin{matrix} (0.13, 0.14, 0.15), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$	$〈 \begin{matrix} (0.7, 0.8, 0.9), \\ [0.11, 0.13], 0.15 \end{matrix} 〉$
B ₃	$〈 \begin{matrix} (0.4, 0.5, 0.8), \\ [0.4, 0.6], 0.8 \end{matrix} 〉$	$〈 \begin{matrix} (0.1, 0.2, 0.3), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$	$〈 \begin{matrix} (0.8, 0.9, 0.10), \\ [0.13, 0.15], 0.17 \end{matrix} 〉$

Table 2

The crisp weighted possibility mean values

	C ₁	C ₂	C ₃
B ₁	〈 [0.05, 0.06] , 0.07〉	〈 [0.171, 0.019] , 0.0228〉	〈 [0.24, 0.27] , 0.03〉
B ₂	〈 [0.21, 0.24] , 0.27〉	〈 [0.072, 0.081] , 0.009〉	〈 [0.055, 0.065] , 0.075〉
B ₃	〈 [0.104, 0.156] , 0.208〉	〈 [0.05, 0.06] , 0.07〉	〈 [0.0923, 0.1065] , 0.1207〉

Table 3

TCFN-WA operator

	C ₁	C ₂	C ₃
B ₁	$〈 \begin{matrix} (0.0196, 0.0392, 0.0588), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$	$〈 \begin{matrix} (0.3430, 0.0381, 0.0419), \\ [0.9, 0.10], 0.12 \end{matrix} 〉$	$〈 \begin{matrix} (0.1690, 0.2113, 0.2536), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$
B ₂	$〈 \begin{matrix} (0.784, 0.098, 0.1176), \\ [0.7, 0.8], 0.9 \end{matrix} 〉$	$〈 \begin{matrix} (0.495, 0.533, 0.0571), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$	$〈 \begin{matrix} (0.2958, 0.3382, 0.3804), \\ [0.11, 0.13], 0.15 \end{matrix} 〉$
B ₃	$〈 \begin{matrix} (0.0784, 0.098, 0.1568), \\ [0.4, 0.6], 0.8 \end{matrix} 〉$	$〈 \begin{matrix} (0.0381, 0.0762, 0.1143), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$	$〈 \begin{matrix} (0.3382, 0.3804, 0.0422), \\ [0.13, 0.15], 0.17 \end{matrix} 〉$

Table 4

The normalized group decision values

	C ₁	C ₂	C ₃
B ₁	$〈 \begin{matrix} (0.9804, 0.9608, 0.9412), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$	$〈 \begin{matrix} (0.657, 0.619, 0.9581), \\ [0.9, 0.10], 0.12 \end{matrix} 〉$	$〈 \begin{matrix} (0.831, 0.7887, 0.7464), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$
B ₂	$〈 \begin{matrix} (0.9216, 0.902, 0.8824), \\ [0.7, 0.8], 0.9 \end{matrix} 〉$	$〈 \begin{matrix} (0.9905, 0.9467, 0.429), \\ [0.8, 0.9], 0.10 \end{matrix} 〉$	$〈 \begin{matrix} (0.7042, 0.6618, 0.6196), \\ [0.11, 0.13], 0.15 \end{matrix} 〉$
B ₃	$〈 \begin{matrix} (0.9216, 0.902, 0.8432), \\ [0.4, 0.6], 0.8 \end{matrix} 〉$	$〈 \begin{matrix} (0.9619, 0.9238, 0.8857), \\ [0.5, 0.6], 0.7 \end{matrix} 〉$	$〈 \begin{matrix} (0.6618, 0.6196, 0.9578), \\ [0.13, 0.15], 0.17 \end{matrix} 〉$

Table 5

Ranking of alternatives

τ	C (B₁)	C (B₂)	C (B₃)	Ranking	Optimal solution
0.1	0.5930	1	0	B₃ > B₁ > B₂	B ₃

Table 6

Ranking of alternatives for different τ values

τ	C (B₁)	C (B₂)	C (B₃)	Ranking	Optimal solution
0.1	0.5930	1	0	B₃ > B₁ > B₂	B ₃
0.2	0.5635	1	0	B₃ > B₁ > B₂	B ₃
0.3	0.5342	1	0	B₃ > B₁ > B₂	B ₃
0.4	0.5046	1	0	B₃ > B₁ > B₂	B ₃
0.5	0.4751	1	0	B₃ > B₁ > B₂	B ₃
0.6	0.4457	1	0	B₃ > B₁ > B₂	B ₃
0.7	0.4155	1	0	B₃ > B₁ > B₂	B ₃
0.8	0.3868	1	0	B₃ > B₁ > B₂	B ₃
0.9	0.3573	1	0	B₃ > B₁ > B₂	B ₃
1	0.3279	1	0	B₃ > B₁ > B₂	B ₃

Table 7

Fuzzy-Vikor

		C ₁	C ₂	C ₃
	B ₁	〈0.7〉	〈0.12〉	〈0.10〉
V =	B ₂	〈0.9〉	〈0.10〉	〈0.15〉
	B ₃	〈0.8〉	〈0.7〉	〈0.17〉

Table 8

Ranking of alternatives

τ	C (B₁)	C (B₂)	C (B₃)	Ranking	Optimal solution
0.5	0	0.0455	1	B₁ > B₂ > B₃	B ₁

Table 9

Intuitionstic Fuzzy-Vikor

		C ₁	C ₂	C ₃
	B ₁	[0.8, 0.10]	[0.12, 0.14]	[0.2, 0.4]
V =	B ₂	[0.3, 0.5]	[0.1, 0.6]	[0.9, 0.11]
	B ₃	[0.7, 0.9]	[0.4, 0.7]	[0.13, 0.15]

Table 10

Ranking of alternatives

τ	C (B₁)	C (B₂)	C (B₃)	Ranking	Optimal solution
0.5	1	0.0146	0.6306	B₂ > B₃ > B₁	B ₂

Table 11

Comparison analysis with existing methods

Method	Ranking
The extended VIKOR method using TCFNS	B₃ > B₁ > B₂
Fuzzy-VIKOR [2]	B₁ > B₂ > B₃
Interval-Value Fuzzy VIKOR [35]	B₁ > B₃ > B₂
Intuitionstic Fuzzy VIKOR [15]	B₂ > B₃ > B₁
Triangular Fuzzy VIKOR [13]	B₁ > B₂ > B₃

Step 1. Ascertain the best rating f⁺ and the worst rating f^- for all the criteria. If the criterion i symbolizes a benefit, then: $\begin{matrix} f^{+} = max f_{ij}, f^{-} = min f_{ij} . \\ f_{1}^{+} = [0.12, 0.14], f_{2}^{+} = [0.9, 0.11], \\ f_{3}^{+} = [0.13, 0.15], \\ f_{1}^{-} = [0.2, 0.4], f_{2}^{-} = [0.1, 0.5], \\ f_{3}^{-} = [0.4, 0.7] . \end{matrix}$

Step 2. Calculate the S_j and R_j values for j = 1, …, m, which symbolize the average and the worst group scores for the alternative B_j, respectively, with the relations $\begin{matrix} S_{1} = 3.1176, S_{2} = 1.9159, S_{3} = 2.6097, \\ R_{1} = 1.0941, R_{2} = 0.5008, R_{3} = 0.4829 . \end{matrix}$

Based on the above value, $\begin{matrix} S^{+} = 1.9159, S^{-} = 3.1176, \\ R^{+} = 0.4829, R^{-} = 1.0941 . \end{matrix}$

We have here, so the relative importance weights of the criteria set by the DM. The smaller values of S_j and R_j correspond to the better average and the worse group scores for the alternative B_j respectively.

Step 3. Calculate the Q_j values for j = 1, …, m with the relation.

7 Conclusion

In this Paper, we initiated the concept of triangular Cubic fuzzy numbers and defined some operational laws. The concept of triangular Cubic fuzzy number is the generalization of Cubic number, triangular intuitionistic fuzzy numbers, triangular cubic fuzzy numbers and interval-valued fuzzy numbers. As investigated in introduction that triangular cubic fuzzy numbers becomes Cubic fuzzy number when remove the triangular numbers form it, triangular Cubic fuzzy numbers become triangular intuitionistic fuzzy numbers if we take only fuzzy number instead of interval in the membership degree, triangular Cubic fuzzy number become triangular interval valued fuzzy number if we remove non-membership degree from it and triangular Cubic fuzzy number becomes triangular fuzzy numbers if we remove the non-membership degree and take fuzzy number instead of interval in the membership degree. The triangular Cubic fuzzy information is more abundant and flexible than Cubic sets, TIFSs, TFS, TIVFSs. We proposed a new decision method to solve the MCDM problems. We assumed that the ratings of alternatives on the given attributes are expressed using triangular Cubic fuzzy numbers (TCFNs) and the weight of attributes is completely unknown. We proposed triangular Cubic fuzzy weighted average operator and investigated some of its properties. To find the unknown attribute weight we developed a non-linear programming model based on maximizing deviation method. The group decision matrix is obtained by an appropriate aggregated process that combines the information provided by all the decision-makers. Since VIKOR method is considered a useful technique to treat the MCDM problems that have non-commensurable and conflicting attributes based on providing a maximum group utility for the majority and a minimum individual regret for the opponent, we extended the classical VIKOR method to triangular Cubic fuzzy numbers. To demonstrate the proposed method real-world MCDM problem was given. Finally we compared the proposed method to the existing methods, which shows the triangular Cubic fuzzy numbers are more flexible to deal uncertainties and fuzziness.

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