CODAS Method for Multiple Attribute Group Decision Making Under 2-Tuple Linguistic Neutrosophic Environment

Abstract

In this paper, we present the 2-tuple linguistic neutrosophic CODAS model based on the traditional fuzzy CODAS (combinative distance-based assessment) model and some fundamental theories of 2-tuple linguistic neutrosophic information. Firstly, we briefly review the definition of 2-tuple linguistic neutrosophic sets (2TLNSs) and introduce the score function, the accuracy function, operation laws and some aggregation operators of 2TLNNs. Then, the calculation steps of traditional fuzzy CODAS model are briefly presented. Furthermore, by combining the traditional fuzzy CODAS model with 2TLNNs information, the 2-tuple linguistic neutrosophic CODAS model is established and the computing steps for multiple attribute group decision making (MAGDM) are simply depicted. Our presented model is more accurate and effective for considering the combinative form of two distance measurements, including fuzzy weighted Hamming distance (HD) and fuzzy weighted Euclidean distance (ED). Finally, a numerical example for safety assessment of construction project has been given to illustrate this new model and some comparisons between 2TLNNs CODAS model and two 2TLNNs aggregation operators are also made to further illustrate the advantages of the new method.

Keywords

multiple attribute group decision making (MAGDM) problems 2-tuple linguistic neutrosophic sets (2TLNSs)CODAS model 2-tuple linguistic neutrosophic number weighted average (2TLNNWA) operators 2-tuple linguistic neutrosophic number weighted geometric (2TLNNWG) operators 2TLNNs CODAS model construction project

1 Introduction

Due to the indeterminacy of DM’s and the decision-making issues, we cannot always give accurate evaluation values of alternatives to select the best project in real MADM problems (Wang et al., 2019a; Wang, 2019; Wu et al., 2019a, 2019b). To conquer this disadvantage, fuzzy set theory which was defined by Zadeh (1965) originally used the membership function to describe the estimation results rather than exact real-numbers. Atanassov (1986, 1989) presented another measurement index which named non-membership function as a complement. Smarandache (1999, 2003) introduced the neutrosophic set (NS). Then, Wang et al. (2010) introduced the definition and some operational rules of single-valued neutrosophic sets (SVNSs), where the evaluation information is depicted by truth membership degree, indeterminacy membership degree and falsity membership degree. Obviously, the SVNSs and their extensions can describe the fuzzy and uncertainty degree of a decision maker and can be more suitable for actual multiple attribute decision making problems (Wang et al., 2019c, 2019e, 2019h, 2019j; Wei et al., 2019e, 2019f).

However, the SVNSs can only represent quantitative decision making information and fail to depict the qualitative decision making information. As we all know, the 2-tuple linguistic set (2TLS) (Herrera and Martinez, 2001) can eliminate this limitation. In order to consider both qualitative and quantitative decision making information, Wang et al. (2018a) defined the 2-tuple linguistic neutrosophic sets (2TLNSs), where the truth membership function, indeterminacy membership function and falsity membership function are presented by 2TLNNs. Thus, the 2TLNNs are considered a useful tool to deal with practical MADM applications. In real decision making problems, finding a good way to denote evaluation information is only one aspect (Deng and Gao, 2019; Gao et al., 2019; Li and Lu, 2019; Lu and Wei, 2019). It is also important to know how to deal with this information. To date, The CODAS (combinative distance-based assessment) method, which was originally defined by Keshavarz Ghorabaee et al. (2016), use the combinative form of two distance measurements, including Euclidean and Taxicab distances, which present accurate values to compute the assessment results of alternatives. However, this model cannot be applied in fuzzy environment. To overcome this disadvantage, Keshavarz Ghorabaee et al. (2017) extended the CODAS method to fuzzy environment and proposed a fuzzy CODAS model which used fuzzy weighted Hamming distance (HD) and fuzzy weighted Euclidean distance (ED) rather than the crisp distances. Thus, we can easily find that CODAS method can handle fuzzy decision making problems effectively.

Motivated by the 2-tuple linguistic neutrosophic sets (2TLNSs) and the traditional CODAS model, the research question and goal of this paper is to build an extended CODAS model to deal with the 2-tuple linguistic neutrosophic decision making problems. The main novelty and contribution of this paper is the proposition of the 2TLN CODAS model. On the one hand, the 2-tuple linguistic neutrosophic number can express the qualitative and quantitative fuzzy decision making information, on the other hand, the CODAS model has important merits mentioned above. Thus, we can derive accuracy assessment results for construction project by utilizing the 2TLN CODAS model. In order to elaborate the process of putting forward the 2TLN CODAS model, this article is structured in the following way: some related work about 2TLNSs and the CODAS method are given in Section 2. The definition, the score function, the accuracy function, operation rules and some aggregation operators of 2TLNNSs are briefly introduced in Section 3. The computing steps of traditional fuzzy CODAS model are briefly presented in Section 4. By combining the fuzzy traditional CODAS model with 2TLNNs information, the 2-tuple linguistic neutrosophic CODAS model is established and the computing steps for MAGDM problems are simply depicted in Section 5. A numerical example for safety assessment of construction project is given to illustrate this new model and some comparisons between 2-tuple linguistic neutrosophic CODAS model and two 2TLNNs aggregation operators are also made to further illustrate advantages of the new method in Section 6. Section 7 gives some conclusions of our work.

2 Related Work

Previously, a lot of decision-making models such as the VIKOR method (He et al., 2019b; Opricovic and Tzeng, 2004; Wang et al., 2018b), the ELECTRE method (Rashid et al., 2018), the TOPSIS method (Chen, 2000; Lu et al., 2019b), the PROMETHEE method (Balali et al., 2014), the MABAC method (Pamucar and Cirovic, 2015), the EDAS method (Keshavarz Ghorabaee et al., 2015; Wang et al., 2019i; Zhang et al., 2019) and the TODIM method (Gomes and Lima, 1979) have been studied extensively by numerous researchers. Compared with the existing literature, the CODAS model has the advantage of taking the combinative form of ED and HD into account with respect to the intangibility of decision maker (DM) and the uncertainty of decision-making environment to obtain more accurate and effective aggregation results. Since the CODAS method was proposed, a large number of scholars have studied it. Pamucar et al. (2018) presented a linguistic neutrosophic CODAS model. Badi et al. (2018) studied the site selection of desalination plant in Libya by using CODAS method. Bolturk (2018) proposed an extended CODAS model to deal with Pythagorean fuzzy decision making problems and studied its application to supplier selection. Based on interval-valued intuitionistic fuzzy information, Bolturk and Kahraman (2018a) developed a novel CODAS model. To handle renewable energy selection, Bolturk and Karasan (2018b) proposed the interval-valued neutrosophic CODAS model. According to novel information measure, Peng and Garg (2018) studied the CODAS method and presented some novel algorithms under the interval-valued fuzzy soft set. Ren (2018) established the intuitionistic fuzzy CODAS model for MADM. Karasan et al. (2019) developed an integrated methodology based on the neutrosophic CODAS model.

As for the 2-tuple linguistic neutrosophic sets, based on the Hamy mean (HM) operator, Wu et al. (2018b) proposed some 2-tuple linguistic neutrosophic Hamy mean (2TLNNHM) operators and 2-tuple linguistic neutrosophic dual Hamy mean (2TLNNDHM) operators to fuse 2TLNNs. Wang et al. (2019b) developed some 2-tuple linguistic neutrosophic Muirhead mean (2TLNNMM) operators for MADM. Considering the Dombi operation laws and BM operators, Wei et al. (2019b) presented some novel aggregation operators. Wang et al. (2019h) combined the EDAS method with the 2-tuple linguistic neutrosophic set to build an extended EDAS model for MADM. Based on the single-valued neutrosophic 2-tuple linguistic set, Wang et al. (2019d) proposed some Muirhead mean (MM) aggregation operators, Wu et al. (2018a) defined some Hamcher aggregation operators, Ju et al. (2018) extended it to interval-valued environment and developed some MSM operators. Wang et al. (2018b) proposed the 2-tuple linguistic neutrosophic TODIM model. Thereafter, the 2TLNSs have been widely studied in MADM issues.

However, it is clear that there are no studies about the CODAS model with 2TLNNs information. Some scholars studied the CODAS model under neutrosophic and linguistic neutrosophic environment, but both of them cannot represent decision information in a convenient way. At the same time, in the area of 2TLNSs, the research mainly focuses on the aggregation operators, but there is a lack of research on 2TLN models. Hence, it is necessary to discuss the 2-tuple linguistic neutrosophic CODAS model. The goal of this paper is to develop a novel CODAS method based on the conventional CODAS model and 2-tuple linguistic neutrosophic information to study MADM problems more effectively.

3 Preliminaries

3.1 2-Tuple Linguistic Neutrosophic Sets

Wang et al. (2018a) initially proposed the 2-tuple linguistic neutrosophic sets (2TLNSs), which consider the important characteristics of 2-tuple linguistic variables and single-valued neutrosophic sets (SVNSs), hence, can be more effective and accurate to evaluate the alternatives in multiple attribute decision making problems. To combine the 2TLSs and SVNSs, the definition of 2TLNSs can be expressed as follows.

Definition 1.
Let
$δ_{1}, δ_{2}, \dots, δ_{k}$ ${\delta _{1}},{\delta _{2}},\dots ,{\delta _{k}}$
be a linguistic term set. Any label
$δ_{i}$ ${\delta _{i}}$
shows a possible linguistic scale, and
$δ = {δ_{0} = exceedingly terrible, δ_{1} = very terrible, δ_{2} = terrible, δ_{3} = medium, δ_{4} = well, δ_{5} = very well, δ_{6} = exceedingly well}$ $ \delta =\{{\delta _{0}}=\mathit{exceedingly}\hspace{2.5pt}\mathit{terrible},{\delta _{1}}=\mathit{very}\hspace{2.5pt}\mathit{terrible},{\delta _{2}}=\mathit{terrible},{\delta _{3}}=\mathit{medium},{\delta _{4}}=\mathit{well},{\delta _{5}}=\mathit{very}\hspace{2.5pt}\mathit{well},{\delta _{6}}=\mathit{exceedingly}\hspace{2.5pt}\mathit{well}\}$
, then we can describe the 2TLNSs as:
$\begin{matrix} δ = ⟨ (s_{t}, α,), (s_{i}, β), (s_{f}, χ) ⟩, \end{matrix}$ \[ \delta =\big\langle ({s_{t}},\alpha ,),({s_{i}},\beta ),({s_{f}},\chi )\big\rangle ,\]
(1)
where
$Δ^{- 1} (s_{t}, α,), Δ^{- 1} (s_{i}, β)$ ${\Delta ^{-1}}({s_{t}},\alpha ,),{\Delta ^{-1}}({s_{i}},\beta )$
and
$Δ^{- 1} (s_{f}, χ) \in [0, k]$ ${\Delta ^{-1}}({s_{f}},\chi )\in [0,k]$
represent the truth membership function, the indeterminacy membership function and the falsity membership function, which are expressed by 2-tuple linguistic variables and satisfy the condition
$0 ⩽ Δ^{- 1} (s_{t}, ϕ) + Δ^{- 1} (s_{f}, φ) + Δ^{- 1} (s_{f}, γ) ⩽ 3 k$ $ 0\leqslant {\Delta ^{-1}}({s_{t}},\phi )+{\Delta ^{-1}}({s_{f}},\varphi )+{\Delta ^{-1}}({s_{f}},\gamma )\leqslant 3k$
.
Definition 2 (See Wang et al., 2018a).

Let

δ_{1} = ⟨ (s_{t_{1}}, α_{1},), (s_{i_{1}}, β_{1}), (s_{f_{1}}, χ_{1}) ⟩

${\delta _{1}}=\langle ({s_{{t_{1}}}},{\alpha _{1}},),({s_{{i_{1}}}},{\beta _{1}}),({s_{{f_{1}}}},{\chi _{1}})\rangle $

and

δ_{2} = ⟨ (s_{t_{2}}, α_{2},), (s_{i_{2}}, β_{2}), (s_{f_{2}}, χ_{2}) ⟩

${\delta _{2}}=\langle ({s_{{t_{2}}}},{\alpha _{2}},),({s_{{i_{2}}}},{\beta _{2}}),({s_{{f_{2}}}},{\chi _{2}})\rangle $

be two 2-tuple linguistic neutrosophic numbers (2TLNNs), the operation formula of them can be defined:

\begin{matrix} (1) δ_{1} \oplus δ_{2} = \{\begin{array}{l} Δ (k (\frac{Δ^{- 1} (s_{t_{1}}, α_{1},)}{k} + \frac{Δ^{- 1} (s_{t_{2}}, α_{2},)}{k} - \frac{Δ^{- 1} (s_{t_{1}}, α_{1},)}{k} \cdot \frac{Δ^{- 1} (s_{t_{2}}, α_{2},)}{k})), \\ Δ (k (\frac{Δ^{- 1} (s_{i_{1}}, β_{1})}{k} \cdot \frac{Δ^{- 1} (s_{i_{2}}, β_{2})}{k})), \\ Δ (k (\frac{Δ^{- 1} (s_{f_{1}}, χ_{1})}{k} \cdot \frac{Δ^{- 1} (s_{f_{1}}, χ_{1})}{k})) \end{array}\}; \end{matrix}

\[ (1)\hspace{2.5pt}{\delta _{1}}\oplus {\delta _{2}}=\left\{\begin{array}{l}\Delta \big(k\big(\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}},)}{k}+\frac{{\Delta ^{-1}}({s_{{t_{2}}}},{\alpha _{2}},)}{k}-\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}},)}{k}\cdot \frac{{\Delta ^{-1}}({s_{{t_{2}}}},{\alpha _{2}},)}{k}\big)\big),\\ {} \Delta \big(k\big(\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})}{k}\cdot \frac{{\Delta ^{-1}}({s_{{i_{2}}}},{\beta _{2}})}{k}\big)\big),\\ {} \Delta \big(k\big(\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})}{k}\cdot \frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})}{k}\big)\big)\end{array}\right\};\]

\begin{matrix} (2) δ_{1} \otimes δ_{2} = \{\begin{array}{l} Δ (k (\frac{Δ^{- 1} (s_{t_{1}}, α_{1},)}{k} \cdot \frac{Δ^{- 1} (s_{t_{2}}, α_{2},)}{k})), \\ Δ (k (\frac{Δ^{- 1} (s_{i_{1}}, β_{1})}{k} + \frac{Δ^{- 1} (s_{i_{2}}, β_{2})}{k} - \frac{Δ^{- 1} (s_{i_{1}}, β_{1})}{k} \cdot \frac{Δ^{- 1} (s_{i_{2}}, β_{2})}{k})), \\ Δ (k (\frac{Δ^{- 1} (s_{f_{1}}, χ_{1})}{k} + \frac{Δ^{- 1} (s_{f_{2}}, χ_{2})}{k} - \frac{Δ^{- 1} (s_{f_{1}}, χ_{1})}{k} \cdot \frac{Δ^{- 1} (s_{f_{2}}, χ_{2})}{k})) \end{array}\}; \end{matrix}

\[ (2)\hspace{2.5pt}{\delta _{1}}\otimes {\delta _{2}}=\left\{\begin{array}{l}\Delta \big(k\big(\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}},)}{k}\cdot \frac{{\Delta ^{-1}}({s_{{t_{2}}}},{\alpha _{2}},)}{k}\big)\big),\\ {} \Delta \big(k\big(\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})}{k}+\frac{{\Delta ^{-1}}({s_{{i_{2}}}},{\beta _{2}})}{k}-\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})}{k}\cdot \frac{{\Delta ^{-1}}({s_{{i_{2}}}},{\beta _{2}})}{k}\big)\big),\\ {} \Delta \big(k\big(\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})}{k}+\frac{{\Delta ^{-1}}({s_{{f_{2}}}},{\chi _{2}})}{k}-\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})}{k}\cdot \frac{{\Delta ^{-1}}({s_{{f_{2}}}},{\chi _{2}})}{k}\big)\big)\end{array}\right\};\]

\begin{matrix} (3) λ δ_{1} = \{\begin{array}{l} Δ (k (1 - {(1 - \frac{Δ^{- 1} (s_{t_{1}}, α_{1},)}{k})}^{λ})), \\ Δ (k {(\frac{Δ^{- 1} (s_{i_{1}}, β_{1})}{k})}^{λ}), \\ Δ (k {(\frac{Δ^{- 1} (s_{f_{1}}, χ_{1})}{k})}^{λ}) \end{array}\}, λ > 0; \end{matrix}

\[ (3)\hspace{2.5pt}\lambda {\delta _{1}}=\left\{\begin{array}{l}\Delta \big(k\big(1-{\big(1-\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}},)}{k}\big)^{\lambda }}\big)\big),\\ {} \Delta \big(k{\big(\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})}{k}\big)^{\lambda }}\big),\\ {} \Delta \big(k{\big(\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})}{k}\big)^{\lambda }}\big)\end{array}\right\},\hspace{1em}\lambda >0;\]

\begin{matrix} (4) δ_{1}^{λ} = \{\begin{array}{l} Δ (k {(\frac{Δ^{- 1} (s_{t_{1}}, α_{1},)}{k})}^{λ}), \\ Δ (k (1 - {(1 - \frac{Δ^{- 1} (s_{i_{1}}, β_{1})}{k})}^{λ})), \\ Δ (k (1 - {(1 - \frac{Δ^{- 1} (s_{f_{1}}, χ_{1})}{k})}^{λ})) \end{array}\}, λ > 0 . \end{matrix}

\[ (4)\hspace{2.5pt}{\delta _{\mathrm{1}}^{\lambda }}=\left\{\begin{array}{l}\Delta \big(k{\big(\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}},)}{k}\big)^{\lambda }}\big),\\ {} \Delta \big(k\big(1-{\big(1-\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})}{k}\big)^{\lambda }}\big)\big),\\ {} \Delta \big(k\big(1-{(1-\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})}{k})^{\lambda }}\big)\big)\end{array}\right\},\hspace{1em}\lambda >0.\]

According to Definition 2, it is clear that the operation laws have the following properties.

\begin{array}{l} δ_{1} \oplus δ_{2} = δ_{2} \oplus δ_{1}, δ_{1} \otimes δ_{2} = δ_{2} \otimes δ_{1}, {({(δ_{1})}^{λ_{1}})}^{λ_{2}} = {(δ_{1})}^{λ_{1} λ_{2}}, \end{array}

\[ {\delta _{1}}\oplus {\delta _{2}}={\delta _{2}}\oplus {\delta _{1}},\hspace{2em}{\delta _{1}}\otimes {\delta _{2}}={\delta _{2}}\otimes {\delta _{1}},\hspace{2em}{\big({({\delta _{1}})^{{\lambda _{1}}}}\big)^{{\lambda _{2}}}}={({\delta _{1}})^{{\lambda _{1}}{\lambda _{2}}}},\]

(2)

\begin{array}{l} λ (δ_{1} \oplus δ_{2}) = λ δ_{1} \oplus λ δ_{2}, {(δ_{1} \otimes δ_{2})}^{λ} = {(δ_{1})}^{λ} \otimes {(δ_{2})}^{λ}, \end{array}

\[ \lambda ({\delta _{1}}\oplus {\delta _{2}})=\lambda {\delta _{1}}\oplus \lambda {\delta _{2}},\hspace{2em}{({\delta _{1}}\otimes {\delta _{2}})^{\lambda }}={({\delta _{1}})^{\lambda }}\otimes {({\delta _{2}})^{\lambda }},\]

(3)

\begin{array}{l} λ_{1} δ_{1} \oplus λ_{2} δ_{1} = (λ_{1} + λ_{2}) δ_{1}, {(δ_{1})}^{λ_{1}} \otimes {(δ_{1})}^{λ_{2}} = {(δ_{1})}^{(λ_{1} + λ_{2})} . \end{array}

\[ {\lambda _{1}}{\delta _{1}}\oplus {\lambda _{2}}{\delta _{1}}=({\lambda _{1}}+{\lambda _{2}}){\delta _{1}},\hspace{2em}{({\delta _{1}})^{{\lambda _{1}}}}\otimes {({\delta _{1}})^{{\lambda _{2}}}}={({\delta _{1}})^{({\lambda _{1}}+{\lambda _{2}})}}.\]

(4)

Definition 3 (See Wang et al., 2018b).

Let

δ = ⟨ (s_{t}, α,), (s_{i}, β), (s_{f}, χ) ⟩

$ \delta =\langle ({s_{t}},\alpha ,),({s_{i}},\beta ),({s_{f}},\chi )\rangle $

be a 2TLNN, the score and accuracy functions of δ can be expressed:

\begin{array}{l} s (δ) = \frac{(2 k + Δ^{- 1} (s_{t}, α,) - Δ^{- 1} (s_{i}, β) - Δ^{- 1} (s_{f}, χ))}{3 k}, s (δ) \in [0, 1], \end{array}

\[ s(\delta )=\frac{(2k+{\Delta ^{-1}}({s_{t}},\alpha ,)-{\Delta ^{-1}}({s_{i}},\beta )-{\Delta ^{-1}}({s_{f}},\chi ))}{3k},\hspace{1em}s(\delta )\in [0,1],\]

(5)

\begin{array}{l} h (δ) = \frac{1}{k} (Δ^{- 1} (s_{t}, α,) - Δ^{- 1} (s_{f}, χ)), h (δ) \in [- 1, 1] . \end{array}

\[ h(\delta )=\frac{1}{k}\big({\Delta ^{-1}}({s_{t}},\alpha ,)-{\Delta ^{-1}}({s_{f}},\chi )\big),\hspace{1em}h(\delta )\in [-1,1].\]

(6)

For two 2TLNNs

δ_{1}

${\delta _{1}}$

and

δ_{2}

${\delta _{2}}$

, based on Definition 3, then

(1)

if s (δ_{1}) ≺ s (δ_{2}), then δ_{1} ≺ δ_{2}

$ \text{if}\hspace{5pt}s({\delta _{1}})\prec s({\delta _{2}}),\hspace{2.5pt}\text{then}\hspace{5pt}{\delta _{1}}\prec {\delta _{2}}$

;

(2)

if s (δ_{1}) ≻ s (δ_{2}), then δ_{1} ≻ δ_{2}

$ \text{if}\hspace{5pt}s({\delta _{1}})\succ s({\delta _{2}}),\hspace{2.5pt}\text{then}\hspace{5pt}{\delta _{1}}\succ {\delta _{2}}$

;

(3)

if s (δ_{1}) = s (δ_{2}), h (δ_{1}) ≺ h (δ_{2}), then δ_{1} ≺ δ_{2}

$ \text{if}\hspace{5pt}s({\delta _{1}})=s({\delta _{2}}),\hspace{2.5pt}h({\delta _{1}})\prec h({\delta _{2}}),\hspace{2.5pt}\text{then}\hspace{5pt}{\delta _{1}}\prec {\delta _{2}}$

;

(4)

if s (δ_{1}) = s (δ_{2}), h (δ_{1}) ≻ h (δ_{2}), then δ_{1} ≻ δ_{2}

$ \text{if}\hspace{5pt}s({\delta _{1}})=s({\delta _{2}}),\hspace{2.5pt}h({\delta _{1}})\succ h({\delta _{2}}),\hspace{2.5pt}\text{then}\hspace{5pt}{\delta _{1}}\succ {\delta _{2}}$

;

(5)

if s (δ_{1}) = s (δ_{2}), h (δ_{1}) = h (δ_{2}), then δ_{1} = δ_{2}

$ \text{if}\hspace{5pt}s({\delta _{1}})=s({\delta _{2}}),\hspace{2.5pt}h({\delta _{1}})=h({\delta _{2}}),\hspace{2.5pt}\text{then}\hspace{5pt}{\delta _{1}}={\delta _{2}}$

3.2 The Distance Measurement of 2TLNNs

Definition 4.
Let
$δ_{1} = {(s_{t_{1}}, α_{1}), (s_{i_{1}}, β_{1}), (s_{f_{1}}, χ_{1})}$ ${\delta _{1}}=\{({s_{{t_{1}}}},{\alpha _{1}}),({s_{{i_{1}}}},{\beta _{1}}),({s_{{f_{1}}}},{\chi _{1}})\}$
and
$δ_{2} = {(s_{t_{2}}, α_{2}), (s_{i_{2}}, β_{2}), (s_{f_{2}} χ_{2})}$ ${\delta _{2}}=\{({s_{{t_{2}}}},{\alpha _{2}}),({s_{{i_{2}}}},{\beta _{2}}),({s_{{f_{2}}}}{\chi _{2}})\}$
be two 2TLNNs, then we can get the normalized Hamming distance:
$\begin{matrix} d^{H} (δ_{1}, δ_{2}) = \frac{1}{3} (\begin{array}{l} | \frac{Δ^{- 1} (s_{t_{1}}, α_{1}) - Δ^{- 1} (s_{t_{2}}, α_{2})}{k} | + | \frac{Δ^{- 1} (s_{i_{1}}, β_{1}) - Δ^{- 1} (s_{i_{2}}, β_{2})}{k} | \\ + | \frac{Δ^{- 1} (s_{f_{1}}, χ_{1}) - Δ^{- 1} (s_{f_{2}}, χ_{2})}{k} | \end{array}) . \end{matrix}$ \[ {d^{H}}({\delta _{1}},{\delta _{2}})=\frac{1}{3}\left(\begin{array}{l}\Big|\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}})-{\Delta ^{-1}}({s_{{t_{2}}}},{\alpha _{2}})}{k}\Big|+\Big|\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})-{\Delta ^{-1}}({s_{{i_{2}}}},{\beta _{2}})}{k}\Big|\\ {} +\Big|\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})-{\Delta ^{-1}}({s_{{f_{2}}}},{\chi _{2}})}{k}\Big|\end{array}\right).\]
(7)

Definition 5.
Let
$δ_{1} = {(s_{t_{1}}, α_{1}), (s_{i_{1}}, β_{1}), (s_{f_{1}}, χ_{1})}$ ${\delta _{1}}=\{({s_{{t_{1}}}},{\alpha _{1}}),({s_{{i_{1}}}},{\beta _{1}}),({s_{{f_{1}}}},{\chi _{1}})\}$
and
$δ_{2} = {(s_{t_{2}}, α_{2}), (s_{i_{2}}, β_{2}), (s_{f_{2}} χ_{2})}$ ${\delta _{2}}=\{({s_{{t_{2}}}},{\alpha _{2}}),({s_{{i_{2}}}},{\beta _{2}}),({s_{{f_{2}}}}{\chi _{2}})\}$
be two 2TLNNs, then we can get the normalized Euclidean distance:
$\begin{matrix} d^{E} (δ_{1}, δ_{2}) = \sqrt{\frac{1}{3} (\begin{array}{l} {| \frac{Δ^{- 1} (s_{t_{1}}, α_{1}) - Δ^{- 1} (s_{t_{2}}, α_{2})}{k} |}^{2} + {| \frac{Δ^{- 1} (s_{i_{1}}, β_{1}) - Δ^{- 1} (s_{i_{2}}, β_{2})}{k} |}^{2} \\ + {| \frac{Δ^{- 1} (s_{f_{1}}, χ_{1}) - Δ^{- 1} (s_{f_{2}}, χ_{2})}{k} |}^{2} \end{array})} . \end{matrix}$ \[ {d^{E}}({\delta _{1}},{\delta _{2}})=\sqrt{\frac{1}{3}\left(\begin{array}{l}{\Big|\frac{{\Delta ^{-1}}({s_{{t_{1}}}},{\alpha _{1}})-{\Delta ^{-1}}({s_{{t_{2}}}},{\alpha _{2}})}{k}\Big|^{2}}+{\Big|\frac{{\Delta ^{-1}}({s_{{i_{1}}}},{\beta _{1}})-{\Delta ^{-1}}({s_{{i_{2}}}},{\beta _{2}})}{k}\Big|^{2}}\\ {} +{\Big|\frac{{\Delta ^{-1}}({s_{{f_{1}}}},{\chi _{1}})-{\Delta ^{-1}}({s_{{f_{2}}}},{\chi _{2}})}{k}\Big|^{2}}\end{array}\right)}.\]
(8)

3.3 The 2TLNNWA and 2TLNNWG Operators

Definition 6 (See Wang et al., 2018a).

Let

δ_{j} = {(s_{t_{j}}, α_{j}), (s_{i_{j}}, β_{j}), (s_{f_{j}}, χ_{j})}

${\delta _{j}}=\{({s_{{t_{j}}}},{\alpha _{j}}),({s_{{i_{j}}}},{\beta _{j}}),({s_{{f_{j}}}},{\chi _{j}})\}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

be a set of 2TLNNs, the 2TLNNWA and 2TLNNWG operators can be presented:

\begin{array}{l} 2 TLNNWA (δ_{1}, δ_{2}, \dots, δ_{n}) \\ = w_{1} δ_{1} \oplus w_{2} δ_{2} \otimes \dots \oplus w_{n} δ_{n} = ⨁_{j = 1}^{n} w_{j} δ_{j} \\ = ⟨\begin{array}{l} Δ (k (1 - \prod_{j = 1}^{n} {(1 - \frac{Δ^{- 1} (s_{t_{j}}, α_{j})}{k})}^{w_{j}})), Δ (k \prod_{j = 1}^{n} {(\frac{Δ^{- 1} (s_{i_{j}}, β_{j})}{k})}^{w_{j}}), \\ Δ (k \prod_{j = 1}^{n} {(\frac{Δ^{- 1} (s_{f_{j}}, χ_{j})}{k})}^{w_{j}}) . \end{array}⟩ \end{array}

\[\begin{array}{l}\displaystyle \mathrm{2}\mathrm{TLNNWA}({\delta _{1}},{\delta _{2}},\dots ,{\delta _{n}})\\ {} \displaystyle \hspace{1em}={w_{1}}{\delta _{1}}\oplus {w_{2}}{\delta _{2}}\otimes \cdots \oplus {w_{n}}{\delta _{n}}={\underset{j=1}{\overset{n}{\bigoplus }}}{w_{j}}{\delta _{j}}\\ {} \displaystyle \hspace{1em}=\left\langle \begin{array}{l}\Delta \Big(k\Big(1-{\textstyle\textstyle\prod _{j=1}^{n}}{\Big(1-\frac{{\Delta ^{-1}}({s_{{t_{j}}}},{\alpha _{j}})}{k}\Big)^{{w_{j}}}}\Big)\Big),\Delta \Big(k{\textstyle\textstyle\prod _{j=1}^{n}}{\Big(\frac{{\Delta ^{-1}}({s_{{i_{j}}}},{\beta _{j}})}{k}\Big)^{{w_{j}}}}\Big),\\ {} \Delta \Big(k{\textstyle\textstyle\prod _{j=1}^{n}}{\Big(\frac{{\Delta ^{-1}}({s_{{f_{j}}}},{\chi _{j}})}{k}\Big)^{{w_{j}}}}\Big).\end{array}\right\rangle \end{array}\]

(9)

and

\begin{array}{l} 2 TLNNWG (δ_{1}, δ_{2}, \dots, δ_{n}) \\ = {(δ_{1})}^{w_{1}} \otimes {(δ_{2})}^{w_{2}} \otimes \dots \otimes {(δ_{n})}^{w_{n}} = ⨂_{j = 1}^{n} {(δ_{j})}^{w_{j}} \\ = ⟨\begin{array}{l} Δ (k \prod_{j = 1}^{n} {(\frac{Δ^{- 1} (s_{t_{j}}, α_{j})}{k})}^{w_{j}}), Δ (k (1 - \prod_{j = 1}^{n} {(1 - \frac{Δ^{- 1} (s_{i_{j}}, β_{j})}{k})}^{w_{j}})), \\ Δ (k (1 - \prod_{j = 1}^{n} {(1 - \frac{Δ^{- 1} (s_{f_{j}}, χ_{j})}{k})}^{w_{j}})) . \end{array}⟩, \end{array}

\[\begin{array}{l}\displaystyle \mathrm{2}\mathrm{TLNNWG}({\delta _{1}},{\delta _{2}},\dots ,{\delta _{n}})\\ {} \displaystyle \hspace{1em}={({\delta _{1}})^{{w_{1}}}}\otimes {({\delta _{2}})^{{w_{2}}}}\otimes \cdots \otimes {({\delta _{n}})^{{w_{n}}}}={\underset{j=1}{\overset{n}{\bigotimes }}}{({\delta _{j}})^{{w_{j}}}}\\ {} \displaystyle \hspace{1em}=\left\langle \begin{array}{l}\Delta \Big(k{\textstyle\textstyle\prod _{j=1}^{n}}{\Big(\frac{{\Delta ^{-1}}({s_{{t_{j}}}},{\alpha _{j}})}{k}\Big)^{{w_{j}}}}\Big),\Delta \Big(k\Big(1-{\textstyle\textstyle\prod _{j=1}^{n}}{\Big(1-\frac{{\Delta ^{-1}}({s_{{i_{j}}}},{\beta _{j}})}{k}\Big)^{{w_{j}}}}\Big)\Big),\\ {} \Delta \Big(k\Big(1-{\textstyle\textstyle\prod _{j=1}^{n}}{(1-\frac{{\Delta ^{-1}}({s_{{f_{j}}}},{\chi _{j}})}{k})^{{w_{j}}}}\Big)\Big).\end{array}\right\rangle ,\end{array}\]

(10)

where

w_{j}

${w_{j}}$

is weighting vector of

δ_{j}

${\delta _{j}}$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

, which satisfies

0 ⩽ w_{j} ⩽ 1

$ 0\leqslant {w_{j}}\leqslant 1$

\sum_{j = 1}^{n} w_{j} = 1

${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$

4 The Traditional Fuzzy CODAS Model

The CODAS (combinative distance-based assessment) method, which was originally defined by Keshavarz Ghorabaee et al. (2016), uses the combinative form of two distance measurements, including Euclidean and Taxicab distances, which present accurate values to compute the assessment results of alternatives. However, this model cannot be applied in fuzzy environment. To overcome this disadvantage, Keshavarz Ghorabaee et al. (2017) extended the CODAS method to fuzzy environment and proposed the fuzzy CODAS model which used fuzzy weighted Hamming distance (HD) and fuzzy weighted Euclidean distance (ED) rather than the crisp distances. Suppose there are alternatives

{ϕ_{1}, ϕ_{2}, \dots, ϕ_{m}}

$ \{{\phi _{1}},{\phi _{2}},\dots ,{\phi _{m}}\}$

, n attributes

{O_{1}, O_{2}, \dots, O_{n}}

$ \{{O_{1}},{O_{2}},\dots ,{O_{n}}\}$

and t experts

{d_{1}, d_{2}, \dots, d_{t}}

$ \{{d_{1}},{d_{2}},\dots ,{d_{t}}\}$

, then the decision making steps are expressed as follows.

Step 1.

Construct the evaluation matrix

R = {[ϕ_{i j}^{t}]}_{m \times n}

$ R={[{\phi _{ij}^{t}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

and calculate the average results matrix

\begin{matrix} r = {[ϕ_{i j}]}_{m \times n}, i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}

\[ r={[{\phi _{ij}}]_{m\times n}},\hspace{1em}i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n\]

which can be depicted as follows:

\begin{matrix} R = {[ϕ_{i j}^{t}]}_{m \times n} = \begin{array}{cc} \begin{array}{cccc} O_{1} & O_{2} & \dots & O_{n} \end{array} \\ \begin{array}{c} ϕ_{1} \\ ϕ_{2} \\ ⋮ \\ ϕ_{m} \end{array} & [\begin{array}{cccc} ϕ_{11}^{t} & ϕ_{12}^{t} & \dots & ϕ_{1 n}^{t} \\ ϕ_{21}^{t} & ϕ_{22}^{t} & \dots & ϕ_{2 n}^{t} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ϕ_{m 1}^{t} & ϕ_{m 2}^{t} & \dots & ϕ_{m n}^{t} \end{array}] \end{array}, \end{matrix}

\[ R={\big[{\phi _{ij}^{t}}\big]_{m\times n}}=\begin{array}{c@{\hskip4.0pt}c}& \begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{\mathrm{O}_{1}}& {\mathrm{O}_{2}}& \dots & {\mathrm{O}_{n}}\end{array}\\ {} \begin{array}{c}{\phi _{1}}\\ {} {\phi _{2}}\\ {} \vdots \\ {} {\phi _{m}}\end{array}& \left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{\phi _{11}^{t}}& {\phi _{12}^{t}}& \dots & {\phi _{1n}^{t}}\\ {} {\phi _{21}^{t}}& {\phi _{22}^{t}}& \dots & {\phi _{2n}^{t}}\\ {} \vdots & \vdots & \vdots & \vdots \\ {} {\phi _{m1}^{t}}& {\phi _{m2}^{t}}& \dots & {\phi _{mn}^{t}}\end{array}\right]\end{array},\]

(11)

\begin{matrix} r = {[ϕ_{i j}]}_{m \times n} = \begin{array}{cc} \begin{array}{cccc} O_{1} & O_{2} & \dots & O_{n} \end{array} \\ \begin{array}{c} ϕ_{1} \\ ϕ_{2} \\ ⋮ \\ ϕ_{m} \end{array} & [\begin{array}{cccc} ϕ_{11} & ϕ_{12} & \dots & ϕ_{1 n} \\ ϕ_{21} & ϕ_{22} & \dots & ϕ_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ϕ_{m 1} & ϕ_{m 2} & \dots & ϕ_{m n} \end{array}] \end{array}, \end{matrix}

\[ r={[{\phi _{ij}}]_{m\times n}}=\begin{array}{c@{\hskip4.0pt}c}& \begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{\mathrm{O}_{1}}& {\mathrm{O}_{2}}& \dots & {\mathrm{O}_{n}}\end{array}\\ {} \begin{array}{c}{\phi _{1}}\\ {} {\phi _{2}}\\ {} \vdots \\ {} {\phi _{m}}\end{array}& \left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{\phi _{11}}& {\phi _{12}}& \dots & {\phi _{1n}}\\ {} {\phi _{21}}& {\phi _{22}}& \dots & {\phi _{2n}}\\ {} \vdots & \vdots & \vdots & \vdots \\ {} {\phi _{m1}}& {\phi _{m2}}& \dots & {\phi _{mn}}\end{array}\right]\end{array},\]

(12)

\begin{matrix} ϕ_{i j} = ϕ_{i j}^{1} \oplus ϕ_{i j}^{2} \oplus \dots \oplus ϕ_{i j}^{t}, \end{matrix}

\[ {\phi _{ij}}=\hspace{0.2778em}{\phi _{ij}^{1}}\oplus {\phi _{ij}^{2}}\oplus \cdots \oplus {\phi _{ij}^{t}},\]

(13)

where

ϕ_{i j}^{t}

${\phi _{ij}^{t}}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) denotes the evaluation information of alternative

ϕ_{i}

${\phi _{i}}$

(i = 1, 2, \dots, m)

$(i=1,2,\dots ,m)$

on attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

by expert

d^{t}

${d^{t}}$

and

ϕ_{i j}

${\phi _{ij}}$

means the average values of alternative

ϕ_{i}

${\phi _{i}}$

with respect to attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

Step 2.

Obtain the attribute’s fuzzy weighting vector

W^{t}

${W^{t}}$

which is given by each expert with respect to all attributes and compute the average fuzzy weighting vector W as follows:

\begin{array}{l} W^{t} = {[w_{j}^{t}]}_{1 \times n}, \end{array}

\[ {W^{t}}={\big[{w_{j}^{t}}\big]_{1\times n}},\]

(14)

\begin{array}{l} W = {[w_{j}]}_{1 \times n}, \end{array}

\[ W={[{w_{j}}]_{1\times n}},\]

(15)

\begin{array}{l} w_{j} = w_{j}^{1} \oplus w_{j}^{2} \oplus \dots \oplus w_{j}^{t}, \end{array}

\[ {w_{j}}={w_{j}^{1}}\oplus {w_{j}^{2}}\oplus \cdots \oplus {w_{j}^{t}},\]

(16)

where

w_{j}^{t}

${w_{j}^{t}}$

denotes the fuzzy weight of attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

by expert

d^{t}

${d^{t}}$

and

w_{j}

${w_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

means the average fuzzy weight values of a attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

Step 3.

Normalize the average results matrix

r = {[ϕ_{i j}]}_{m \times n}

$ r={[{\phi _{ij}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

based on the type of each attributes using the following formulae.

For benefit attributes:

\begin{matrix} N_{i j} = ϕ_{i j} / {max}_{i} (ϕ_{i j}), i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}

\[ {N_{ij}}={\phi _{ij}}/{\max _{i}}({\phi _{ij}}),\hspace{1em}i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n.\]

(17)

For cost attributes:

\begin{matrix} N_{i j} = 1 - ϕ_{i j} / {max}_{i} (ϕ_{i j}), i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}

\[ {N_{ij}}=1-{\phi _{ij}}/{\max _{i}}({\phi _{ij}}),\hspace{1em}i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n.\]

(18)

Step 4.

According to the normalized average matrix

N_{i j}

${N_{ij}}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) and average fuzzy weighting vector

w_{j}

${w_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

, the fuzzy weighted normalized average matrix

{WN}_{i j}

${\mathit{WN}_{ij}}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) can be computed as:

\begin{matrix} {WN}_{i j} = w_{j} \otimes N_{i j} (i = 1, 2, \dots, m, j = 1, 2, \dots, n) . \end{matrix}

\[ {\mathit{WN}_{ij}}={w_{j}}\otimes {N_{ij}}\hspace{1em}(i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n).\]

(19)

Step 5.

Determine the fuzzy negative solution (NS) based on the equation (20):

\begin{matrix} {NS}_{j} = min_{i} ({WN}_{i j}) (i = 1, 2, \dots, m, j = 1, 2, \dots, n) . \end{matrix}

\[ {\mathit{NS}_{j}}=\underset{i}{\min }({\mathit{WN}_{ij}})\hspace{1em}(i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n).\]

(20)

Step 6.

Calculate the fuzzy weighted Hamming distance

({HD}_{i})

$({\mathit{HD}_{i}})$

and fuzzy weighted Euclidean distance

({ED}_{i})

$({\mathit{ED}_{i}})$

between each alternatives and the negative solution (NS) according to Definition 4 and Definition 5:

\begin{array}{l} {HD}_{i} = \sum_{j = 1}^{n} d^{H} ({WN}_{i j}, {NS}_{j}), \end{array}

\[ {\mathit{HD}_{i}}={\sum \limits_{j=1}^{n}}{d^{H}}({\mathit{WN}_{ij}},{\mathit{NS}_{j}}),\]

(21)

\begin{array}{l} {ED}_{i} = \sum_{j = 1}^{n} d^{E} ({WN}_{i j}, {NS}_{j}) . \end{array}

\[ {\mathit{ED}_{i}}={\sum \limits_{j=1}^{n}}{d^{E}}({\mathit{WN}_{ij}},{\mathit{NS}_{j}}).\]

(22)

Step 7.

Determine the relative assessment (RA) matrix which is presented as follows:

\begin{array}{l} RA = {[p_{i l}]}_{m \times m}, \end{array}

\[ \mathit{RA}={[{p_{il}}]_{m\times m}},\]

(23)

\begin{array}{l} p_{i l} = ({ED}_{i} - {ED}_{l}) + (λ ({ED}_{i} - {ED}_{l}) \times ({HD}_{i} - {HD}_{l})), \end{array}

\[ {p_{il}}=({\mathit{ED}_{i}}-{\mathit{ED}_{l}})+\big(\lambda ({\mathit{ED}_{i}}-{\mathit{ED}_{l}})\times ({\mathit{HD}_{i}}-{\mathit{HD}_{l}})\big),\]

(24)

where

i, l = 1, 2, \dots, m

$ i,l=1,2,\dots ,m$

and λ is a threshold function that can be defined:

\begin{matrix} λ (x) = \{\begin{array}{ll} 1 & if | x | ⩾ θ, \\ 0 & if | x | < θ . \end{array} \end{matrix}

\[ \lambda (x)=\left\{\begin{array}{l@{\hskip4.0pt}l}1\hspace{1em}& \text{if}\hspace{5pt}|x|\geqslant \theta ,\\ {} 0\hspace{1em}& \text{if}\hspace{5pt}|x|<\theta .\end{array}\right.\]

(25)

The threshold parameter θ of this function can be set by the decision maker. In our paper, we let

θ = 0.02

$ \theta =0.02$

for the calculations.

Step 8.

Compute the values of assessment score (AS) based on each alternative’s using the following equation:

\begin{matrix} A S_{i} = \sum_{l = 1}^{m} p_{i l} . \end{matrix}

\[ A{S_{i}}={\sum \limits_{l=1}^{m}}{p_{il}}.\]

(26)

Step 9.

According to the calculation results of

{AS}_{i}

${\mathit{AS}_{i}}$

, we can rank all the alternatives. The bigger the value of

{AS}_{i}

${\mathit{AS}_{i}}$

is, the better alternative will be selected.

5 The CODAS Model with 2-Tuple Linguistic Neutrosophic Information

By combining the CODAS method with 2-tuple linguistic neutrosophic information we can build the 2-tuple linguistic neutrosophic CODAS model where all the evaluation information and attribute’s weighting vector are presented with 2-tuple linguistic neutrosophic numbers (2TLNNs). Suppose there are m alternatives

{ϕ_{1}, ϕ_{2}, \dots, ϕ_{m}}

$ \{{\phi _{1}},{\phi _{2}},\dots ,{\phi _{m}}\}$

, n attributes

{O_{1}, O_{2}, \dots, O_{n}}

$ \{{O_{1}},{O_{2}},\dots ,{O_{n}}\}$

and t experts

{d_{1}, d_{2}, \dots, d_{t}}

$ \{{d_{1}},{d_{2}},\dots ,{d_{t}}\}$

, let expert’s weighting vector be

{a_{1}, a_{2}, \dots, a_{t}}

$ \{{a_{1}},{a_{2}},\dots ,{a_{t}}\}$

, then the decision making steps are expressed as follows.

Step 1.

Construct the 2-tuple linguistic neutrosophic evaluation matrix

R = {[ϕ_{i j}^{t}]}_{m \times n}

$ R={[{\phi _{ij}^{t}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

and calculate the average results matrix

\begin{matrix} r = {[ϕ_{i j}]}_{m \times n}, i = 1, 2, \dots, m, j = 1, 2, \dots, n, \end{matrix}

\[ r={[{\phi _{ij}}]_{m\times n}},\hspace{1em}i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n,\]

which can be depicted as follows:

\begin{matrix} R = {[ϕ_{i j}^{t}]}_{m \times n} = \begin{array}{cc} \begin{array}{cccc} O_{1} & O_{2} & \dots & O_{n} \end{array} \\ \begin{array}{c} ϕ_{1} \\ ϕ_{2} \\ ⋮ \\ ϕ_{m} \end{array} & [\begin{array}{cccc} ϕ_{11}^{t} & ϕ_{12}^{t} & \dots & ϕ_{1 n}^{t} \\ ϕ_{21}^{t} & ϕ_{22}^{t} & \dots & ϕ_{2 n}^{t} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ϕ_{m 1}^{t} & ϕ_{m 2}^{t} & \dots & ϕ_{m n}^{t} \end{array}] \end{array}, \end{matrix}

(27)

\begin{matrix} r = {[ϕ_{i j}]}_{m \times n} = \begin{array}{cc} \begin{array}{cccc} O_{1} & O_{2} & \dots & O_{n} \end{array} \\ \begin{array}{c} ϕ_{1} \\ ϕ_{2} \\ ⋮ \\ ϕ_{m} \end{array} & [\begin{array}{cccc} ϕ_{11} & ϕ_{12} & \dots & ϕ_{1 n} \\ ϕ_{21} & ϕ_{22} & \dots & ϕ_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ϕ_{m 1} & ϕ_{m 2} & \dots & ϕ_{m n} \end{array}] \end{array} . \end{matrix}

(28)

Based on the assessment information and expert’s weighting vector

{a_{1}, a_{2}, \dots, a_{t}}

$ \{{a_{1}},{a_{2}},\dots ,{a_{t}}\}$

, the

\begin{matrix} r = {[ϕ_{i j}]}_{m \times n} \end{matrix}

\[ r={[{\phi _{ij}}]_{m\times n}}\]

can be calculated as in Wang et al. (2018c) :

\begin{array}{rcl} ϕ_{i j} & = & a_{1} ϕ_{i j}^{1} \oplus a_{2} ϕ_{i j}^{2} \oplus \dots \oplus a_{t} ϕ_{i j}^{t} \\ = & \{\begin{array}{l} Δ (k (1 - \prod_{d = 1}^{t} {(1 - \frac{Δ^{- 1} {(s_{t_{i j}}, α_{i j})}^{t}}{k})}^{a_{t}})), \\ Δ (k \prod_{d = 1}^{t} {(\frac{Δ^{- 1} {(s_{i_{i j}}, β_{i j})}^{t}}{k})}^{a_{t}}), \\ Δ (k \prod_{d = 1}^{t} {(\frac{Δ^{- 1} {(s_{f_{i j}}, χ_{i j})}^{t}}{k})}^{a_{t}}) \end{array}\}, \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\phi _{ij}}& \displaystyle =& \displaystyle {a_{1}}{\phi _{ij}^{1}}\oplus {a_{2}}{\phi _{ij}^{2}}\oplus \cdots \oplus {a_{t}}{\phi _{ij}^{t}}\\ {} & \displaystyle =& \displaystyle \left\{\begin{array}{l}\Delta \Big(k\Big(1-{\textstyle\textstyle\prod _{d=1}^{t}}{(1-\frac{{\Delta ^{-1}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{t}}}{k})^{{a_{t}}}}\Big)\Big),\\ {} \Delta \Big(k{\textstyle\textstyle\prod _{d=1}^{t}}{\Big(\frac{{\Delta ^{-1}}{({s_{{i_{ij}}}},{\beta _{ij}})^{t}}}{k}\Big)^{{a_{t}}}}\Big),\\ {} \Delta \Big(k{\textstyle\textstyle\prod _{d=1}^{t}}{\Big(\frac{{\Delta ^{-1}}{({s_{{f_{ij}}}},{\chi _{ij}})^{t}}}{k}\Big)^{{a_{t}}}}\Big)\end{array}\right\},\end{array}\]

(29)

where

ϕ_{i j}^{t} = {{(s_{t_{i j}}, α_{i j})}^{t}, {(s_{i_{i j}}, β_{i j})}^{t}, {(s_{f_{i j}}, χ_{i j})}^{t}}

${\phi _{ij}^{t}}=\{{({s_{{t_{ij}}}},{\alpha _{ij}})^{t}},{({s_{{i_{ij}}}},{\beta _{ij}})^{t}},{({s_{{f_{ij}}}},{\chi _{ij}})^{t}}\}$

, (

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) denotes the 2-tuple linguistic neutrosophic information of alternative

ϕ_{i}

${\phi _{i}}$

(i = 1, 2, \dots, m)

$(i=1,2,\dots ,m)$

on attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

by expert

d^{t}

${d^{t}}$

and

ϕ_{i j} = {(s_{t_{i j}}, α_{i j}), (s_{i_{i j}}, β_{i j}), (s_{f_{i j}}, χ_{i j})}

${\phi _{ij}}=\{({s_{{t_{ij}}}},{\alpha _{ij}}),({s_{{i_{ij}}}},{\beta _{ij}}),({s_{{f_{ij}}}},{\chi _{ij}})\}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) means the average 2TLNNs values of alternative with respect to attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

Step 2.

Obtain the attribute’s fuzzy weighting vector

W^{t}

${W^{t}}$

which is given by each expert with respect to all attributes and compute the average fuzzy weighting vector W as follows:

\begin{array}{l} W^{t} = {[w_{j}^{t}]}_{1 \times n} = {{(s_{t_{j}}, α_{j})}^{t}, {(s_{i_{j}}, β_{j})}^{t}, {(s_{f_{j}}, χ_{j})}^{t}}_{1 \times n}, \end{array}

\[ {W^{t}}={\big[{w_{j}^{t}}\big]_{1\times n}}={\big\{{({s_{{t_{j}}}},{\alpha _{j}})^{t}},{({s_{{i_{j}}}},{\beta _{j}})^{t}},{({s_{{f_{j}}}},{\chi _{j}})^{t}}\big\}_{1\times n}},\]

(30)

\begin{array}{l} W = {[w_{j}]}_{1 \times n} = {(s_{t_{j}}, α_{j}), (s_{i_{j}}, β_{j}), (s_{f_{j}}, χ_{j})}_{1 \times n} . \end{array}

\[ W={[{w_{j}}]_{1\times n}}={\big\{({s_{{t_{j}}}},{\alpha _{j}}),({s_{{i_{j}}}},{\beta _{j}}),({s_{{f_{j}}}},{\chi _{j}})\big\}_{1\times n}}.\]

(31)

Based on the operation rules of 2TLNNs, the

W = {[w_{j}]}_{1 \times n}

$ W={[{w_{j}}]_{1\times n}}$

can be calculated as:

\begin{array}{rcl} w_{j} & = & a_{1} w_{j}^{1} \oplus a_{2} w_{j}^{2} \oplus \dots \oplus a_{t} w_{j}^{t} \\ = & \{\begin{array}{l} Δ (k (1 - \prod_{d = 1}^{t} {(1 - \frac{Δ^{- 1} {(s_{t_{j}}, α_{j})}^{t}}{k})}^{a_{t}})), \\ Δ (k \prod_{d = 1}^{t} {(\frac{Δ^{- 1} {(s_{i_{j}}, β_{j})}^{t}}{k})}^{a_{t}}), Δ (k \prod_{d = 1}^{t} {(\frac{Δ^{- 1} {(s_{f_{j}}, χ_{j})}^{t}}{k})}^{a_{t}}) \end{array}\}, \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {w_{j}}& \displaystyle =& \displaystyle {a_{1}}{w_{j}^{1}}\oplus {a_{2}}{w_{j}^{2}}\oplus \cdots \oplus {a_{t}}{w_{j}^{t}}\\ {} & \displaystyle =& \displaystyle \left\{\begin{array}{l}\Delta \Big(k\Big(1-{\textstyle\textstyle\prod _{d=1}^{t}}{\Big(1-\frac{{\Delta ^{-1}}{({s_{{t_{j}}}},{\alpha _{j}})^{t}}}{k}\Big)^{{a_{t}}}}\Big)\Big),\\ {} \Delta \Big(k{\textstyle\textstyle\prod _{d=1}^{t}}{\Big(\frac{{\Delta ^{-1}}{({s_{{i_{j}}}},{\beta _{j}})^{t}}}{k}\Big)^{{a_{t}}}}\Big),\Delta \Big(k{\textstyle\textstyle\prod _{d=1}^{t}}{\Big(\frac{{\Delta ^{-1}}{({s_{{f_{j}}}},{\chi _{j}})^{t}}}{k}\Big)^{{a_{t}}}}\Big)\end{array}\right\},\end{array}\]

(32)

where

w_{j}^{t} = w_{j} = {{(s_{t_{j}}, α_{j})}^{t}, {(s_{i_{j}}, β_{j})}^{t}, {(s_{f_{j}}, χ_{j})}^{t}}

${w_{j}^{t}}={w_{j}}=\{{({s_{{t_{j}}}},{\alpha _{j}})^{t}},{({s_{{i_{j}}}},{\beta _{j}})^{t}},{({s_{{f_{j}}}},{\chi _{j}})^{t}}\}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

denotes the fuzzy weight of attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

by expert

d^{t}

${d^{t}}$

and

w_{j} = {(s_{t_{j}}, α_{j}), (s_{i_{j}}, β_{j}), (s_{f_{j}}, χ_{j})}

${w_{j}}=\{({s_{{t_{j}}}},{\alpha _{j}}),({s_{{i_{j}}}},{\beta _{j}}),({s_{{f_{j}}}},{\chi _{j}})\}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

means the average fuzzy weight values of attribute

O_{j}

${O_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

Step 3.

Normalize the average results matrix

r = {[ϕ_{i j}]}_{m \times n}

$ r={[{\phi _{ij}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

based on the type of each attributes using the following formulae.

For benefit attributes:

\begin{array}{rcl} N_{i j} & = & ϕ_{i j} = {{(s_{t_{i j}}, α_{i j})}^{'}, {(s_{i_{i j}}, β_{i j})}^{'}, {(s_{f_{i j}}, χ_{i j})}^{'}} \\ = & {(s_{t_{i j}}, α_{i j}), (s_{i_{i j}}, β_{i j}), (s_{f_{i j}}, χ_{i j})}, i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {N_{ij}}& \displaystyle =& \displaystyle {\phi _{ij}}=\big\{{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime }},{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime }},{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime }}\big\}\\ {} & \displaystyle =& \displaystyle \big\{({s_{{t_{ij}}}},{\alpha _{ij}}),({s_{{i_{ij}}}},{\beta _{ij}}),({s_{{f_{ij}}}},{\chi _{ij}})\big\},\hspace{1em}i=1,2,\dots ,m,\hspace{2.5pt}j=1,2,\dots ,n.\end{array}\]

(33)

For cost attributes:

\begin{array}{l} N_{i j} = k - ϕ_{i j} = \{\begin{array}{c} {(s_{t_{i j}}, α_{i j})}^{'}, \\ {(s_{i_{i j}}, β_{i j})}^{'}, \\ {(s_{f_{i j}}, χ_{i j})}^{'} \end{array}\} = \{\begin{array}{c} Δ (k - Δ^{- 1} (s_{t_{i j}}, α_{i j})), \\ Δ (k - Δ^{- 1} (s_{i_{i j}}, β_{i j})), \\ Δ (k - Δ^{- 1} (s_{f_{i j}}, χ_{i j})), \end{array}\}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{array}

\[\begin{array}{l}\displaystyle {N_{ij}}=k-{\phi _{ij}}=\left\{\begin{array}{c}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime }},\\ {} {({s_{{i_{ij}}}},{\beta _{ij}})^{\prime }},\\ {} {({s_{{f_{ij}}}},{\chi _{ij}})^{\prime }}\end{array}\right\}=\left\{\begin{array}{c}\Delta (k-{\Delta ^{-1}}({s_{{t_{ij}}}},{\alpha _{ij}})),\\ {} \Delta (k-{\Delta ^{-1}}({s_{{i_{ij}}}},{\beta _{ij}})),\\ {} \Delta (k-{\Delta ^{-1}}({s_{{f_{ij}}}},{\chi _{ij}})),\end{array}\right\},\\ {} \displaystyle \hspace{1em}i=1,2,\dots ,m,j=1,2,\dots ,n.\end{array}\]

(34)

Step 4.

According to the normalized average matrix

N_{i j} = {{(s_{t_{i j}}, α_{i j})}^{'}, {(s_{i_{i j}}, α_{i j})}^{'}, {(s_{f_{i j}}, α_{i j})}^{'}}

${N_{ij}}=\{{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime }},{({s_{{i_{ij}}}},{\alpha _{ij}})^{\prime }},{({s_{{f_{ij}}}},{\alpha _{ij}})^{\prime }}\}$

and average fuzzy weighting vector

w_{j} = {(s_{t_{j}}, α_{j}), (s_{i_{j}}, β_{j}), (s_{f_{j}}, χ_{j})}

${w_{j}}=\{({s_{{t_{j}}}},{\alpha _{j}}),({s_{{i_{j}}}},{\beta _{j}}),({s_{{f_{j}}}},{\chi _{j}})\}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

the fuzzy weighted normalized average matrix

{WN}_{i j} = {{(s_{t_{i j}}, α_{i j})}^{″}, {(s_{i_{i j}}, β_{i j})}^{″}, {(s_{f_{i j}}, χ_{i j})}^{″}}

${\mathit{WN}_{ij}}=\{{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }},{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }},{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}\}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) can be computed as:

\begin{array}{rcl} W N_{i j} & = & w_{j} \otimes N_{i j} = {{(s_{t_{i j}}, α_{i j})}^{″}, {(s_{i_{i j}}, β_{i j})}^{″}, {(s_{f_{i j}}, χ_{i j})}^{″}} \\ = & \{\begin{array}{l} Δ (k (\frac{Δ^{- 1} {(s_{t_{i j}}, α_{i j})}^{'}}{k} \cdot \frac{Δ^{- 1} (s_{t_{j}}, α_{j})}{k})), \\ Δ (1 - (1 - \frac{Δ^{- 1} {(s_{i_{i j}}, β_{i j})}^{'}}{k}) \cdot (1 - \frac{Δ^{- 1} (s_{i_{j}}, β_{j})}{k})), \\ Δ (1 - (1 - \frac{Δ^{- 1} {(s_{f_{i j}}, χ_{i j})}^{'}}{k}) \cdot (1 - \frac{Δ^{- 1} (s_{f_{j}}, χ_{j})}{k})) \end{array}\}, \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle W{N_{ij}}& \displaystyle =& \displaystyle {w_{j}}\otimes {N_{ij}}=\big\{{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }},{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }},{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}\big\}\\ {} & \displaystyle =& \displaystyle \left\{\begin{array}{l}\Delta \Big(k\Big(\frac{{\Delta ^{-1}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime }}}{k}\cdot \frac{{\Delta ^{-1}}({s_{{t_{j}}}},{\alpha _{j}})}{k}\Big)\Big),\\ {} \Delta \Big(1-\Big(1-\frac{{\Delta ^{-1}}{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime }}}{k}\Big)\cdot \Big(1-\frac{{\Delta ^{-1}}({s_{{i_{j}}}},{\beta _{j}})}{k}\Big)\Big),\\ {} \Delta \Big(1-\Big(1-\frac{{\Delta ^{-1}}{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime }}}{k}\Big)\cdot \Big(1-\frac{{\Delta ^{-1}}({s_{{f_{j}}}},{\chi _{j}})}{k}\Big)\Big)\end{array}\right\},\end{array}\]

(35)

where

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

Step 5.

Determine the fuzzy negative solution (NS) based on the equation (20)

\begin{array}{l} {NS}_{j} = min_{i} (W N_{i j}) = {{min}_{i} {(s_{t_{i j}}, α_{i j})}^{″}, {max}_{i} {(s_{i_{i j}}, β_{i j})}^{″}, {max}_{i} {(s_{f_{i j}}, χ_{i j})}^{″}}, \end{array}

\[ {\mathit{NS}_{j}}=\underset{i}{\min }(W{N_{ij}})=\big\{{\min _{i}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }},{\max _{i}}{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }},{\max _{i}}{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}\big\},\]

(36)

where

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

Step 6.

Calculate the fuzzy weighted Hamming distance and fuzzy weighted Euclidean distance between each alternatives and the negative solution (NS) according to Definition 4 and Definition 5:

\begin{array}{rcl} {HD}_{i} & = & \sum_{j = 1}^{n} d^{H} ({WN}_{i j}, {NS}_{j}) \\ = & \sum_{j = 1}^{n} (\frac{1}{3} \{\begin{array}{l} | \frac{Δ^{- 1} {(s_{t_{i j}}, α_{i j})}^{″} - {min}_{i} Δ^{- 1} {(s_{t_{i j}}, α_{i j})}^{″}}{k} | \\ + | \frac{Δ^{- 1} {(s_{i_{i j}}, β_{i j})}^{″} - {max}_{i} Δ^{- 1} {(s_{i_{i j}}, β_{i j})}^{″}}{k} | \\ + | \frac{Δ^{- 1} {(s_{f_{i j}}, χ_{i j})}^{″} - {max}_{i} Δ^{- 1} {(s_{f_{i j}}, χ_{i j})}^{″}}{k} | \end{array}\}), \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\mathit{HD}_{i}}& \displaystyle =& \displaystyle {\sum \limits_{j=1}^{n}}{d^{H}}({\mathit{WN}_{ij}},{\mathit{NS}_{j}})\\ {} & \displaystyle =& \displaystyle {\sum \limits_{j=1}^{n}}\left(\frac{1}{3}\left\{\begin{array}{l}\Big|\frac{{\Delta ^{-1}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }}-{\min _{i}}{\Delta ^{-1}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }}}{k}\Big|\\ {} +\Big|\frac{{\Delta ^{-1}}{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }}-{\max _{i}}{\Delta ^{-1}}{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }}}{k}\Big|\\ {} +\Big|\frac{{\Delta ^{-1}}{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}-{\max _{i}}{\Delta ^{-1}}{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}}{k}\Big|\end{array}\right\}\right),\end{array}\]

(37)

\begin{array}{rcl} {ED}_{i} & = & \sum_{j = 1}^{n} d^{E} ({WN}_{i j}, {NS}_{j}) \\ = & \sqrt{\sum_{j = 1}^{n} (\frac{1}{3} \{\begin{array}{l} {| \frac{Δ^{- 1} {(s_{t_{i j}}, α_{i j})}^{″} - {min}_{i} Δ^{- 1} {(s_{t_{i j}}, α_{i j})}^{″}}{k} |}^{2} \\ + {| \frac{Δ^{- 1} {(s_{i_{i j}}, β_{i j})}^{″} - {max}_{i} Δ^{- 1} {(s_{i_{i j}}, β_{i j})}^{″}}{k} |}^{2} \\ + {| \frac{Δ^{- 1} {(s_{f_{i j}}, χ_{i j})}^{″} - {max}_{i} Δ^{- 1} {(s_{f_{i j}}, χ_{i j})}^{″}}{k} |}^{2} \end{array}\})} . \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\mathit{ED}_{i}}& \displaystyle =& \displaystyle {\sum \limits_{j=1}^{n}}{d^{E}}({\mathit{WN}_{ij}},{\mathit{NS}_{j}})\\ {} & \displaystyle =& \displaystyle \sqrt{{\sum \limits_{j=1}^{n}}\left(\frac{1}{3}\left\{\begin{array}{l}{\Big|\frac{{\Delta ^{-1}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }}-{\min _{i}}{\Delta ^{-1}}{({s_{{t_{ij}}}},{\alpha _{ij}})^{\prime\prime }}}{k}\Big|^{2}}\\ {} +{\Big|\frac{{\Delta ^{-1}}{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }}-{\max _{i}}{\Delta ^{-1}}{({s_{{i_{ij}}}},{\beta _{ij}})^{\prime\prime }}}{k}\Big|^{2}}\\ {} +{\Big|\frac{{\Delta ^{-1}}{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}-{\max _{i}}{\Delta ^{-1}}{({s_{{f_{ij}}}},{\chi _{ij}})^{\prime\prime }}}{k}\Big|^{2}}\end{array}\right\}\right)}.\end{array}\]

(38)

Step 7.

Determine the relative assessment (RA) matrix which is presented as follows.

\begin{array}{l} RA = {[p_{i l}]}_{m \times m}, \end{array}

\[ \mathit{RA}={[{p_{il}}]_{m\times m}},\]

(39)

\begin{array}{l} p_{i l} = ({ED}_{i} - {ED}_{l}) + (λ ({ED}_{i} - {ED}_{l}) \times ({HD}_{i} - {HD}_{l})), \end{array}

\[ {p_{il}}=({\mathit{ED}_{i}}-{\mathit{ED}_{l}})+\big(\lambda ({\mathit{ED}_{i}}-{\mathit{ED}_{l}})\times ({\mathit{HD}_{i}}-{\mathit{HD}_{l}})\big),\]

(40)

where

i, l = 1, 2, \dots, m

$ i,l=1,2,\dots ,m$

and λ is a threshold function which can be defined:

\begin{matrix} λ (x) = \{\begin{array}{ll} 1 & if | x | ⩾ θ, \\ 0 & if | x | < θ . \end{array} \end{matrix}

\[ \lambda (x)=\left\{\begin{array}{l@{\hskip4.0pt}l}1\hspace{1em}& \text{if}\hspace{5pt}|x|\geqslant \theta ,\\ {} 0\hspace{1em}& \text{if}\hspace{5pt}|x|<\theta .\end{array}\right.\]

(41)

The threshold parameter θ of this function can be set by the decision maker. In our paper, we let

θ = 0.02

$ \theta =0.02$

for the calculations.

Step 8.

Compute the values of assessment score (AS) based on each alternative’s

p_{i l}

${p_{il}}$

using the following equation:

\begin{matrix} A S_{i} = \sum_{l = 1}^{m} p_{i l} . \end{matrix}

\[ A{S_{i}}={\sum \nolimits_{l=1}^{m}}{p_{il}}.\]

(42)

Step 9.

According to the calculation results of

{AS}_{i}

${\mathit{AS}_{i}}$

, we can rank all the alternatives. The bigger the value of

{AS}_{i}

${\mathit{AS}_{i}}$

is, the better alternative will be selected.

Thus, the decision making model can be described as: Step 1.

Construct the 2-tuple linguistic neutrosophic evaluation matrix and calculate the average results matrix by using the equation (29);

Step 2.

Obtain the attribute’s fuzzy weighting vector

W^{t}

${W^{t}}$

and compute the average fuzzy weighting vector W by using the equation (32);

Step 3.

Normalize the average results matrix based on the type of each attributes by using the equations (33) and (34);

Step 4.

According to the normalized average matrix and average fuzzy weighting vector, compute the fuzzy weighted normalized average matrix

{WN}_{i j}

${\mathit{WN}_{ij}}$

by using the equation (35);

Step 5.

Determine the fuzzy negative solution (NS) by using the equation (36);

Step 6.

Calculate the fuzzy weighted Hamming distance and the fuzzy weighted Euclidean distance between each alternatives and the NS by using the equations (37) and (38);

Step 7.

Determine the relative assessment (RA) matrix by using the equation (40);

Step 8.

Compute the values of assessment score (AS) by using the equation (42);

Step 9.

According to the calculation results of

A S_{i}

$ A{S_{i}}$

, rank all the alternatives.

6 The Numerical Example

6.1 Numerical for 2TLNNs MAGDM Problems

The safety assessment of a construction project could be considered an MAGDM issue (Tang and Wei, 2019a; Tang et al., 2019; Wang et al., 2019f, 2019g). With the gradual progress of urbanization in China, the number of construction projects under development has increased. Thus, it is very important to evaluate the safety of construction projects. In this section, we provide a numerical example to select the best construction projects by using the CODAS model with 2-tuple linguistic neutrosophic information. In order to choose a suitable construction scheme for construction, assume there are five possible construction projects

ϕ_{i}

${\phi _{i}}$

(i = 1, 2, 3, 4, 5)

$(i=1,2,3,4,5)$

, which are provided by five famous construction companies with different construction advantages. In order to select the best construction project, invite some experts with experience of construction engineering and fuzzy set theory to construct the evaluation system in order to assess these construction projects. The evaluation index includes: (1)

O_{1}

${O_{1}}$

is the human factor in construction projects; (2)

O_{2}

${O_{2}}$

is the energy cost factor; (3)

O_{3}

${O_{3}}$

is the building materials and equipment factor; (4)

O_{4}

${O_{4}}$

is the environmental factor. The five possible construction projects

ϕ_{i}

${\phi _{i}}$

(i = 1, 2, 3, 4, 5)

$(i=1,2,3,4,5)$

are to be evaluated with 2TLNNs according to the four criteria by three experts

d^{t}

${d^{t}}$

(according to the professional years and the degree of authority of the expert, the weight of the expert is determined as

(0.3, 0.4, 0.3)

$(0.3,0.4,0.3)$

Step 1.

Construct the 2-tuple linguistic neutrosophic evaluation matrix

R = {[ϕ_{i j}^{t}]}_{m \times n}

$ R={[{\phi _{ij}^{t}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

. Then according to equation (29) and expert’s weights, we can obtain the average results matrix

r = {[ϕ_{i j}]}_{m \times n}

$ r={[{\phi _{ij}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

as follows. (Take

ϕ_{11}

${\phi _{11}}$

for example.)

According to Table 1, we can derive

ϕ_{11}^{1} = {(s_{4}, 0), (s_{3}, 0), (s_{2}, 0)}

${\phi _{11}^{1}}=\{({s_{4}},0),({s_{3}},0),({s_{2}},0)\}$

ϕ_{11}^{2} = {(s_{5}, 0), (s_{2}, 0), (s_{1}, 0)}

${\phi _{11}^{2}}=\{({s_{5}},0),({s_{2}},0),({s_{1}},0)\}$

ϕ_{11}^{3} = {(s_{4}, 0), (s_{3}, 0), (s_{2}, 0)}

${\phi _{11}^{3}}=\{({s_{4}},0),({s_{3}},0),({s_{2}},0)\}$

, then we can get (Tables 2, 3, 4)

\begin{array}{rcl} ϕ_{11} & = & a_{1} ϕ_{11}^{1} \oplus a_{2} ϕ_{11}^{2} \oplus a_{3} ϕ_{11}^{3} \\ = & \{\begin{array}{l} Δ (6 (1 - {(1 - \frac{4}{6})}^{0.3} \times {(1 - \frac{5}{6})}^{0.4} \times {(1 - \frac{4}{6})}^{0.3})), \\ Δ (6 {(\frac{3}{6})}^{0.3} \times {(\frac{2}{6})}^{0.4} \times {(\frac{3}{6})}^{0.3}), Δ (6 {(\frac{2}{6})}^{0.3} \times {(\frac{1}{6})}^{0.4} \times {(\frac{2}{6})}^{0.3}) \end{array}\} \\ = & {(s_{4}, 0.4843), (s_{3}, - 0.4492), (s_{2}, - 0.4843)} . \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\phi _{11}}& \displaystyle =& \displaystyle {a_{1}}{\phi _{11}^{1}}\oplus {a_{2}}{\phi _{11}^{2}}\oplus {a_{3}}{\phi _{11}^{3}}\\ {} & \displaystyle =& \displaystyle \left\{\begin{array}{l}\Delta \big(6\big(1-{\big(1-\frac{4}{6}\big)^{0.3}}\times {\big(1-\frac{5}{6}\big)^{0.4}}\times {\big(1-\frac{4}{6}\big)^{0.3}}\big)\big),\\ {} \Delta \big(6{\big(\frac{3}{6}\big)^{0.3}}\times {\big(\frac{2}{6}\big)^{0.4}}\times {\big(\frac{3}{6}\big)^{0.3}}\big),\Delta \big(6{\big(\frac{2}{6}\big)^{0.3}}\times {\big(\frac{1}{6}\big)^{0.4}}\times {\big(\frac{2}{6}\big)^{0.3}}\big)\end{array}\right\}\\ {} & \displaystyle =& \displaystyle \big\{({s_{4}},0.4843),({s_{3}},-0.4492),({s_{2}},-0.4843)\big\}.\end{array}\]

Similarly, we can obtain other average results and all the average results are listed in Table 5.

Table 1
Linguistic scale.

Linguistic terms	2-tuple linguistic neutrosophic numbers
Exceedingly Terrible – ET	${(s_{0}, 0), (s_{5}, 0), (s_{6}, 0)}$ $ \{({s_{0}},0),({s_{5}},0),({s_{6}},0)\}$
Very Terrible – VT	${(s_{1}, 0), (s_{4}, 0), (s_{5}, 0)}$ $ \{({s_{1}},0),({s_{4}},0),({s_{5}},0)\}$
Terrible – T	${(s_{2}, 0), (s_{3}, 0), (s_{4}, 0)}$ $ \{({s_{2}},0),({s_{3}},0),({s_{4}},0)\}$
Medium – M	${(s_{3}, 0), (s_{3}, 0), (s_{3}, 0)}$ $ \{({s_{3}},0),({s_{3}},0),({s_{3}},0)\}$
Well – W	${(s_{4}, 0), (s_{3}, 0), (s_{2}, 0)}$ $ \{({s_{4}},0),({s_{3}},0),({s_{2}},0)\}$
Very Well – VW	${(s_{5}, 0), (s_{2}, 0), (s_{1}, 0)}$ $ \{({s_{5}},0),({s_{2}},0),({s_{1}},0)\}$
Exceedingly Well – EW	${(s_{6}, 0), (s_{1}, 0), (s_{0}, 0)}$ $ \{({s_{6}},0),({s_{1}},0),({s_{0}},0)\}$

Table 2

Evaluation information by

d^{1}

${d^{1}}$

	$O_{1}$ ${O_{1}}$ (benefit)	$O_{2}$ ${O_{2}}$ (cost)	$O_{3}$ ${O_{3}}$ (benefit)	$O_{4}$ ${O_{4}}$ (benefit)
$ϕ_{1}$ ${\phi _{1}}$	W	M	VT	VW
$ϕ_{2}$ ${\phi _{2}}$	M	T	VW	W
$ϕ_{3}$ ${\phi _{3}}$	W	VW	VT	M
$ϕ_{4}$ ${\phi _{4}}$	VT	VW	M	VT
$ϕ_{5}$ ${\phi _{5}}$	VW	VT	M	T

Table 3

Evaluation information by

d^{2}

${d^{2}}$

	$O_{1}$ ${O_{1}}$ (benefit)	$O_{2}$ ${O_{2}}$ (cost)	$O_{3}$ ${O_{3}}$ (benefit)	$O_{4}$ ${O_{4}}$ (benefit)
$ϕ_{1}$ ${\phi _{1}}$	VW	W	VT	M
$ϕ_{2}$ ${\phi _{2}}$	M	T	VW	W
$ϕ_{3}$ ${\phi _{3}}$	VT	M	VW	T
$ϕ_{4}$ ${\phi _{4}}$	T	W	M	VW
$ϕ_{5}$ ${\phi _{5}}$	M	W	VT	VW

Table 4

Evaluation information by

d^{3}

${d^{3}}$

	$O_{1}$ ${O_{1}}$ (benefit)	$O_{2}$ ${O_{2}}$ (cost)	$O_{3}$ ${O_{3}}$ (benefit)	$O_{4}$ ${O_{4}}$ (benefit)
$ϕ_{1}$ ${\phi _{1}}$	W	VW	T	M
$ϕ_{2}$ ${\phi _{2}}$	VT	VT	M	W
$ϕ_{3}$ ${\phi _{3}}$	M	T	VW	W
$ϕ_{4}$ ${\phi _{4}}$	W	VW	VT	T
$ϕ_{5}$ ${\phi _{5}}$	W	VW	VT	T

Table 5

The average results matrix.

	$O_{1}$ ${O_{1}}$	$O_{2}$ ${O_{2}}$
$ϕ_{1}$ ${\phi _{1}}$	${(s_{4}, 0.4843), (s_{3}, - 0.4492), (s_{2}, - 0.4843)}$ $ \{({s_{4}},\hspace{0.2778em}0.4843),({s_{3}},-0.4492),({s_{2}},-0.4843)\}$	${(s_{4}, 0.1654), (s_{3}, - 0.3436), (s_{2}, - 0.1654)}$ $ \{({s_{4}},\hspace{0.2778em}0.1654),({s_{3}},-0.3436),({s_{2}},-0.1654)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{3}, - 0.4968), (s_{3}, 0.2704), (s_{3}, 0.4968)}$ $ \{({s_{3}},\hspace{0.2778em}-0.4968),({s_{3}},0.2704),({s_{3}},0.4968)\}$	${(s_{2}, - 0.2769), (s_{3}, 0.2704), (s_{4}, 0.2769)}$ $ \{({s_{2}},\hspace{0.2778em}-0.2769),({s_{3}},0.2704),({s_{4}},0.2769)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{3}, - 0.2586), (s_{3}, 0.3659), (s_{3}, 0.2586)}$ $ \{({s_{3}},\hspace{0.2778em}-0.2586),({s_{3}},0.3659),({s_{3}},0.2586)\}$	${(s_{4}, - 0.3522), (s_{3}, - 0.3436), (s_{2}, 0.3522)}$ $ \{({s_{4}},\hspace{0.2778em}-0.3522),({s_{3}},-0.3436),({s_{2}},0.3522)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{3}, - 0.4740), (s_{3}, 0.2704), (s_{3}, 0.4740)}$ $ \{({s_{3}},\hspace{0.2778em}-0.4740),({s_{3}},0.2704),({s_{3}},0.4740)\}$	${(s_{5}, - 0.3195), (s_{2}, 0.3522), (s_{1}, 0.3195)}$ $ \{({s_{5}},\hspace{0.2778em}-0.3195),({s_{2}},0.3522),({s_{1}},0.3195)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{4}, - 0.1577), (s_{3}, - 0.3436), (s_{2}, 0.1577)}$ $ \{({s_{4}},\hspace{0.2778em}-0.1577),({s_{3}},-0.3436),({s_{2}},0.1577)\}$	${(s_{3}, 0.3672), (s_{3}, 0.2704), (s_{3}, - 0.3672)}$ $ \{({s_{3}},\hspace{0.2778em}0.3672),({s_{3}},0.2704),({s_{3}},-0.3672)\}$
	$O_{3}$ ${O_{3}}$	$O_{4}$ ${O_{4}}$
$ϕ_{1}$ ${\phi _{1}}$	${(s_{1}, 0.3238), (s_{4}, - 0.3307), (s_{5}, - 0.3238)}$ $ \{({s_{1}},\hspace{0.2778em}0.3238),({s_{4}},-0.3307),({s_{5}},-0.3238)\}$	${(s_{4}, - 0.1577), (s_{3}, - 0.3436), (s_{2}, 0.1577)}$ $ \{({s_{4}},\hspace{0.2778em}-0.1577),({s_{3}},-0.3436),({s_{2}},0.1577)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{5}, - 0.3904), (s_{2}, 0.2587), (s_{1}, 0.3904)}$ $ \{({s_{5}},\hspace{0.2778em}-0.3904),({s_{2}},0.2587),({s_{1}},0.3904)\}$	${(s_{4}, 0.0000), (s_{3}, 0.0000), (s_{2}, 0.0000)}$ $ \{({s_{4}},\hspace{0.2778em}\hspace{0.2778em}0.0000),({s_{3}},\hspace{0.2778em}\hspace{0.2778em}0.0000),({s_{2}},\hspace{0.2778em}\hspace{0.2778em}0.0000)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{4}, 0.3793), (s_{2}, 0.4623), (s_{2}, - 0.3793)}$ $ \{({s_{4}},\hspace{0.2778em}0.3793),({s_{2}},0.4623),({s_{2}},-0.3793)\}$	${(s_{3}, 0.0199), (s_{3}, 0.0000), (s_{3}, - 0.0199)}$ $ \{({s_{3}},\hspace{0.2778em}0.0199),({s_{3}},0.0000),({s_{3}},-0.0199)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{3}, - 0.4968), (s_{3}, 0.2704), (s_{3}, 0.4968)}$ $ \{({s_{3}},\hspace{0.2778em}-0.4968),({s_{3}},0.2704),({s_{3}},0.4968)\}$	${(s_{4}, - 0.4565), (s_{3}, - 0.2192), (s_{2}, 0.4565)}$ $ \{({s_{4}},\hspace{0.2778em}-0.4565),({s_{3}},-0.2192),({s_{2}},0.4565)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{2}, - 0.0118), (s_{3}, 0.3659), (s_{4}, 0.0118)}$ $ \{({s_{2}},\hspace{0.2778em}-0.0118),({s_{3}},0.3659),({s_{4}},0.0118)\}$	${(s_{4}, - 0.1074), (s_{3}, - 0.4492), (s_{2}, 0.1074)}$ $ \{({s_{4}},\hspace{0.2778em}-0.1074),({s_{3}},-0.4492),({s_{2}},0.1074)\}$

Step 2.

Obtain the attribute’s fuzzy weighting vector

W^{t}

${W^{t}}$

which is given by each expert with respect to all attributes and compute the average fuzzy weighting vector W as follows (Table 6). According to expert’s weighting vector and the equation (32), the attribute’s weighting vector can be calculated as (take

W_{1}

${W_{1}}$

for example).

Table 6

Attribute’s weighting vector given by decision-maker.

	$O_{1}$ ${O_{1}}$ (benefit)	$O_{2}$ ${O_{2}}$ (cost)	$O_{3}$ ${O_{3}}$ (benefit)	$O_{4}$ ${O_{4}}$ (benefit)
$d^{1}$ ${d^{1}}$	VT	W	VW	T
$d^{2}$ ${d^{2}}$	M	VT	W	M
$d^{3}$ ${d^{3}}$	VW	W	M	VT

Accordingto Table 1, we can derive

W^{1} = {(s_{1}, 0), (s_{4}, 0), (s_{5}, 0)}

${W^{1}}=\{({s_{1}},0),({s_{4}},0),({s_{5}},0)\}$

W^{2} = {(s_{3}, 0), (s_{3}, 0), (s_{3}, 0)}

${W^{2}}=\{({s_{3}},0),({s_{3}},0),({s_{3}},0)\}$

W^{3} = {(s_{5}, 0), (s_{2}, 0), (s_{1}, 0)}

${W^{3}}=\{({s_{5}},0),({s_{2}},0),({s_{1}},0)\}$

, then we can get

\begin{array}{rcl} W_{1} & = & a_{1} W_{}^{1} \oplus a_{2} W_{}^{2} \oplus a_{3} W_{}^{3} \\ = & \{\begin{array}{l} Δ (6 (1 - {(1 - \frac{1}{6})}^{0.3} \times {(1 - \frac{3}{6})}^{0.4} \times {(1 - \frac{5}{6})}^{0.3})), \\ Δ (6 {(\frac{4}{6})}^{0.3} \times {(\frac{3}{6})}^{0.4} \times {(\frac{2}{6})}^{0.3}), Δ (6 {(\frac{5}{6})}^{0.3} \times {(\frac{3}{6})}^{0.4} \times {(\frac{1}{6})}^{0.3}) \end{array}\} \\ = & {(s_{3}, 0.4850), (s_{3}, - 0.1042), (s_{3}, 0.4850)} . \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {W_{1}}& \displaystyle =& \displaystyle {a_{1}}{W_{}^{1}}\oplus {a_{2}}{W_{}^{2}}\oplus {a_{3}}{W_{}^{3}}\\ {} & \displaystyle =& \displaystyle \left\{\begin{array}{l}\Delta \big(6\big(1-{\big(1-\frac{1}{6}\big)^{0.3}}\times {\big(1-\frac{3}{6}\big)^{0.4}}\times {\big(1-\frac{5}{6}\big)^{0.3}}\big)\big),\\ {} \Delta \big(6{\big(\frac{4}{6}\big)^{0.3}}\times {\big(\frac{3}{6}\big)^{0.4}}\times {\big(\frac{2}{6}\big)^{0.3}}\big),\Delta \big(6{\big(\frac{5}{6}\big)^{0.3}}\times {\big(\frac{3}{6}\big)^{0.4}}\times {\big(\frac{1}{6}\big)^{0.3}}\big)\end{array}\right\}\\ {} & \displaystyle =& \displaystyle \big\{({s_{3}},\hspace{0.2778em}0.4850),({s_{3}},-0.1042),({s_{3}},0.4850)\big\}.\end{array}\]

So the attribute weights are derived as:

\begin{matrix} W = \{\begin{array}{l} {(s_{3}, 0.4850), (s_{3}, - 0.1042), (s_{3}, 0.4850)}, \\ {(s_{3}, 0.1146), (s_{3}, 0.3659), (s_{3}, - 0.1146)}, \\ {(s_{4}, 0.1654), (s_{3}, - 0.3436), (s_{2}, - 0.1654)}, \\ {(s_{2}, 0.1880), (s_{3}, 0.2704), (s_{4}, - 0.1880)} . \end{array}\} . \end{matrix}

\[ W=\left\{\begin{array}{l}\{({s_{3}},0.4850),({s_{3}},-0.1042),({s_{3}},0.4850)\},\\ {} \{({s_{3}},0.1146),({s_{3}},0.3659),({s_{3}},-0.1146)\},\\ {} \{({s_{4}},0.1654),({s_{3}},-0.3436),({s_{2}},-0.1654)\},\\ {} \{({s_{2}},0.1880),({s_{3}},0.2704),({s_{4}},-0.1880)\}.\end{array}\right\}.\]

Step 3.

Normalize the average results matrix

r = {[ϕ_{i j}]}_{m \times n}

$ r={[{\phi _{ij}}]_{m\times n}}$

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

based on the type of each attributes by formulae (33) and (34).

Table 7

The normalized decision-making matrix

N_{i j}

${N_{ij}}$

	$O_{1}$ ${O_{1}}$	$O_{2}$ ${O_{2}}$
$ϕ_{1}$ ${\phi _{1}}$	${(s_{4}, 0.4843), (s_{3}, - 0.4492), (s_{2}, - 0.4843)}$ $ \{({s_{4}},0.4843),({s_{3}},-0.4492),({s_{2}},-0.4843)\}$	${(s_{2}, - 0.1654), (s_{3}, 0.3436), (s_{4}, 0.1654)}$ $ \{({s_{2}},-0.1654),({s_{3}},0.3436),({s_{4}},0.1654)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{3}, - 0.4968), (s_{3}, 0.2704), (s_{3}, 0.4968)}$ $ \{({s_{3}},-0.4968),({s_{3}},0.2704),({s_{3}},0.4968)\}$	${(s_{4}, 0.2769), (s_{3}, - 0.2704), (s_{2}, - 0.2769)}$ $ \{({s_{4}},0.2769),({s_{3}},-0.2704),({s_{2}},-0.2769)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{3}, - 0.2586), (s_{3}, 0.3659), (s_{3}, 0.2586)}$ $ \{({s_{3}},-0.2586),({s_{3}},0.3659),({s_{3}},0.2586)\}$	${(s_{2}, 0.3522), (s_{3}, 0.3436), (s_{4}, - 0.3522)}$ $ \{({s_{2}},0.3522),({s_{3}},0.3436),({s_{4}},-0.3522)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{3}, - 0.4740), (s_{3}, 0.2704), (s_{3}, 0.4740)}$ $ \{({s_{3}},-0.4740),({s_{3}},0.2704),({s_{3}},0.4740)\}$	${(s_{1}, 0.3195), (s_{4}, - 0.3522), (s_{5}, - 0.3195)}$ $ \{({s_{1}},0.3195),({s_{4}},-0.3522),({s_{5}},-0.3195)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{4}, - 0.1577), (s_{3}, - 0.3436), (s_{2}, 0.1577)}$ $ \{({s_{4}},-0.1577),({s_{3}},-0.3436),({s_{2}},0.1577)\}$	${(s_{3}, - 0.3672), (s_{3}, - 0.2704), (s_{3}, 0.3672)}$ $ \{({s_{3}},-0.3672),({s_{3}},-0.2704),({s_{3}},0.3672)\}$
	$O_{3}$ ${O_{3}}$	$O_{4}$ ${O_{4}}$
$ϕ_{1}$ ${\phi _{1}}$	${(s_{1}, 0.3238), (s_{4}, - 0.3307), (s_{5}, - 0.3238)}$ $ \{({s_{1}},0.3238),({s_{4}},-0.3307),({s_{5}},-0.3238)\}$	${(s_{4}, - 0.1577), (s_{3}, - 0.3436), (s_{2}, 0.1577)}$ $ \{({s_{4}},-0.1577),({s_{3}},-0.3436),({s_{2}},0.1577)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{5}, - 0.3904), (s_{2}, 0.2587), (s_{1}, 0.3904)}$ $ \{({s_{5}},-0.3904),({s_{2}},0.2587),({s_{1}},0.3904)\}$	${(s_{4}, 0.0000), (s_{3}, 0.0000), (s_{2}, 0.0000)}$ $ \{({s_{4}},\hspace{0.2778em}0.0000),({s_{3}},\hspace{0.2778em}0.0000),({s_{2}},\hspace{0.2778em}0.0000)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{4}, 0.3793), (s_{2}, 0.4623), (s_{2}, - 0.3793)}$ $ \{({s_{4}},0.3793),({s_{2}},0.4623),({s_{2}},-0.3793)\}$	${(s_{3}, 0.0199), (s_{3}, 0.0000), (s_{3}, - 0.0199)}$ $ \{({s_{3}},0.0199),({s_{3}},0.0000),({s_{3}},-0.0199)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{3}, - 0.4968), (s_{3}, 0.2704), (s_{3}, 0.4968)}$ $ \{({s_{3}},-0.4968),({s_{3}},0.2704),({s_{3}},0.4968)\}$	${(s_{4}, - 0.4565), (s_{3}, - 0.2192), (s_{2}, 0.4565)}$ $ \{({s_{4}},-0.4565),({s_{3}},-0.2192),({s_{2}},0.4565)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{2}, - 0.0118), (s_{3}, 0.3659), (s_{4}, 0.0118)}$ $ \{({s_{2}},-0.0118),({s_{3}},0.3659),({s_{4}},0.0118)\}$	${(s_{4}, - 0.1074), (s_{3}, - 0.4492), (s_{2}, 0.1074)}$ $ \{({s_{4}},-0.1074),({s_{3}},-0.4492),({s_{2}},0.1074)\}$

Step 4.

According to the normalized average matrix (

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) and average fuzzy weighting vector

w_{j}

${w_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\dots ,n)$

, the fuzzy weighted normalized average matrix

W N_{i j}

$ W{N_{ij}}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) can be computed as (take

{WN}_{11}

${\mathit{WN}_{11}}$

for example):

\begin{array}{rcl} {WN}_{11} & = & w_{j} \otimes N_{i j} \\ = & \{\begin{array}{l} Δ (6 (\frac{4.4843}{6} \cdot \frac{3.4850}{6})), Δ (1 - (1 - \frac{2.5508}{6}) \cdot (1 - \frac{2.8958}{6})), \\ Δ (1 - (1 - \frac{1.5157}{6}) \cdot (1 - \frac{3.4850}{6})) \end{array}\} \\ = & {(s_{3}, - 0.3954), (s_{4}, 0.2156), (s_{3}, 0.3954)} . \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\mathit{WN}_{11}}& \displaystyle =& \displaystyle {w_{j}}\otimes {N_{ij}}\\ {} & \displaystyle =& \displaystyle \left\{\begin{array}{l}\Delta \Big(6\Big(\frac{4.4843}{6}\cdot \frac{3.4850}{6}\Big)\Big),\Delta \Big(1-\Big(1-\frac{2.5508}{6}\Big)\cdot \Big(1-\frac{2.8958}{6}\Big)\Big),\\ {} \Delta \Big(1-\Big(1-\frac{1.5157}{6}\Big)\cdot \Big(1-\frac{3.4850}{6}\Big)\Big)\end{array}\right\}\\ {} & \displaystyle =& \displaystyle \big\{({s_{3}},-0.3954),({s_{4}},0.2156),({s_{3}},0.3954)\big\}.\end{array}\]

Thus, the weighted normalized average matrix

{WN}_{i j}

${\mathit{WN}_{ij}}$

(

i = 1, 2, \dots, m

$ i=1,2,\dots ,m$

j = 1, 2, \dots, n

$ j=1,2,\dots ,n$

) is derived as follows (Table 8).

Table 8

The fuzzy weighted normalized average matrix

N_{i j}

${N_{ij}}$

	$O_{1}$ ${O_{1}}$	$O_{2}$ ${O_{2}}$
$ϕ_{1}$ ${\phi _{1}}$	${(s_{3}, - 0.3954), (s_{4}, 0.2156), (s_{3}, 0.3954)}$ $ \{({s_{3}},-0.3954),({s_{4}},0.2156),({s_{3}},0.3954)\}$	${(s_{1}, - 0.0476), (s_{5}, - 0.1662), (s_{5}, 0.0476)}$ $ \{({s_{1}},-0.0476),({s_{5}},-0.1662),({s_{5}},0.0476)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{1}, 0.4539), (s_{5}, - 0.4122), (s_{4}, - 0.4539)}$ $ \{({s_{1}},0.4539),({s_{5}},-0.4122),({s_{4}},-0.4539)\}$	${(s_{2}, 0.2202), (s_{5}, - 0.4358), (s_{4}, - 0.2202)}$ $ \{({s_{2}},0.2202),({s_{5}},-0.4358),({s_{4}},-0.2202)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{2}, - 0.4077), (s_{5}, - 0.3628), (s_{4}, 0.4077)}$ $ \{({s_{2}},-0.4077),({s_{5}},-0.3628),({s_{4}},0.4077)\}$	${(s_{1}, 0.2210), (s_{5}, - 0.1662), (s_{5}, - 0.2210)}$ $ \{({s_{1}},0.2210),({s_{5}},-0.1662),({s_{5}},-0.2210)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{1}, 0.4672), (s_{5}, - 0.4122), (s_{5}, - 0.4672)}$ $ \{({s_{1}},0.4672),({s_{5}},-0.4122),({s_{5}},-0.4672)\}$	${(s_{1}, - 0.3150), (s_{5}, - 0.0327), (s_{5}, 0.3150)}$ $ \{({s_{1}},-0.3150),({s_{5}},-0.0327),({s_{5}},0.3150)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{2}, 0.2317), (s_{4}, 0.2702), (s_{4}, - 0.2317)}$ $ \{({s_{2}},0.2317),({s_{4}},0.2702),({s_{4}},-0.2317)\}$	${(s_{1}, 0.3667), (s_{5}, - 0.4358), (s_{5}, - 0.3667)}$ $ \{({s_{1}},0.3667),({s_{5}},-0.4358),({s_{5}},-0.3667)\}$
	$O_{3}$ ${O_{3}}$	$O_{4}$ ${O_{4}}$
$ϕ_{1}$ ${\phi _{1}}$	${(s_{1}, - 0.0810), (s_{5}, - 0.2988), (s_{5}, 0.0810)}$ $ \{({s_{1}},-0.0810),({s_{5}},-0.2988),({s_{5}},0.0810)\}$	${(s_{1}, 0.4011), (s_{4}, 0.4789), (s_{5}, - 0.4011)}$ $ \{({s_{1}},0.4011),({s_{4}},0.4789),({s_{5}},-0.4011)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{3}, 0.2001), (s_{4}, - 0.0849), (s_{3}, - 0.2001)}$ $ \{({s_{3}},0.2001),({s_{4}},-0.0849),({s_{3}},-0.2001)\}$	${(s_{1}, 0.4586), (s_{5}, - 0.3648), (s_{5}, - 0.4586)}$ $ \{({s_{1}},\hspace{0.2778em}0.4586),({s_{5}},-0.3648),({s_{5}},-0.4586)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{3}, 0.0403), (s_{4}, 0.0286), (s_{3}, - 0.0403)}$ $ \{({s_{3}},0.0403),({s_{4}},0.0286),({s_{3}},-0.0403)\}$	${(s_{1}, 0.1011), (s_{5}, - 0.3648), (s_{5}, - 0.1011)}$ $ \{({s_{1}},0.1011),({s_{5}},-0.3648),({s_{5}},-0.1011)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{2}, - 0.2622), (s_{4}, 0.4789), (s_{4}, 0.2622)}$ $ \{({s_{2}},-0.2622),({s_{4}},0.4789),({s_{4}},0.2622)\}$	${(s_{1}, 0.2922), (s_{5}, - 0.4645), (s_{5}, - 0.2922)}$ $ \{({s_{1}},0.2922),({s_{5}},-0.4645),({s_{5}},-0.2922)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{1}, 0.3802), (s_{5}, - 0.4679), (s_{5}, - 0.3802)}$ $ \{({s_{1}},0.3802),({s_{5}},-0.4679),({s_{5}},-0.3802)\}$	${(s_{1}, 0.4195), (s_{4}, 0.4309), (s_{5}, - 0.4195)}$ $ \{({s_{1}},0.4195),({s_{4}},0.4309),({s_{5}},-0.4195)\}$

Step 5.

Determine the fuzzy negative solution (NS):

\begin{matrix} {NS}_{j} = min_{i} (W N_{i j}) = \{\begin{array}{l} {(s_{1}, 0.4539), (s_{5}, - 0.3959), (s_{5}, - 0.4539)}, \\ {(s_{1}, - 0.3150), (s_{5}, - 0.0327), (s_{5}, 0.3150)}, \\ {(s_{1}, - 0.0810), (s_{5}, - 0.2988), (s_{5}, 0.0810)}, \\ {(s_{1}, 0.1011), (s_{5}, - 0.3648), (s_{5}, - 0.1011)} \end{array}\} . \end{matrix}

\[ {\mathit{NS}_{j}}=\underset{i}{\min }(W{N_{ij}})=\left\{\begin{array}{l}\{({s_{1}},0.4539),({s_{5}},-0.3959),({s_{5}},-0.4539)\},\\ {} \{({s_{1}},-0.3150),({s_{5}},-0.0327),({s_{5}},0.3150)\},\\ {} \{({s_{1}},-0.0810),({s_{5}},-0.2988),({s_{5}},0.0810)\},\\ {} \{({s_{1}},0.1011),({s_{5}},-0.3648),({s_{5}},-0.1011)\}\end{array}\right\}.\]

Step 6.

Calculate the fuzzy weighted Hamming distance

({HD}_{i})

$({\mathit{HD}_{i}})$

and fuzzy weighted Euclidean distance

({ED}_{i})

$({\mathit{ED}_{i}})$

between each alternatives and the negative solution (NS) according to Definition 4 and Definition 5.

For example:

\begin{array}{rcl} {HD}_{1} & = & \sum_{j = 1}^{n} d^{H} (W N_{i j}, N S_{j}) \\ = & (\begin{array}{l} \frac{1}{3} {| \frac{2.6046 - 1.4539}{6} | + | \frac{4.2156 - 4.6372}{6} | + | \frac{3.3954 - 4.5461}{6} |} \\ + \frac{1}{3} {| \frac{0.9524 - 0.6850}{6} | + | \frac{4.8338 - 4.9673}{6} | + | \frac{5.0476 - 5.3150}{6} |} \\ + \frac{1}{3} {| \frac{0.9190 - 0.9190}{6} | + | \frac{4.7012 - 4.7012}{6} | + | \frac{5.0810 - 5.0810}{6} |} \\ + \frac{1}{3} {| \frac{1.4011 - 1.1011}{6} | + | \frac{4.4789 - 4.6352}{6} | + | \frac{4.5989 - 4.8989}{6} |} \end{array}) \\ = & 0.2304, \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\mathit{HD}_{1}}& \displaystyle =& \displaystyle {\sum \limits_{j=1}^{n}}{d^{H}}(W{N_{ij}},N{S_{j}})\\ {} & \displaystyle =& \displaystyle \left(\begin{array}{l}\frac{1}{3}\Big\{\Big|\frac{2.6046-1.4539}{6}\Big|+\Big|\frac{4.2156-4.6372}{6}\Big|+\Big|\frac{\mathrm{3.3954}-\mathrm{4.5461}}{6}\Big|\Big\}\\ {} +\frac{1}{3}\Big\{\Big|\frac{\mathrm{0.9524}-\mathrm{0.6850}}{6}\Big|+\Big|\frac{\mathrm{4.8338}-\mathrm{4.9673}}{6}\Big|+\Big|\frac{\mathrm{5.0476}-\mathrm{5.3150}}{6}\Big|\Big\}\\ {} +\frac{1}{3}\Big\{\Big|\frac{\mathrm{0.9190}-\mathrm{0.9190}}{6}\Big|+\Big|\frac{\mathrm{4.7012}-\mathrm{4.7012}}{6}\Big|+\Big|\frac{\mathrm{5.0810}-\mathrm{5.0810}}{6}\Big|\Big\}\\ {} +\frac{1}{3}\Big\{\Big|\frac{\mathrm{1.4011}-\mathrm{1.1011}}{6}\Big|+\Big|\frac{\mathrm{4.4789}-\mathrm{4.6352}}{6}\Big|+\Big|\frac{\mathrm{4.5989}-\mathrm{4.8989}}{6}\Big|\Big\}\end{array}\right)\\ {} & \displaystyle =& \displaystyle 0.2304,\end{array}\]

\begin{array}{rcl} {ED}_{1} & = & \sum_{j = 1}^{n} d^{E} (W N_{i j}, N S_{j}) \\ = & \sqrt{(\begin{array}{l} \frac{1}{3} {{| \frac{2.6046 - 1.4539}{6} |}^{2} + {| \frac{4.2156 - 4.6372}{6} |}^{2} + {| \frac{3.3954 - 4.5461}{6} |}^{2}} \\ + \frac{1}{3} {{| \frac{0.9524 - 0.6850}{6} |}^{2} + {| \frac{4.8338 - 4.9673}{6} |}^{2} + {| \frac{5.0476 - 5.3150}{6} |}^{2}} \\ + \frac{1}{3} {{| \frac{0.9190 - 0.9190}{6} |}^{2} + {| \frac{4.7012 - 4.7012}{6} |}^{2} + {| \frac{5.0810 - 5.0810}{6} |}^{2}} \\ + \frac{1}{3} {{| \frac{1.4011 - 1.1011}{6} |}^{2} + {| \frac{4.4789 - 4.6352}{6} |}^{2} + {| \frac{4.5989 - 4.8989}{6} |}^{2}} \end{array})} \\ = & 0.2439. \end{array}

\[\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}\displaystyle {\mathit{ED}_{1}}& \displaystyle =& \displaystyle {\sum \limits_{j=1}^{n}}{d^{E}}(W{N_{ij}},N{S_{j}})\\ {} & \displaystyle =& \displaystyle \sqrt{\left(\begin{array}{l}\frac{1}{3}\Big\{{\Big|\frac{\mathrm{2.6046}-\mathrm{1.4539}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{4.2156}-\mathrm{4.6372}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{3.3954}-\mathrm{4.5461}}{6}\Big|^{2}}\Big\}\\ {} +\frac{1}{3}\Big\{{\Big|\frac{\mathrm{0.9524}-\mathrm{0.6850}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{4.8338}-\mathrm{4.9673}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{5.0476}-\mathrm{5.3150}}{6}\Big|^{2}}\Big\}\\ {} +\frac{1}{3}\Big\{{\Big|\frac{\mathrm{0.9190}-\mathrm{0.9190}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{4.7012}-\mathrm{4.7012}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{5.0810}-\mathrm{5.0810}}{6}\Big|^{2}}\Big\}\\ {} +\frac{1}{3}\Big\{{\Big|\frac{\mathrm{1.4011}-\mathrm{1.1011}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{4.4789}-\mathrm{4.6352}}{6}\Big|^{2}}+{\Big|\frac{\mathrm{4.5989}-\mathrm{4.8989}}{6}\Big|^{2}}\Big\}\end{array}\right)}\\ {} & \displaystyle =& \displaystyle 0.2439.\end{array}\]

Similarly, we can obtain

\begin{array}{l} {HD}_{1} = 0.2304, {HD}_{2} = 0.5326, {HD}_{3} = 0.3554, {HD}_{4} = 0.1343, \\ {HD}_{5} = 0.3123, {ED}_{1} = 0.2439, {ED}_{2} = 0.5854, {ED}_{3} = 0.3887, \\ {ED}_{4} = 0.1463, {ED}_{5} = 0.3246. \end{array}

\[\begin{array}{l}\displaystyle {\mathit{HD}_{1}}=0.2304,\hspace{2em}{\mathit{HD}_{2}}=0.5326,\hspace{2em}{\mathit{HD}_{3}}=0.3554,\hspace{2em}{\mathit{HD}_{4}}=0.1343,\\ {} \displaystyle {\mathit{HD}_{5}}=0.3123,\hspace{2em}{\mathit{ED}_{1}}=0.2439,\hspace{2em}{\mathit{ED}_{2}}=0.5854,\hspace{2em}{\mathit{ED}_{3}}=0.3887,\\ {} \displaystyle {\mathit{ED}_{4}}=0.1463,\hspace{2em}{\mathit{ED}_{5}}=0.3246.\end{array}\]

Step 7.

Determine the relative assessment (RA) matrix which is presented as follows (Table 9).

Table 9

The relative assessment matrix

{(R A)}_{m \times m}

${(RA)_{m\times m}}$

	$ϕ_{1}$ ${\phi _{1}}$	$ϕ_{2}$ ${\phi _{2}}$	$ϕ_{3}$ ${\phi _{3}}$	$ϕ_{4}$ ${\phi _{4}}$	$ϕ_{5}$ ${\phi _{5}}$
$ϕ_{1}$ ${\phi _{1}}$	0.0000	0.6437	0.2699	−0.1937	0.1626
$ϕ_{2}$ ${\phi _{2}}$	−0.6437	0.0000	−0.3738	−0.8374	−0.4811
$ϕ_{3}$ ${\phi _{3}}$	−0.2699	0.3738	0.0000	−0.4636	−0.1073
$ϕ_{4}$ ${\phi _{4}}$	0.1937	0.8374	0.4636	0.0000	0.3563
$ϕ_{5}$ ${\phi _{5}}$	−0.1626	0.4811	0.1073	−0.3563	0.0000

Step 8.

Compute the values of assessment score (AS) based on each alternative’s

p_{i l}

${p_{il}}$

\begin{array}{l} {AS}_{1} = - 0.8824, {AS}_{2} = 2.3360, {AS}_{3} = 0.4669, \\ {AS}_{4} = - 1.8510, {AS}_{5} = - 0.0695. \end{array}

\[\begin{array}{l}\displaystyle {\mathit{AS}_{1}}=-0.8824,\hspace{2em}{\mathit{AS}_{2}}=2.3360,\hspace{2em}{\mathit{AS}_{3}}=0.4669,\\ {} \displaystyle {\mathit{AS}_{4}}=-1.8510,\hspace{2em}{\mathit{AS}_{5}}=-0.0695.\end{array}\]

Step 9.

According to the calculation results of AS, we can rank all the alternatives. The bigger the value of AS is, the better alternative will be selected. Obviously, the rank of all alternatives is

ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}

${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

and

ϕ_{2}

${\phi _{2}}$

is the best alternative.

6.2 Sensitivity Analysis

To show the influence of the threshold parameter θ which is set by the decision maker, the ordering of the alternatives is shown as follows.

Table 10
Ordering by different parameter θ.

Parameter $A S_{1}$ $ A{S_{1}}$
$A S_{1}$ $ A{S_{1}}$

$A S_{1}$ $ A{S_{1}}$

$A S_{1}$ $ A{S_{1}}$

$A S_{1}$ $ A{S_{1}}$
Ordering

$θ =$ $ \theta =$
0.05 −0.8824 2.3360 0.4669 −1.8510 −0.0695
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

$θ =$ $ \theta =$
0.06 −0.8824 2.3360 0.4669 −1.8510 −0.0695
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

$θ =$ $ \theta =$
0.07 −0.8824 2.3360 0.4238 −1.8510 −0.0264
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

$θ =$ $ \theta =$
0.10 −0.8966 2.3360 0.4238 −1.7549 −0.1083
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

$θ =$ $ \theta =$
0.20 −0.7716 2.1588 0.4760 −1.5769 −0.2863
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

$θ =$ $ \theta =$
0.30 −0.7716 1.9386 0.2549 −1.3558 −0.0660
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

$θ =$ $ \theta =$
0.40 −0.4695 1.6364 0.2549 −1.3558 −0.0660
$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

Parameter	$A S_{1}$ $ A{S_{1}}$	$A S_{1}$ $ A{S_{1}}$	$A S_{1}$ $ A{S_{1}}$	$A S_{1}$ $ A{S_{1}}$	$A S_{1}$ $ A{S_{1}}$	Ordering
$θ =$ $ \theta =$ 0.05	−0.8824	2.3360	0.4669	−1.8510	−0.0695	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
$θ =$ $ \theta =$ 0.06	−0.8824	2.3360	0.4669	−1.8510	−0.0695	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
$θ =$ $ \theta =$ 0.07	−0.8824	2.3360	0.4238	−1.8510	−0.0264	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
$θ =$ $ \theta =$ 0.10	−0.8966	2.3360	0.4238	−1.7549	−0.1083	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
$θ =$ $ \theta =$ 0.20	−0.7716	2.1588	0.4760	−1.5769	−0.2863	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
$θ =$ $ \theta =$ 0.30	−0.7716	1.9386	0.2549	−1.3558	−0.0660	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
$θ =$ $ \theta =$ 0.40	−0.4695	1.6364	0.2549	−1.3558	−0.0660	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

From Table 10, we can easily find that the ordering of alternatives is the same, which indicates that our proposed 2TLNN CODAS model is robust and effective. At the same time, when the threshold parameter

θ ⩽ 0.06

$ \theta \leqslant 0.06$

, the assessment score (AS) remains the same, which indicates that the Hamming distance and the Euclidean distance are considered; when the threshold parameter

θ ⩾ 0.50

$ \theta \geqslant 0.50$

, the assessment score (AS) also remains the same, which indicates only the Euclidean distance are considered. In other words, when the threshold parameter

0.06 ⩽ θ ⩽ 0.50

$ 0.06\leqslant \theta \leqslant 0.50$

, the Euclidean distance is considered absolutely and the Hamming distance is considered partly. In addition, the absolute values of assessment scores become smaller with the increase of the parameter. Thus, the decision maker can obtain different assessment scores by altering the threshold parameter.

6.3 Compare 2TLNNs CODAS Method with Some 2TLNNs Aggregation Operators

In this chapter, we compare our proposed 2-tuple linguistic neutrosophic CODAS method with the 2-tuple linguistic neutrosophic weighted average (2TLNNWA) operator and 2-tuple linguistic neutrosophic weighted geometric (2TLNNWG) operator. For the attribute’s weights that are presented by 2TLNNs we can use the score function to obtain the attribute’s weights with crisp number.

According to the value of the average attribute’s weighting vector W which is listed as:

\begin{matrix} W = \{\begin{array}{l} {(s_{3}, 0.4850), (s_{3}, - 0.1042), (s_{3}, 0.4850)}, \\ {(s_{3}, 0.1146), (s_{3}, 0.3659), (s_{3}, - 0.1146)}, \\ {(s_{4}, 0.1654), (s_{3}, - 0.3436), (s_{2}, - 0.1654)}, \\ {(s_{2}, 0.1880), (s_{3}, 0.2704), (s_{4}, - 0.1880)} \end{array}\} . \end{matrix}

We can obtain the score results as:

\begin{matrix} W_{1} = 0.5597, W_{2} = 0.4924, W_{3} = 0.6486, W_{4} = 0.3948. \end{matrix}

\[ {W_{1}}=0.5597,\hspace{2em}{W_{2}}=0.4924,\hspace{2em}{W_{3}}=0.6486,\hspace{2em}{W_{4}}=0.3948.\]

Then the normalized results

w_{j}

${w_{j}}$

(j = 1, 2, \dots, n)

$(j=1,2,\cdots \hspace{0.1667em},n)$

can be expressed as:

\begin{array}{l} w_{j} = W_{i} / \sum_{i = 1}^{n} W_{i}, \\ w_{1} = 0.2671, w_{2} = 0.2350, w_{3} = 0.3095, w_{4} = 0.1884. \end{array}

\[\begin{array}{l}\displaystyle {w_{j}}={W_{i}}\big/{\sum \limits_{i=1}^{n}}{W_{i}},\\ {} \displaystyle {w_{1}}=0.2671,\hspace{2em}{w_{2}}=0.2350,\hspace{2em}{w_{3}}=0.3095,\hspace{2em}{w_{4}}=0.1884.\end{array}\]

(43)

Based on the attribute’s weight and the results in Table 7, the fused values by 2TLNNWA and 2TLNNWG operators are shown in Table 11.

Table 11

The fused values by using some 2TLNNs aggregation operator.

	2TLNNWA	2TLNNWG
$ϕ_{1}$ ${\phi _{1}}$	${(s_{3}, 0.0885), (s_{3}, 0.0654), (s_{3}, - 0.0885)}$ $ \{({s_{3}},0.0885),({s_{3}},0.0654),({s_{3}},-0.0885)\}$	${(s_{2}, 0.4201), (s_{3}, 0.1435), (s_{4}, - 0.4201)}$ $ \{({s_{2}},0.4201),({s_{3}},0.1435),({s_{4}},-0.4201)\}$
$ϕ_{2}$ ${\phi _{2}}$	${(s_{4}, - 0.0034), (s_{3}, - 0.2500), (s_{2}, 0.0034)}$ $ \{({s_{4}},-0.0034),({s_{3}},-0.2500),({s_{2}},0.0034)\}$	${(s_{4}, - 0.2538), (s_{3}, - 0.1964), (s_{2}, 0.2538)}$ $ \{({s_{4}},-0.2538),({s_{3}},-0.1964),({s_{2}},0.2538)\}$
$ϕ_{3}$ ${\phi _{3}}$	${(s_{3}, 0.3493), (s_{3}, - 0.0147), (s_{3}, - 0.3493)}$ $ \{({s_{3}},0.3493),({s_{3}},-0.0147),({s_{3}},-0.3493)\}$	${(s_{3}, 0.1133), (s_{3}, 0.0367), (s_{3}, - 0.1133)}$ $ \{({s_{3}},0.1133),({s_{3}},0.0367),({s_{3}},-0.1133)\}$
$ϕ_{4}$ ${\phi _{4}}$	${(s_{3}, - 0.4976), (s_{3}, 0.2545), (s_{3}, 0.4976)}$ $ \{({s_{3}},-0.4976),({s_{3}},0.2545),({s_{3}},0.4976)\}$	${(s_{2}, 0.3048), (s_{3}, 0.2810), (s_{4}, - 0.3048)}$ $ \{({s_{2}},0.3048),({s_{3}},0.2810),({s_{4}},-0.3048)\}$
$ϕ_{5}$ ${\phi _{5}}$	${(s_{3}, 0.1103), (s_{3}, - 0.1452), (s_{3}, - 0.1103)}$ $ \{({s_{3}},0.1103),({s_{3}},-0.1452),({s_{3}},-0.1103)\}$	${(s_{3}, - 0.1259), (s_{3}, - 0.1077), (s_{3}, - 0.1259)}$ $ \{({s_{3}},-0.1259),({s_{3}},-0.1077),({s_{3}},-0.1259)\}$

According to the score function of 2TLNNs, we can obtain the alternative score results which are shown in Table 12.

Table 12

Score results of alternatives

ϕ_{i}

${\phi _{i}}$

	2TLNNWA	2TLNNWG
$s (ϕ_{1})$ $ s({\phi _{1}})$	0.5062	0.4276
$s (ϕ_{2})$ $ s({\phi _{2}})$	0.6246	0.5938
$s (ϕ_{3})$ $ s({\phi _{3}})$	0.5396	0.5106
$s (ϕ_{4})$ $ s({\phi _{4}})$	0.4306	0.4071
$s (ϕ_{1})$ $ s({\phi _{1}})$	0.5203	0.4920

The ranking of alternatives by some 2TLNNs aggregation operators are listed as follows (Table 13).

Table 13

Rank of Alternatives by some 2TLNNs aggregation operators.

	order
2TLNNWA	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
2TLNNWG	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$
2TLNNs CODAS model	$ϕ_{2} > ϕ_{3} > ϕ_{5} > ϕ_{1} > ϕ_{4}$ ${\phi _{2}}>{\phi _{3}}>{\phi _{5}}>{\phi _{1}}>{\phi _{4}}$

Comparing the results of the 2-tuple linguistic neutrosophic CODAS model with 2TLNNWA and 2TLNNWG operators, it can be noted that the aggregation results are slightly different in ranking of alternatives and the best alternatives are the same. However, 2-tuple linguistic neutrosophic CODAS model has important characteristics of using the combinative form of two distance measurements, including fuzzy weighted Hamming distance (HD) and fuzzy weighted Euclidean distance (ED) and can be more accurate and effective in the application of MADM problems.

7 Conclusion

In this paper, we present the 2-tuple linguistic neutrosophic CODAS model based on the traditional fuzzy CODAS (combinative distance-based assessment) model and some fundamental theories of 2-tuple linguistic neutrosophic information. Firstly, we briefly review the definition of 2-tuple linguistic neutrosophic sets (2TLNNSs) and introduce the score function, the accuracy function, operation laws and some aggregation operators of 2TLNNs. Then, the calculation steps of traditional fuzzy CODAS model are briefly presented. Furthermore, by combining the traditional fuzzy CODAS model with 2TLNNs information, the 2-tuple linguistic neutrosophic CODAS model is established and the computing steps are simply depicted. Finally, a numerical example for safety assessment of a construction project is given to illustrate this new model and some comparisons between 2TLNNs CODAS model and two 2TLNNs aggregation operators are also made to further illustrate advantages of the new method. In actual decision making applications, our developed model has the advantage of considering the combinative form of two distance measurements, including fuzzy weighted Hamming distance (HD) and fuzzy weighted Euclidean distance (ED). However, it is difficult to obtain assessment information which is expressed by 2TLNNs, so we need to continue to study this problem. In the future, the 2-tuple linguistic neutrosophic CODAS model can be applied to the risk analysis (Wei et al., 2019h, 2018), the MADM problems (Wei et al., 2019a; He et al., 2019a; Tang and Wei, 2019b; Wei, 2019a, 2019b; Wei et al., 2019c) and many other uncertain and fuzzy environments (Lu et al., 2019a; Wei et al., 2019c, 2019d, 2019f).

References

Atanassov

(1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96.

Atanassov

(1989). More on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 33, 37–46.

Badi

Ballem

Shetwan

(2018). Site selection of desalination plant in LIBYA by using combinative distance-based assessment (CODAS) method. International Journal for Quality Research, 12, 609–623.

Balali

Zahraie

Roozbahani

(2014). Integration of ELECTRE III and PROMETHEE II decision-making methods with an interval approach: application in selection of appropriate structural systems. Journal of Computing in Civil Engineering, 28, 297–314.

Bolturk

(2018). Pythagorean fuzzy CODAS and its application to supplier selection in a manufacturing firm. Journal of Enterprise Information Management, 31, 550–564.

Bolturk

Kahraman

(2018a). Interval-valued intuitionistic fuzzy CODAS method and its application to wave energy facility location selection problem. Journal of Intelligent & Fuzzy Systems, 35, 4865–4877.

Bolturk

Karasan

(2018b). Interval Valued Neutrosophic CODAS Method for Renewable Energy Selection, Vol. 11.

Chen

C.T.

(2000). Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets and Systems, 114, 1–9.

Deng

X.M.

Gao

(2019). TODIM method for multiple attribute decision making with 2-tuple linguistic Pythagorean fuzzy information. Journal of Intelligent & Fuzzy Systems, 37, 1769–1780.

10.

Gao

Wei

(2019). Dual hesitant bipolar fuzzy hamacher aggregation operators and their applications to multiple attribute decision making. Journal of Intelligent & Fuzzy Systems, 37, 5755–5766.

11.

Gomes

Lima

(1979). TODIM: basics and application to multicriteria ranking of projects with environmental impacts. Foundations of Computing and Decision Sciences, 16, 113–127.

12.

T.T.

Wei

G.W.

J.P.

Wei

Lin

(2019a). Pythagorean 2-tuple linguistic VIKOR method for evaluating human factors in construction project management. Mathematics, 7, 1149.

13.

T.T.

Wei

G.W.

C.W.

Wei

Lin

(2019b). Pythagorean 2-tuple linguistic taxonomy method for supplier selection in medical instrument industries. International Journal of Environmental Research and Public Health, 16, 4875.

14.

Herrera

Martinez

(2001). The 2-tuple linguistic computational model. Advantages of its linguistic description, accuracy and consistency. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 9, 33–48.

15.

D.W.

Y.B.

Wang

A.H.

(2018). Multiple attribute group decision making based on Maclaurin symmetric mean operator under single-valued neutrosophic interval 2-tuple linguistic environment. Journal of Intelligent & Fuzzy Systems, 34, 2579–2595.

16.

Karasan

Bolturk

Kahraman

(2019). An integrated methodology using neutrosophic CODAS & fuzzy inference system: assessment of livability index of urban districts. Journal of Intelligent & Fuzzy Systems, 36, 5443–5455.

17.

Keshavarz Ghorabaee

Zavadskas

E.K.

Olfat

Turskis

(2015). Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica, 26, 435–451.

18.

Keshavarz Ghorabaee

Zavadskas

E.K.

Turskis

Antucheviciene

(2016). A new combinative distance distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50, 25–44.

19.

Keshavarz Ghorabaee

Amiri

Zavadskas

E.K.

Hooshmand

Antucheviciene

(2017). Fuzzy extension of the CODAS method for multi-criteria market segment evaluation. Journal of Business Economics and Management, 18, 1–19.

20.

Z.X.

(2019). Some novel similarity and distance and measures of Pythagorean fuzzy sets and their applications. Journal of Intelligent & Fuzzy Systems, 37, 1781–1799.

21.

J.P.

Wei

(2019). TODIM method for performance appraisal on social-integration-based rural reconstruction with interval-valued intuitionistic fuzzy information. Journal of Intelligent & Fuzzy Systems, 37, 1731–1740.

22.

J.P.

Tang

X.Y.

Wei

G.W.

Wei

(2019a). Bidirectional project method for dual hesitant Pythagorean fuzzy multiple attribute decision-making and their application to performance assessment of new rural construction. International Journal of Intelligent Systems, 34, 1920–1934.

23.

J.P.

Wei

G.W.

(2019b). TOPSIS method for probabilistic linguistic MAGDM with entropy weight and its application to supplier selection of new agricultural machinery products. Entropy, 21, 953.

24.

Opricovic

Tzeng

G.H.

(2004). Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156, 445–455.

25.

Pamucar

Cirovic

(2015). The selection of transport and handling resources in logistics centers using multi-attributive border approximation area comparison (MABAC). Expert Systems with Applications, 42, 3016–3028.

26.

Pamucar

Badi

Sanja

Obradovic

(2018). A novel approach for the selection of power-generation technology using a linguistic neutrosophic CODAS method: a case study in Libya. Energies, 11.

27.

Peng

X.D.

Garg

(2018). Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure. Computers & Industrial Engineering, 119, 439–452.

28.

Rashid

Faizi

Z.S.

Zafar

(2018). ELECTRE-based outranking method for multi-criteria decision making using hesitant intuitionistic fuzzy linguistic term sets. International Journal of Fuzzy Systems, 20, 78–92.

29.

Ren

J.Z.

(2018). Sustainability prioritization of energy storage technologies for promoting the development of renewable energy: a novel intuitionistic fuzzy combinative distance-based assessment approach. Renewable Energy, 121, 666–676.

30.

Smarandache

(1999). A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic. American Research Press, Rehoboth, NM, USA.

31.

Smarandache

(2003). A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics. American Research Press, Phoenix, AZ, USA.

32.

Tang

X.Y.

Wei

G.W.

(2019a). Dual hesitant Pythagorean fuzzy Bonferroni mean operators in multi-attribute decision making. Archives of Control Sciences, 29, 339–386.

33.

Tang

X.Y.

Wei

G.W.

(2019b). Multiple attribute decision making with dual hesitant Pythagorean fuzzy information. Cognitive Computation, 11, 193–211.

34.

Tang

X.Y.

Wei

G.W.

Gao

(2019). Models for multiple attribute decision making with interval-valued Pythagorean fuzzy muirhead mean operators and their application to green suppliers selection. Informatica, 30, 153–186.

35.

Wang

(2019). Research on the application of the financial investment risk appraisal models with some interval number muirhead mean operators. Journal of Intelligent & Fuzzy Systems, 37, 1741–1752.

36.

Wang

Smarandache

Zhang

Y.Q.

Sunderraman

(2010). Single valued neutrosophic sets. Multispace Multistruct, 4, 410–413.

37.

Wang

Wei

(2018a). Models for Green supplier selection with some 2-tuple linguistic neutrosophic number Bonferroni mean operators. Symmetry-Basel, 10.

38.

Wang

Wei

G.W.

(2018b). An extended VIKOR method for multiple criteria group decision making with triangular fuzzy neutrosophic numbers. Symmetry-Basel, 10, 497.

39.

Wang

Wei

G.W.

(2018c). TODIM method for multiple attribute group decision making under 2-tuple linguistic neutrosophic environment. Symmetry-Basel, 10.

40.

Wang

Gao

(2019a). Approaches to strategic supplier selection under interval neutrosophic environment. Journal of Intelligent & Fuzzy Systems, 37, 1707–1730.

41.

Wang

Gao

Wei

(2019b). Some 2-tuple linguistic neutrosophic number Muirhead mean operators and their applications to multiple attribute decision making. Journal of Experimental & Theoretical Artificial Intelligence, 31, 409–439.

42.

Wang

Gao

Wei

G.W.

(2019c). Some 2-tuple linguistic neutrosophic number Muirhead mean operators and their applications to multiple attribute decision making. Journal of Experimental & Theoretical Artificial Intelligence, 31, 409–439.

43.

Wang

Wei

Lin

Wei

(2019d). Models for MADM with single-valued neutrosophic 2-tuple linguistic muirhead mean operators. Mathematics, 7.

44.

Wang

J.P.

Wei

G.W.

Lin

Wei

(2019e). Models for MADM with single-valued neutrosophic 2-tuple linguistic muirhead mean operators. Mathematics, 7. https://doi.org/10.3390/math7050442.

45.

Wang

Wei

G.W.

J.P.

Alsaadi

F.E.

Hayat

Wei

Zhang

(2019f). Some q-rung orthopair fuzzy Hamy mean operators in multiple attribute decision-making and their application to enterprise resource planning systems selection. International Journal of Intelligent Systems, 34, 2429–2458.

46.

Wang

Wei

G.W.

Wei

(2019g). Dual hesitant q-rung orthopair fuzzy muirhead mean operators in multiple attribute decision making. IEEE Access, 7, 67139–67166.

47.

Wang

Wei

(2019h). EDAS method for multiple criteria group decision making under 2-tuple linguistic neutrosophic environment. Journal of Intelligent & Fuzzy Systems, 37, 1597–1608.

48.

Wang

Wei

G.W.

(2019i). EDAS method for multiple criteria group decision making under 2-tuple linguistic neutrosophic environment. Journal of Intelligent & Fuzzy Systems, 37, 1597–1608.

49.

Wang

Wei

G.W.

Wei

(2019j). The multi-attributive border approximation area comparison (MABAC) for multiple attribute group decision making under 2-tuple linguistic neutrosophic environment. Informatica, 30, 799–818.

50.

Wei

Liu

Cao

(2018). Hot money and China’s stock market volatility: Further evidence using the GARCH-MIDAS model. Physica A: Statistical Mechanics and its Applications, 492, 923–930.

51.

Wei

G.W.

(2019a). 2-tuple intuitionistic fuzzy linguistic aggregation operators in multiple attribute decision making. Iranian Journal of Fuzzy Systems, 16, 159–174.

52.

Wei

G.W.

(2019b). The generalized dice similarity measures for multiple attribute decision making with hesitant fuzzy linguistic information. Economic Research-Ekonomska Istrazivanja, 32, 1498–1520.

53.

Wei

G.W.

Wei

Wang

H.J.

(2019a). Supplier selection of medical consumption products with a probabilistic linguistic MABAC method. International Journal of Environmental Research and Public Health, 16, 5082.

54.

Wei

Wang

(2019b). Models for MADM with 2-tuple linguistic neutrosophic Dombi Bonferroni mean operators. IEEE Access, 7, 108878–108905.

55.

Wei

G.W.

Wang

Wei

(2019c). Similarity measures of spherical fuzzy sets based on cosine function and their applications. IEEE Access, 7, 159069–159080.

56.

Wei

G.W.

Wang

Wei

Zhang

(2019d). Dual hesitant Pythagorean fuzzy hamy mean operators in multiple attribute decision making. IEEE Access, 7, 86697–86716.

57.

Wei

G.W.

Wang

Wei

Zhang

(2019e). Methods for evaluating the technological innovation capability for the high-tech enterprises with generalized interval neutrosophic number Bonferroni mean operators. IEEE Access, 7, 86473–86492.

58.

Wei

G.W.

Wei

Wang

J.P.

(2019f). Models for MADM with 2-tuple linguistic neutrosophic Dombi Bonferroni mean operators. IEEE Access, 7, 108878–108905.

59.

Wei

G.W.

Zhang

S.Q.

J.P.

Wei

(2019g). An extended bidirectional projection method for picture fuzzy MAGDM and its application to safety assessment of construction project. IEEE Access, 7, 166138–166147.

60.

Wei

Qin

Zhu

S.R.

Wei

(2019h). Oil price fluctuation, stock market and macroeconomic fundamentals: Evidence from China before and after the financial crisis. Finance Research Letters, 30, 23–29.

61.

Zhou

L.G.

Chen

H.Y.

Guan

X.J.

(2018a). Some new Hamacher aggregation operators under single-valued neutrosophic 2-tuple linguistic environment and their applications to multi-attribute group decision making. Computers & Industrial Engineering, 116, 144–162.

62.

Wang

Wei

(2018b). Research on construction engineering project risk assessment with some 2-tuple linguistic neutrosophic Hamy mean operators. Sustainability, 10.

63.

L.P.

Gao

Wei

(2019a). VIKOR method for financing risk assessment of rural tourism projects under interval-valued intuitionistic fuzzy environment. Journal of Intelligent & Fuzzy Systems, 37, 2001–2008.

64.

L.P.

Wang

Gao

(2019b). Models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. Journal of Intelligent & Fuzzy Systems, 36, 5693–5709.

65.

Zadeh

L.A.

(1965). Fuzzy Sets. Information and Control, 8, 338–356.

66.

Zhang

S.Q.

Wei

G.W.

Gao

Wei

(2019). EDAS method for multiple criteria group decision making with picture fuzzy information and its application to green suppliers selections. Technological and Economic Development of Economy, 26, 1123–1138.

CODAS Method for Multiple Attribute Group Decision Making Under 2-Tuple Linguistic Neutrosophic Environment

Abstract

Keywords

1 Introduction

2 Related Work

3 Preliminaries

3.1 2-Tuple Linguistic Neutrosophic Sets

Definition 6 (See Wang et al., 2018a).

6.1 Numerical for 2TLNNs MAGDM Problems

Table 1 Linguistic scale.

References

Table 1
Linguistic scale.