1 Introduction
2 Related Literature
2.1 Green Supplier Selection Methods
2.2 Interval 2Tuple Linguistic Aggregation Operators
3 Preliminaries
3.1 2Tuple Linguistic Variables
Definition 1.
Definition 2.
3.2 Interval 2Tuple Linguistic Variables
Definition 3.
(6)
\[ \Delta [{\beta _{1}},{\beta _{2}}]=\big[({s_{i}},{\alpha _{i}}),({s_{j}},{\alpha _{j}})\big]\hspace{1em}\text{with}\hspace{2.5pt}\left\{\begin{array}{l@{\hskip4.0pt}l}{s_{i}},\hspace{1em}& i=\text{round}({\beta _{1}}\cdot g),\\ {} {s_{j}},\hspace{1em}& j=\text{round}\hspace{2.5pt}({\beta _{2}}\cdot g),\\ {} {\alpha _{i}}={\beta _{1}}\frac{i}{g},\hspace{1em}& {\alpha _{i}}\in \big[\frac{1}{2g},\frac{1}{2g}\big),\\ {} {\alpha _{j}}={\beta _{2}}\frac{j}{g},\hspace{1em}& {\alpha _{j}}\in \big[\frac{1}{2g},\frac{1}{2g}\big).\end{array}\right.\](7)
\[ {\Delta ^{1}}\big[({s_{i}},{\alpha _{i}}),({s_{j}},{\alpha _{j}})\big]=\bigg[\frac{i}{g}+{\alpha _{i}},\frac{j}{g}+{\alpha _{j}}\bigg]=[{\beta _{1}},{\beta _{2}}].\]Definition 4.
(8)
\[ p({\tilde{a}_{1}}\geqslant {\tilde{a}_{2}})=\max \bigg\{1\max \bigg(\frac{{\delta _{2}}{\beta _{1}}}{h({\tilde{a}_{1}})+h({\tilde{a}_{2}})},0\bigg),0\bigg\}.\]
(1) $0\leqslant p({\tilde{a}_{1}}\geqslant {\tilde{a}_{2}})\leqslant 1$, $0\leqslant p({\tilde{a}_{2}}\geqslant {\tilde{a}_{1}})\leqslant 1$;

(2) $p({\tilde{a}_{1}}\geqslant {\tilde{a}_{2}})+p({\tilde{a}_{2}}\geqslant {\tilde{a}_{1}})=1$. Especially, $p({\tilde{a}_{1}}\geqslant {\tilde{a}_{1}})=p({\tilde{a}_{2}}\geqslant {\tilde{a}_{2}})=0.5$.
4 Interval 2Tuple Linguistic Hybrid Aggregation Operators
4.1 Interval 2Tuple Hybrid Averaging Operators
Definition 5.
(10)
\[\begin{array}{l}\displaystyle {\text{ITWA}_{w}}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{i=1}{\overset{n}{\bigoplus }}}({w_{i}}{\tilde{a}_{i}})\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\sum \limits_{i=1}^{n}}{w_{i}}{\Delta ^{1}}({s_{i}},{\alpha _{i}}),{\sum \limits_{i=1}^{n}}{w_{i}}{\Delta ^{1}}({t_{i}},{\varepsilon _{i}})\Bigg].\end{array}\]Definition 6.
(11)
\[\begin{array}{l}\displaystyle {\text{ITOWA}_{\omega }}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{j=1}{\overset{n}{\bigoplus }}}({\omega _{j}}{\tilde{a}_{\sigma (j)}})\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\sum \limits_{j=1}^{n}}{\omega _{j}}{\Delta ^{1}}({s_{\sigma (j)}},{\alpha _{\sigma (j)}}),{\sum \limits_{j=1}^{n}}{\omega _{j}}{\Delta ^{1}}({t_{\sigma (j)}},{\varepsilon _{\sigma (j)}})\Bigg],\end{array}\]Definition 7.
(12)
\[\begin{array}{l}\displaystyle {\text{ITHA}_{\omega ,w}}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{j=1}{\overset{n}{\bigoplus }}}({\omega _{j}}{\dot{\tilde{a}}_{\sigma (j)}})\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\sum \limits_{j=1}^{n}}{\omega _{j}}{\Delta ^{1}}({\dot{s}_{\sigma (j)}},{\dot{\alpha }_{\sigma (j)}}),{\sum \limits_{j=1}^{n}}{\omega _{j}}{\Delta ^{1}}({\dot{t}_{\sigma (j)}},{\dot{\varepsilon }_{\sigma (j)}})\Bigg].\end{array}\]Example 1.
Definition 8.
(13)
\[\begin{array}{l}\displaystyle {\text{ITOWAWA}_{\omega ,w}}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{j=1}{\overset{n}{\bigoplus }}}({v_{j}}{\tilde{a}_{\sigma (j)}})\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\sum \limits_{j=1}^{n}}{v_{j}}{\Delta ^{1}}({s_{\sigma (j)}},{\alpha _{\sigma (j)}}),{\sum \limits_{j=1}^{n}}{v_{j}}{\Delta ^{1}}({t_{\sigma (j)}},{\varepsilon _{\sigma (j)}})\Bigg],\end{array}\]Definition 9.
(14)
\[ {\text{ITOWAWA}_{\omega ,w}}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})=\phi {\underset{j=1}{\overset{n}{\bigoplus }}}({\omega _{j}}{\tilde{a}_{\sigma (j)}})\oplus (1\phi ){\underset{i=1}{\overset{n}{\bigoplus }}}({w_{i}}{\tilde{a}_{i}}),\]Example 2.
4.2 Interval 2Tuple Hybrid Geometric Operators
Definition 10.
(15)
\[\begin{array}{l}\displaystyle {\text{ITWG}_{w}}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{i=1}{\overset{n}{\bigotimes }}}{({\tilde{a}_{i}})^{{w_{i}}}}\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\prod \limits_{i=1}^{n}}{\big({\Delta ^{1}}({s_{i}},{\alpha _{i}})\big)^{{w_{i}}}},{\prod \limits_{i=1}^{n}}{\big({\Delta ^{1}}({t_{i}},{\varepsilon _{i}})\big)^{{w_{i}}}}\Bigg].\end{array}\]Definition 11.
(16)
\[\begin{array}{l}\displaystyle {\text{ITOWG}_{\omega }}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{i=1}{\overset{n}{\bigotimes }}}{({\tilde{a}_{\sigma (j)}})^{{\omega _{j}}}}\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\prod \limits_{j=1}^{n}}{\big({\Delta ^{1}}({s_{\sigma (j)}},{\alpha _{\sigma (j)}})\big)^{{\omega _{j}}}},{\prod \limits_{j=1}^{n}}{\big({\Delta ^{1}}({t_{\sigma (j)}},{\varepsilon _{\sigma (j)}})\big)^{{\omega _{j}}}}\Bigg],\end{array}\]Definition 12.
(17)
\[\begin{array}{l}\displaystyle {\text{ITHG}_{\omega ,w}}({\tilde{a}_{1}},{\tilde{a}_{2}},\dots ,{\tilde{a}_{n}})={\underset{j=1}{\overset{n}{\bigotimes }}}{({\dot{\tilde{a}}_{\sigma (j)}})^{{\omega _{j}}}}\\ {} \displaystyle \hspace{1em}=\Delta \Bigg[{\prod \limits_{j=1}^{n}}{\big({\Delta ^{1}}({\dot{s}_{\sigma (j)}},{\dot{\alpha }_{\sigma (j)}})\big)^{{\omega _{j}}}},{\prod \limits_{j=1}^{n}}{\big({\Delta ^{1}}({\dot{t}_{\sigma (j)}},{\dot{\varepsilon }_{\sigma (j)}})\big)^{{\omega _{j}}}}\Bigg],\end{array}\]Example 3.
5 The Proposed Green Supplier Selection Approach

1. A certain rating such as Poor can be denoted as $[({s_{1}},0),({s_{1}},0)]$.

2. An interval grade, e.g. PoorMedium, can be expressed as $[({s_{1}},0),({s_{2}},0)]$. This means that the rating of an alternative concerning the criterion under consideration is between Poor and Medium.

3. If a decision maker is not willing to or cannot provide a judgement for an alternative concerning the criterion under consideration, then the assessment could be anywhere between Very poor and Very good and thus can be written as $[({s_{0}},0),({s_{4}},0)]$.
(18)
\[\begin{array}{l}\displaystyle {\tilde{r}_{i}^{k}}=\big[\big({s_{i}^{k}},{\alpha _{i}^{k}}\big),\big({t_{i}^{k}},{\varepsilon _{i}^{k}}\big)\big]={\text{ITWA}_{w}}\big({\tilde{r}_{i1}^{k}},{\tilde{r}_{i2}^{k}},\dots ,{\tilde{r}_{in}^{k}}\big),\\ {} \displaystyle \hspace{1em}i=1,2,\dots ,m,\hspace{2.5pt}k=1,2,\dots ,l.\end{array}\](19)
\[ {\tilde{r}_{i}}=\big[({s_{i}},{\alpha _{i}}),({t_{i}},{\varepsilon _{i}})\big]={\text{ITHA}_{\omega ,\lambda }}\big({\tilde{r}_{i}^{1}},{\tilde{r}_{i}^{2}},\dots ,{\tilde{r}_{i}^{l}}\big),\hspace{1em}i=1,2,\dots ,m,\](20)
\[ {\tilde{r}_{i}}=\big[({s_{i}},{\alpha _{i}}),({t_{i}},{\varepsilon _{i}})\big]={\text{ITOWAWA}_{\omega ,\lambda }}\big({\tilde{r}_{i}^{1}},{\tilde{r}_{i}^{2}},\dots ,{\tilde{r}_{i}^{l}}\big),\hspace{1em}i=1,2,\dots ,m,\](21)
\[ {\tilde{r}_{i}}=\big[({s_{i}},{\alpha _{i}}),({t_{i}},{\varepsilon _{i}})\big]={\text{ITHG}_{\omega ,\lambda }}\big({\tilde{r}_{i}^{1}},{\tilde{r}_{i}^{2}},\dots ,{\tilde{r}_{i}^{l}}\big),\hspace{1em}i=1,2,\dots ,m,\]6 Illustrative Example
6.1 Application of the Proposed Approach
Table 1
Decision makers  Alternatives  Criteria  
C_{1}  C_{2}  C_{3}  C_{4}  
DM_{1}  A_{1}  G  G  G  MG 
A_{2}  G  M  MG  G  
A_{3}  MG  G  MG  
A_{4}  G  G  MG  VG  
A_{5}  GVG  VG  G  G  
DM_{2}  A_{1}  VG  M  MG  MGG 
A_{2}  G  MMG  MG  M  
A_{3}  G  MGG  G  
A_{4}  MG  MG  G  G  
A_{5}  GVG  G  G  VG  
DM_{3}  A_{1}  VG  MG  MGVG  G 
A_{2}  G  M  MG  MMG  
A_{3}  MG  G  G  MG  
A_{4}  M  M  G  
A_{5}  GEG  G  GVG  G  
DM_{4}  A_{1}  G  M  GVG  G 
A_{2}  MG  MG  G  M  
A_{3}  G  VG  GVG  G  
A_{4}  G  G  G  
A_{5}  GVG  VG  VG  G 
Table 2
Decision makers  Candidates  Criteria  
C_{1}  C_{2}  C_{3}  C_{4}  
DM_{1}  A_{1}  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{2}},0),({a_{3}},0)]$ 
A_{2}  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{2}},0),({a_{2}},0)]$  $[({a_{2}},0),({a_{3}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  
A_{3}  $[({a_{2}},0),({a_{3}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{0}},0),({a_{4}},0)]$  $[({a_{2}},0),({a_{3}},0)]$  
A_{4}  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{2}},0),({a_{3}},0)]$  $[({a_{4}},0),({a_{4}},0)]$  
A_{5}  $[({a_{3}},0),({a_{4}},0)]$  $[({a_{4}},0),({a_{4}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  $[({a_{3}},0),({a_{3}},0)]$  
DM_{2}  A_{1}  $[({b_{6}},0),({b_{6}},0)]$  $[({b_{3}},0),({b_{3}},0)]$  $[({b_{4}},0),({b_{4}},0)]$  $[({b_{4}},0),({b_{5}},0)]$ 
A_{2}  $[({b_{5}},0),({b_{5}},0)]$  $[({b_{3}},0),({b_{4}},0)]$  $[({b_{4}},0),({b_{4}},0)]$  $[({b_{3}},0),({b_{3}},0)]$  
A_{3}  $[({b_{0}},0),({b_{6}},0)]$  $[({b_{5}},0),({b_{5}},0)]$  $[({b_{4}},0),({b_{5}},0)]$  $[({b_{5}},0),({b_{5}},0)]$  
A_{4}  $[({b_{4}},0),({b_{4}},0)]$  $[({b_{4}},0),({b_{4}},0)]$  $[({b_{5}},0),({b_{5}},0)]$  $[({b_{5}},0),({b_{5}},0)]$  
A_{5}  $[({b_{5}},0),({b_{6}},0)]$  $[({b_{5}},0),({b_{5}},0)]$  $[({b_{5}},0),({b_{5}},0)]$  $[({b_{6}},0),({b_{6}},0)]$  
DM_{3}  A_{1}  $[({c_{7}},0),({c_{7}},0)]$  $[({c_{5}},0),({c_{5}},0)]$  $[({c_{5}},0),({c_{7}},0)]$  $[({c_{6}},0),({c_{6}},0)]$ 
A_{2}  $[({c_{6}},0),({c_{6}},0)]$  $[({c_{4}},0),({c_{4}},0)]$  $[({c_{5}},0),({c_{5}},0)]$  $[({c_{4}},0),({c_{5}},0)]$  
A_{3}  $[({c_{4}},0),({c_{6}},0)]$  $[({c_{6}},0),({c_{6}},0)]$  $[({c_{6}},0),({c_{6}},0)]$  $[({c_{5}},0),({c_{5}},0)]$  
A_{4}  $[({c_{4}},0),({c_{4}},0)]$  $[({c_{0}},0),({c_{8}},0)]$  $[({c_{4}},0),({c_{4}},0)]$  $[({c_{6}},0),({c_{6}},0)]$  
A_{5}  $[({c_{6}},0),({c_{8}},0)]$  $[({c_{6}},0),({c_{6}},0)]$  $[({c_{6}},0),({c_{7}},0)]$  $[({c_{6}},0),({c_{6}},0)]$  
DM_{4}  A_{1}  $[({d_{3}},0),({d_{3}},0)]$  $[({d_{2}},0),({d_{2}},0)]$  $[({d_{3}},0),({d_{4}},0)]$  $[({d_{3}},0),({d_{3}},0)]$ 
A_{2}  $[({d_{2}},0),({d_{3}},0)]$  $[({d_{2}},0),({d_{3}},0)]$  $[({d_{3}},0),({d_{3}},0)]$  $[({d_{2}},0),({d_{2}},0)]$  
A_{3}  $[({d_{3}},0),({d_{3}},0)]$  $[({d_{4}},0),({d_{4}},0)]$  $[({d_{3}},0),({d_{4}},0)]$  $[({d_{3}},0),({d_{3}},0)]$  
A_{4}  $[({d_{3}},0),({d_{3}},0)]$  $[({d_{3}},0),({d_{3}},0)]$  $[({d_{3}},0),({d_{3}},0)]$  $[({d_{0}},0),({d_{4}},0)]$  
A_{5}  $[({d_{3}},0),({d_{4}},0)]$  $[({d_{4}},0),({d_{4}},0)]$  $[({d_{4}},0),({d_{4}},0)]$  $[({d_{3}},0),({d_{3}},0)]$ 
Table 3
Alternatives  Decision makers  
DM_{1}  DM_{2}  DM_{3}  DM_{4}  
A_{1}  $\Delta [0.7025,0.7500]$  $\Delta [0.6983,0.7300]$  $\Delta [0.7063,0.7838]$  $\Delta [0.6825,0.7600]$ 
A_{2}  $\Delta [0.6050,0.6825]$  $\Delta [0.6283,0.6733]$  $\Delta [0.5963,0.6200]$  $\Delta [0.5775,0.7025]$ 
A_{3}  $\Delta [0.4125,0.8275]$  $\Delta [0.5900,0.8717]$  $\Delta [0.6688,0.7263]$  $\Delta [0.8175,0.8950]$ 
A_{4}  $\Delta [0.7200,0.7975]$  $\Delta [0.7500,0.7500]$  $\Delta [0.4125,0.6825]$  $\Delta [0.6075,0.7975]$ 
A_{5}  $\Delta [0.8175,0.8750]$  $\Delta [0.8650,0.9033]$  $\Delta [0.7500,0.8463]$  $\Delta [0.8950,0.9525]$ 
Table 4
Alternatives  
A_{1}  A_{2}  A_{3}  A_{4}  A_{5}  
By ITHA  $\Delta [0.4864,0.5297]$  $\Delta [0.4190,0.4589]$  $\Delta [0.4569,0.5664]$  $\Delta [0.4022,0.5138]$  $\Delta [0.5713,0.6187]$ 
By ITOWAWA  $\Delta [0.5901,0.6429]$  $\Delta [0.5069,0.5705]$  $\Delta [0.5509,0.6999]$  $\Delta [0.5210,0.6424]$  $\Delta [0.7078,0.7603]$ 
By ITHG  $\Delta [0.7788,0.8262]$  $\Delta [0.7017,0.7473]$  $\Delta [0.7328,0.8931]$  $\Delta [0.6976,0.8222]$  $\Delta [0.8897,0.9356]$ 
Table 5
${\tilde{r}_{1}}$  ${\tilde{r}_{2}}$  ${\tilde{r}_{3}}$  ${\tilde{r}_{4}}$  ${\tilde{r}_{5}}$  Ranking  
ITHA  2.799  0.888  2.753  1.560  4.500  ${\tilde{r}_{5}}\succ {\tilde{r}_{1}}\succ {\tilde{r}_{3}}\succ {\tilde{r}_{4}}\succ {\tilde{r}_{2}}$ 
ITOWAWA  2.656  0.860  2.613  1.871  4.500  ${\tilde{r}_{5}}\succ {\tilde{r}_{1}}\succ {\tilde{r}_{3}}\succ {\tilde{r}_{4}}\succ {\tilde{r}_{2}}$ 
ITHG  2.697  0.863  2.682  1.774  4.484  ${\tilde{r}_{5}}\succ {\tilde{r}_{1}}\succ {\tilde{r}_{3}}\succ {\tilde{r}_{4}}\succ {\tilde{r}_{2}}$ 
Table 6
Alternatives  
A_{1}  A_{2}  A_{3}  A_{4}  A_{5}  
ITHA  2  5  3  4  1 
ITOWAWA  2  5  3  4  1 
ITHG  2  5  3  4  1 
6.2 Comparative Analysis
Table 7
Alternatives  Fuzzy TOPSIS  COPRASG  ITLVIKOR  The proposed method  
$C{C_{i}}$  Ranking  $R{S_{i}}$  Ranking  ${Q_{i}}$  Ranking  
A_{1}  0.493  2  0.201  2  0.713  3  2 
A_{2}  0.123  5  0.181  5  0.988  5  5 
A_{3}  0.475  3  0.200  3  0.512  2  3 
A_{4}  0.310  4  0.190  4  0.909  4  4 
A_{5}  1.000  1  0.229  1  0.000  1  1 