## Appendix A

In what follows, the steps of the green supplier selection problem, solved by employing the presented approach, are described in detail.

**Step 1:** The collection of green suppliers and criteria are defined.

To choose the most suitable green supplier, a group of experts is formed, and it includes three experts $({E}_{1},{E}_{2},{E}_{3})$ who come from different departments in an international automobile company. Through preliminary screening, the group of experts identifies four potential green suppliers, $({A}_{1},{A}_{2},{A}_{3},{A}_{4})$, for further assessment. Four criteria to be taken into account in this green supplier selection process are: environment management systems $({C}_{1})$, price $({C}_{2})$, quality $({C}_{3})$, and service level $({C}_{4})$.

**Step 2:** The hesitant fuzzy linguistic evaluation matrices are provided by the group of experts.

Linguistic term set

$S=\{{s}_{0}=extremly\phantom{\rule{0.277778em}{0ex}}bad,\phantom{\rule{0.277778em}{0ex}}{s}_{1}=very\phantom{\rule{0.277778em}{0ex}}bad,\phantom{\rule{0.277778em}{0ex}}{s}_{2}=bad,\phantom{\rule{0.277778em}{0ex}}{s}_{3}=slightly\phantom{\rule{0.277778em}{0ex}}bad,\phantom{\rule{0.277778em}{0ex}}{s}_{4}=medium,\phantom{\rule{0.277778em}{0ex}}{s}_{5}=slightly\phantom{\rule{0.277778em}{0ex}}good,\phantom{\rule{0.277778em}{0ex}}{s}_{6}=good,\phantom{\rule{0.277778em}{0ex}}{s}_{7}=very\phantom{\rule{0.277778em}{0ex}}good,\phantom{\rule{0.277778em}{0ex}}{s}_{8}=extremely\phantom{\rule{0.277778em}{0ex}}good\}$ is used by experts to assess each green supplier with regard to each criterion. The assessment results provided by experts are shown in

Table A1.

**Table A1.**
Hesitant fuzzy linguistic evaluation matrices.

**Table A1.**
Hesitant fuzzy linguistic evaluation matrices.

Expert | Alternatives | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|

${E}_{1}$ | ${A}_{1}$ | $\left\{{s}_{7}\right\}$ | $\left\{{s}_{1}\right\}$ | $\{{s}_{5},{s}_{6}\}$ | $\left\{{s}_{4}\right\}$ |

| ${A}_{2}$ | $\{{s}_{4},{s}_{5}\}$ | $\left\{{s}_{3}\right\}$ | $\left\{{s}_{6}\right\}$ | $\left\{{s}_{6}\right\}$ |

| ${A}_{3}$ | $\left\{{s}_{5}\right\}$ | $\{{s}_{0},{s}_{1}\}$ | $\left\{{s}_{6}\right\}$ | $\left\{{s}_{6}\right\}$ |

| ${A}_{4}$ | $\left\{{s}_{6}\right\}$ | $\left\{{s}_{1}\right\}$ | $\left\{{s}_{7}\right\}$ | $\{{s}_{6},{s}_{7},{s}_{8}\}$ |

${E}_{2}$ | ${A}_{1}$ | $\{{s}_{7},{s}_{8}\}$ | $\left\{{s}_{2}\right\}$ | $\left\{{s}_{5}\right\}$ | $\{{s}_{5},{s}_{6}\}$ |

| ${A}_{2}$ | $\left\{{s}_{7}\right\}$ | $\{{s}_{1},{s}_{2},{s}_{3}\}$ | $\left\{{s}_{5}\right\}$ | $\left\{{s}_{5}\right\}$ |

| ${A}_{3}$ | $\left\{{s}_{8}\right\}$ | $\left\{{s}_{0}\right\}$ | $\{{s}_{5},{s}_{6}\}$ | $\left\{{s}_{3}\right\}$ |

| ${A}_{4}$ | $\left\{{s}_{5}\right\}$ | $\{{s}_{0},{s}_{1}\}$ | $\left\{{s}_{7}\right\}$ | $\left\{{s}_{5}\right\}$ |

${E}_{3}$ | ${A}_{1}$ | $\{{s}_{6},{s}_{7}\}$ | $\left\{{s}_{1}\right\}$ | $\left\{{s}_{6}\right\}$ | $\left\{{s}_{5}\right\}$ |

| ${A}_{2}$ | $\left\{{s}_{4}\right\}$ | $\left\{{s}_{2}\right\}$ | $\{{s}_{6},{s}_{7}\}$ | $\left\{{s}_{6}\right\}$ |

| ${A}_{3}$ | $\left\{{s}_{6}\right\}$ | $\{{s}_{0},{s}_{1}\}$ | $\left\{{s}_{5}\right\}$ | $\{{s}_{6},{s}_{7}\}$ |

| ${A}_{4}$ | $\left\{{s}_{7}\right\}$ | $\{{s}_{1},{s}_{2}\}$ | $\left\{{s}_{7}\right\}$ | $\left\{{s}_{7}\right\}$ |

**Step 3:** Normalize the hesitant fuzzy linguistic evaluation matrices.

In these criteria, criteria

${C}_{1}$,

${C}_{3}$, and

${C}_{4}$ are benefit criteria, while criterion

${C}_{2}$ is a cost criterion. Therefore, the evaluation value of criterion

${C}_{2}$ should be normalized according to Equation (

13).

**Step 4:** Perform the consensus reaching process.

The consensus threshold and the maximum number of iterations are designated as $\phi =0.85$ and $\gamma =3$ by the group of experts, respectively, before implementing the consensus reaching process. Then, the consensus reaching process is performed.

**Step 4.1:** Let $\kappa =0$ and ${\overline{H}}_{{S}_{ij,\kappa}}^{p}={\overline{H}}_{{S}_{ij}}^{p}(p=1,2,\dots ,q)$.

**Step 4.2:** Covert the hesitant fuzzy linguistic evaluation matrices

${\overline{H}}_{{S}_{ij,\kappa}}^{p}$ into hesitant 2-tuple linguistic assessment matrices

${\overline{H}}_{{T}_{ij,\kappa}}^{p}$. The obtained results are shown in

Table A2.

**Table A2.**
Hesitant 2-tuple linguistic assessment matrices

**Table A2.**
Hesitant 2-tuple linguistic assessment matrices

Expert | Alternatives | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|

${E}_{1}$ | ${A}_{1}$ | $\{({s}_{7},0),({s}_{7},0)\}$ | $\left\{({s}_{7},0)\right\}$ | $\{({s}_{5},0),({s}_{6},0)\}$ | $\{({s}_{4},0),({s}_{4},0)\}$ |

| ${A}_{2}$ | $\{({s}_{4},0),({s}_{5},0)\}$ | $\{({s}_{5},0),({s}_{5},0),({s}_{5},0)\}$ | $\{({s}_{6},0),({s}_{6},0)\}$ | $\left\{({s}_{6},0)\right\}$ |

| ${A}_{3}$ | $\left\{({s}_{5},0)\right\}$ | $\{({s}_{7},0),({s}_{8},0)\}$ | $\{({s}_{6},0),({s}_{6},0)\}$ | $\{({s}_{6},0),({s}_{6},0)\}$ |

| ${A}_{4}$ | $\left\{({s}_{6},0)\right\}$ | $\{({s}_{7},0),({s}_{7},0)\}$ | $\left\{({s}_{7},0)\right\}$ | $\{({s}_{6},0),({s}_{7},0),({s}_{8},0)\}$ |

${E}_{2}$ | ${A}_{1}$ | $\{({s}_{7},0),({s}_{8},0)\}$ | $\left\{({s}_{6},0)\right\}$ | $\{({s}_{5},0),({s}_{5},0)\}$ | $\{({s}_{5},0),({s}_{6},0)\}$ |

| ${A}_{2}$ | $\{({s}_{7},0),({s}_{7},0)\}$ | $\{({s}_{5},0),({s}_{6},0),({s}_{7},0)\}$ | $\{({s}_{5},0),({s}_{5},0)\}$ | $\left\{({s}_{5},0)\right\}$ |

| ${A}_{3}$ | $\left\{({s}_{8},0)\right\}$ | $\{({s}_{8},0),({s}_{8},0)\}$ | $\{({s}_{5},0),({s}_{6},0)\}$ | $\{({s}_{3},0),({s}_{3},0)\}$ |

| ${A}_{4}$ | $\left\{({s}_{5},0)\right\}$ | $\{({s}_{7},0),({s}_{8},0)\}$ | $\left\{({s}_{7},0)\right\}$ | $\{({s}_{5},0),({s}_{5},0),({s}_{5},0)\}$ |

${E}_{3}$ | ${A}_{1}$ | $\{({s}_{6},0),({s}_{7},0)\}$ | $\left\{({s}_{7},0)\right\}$ | $\{({s}_{6},0),({s}_{6},0)\}$ | $\{({s}_{5},0),({s}_{5},0)\}$ |

| ${A}_{2}$ | $\{({s}_{4},0),({s}_{4},0)\}$ | $\{({s}_{6},0),({s}_{6},0),({s}_{6},0)\}$ | $\{({s}_{6},0),({s}_{7},0)\}$ | $\left\{({s}_{6},0)\right\}$ |

| ${A}_{3}$ | $\left\{({s}_{6},0)\right\}$ | $\{({s}_{7},0),({s}_{8},0)\}$ | $\{({s}_{5},0),({s}_{5},0)\}$ | $\{({s}_{6},0),({s}_{7},0)\}$ |

| ${A}_{4}$ | $\left\{({s}_{7},0)\right\}$ | $\{({s}_{6},0),({s}_{7},0)\}$ | $\left\{({s}_{7},0)\right\}$ | $\{({s}_{7},0),({s}_{7},0),({s}_{7},0)\}$ |

**Step 4.3:** Based on Equations (

11)–(

13), the values of

$CD{E}_{ij,0}^{p}$,

$CD{A}_{i,0}^{p}$, and

$CD{I}_{0}^{p}$ are obtained.

From the results, we can see that $CD{I}_{0}^{2}<0.85$. Then, go on the next step.

**Step 4.4:** The elements that need to be revised are obtained by Equations (

14)–(

16), that is,

$AES=\{(2,3,1),(2,3,4)\}$.

**Step 4.5:** According to Equation (

17), the modification suggestions are generated. Then, let

$\kappa =1$, and go to Step 4.2.

First iteration.

**Step 4.2:** Transform the hesitant fuzzy linguistic evaluation matrices ${\overline{H}}_{{S}_{ij,1}}^{p}$ into hesitant 2-tuple linguistic assessment matrices ${\overline{H}}_{{T}_{ij,1}}^{p}$.

**Step 4.3:** The values of

$CD{E}_{ij,1}^{p}$,

$CD{A}_{i,1}^{p}$, and

$CD{I}_{1}^{p}$ are obtained according to Equations (

11)–(

13).

Based on the above results, we know that $CD{I}_{1}^{2}<0.85$. Then, go to the next step.

**Step 4.4:** The elements that need to be modified are obtained based on Equations (

14)–(

16), that is,

$AES=\{(2,2,1),(2,2,3)\}$.

**Step 4.5:** According to Equation (

17), the modification suggestions are generated. Then, let

$\kappa =2$, and go to Step 4.2.

Second iteration.

**Step 4.2:** Transform the hesitant fuzzy linguistic evaluation matrices ${\overline{H}}_{{S}_{ij,2}}^{p}$ into hesitant 2-tuple linguistic assessment matrices ${\overline{H}}_{{T}_{ij,2}}^{p}$.

**Step 4.3:** The values of

$CD{E}_{ij,2}^{p}$,

$CD{A}_{i,2}^{p}$, and

$CD{I}_{2}^{p}$ are obtained based on Equations (

11)–(

13).

According to the results, we know that $CD{I}_{2}^{p}>0.85$ for any p. Then, go to Step 4.6.

**Step 4.6:** Output the revised individual hesitant 2-tuple linguistic evaluation matrices ${H}_{{T}_{ij}}^{p}$, the consensus measure $CD{I}^{p}=0.8620$, and the number of iterations $\kappa =2$.

**Step 5:** Construct the group evaluation matrix.

**Step 5.1:** According to Equations (

18) and (

19), we obtain

${T}_{ij}^{p}$ and

${W}_{ij}^{p}$ as follows:

**Step 5.2:** The group evaluation matrix is obtained by Equation (

21), and the results are shown in

Table A3.

**Table A3.**
Group evaluation matrix.

**Table A3.**
Group evaluation matrix.

${\mathit{A}}_{\mathit{i}}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|

${A}_{1}$ | $\left\{\begin{array}{c}({s}_{7},-0.0380),\hfill \\ ({s}_{7},0),\hfill \\ ({s}_{7},0.0025),\hfill \\ ({s}_{7},0.0406)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{7},-0.0432)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{5},0.0254),\hfill \\ ({s}_{6},-0.0406)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{4},0.0572),\hfill \\ ({s}_{5},-0.0339)\hfill \end{array}\right\}$ |

${A}_{2}$ | $\left\{\begin{array}{c}({s}_{5},-0.0541),\hfill \\ ({s}_{5},0.0089)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{6},-0.0597),\hfill \\ ({s}_{6},-0.0224),\hfill \\ ({s}_{5},0.0280)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{6},0),\hfill \\ ({s}_{6},0.0304)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{6},-0.0422)\hfill \end{array}\right\}$ |

${A}_{3}$ | $\left\{\begin{array}{c}({s}_{6},-0.0216)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{7},0.0408),\hfill \\ ({s}_{8},-0.0435),\hfill \\ ({s}_{8},-0.0408),\hfill \\ ({s}_{8},0)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{5},0.0552),\hfill \\ ({s}_{6},-0.0284)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{5},0.0368),\hfill \\ ({s}_{5},0.0588)\hfill \end{array}\right\}$ |

${A}_{4}$ | $\left\{\begin{array}{c}({s}_{6},-0.0158)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{7},-0.0380),\hfill \\ ({s}_{7},0),\hfill \\ ({s}_{7},0.0025),\hfill \\ ({s}_{7},0.0406)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{7},0)\hfill \end{array}\right\}$ | $\left\{\begin{array}{c}({s}_{6},-0.0169),\hfill \\ ({s}_{6},0.0347),\hfill \\ ({s}_{7},-0.0387)\hfill \end{array}\right\}$ |

**Step 6:** Obtain the collective evaluation value of each green supplier.

**Step 6.1:** Calculate the fuzzy measures of criteria and the value of

${W}_{ij}$. We suppose that

$\mu ({C}_{1})=0.5$,

$\mu ({C}_{2})=0.2$,

$\mu ({C}_{3})=0.4$, and

$\mu ({C}_{4})=0.3$. Then,

$\lambda =-0.65$ is obtained based on Equation (

23). According to the three axioms of the fuzzy measure

$\mu $ [

71], the fuzzy measures of criteria set are obtained, that is,

$\mu ({C}_{1},{C}_{2})=0.635$,

$\mu ({C}_{1},{C}_{3})=0.77$,

$\mu ({C}_{1},{C}_{4})=0.7025$,

$\mu ({C}_{2},{C}_{3})=0.548$,

$\mu ({C}_{2},{C}_{4})=0.461$,

$\mu ({C}_{3},{C}_{4})=0.622$,

$\mu ({C}_{1},{C}_{2},{C}_{3})=0.8699$,

$\mu ({C}_{1},{C}_{2},{C}_{4})=\mathrm{0.0.8112}$,

$\mu ({C}_{1},{C}_{3},{C}_{4})=0.9199$,

$\mu ({C}_{2},{C}_{3},{C}_{4})=0.7411$,

$\mu ({C}_{1},{C}_{2},{C}_{3},{C}_{4})=1$. Further, the matrix

${W}_{ij}$ is obtained based on Equation (

24).

**Step 6.2:** The comprehensive evaluation value of each green supplier is aggregated by Equation (

25).

${A}_{1}=\left\{\begin{array}{c}({s}_{6},0.0061),({s}_{6},0.0105),({s}_{6},0.0199),({s}_{6},0.0243),({s}_{6},0.0251),({s}_{6},0.0295),({s}_{6},0.0389),\hfill \\ ({s}_{6},0.0433),({s}_{6},0.0263),({s}_{6},0.0307),({s}_{6},0.0402),({s}_{6},0.0446),({s}_{6},0.0454),({s}_{6},0.0498),\hfill \\ ({s}_{6},0.0592),({s}_{7},-0.0614)\hfill \end{array}\right\}$

${A}_{2}=\left\{\begin{array}{c}({s}_{5},0.0577),({s}_{6},-0.0551),({s}_{6},-0.0510),({s}_{6},-0.0388),({s}_{5},0.0622),({s}_{6},-0.0507),\hfill \\ ({s}_{6},-0.0465),({s}_{6},-0.0344),({s}_{6},-0.0584),({s}_{6},-0.0462),({s}_{6},-0.0421),({s}_{6},-0.0299)\hfill \end{array}\right\}$

${A}_{3}=\left\{\begin{array}{c}({s}_{6},0.0225),({s}_{6},0.0253),({s}_{6},0.0056),({s}_{6},0.0085),({s}_{6},0.0040),({s}_{6},0.0069),({s}_{6},0.0138),\hfill \\ ({s}_{6},0.0166),({s}_{6},0.0046),({s}_{6},0.0074),({s}_{6},0.0143),({s}_{6},0.0172),({s}_{6},0.0127),({s}_{6},0.0156),\hfill \\ ({s}_{6},-0.0041),({s}_{6},-0.0012)\hfill \end{array}\right\}$

${A}_{4}=\left\{\begin{array}{c}({s}_{6},0.0535),({s}_{6},0.0611),({s}_{6},0.0616),({s}_{7},-0.0557),({s}_{7},-0.0615),({s}_{7},-0.0539),\hfill \\ ({s}_{7},-0.0534),({s}_{7},-0.0458),({s}_{7},-0.0515),({s}_{7},-0.0439),({s}_{7},-0.0434),({s}_{7},-0.0358)\hfill \end{array}\right\}$

**Step 7:** Determine the ranking order of green suppliers. According to Equation (

8), the comprehensive score values of green suppliers are obtained, that is,

$S({A}_{1})=0.7848$,

$S({A}_{2})=0.7014$,

$S({A}_{3})=0.7606$,

$S({A}_{4})=0.8214$. Then, we obtain the priority order of green suppliers, which is

${A}_{4}>{A}_{1}>{A}_{3}>{A}_{2}$.