# Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method

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## Abstract

**:**

## 1. Introduction

- Although the MCDM method has been applied in many fields, it has not been applied to the selection of BPM tools. Therefore, the first important constitution of this study is to propose criteria for the selection of BPM tools and an MCDM method for the selection of BPM tools.
- In the BPM tool selection process for a company, there will be direct and indirect interdependence between all the criteria. Therefore, the second important conclusion of this paper is to use DEMATEL analysis to fix the direct and indirect influence problem between criteria in the BPM tool selection process.
- When experts define the affecting rank between BPM tool selection criteria, there is uncertainty here because experts cannot clearly determine the impact of a specific scale value. Therefore, the third objective of this paper is to approach the hybrid fuzzy (uncertainty) decision-making issue.

## 2. Literature Review

- Namely, early or activity-centered languages: NIAM, IDEFx, MERISE, IEM, GRAI, and so on.
- Business process-centered languages: CIMOSA, ARIS, IEM, BPMN, EPC, Petrinet, and so on.
- Enterprise knowledge-centered languages: 4EM, DEMO, etc.

- The pre- and post-steps of the TOPSIS method are logical and easy to comprehend.
- The calculation steps at a glance.
- The method can use straightforward mathematical criteria descriptions to find optimal candidate options.
- The significance weights are considered in the decision-making process.

## 3. Business Process Modeling Tool Selection Methodology

#### 3.1. Determining the Criteria to Be Used in Evaluation for BPM Tool Alternatives

_{1}), Economical parameters (C

_{2}), Time parameters (C

_{3}), and Safety parameters(C

_{4})), and every type of criterion also has different corresponding sub-criteria. A detailed description of all these criteria can be seen as follows:

- Efficiency parameters (C
_{1}) are related to the factors affecting BPM efficiency. Efficiency parameters include Expressiveness, Readability, Usability, Formality, and Ease of Learning.- Expressiveness (C
_{11}): This parameter checks whether the modeling tool can express various kinds of organizational environments on the basis of informational, structural, behavioral, and functional perspectives [62]. - Readability (C
_{12}): This parameter checks whether the model is simple to comprehend for stakeholders. - Usability (C
_{13}): This parameter checks whether the modeling tool is easy to apply and install. - Formality (C
_{14}): This parameter checks whether the model has ambiguities and inaccuracies in model interpretation. - Ease of Learning (C
_{15}): This parameter checks whether the modeling tool and language are easy for the company modelers to learn.

- Economical parameters (C
_{2}) related to the various expenses incurred when using the tool. Economical parameters include application and installment costs, operating costs, and training costs.- Application and installment cost (C
_{21}): The cost of parameter configuration, application, and installation when starting to use the modeling tool in the company. - Operating cost (C
_{22}): The rental cost of the modeling tool and the salary of the company modelers. - Training cost (C
_{23}): Training costs of modeling tools for company modelers.

- Time parameters (C
_{3}) cluster includes application and installment time, operating time, and training time.- Application and installment time (C
_{31}): The time consumption of parameter configuration, application, and installation when starting to use the modeling tool in the company. - Operating time (C
_{32}): The time consumed by company modelers operating the modeling tool. - Training time (C
_{33}): The time consumed by company modelers to learn the modeling language and tool.

- Safety parameters (C
_{4}) include all parameters that the BPM tool can affect the safety of the IS of the company. Safety parameters include Internal safety and External safety.- Internal safety (C
_{41}): The safety of BPM tools inside the company, such as the software freezes, disappearance, and error storing modeling data. - External safety (C
_{42}): The safety of the BPM tool outside of the company, for example, if the tool is vulnerable to network intrusion and whether the modeling data are easily leaked to the outside through the tool.

#### 3.2. Determining the Fuzzy Weight for the Criteria

_{A}(x) of TFN is expressed as Equation (1) and Figure 2.

_{A}(y) value will vary between 0 and 1.

- Step 1: Collect the opinions of company experts for direct influence between criteria.

- (1)
- Normalization:$${\mathrm{nf}}_{\mathrm{ij}}{}^{\mathrm{q}}=\left({\mathrm{f}}_{\mathrm{ij}}{}^{\mathrm{q}}-{\mathrm{min}\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)/\left(\mathrm{max}{\mathrm{f}}_{\mathrm{ij}}{}^{\mathrm{q}}-\mathrm{min}{\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$$${\mathrm{ne}}_{\mathrm{ij}}{}^{\mathrm{q}}=\left({\mathrm{e}}_{\mathrm{ij}}{}^{\mathrm{q}}-{\mathrm{min}\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)/\left(\mathrm{max}{\mathrm{f}}_{\mathrm{ij}}{}^{\mathrm{q}}-\mathrm{min}{\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$$${\mathrm{nd}}_{\mathrm{ij}}{}^{\mathrm{q}}=\left({\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}-{\mathrm{min}\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)/\left(\mathrm{max}{\mathrm{f}}_{\mathrm{ij}}{}^{\mathrm{q}}-\mathrm{min}{\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$
- (2)
- Calculate right and left normalized numbers:$${\mathrm{rn}}_{\mathrm{ij}}{}^{\mathrm{q}}={\mathrm{nf}}_{\mathrm{ij}}{}^{\mathrm{q}}/\left(1+{\mathrm{nf}}_{\mathrm{ij}}{}^{\mathrm{q}}-{\mathrm{ne}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$$${\mathrm{ln}}_{\mathrm{ij}}{}^{\mathrm{q}}={\mathrm{ne}}_{\mathrm{ij}}{}^{\mathrm{q}}/\left(1+{\mathrm{ne}}_{\mathrm{ij}}{}^{\mathrm{q}}-{\mathrm{nd}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$
- (3)
- Calculate total generalized crisp numbers:$${\mathrm{tn}}_{\mathrm{ij}}{}^{\mathrm{q}}=\left[{\mathrm{ln}}_{\mathrm{ij}}{}^{\mathrm{q}}\left(1-{\mathrm{ln}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)+{\mathrm{rn}}_{\mathrm{ij}}{}^{\mathrm{q}}\times {\mathrm{rn}}_{\mathrm{ij}}{}^{\mathrm{q}}\right]/\left(1-{\mathrm{ln}}_{\mathrm{ij}}{}^{\mathrm{q}}+{\mathrm{rn}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$
- (4)
- Calculate crisp numbers:$${\mathrm{p}}_{\mathrm{ij}}{}^{\mathrm{q}}=\mathrm{min}{\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}+{\mathrm{tn}}_{\mathrm{ij}}{}^{\mathrm{q}}\times \left(\mathrm{max}{\mathrm{f}}_{\mathrm{ij}}{}^{\mathrm{q}}-\mathrm{min}{\mathrm{d}}_{\mathrm{ij}}{}^{\mathrm{q}}\right)$$

_{mn}] (Equation (9)).

- Step 2: Calculate average fuzzy matrix T = [t
_{nj}] (Equation (10)).

- Step 3: Calculate the normalized direct fuzzy impact matrix G.

- Step 4: Calculate the overall direct and indirect fuzzy impact matrix E.

- Step 5: Calculate the sums of the rows and columns of matrix H.

- Step 6: Calculate the normalized (${\mathrm{ro}}_{\mathrm{n}}+{\mathrm{co}}_{\mathrm{j}}$) value.$${\mathrm{NRC}}_{\mathrm{l}}=\frac{\left({\mathrm{ro}}_{\mathrm{l}}+{\mathrm{co}}_{\mathrm{l}}\right)}{{{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{J}}\left({\mathrm{ro}}_{\mathrm{j}}+{\mathrm{co}}_{\mathrm{j}}\right)}$$

#### 3.3. Determine the TOPSIS Result Value for All the Candidate BPM Tool Alternatives

- Step 7: Establish a decision matrix for alternatives (Equation (18)).$$D=\left[\begin{array}{c}{\mathrm{A}}_{1}\\ {\mathrm{A}}_{2}\\ \vdots \\ \mathrm{A}\\ \vdots \\ {\mathrm{A}}_{\mathrm{I}}\end{array}\right]\left[\begin{array}{cccccc}{\mathrm{C}}_{11}& {\mathrm{C}}_{12}& \cdots & {\mathrm{C}}_{1\mathrm{j}}& \cdots & {\mathrm{C}}_{1\mathrm{J}}\\ {\mathrm{C}}_{21}& {\mathrm{C}}_{22}& \cdots & {\mathrm{C}}_{2\mathrm{j}}& \cdots & {\mathrm{C}}_{2\mathrm{J}}\\ \vdots & \vdots & \cdots & \vdots & \cdots & \vdots \\ {\mathrm{C}}_{\mathrm{i}1}& {\mathrm{C}}_{\mathrm{j}2}& \cdots & {\mathrm{C}}_{\mathrm{i}\mathrm{j}}& \cdots & {\mathrm{C}}_{\mathrm{i}\mathrm{J}}\\ \vdots & \vdots & \cdots & \vdots & \cdots & \vdots \\ {\mathrm{C}}_{\mathrm{I}1}& {\mathrm{C}}_{\mathrm{I}2}& \cdots & {\mathrm{C}}_{\mathrm{I}\mathrm{j}}& \cdots & {\mathrm{C}}_{\mathrm{I}\mathrm{J}}\end{array}\right]$$
_{i}is alternative to i, ${\mathrm{c}}_{\mathrm{ij}}$ is the jth standard number corresponding to the ith alternative (A_{i}), I is the number of alternatives, and J is the number of criteria. - Step 8: Get the normalized decision matrix Z(=z
_{ij}) (Equation (19)).$${\mathrm{z}}_{\mathrm{ij}}=\frac{{\mathrm{c}}_{\mathrm{ij}}}{\sqrt{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{c}}_{\mathrm{ij}}{}^{2}}}$$ - Note. z
_{ij}= Normalized number for jth standard corresponding to ith alternative. I = Sum of candidate options. - Step 9: Obtain the weighted normalized decision matrix X(=x
_{ij}) (Equation (20)).$${\mathrm{x}}_{\mathrm{ij}}={\mathrm{w}}_{\mathrm{j}}\xb7{\mathrm{z}}_{\mathrm{ij}}$$

- Step 10: Decide the P-I and N-I solutions (Equations (21) and (22)).$$\mathrm{P-I}\mathrm{solution}:{{\mathrm{x}}_{\mathrm{j}}}^{*}=\{\begin{array}{c}\underset{\mathrm{i}}{\mathrm{max}}{\mathrm{x}}_{\mathrm{ij}},\mathrm{i}\in {\mathrm{l}}^{\prime}\\ \underset{\mathrm{i}}{\mathrm{min}}{\mathrm{x}}_{\mathrm{ij}},i\in {\mathrm{l}}^{\u2033}\end{array}$$$$\mathrm{N-I}\mathrm{solution}:{\mathrm{x}}_{\mathrm{j}}{}^{0}=\{\begin{array}{c}\underset{\mathrm{i}}{\mathrm{min}}{\mathrm{x}}_{\mathrm{ij}},\mathrm{i}\in {\mathrm{l}}^{\prime}\\ \underset{\mathrm{i}}{\mathrm{max}}{\mathrm{x}}_{\mathrm{ij}},i\in {\mathrm{l}}^{\u2033}\end{array}$$
- Step 11: Calculate the n-dimensional Euclidean distance from each solution to the P-I solution and the N-I solution (Equations (22) and (23)).$$\mathrm{Distance}\mathrm{to}\mathrm{P-I}\mathrm{solution}:{\mathrm{d}}_{\mathrm{i}}{}^{*}=\sqrt{{{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{J}}{\left({\mathrm{x}}_{\mathrm{ij}}-{\mathrm{x}}_{\mathrm{j}}{}^{*}\right)}^{2}}$$$$\mathrm{Distance}\mathrm{to}\mathrm{N-I}\mathrm{solution}:{\mathrm{d}}_{\mathrm{i}}{}^{0}=\sqrt{{{\displaystyle \sum}}_{\mathrm{j}=1}^{\mathrm{J}}{\left({\mathrm{x}}_{\mathrm{ij}}-{\mathrm{x}}_{\mathrm{j}}{}^{0}\right)}^{2}}$$
- Step 12: Calculate the relative closeness to the idea solution (Equation (25)).$${\mathrm{H}}_{\mathrm{i}}{}^{*}=\frac{{\mathrm{d}}_{\mathrm{i}}{}^{0}}{\left({\mathrm{d}}_{\mathrm{i}}{}^{0}+{\mathrm{d}}_{\mathrm{i}}{}^{*}\right)}$$
- Step 13: According to the order of the H
_{i}* number, determine the capability of the alternatives. A higher H_{i}* number indicates a better alternative capability. Then, we can rank the alternatives depending on the H_{i}* numbers for the purpose of showing the performance comparison results for all the alternatives.

## 4. Results

_{1}, C

_{2}, C

_{3}and C

_{4}) and corresponding sub-parameters (C

_{11}, C

_{12}, C

_{13}, C

_{14}, C

_{15}, C

_{21}, C

_{22}, C

_{23}, C

_{31}, C

_{32}, C

_{33}, C

_{41}and C

_{42}) in Table 1.

_{1}to C

_{3}is 0.545 which is obtained from the average fuzzy influence value of 10 experts ($\frac{0.73+0.5+0.27+0.27+0.25+0.73+0.5+0.5+0.73+0.97}{10}$ = 0.545) (Equation (10)). After that, depending on the Equations (11) and (12), we can calculate the normalized direct influence matrix T (Table 10, Table 11, Table 12, Table 13 and Table 14) for all the criterion and sub-criterions.

_{1}to C

_{3}is 0.262, which is obtained from the multiplication between the direct influence rank from C

_{1}to C

_{3}(0.545 in Table 5) and the $\lambda $ value ($\frac{1}{\mathrm{max}1\le \mathrm{n}\le 4{{\displaystyle \sum}}_{\mathrm{j}=1}^{4}{\mathrm{t}}_{\mathrm{nj}}}=\frac{1}{\mathrm{max}\left({{\displaystyle \sum}}_{\mathrm{j}=1}^{4}{\mathrm{t}}_{1\mathrm{j}},{{\displaystyle \sum}}_{\mathrm{j}=1}^{4}{\mathrm{t}}_{2\mathrm{j}},{{\displaystyle \sum}}_{\mathrm{j}=1}^{4}{\mathrm{t}}_{3\mathrm{j}},{{\displaystyle \sum}}_{\mathrm{j}=1}^{4}{\mathrm{t}}_{4\mathrm{j}}\right)}=\frac{1}{\mathrm{max}\left(1.819,1.656,1.679,2.081\right)}=0.481$) for the average direct influence matrix (Table 5). After that, depending on the Equations (13)–(16), we can obtain the total direct and indirect influence matrix E (Table 15, Table 16, Table 17, Table 18 and Table 19) and the corresponding sum of rows (or in Equation (15)) and columns (co in Equation (16)) of the E.

_{n}(highlighted in yellow) means the total given both direct and indirect effects from criteria n to the other criteria (for example, in Table 15, ro

_{1}= 1.484 + 1.852 + 1.614 + 1.696 = 6.646), and co

_{j}(highlighted in pink) means the total received both direct and indirect effects from other criteria to criterion j (for example, in Table 15, co1 = 1.484 + 1.592 + 1.585 + 1.874 = 6.535). After, we obtain the overall direct and indirect impact matrix E with corresponding ro

_{n}and co

_{j}values, when n equals j, we can release the (ro

_{j}+ co

_{j}) (highlighted in lime green) value, which means the centrality of criterion j, for all the criteria (Table 15, Table 16, Table 17, Table 18 and Table 19).

_{1}) is 0.251, which is obtained by dividing ro

_{1}+ co

_{1}(Efficiency parameters (C

_{1}) in Table 15 by the sum of column ro

_{n}+ co

_{j}in Table 15 ($\frac{13.181}{6.646+6.172+6.153+7.317}=0.251$). After that, the final weight of sub-parameter is the multiplication of the weights of the four main and corresponding sub-criteria.

_{11}is 0.009. The number is obtained by multiplying the final weight of sub-criteria C

_{11}(0.251 × 0.194 = 0.049) and the normalized decision matrix value for C

_{11}($0.190=\frac{2}{\sqrt{{{\displaystyle \sum}}_{\mathrm{i}=1}^{24}{\mathrm{y}}_{\mathrm{i}1}{}^{2}}}$).

_{i}* value is obtained by the relative closeness to the ideal solution (Equation (25)). For example, the value for the C

_{i}* value for T1 is 0.405 (${\mathrm{H}}_{\mathrm{i}}{}^{*}=\frac{{\mathrm{d}}_{\mathrm{i}}{}^{0}}{\left({\mathrm{d}}_{\mathrm{i}}{}^{0}+{\mathrm{d}}_{\mathrm{i}}{}^{*}\right)}=\frac{0.06}{\left(0.06+0.088\right)}=0.405$). Meanwhile, in Table 23, di* is the n-dimensional Euclidean distance from each solution in Table 21 to the P-I solution (T

^{+}Table 22), and d

_{i}

^{0}is the n-dimensional Euclidean distance from each solution in Table 21 to the N-I solution (T

^{−}Table 22). For example, the d

_{i}

^{*}value for T1 is 0.88 ($\sqrt{{\displaystyle \sum}_{\mathrm{j}=1}^{13}{\left({\mathrm{x}}_{\mathrm{ij}}-{\mathrm{x}}_{\mathrm{j}}{}^{*}\right)}^{2}}=$ $\sqrt{\begin{array}{c}{\left(0.009-0.023\right)}^{2}+{\left(0.015-0.025\right)}^{2}+{\left(0.011-0.027\right)}^{2}+{\left(0.019-0.025\right)}^{2}+{\left(0.025-0.025\right)}^{2}+{\left(0.012-0.012\right)}^{2}+\\ {\left(0.022-0.011\right)}^{2}+{\left(0.021-0.011\right)}^{2}+{\left(0.003-0.001\right)}^{2}+{\left(0.027-0.009\right)}^{2}+{\left(0.029-0.001\right)}^{2}+{\left(0.014-0.071\right)}^{2}+\\ +{\left(0.013-0.063\right)}^{2}\end{array}}$= 0.88). Meanwhile, the d

_{i}

^{0}value for T1 is 0.06 ($\sqrt{{\displaystyle \sum}_{\mathrm{j}=1}^{13}{\left({\mathrm{x}}_{\mathrm{ij}}-{\mathrm{x}}_{\mathrm{j}}{}^{0}\right)}^{2}}=$ $\sqrt{\begin{array}{c}{\left(0.009-0.009\right)}^{2}+{\left(0.015-0.01\right)}^{2}+{\left(0.011-0.05\right)}^{2}+{\left(0.019-0.012\right)}^{2}+{\left(0.025-0.001\right)}^{2}+{\left(0.012-0.047\right)}^{2}+\\ {\left(0.022-0.045\right)}^{2}+{\left(0.021-0.042\right)}^{2}+{\left(0.003-0.05\right)}^{2}+{\left(0.027-0.046\right)}^{2}+{\left(0.029-0.048\right)}^{2}+{\left(0.014-0.014\right)}^{2}+\\ +{\left(0.013-0.013\right)}^{2}\end{array}}$ = 0.06). From the C

_{i}* value in Table 23, it is possible to find that the alternative T4 obtains the highest value (0.642) (highlighted in green). Here, when considering the four types of criteria (BPM efficiency (C

_{1}), various expenses incurred when using the BPM tool (C

_{2}), application, installment, operating and training time (C

_{3}), and security issues affected by BPM tools (C

_{4})), alternative T4 is the best overall performing BPM tool. Therefore, managers can select BPM tool T4 for the company.

## 5. Discussion and Conclusions

_{1}), economical parameters (C

_{2}), time parameters (C

_{3}) and safety parameters (C

_{4})) and corresponding sub-parameters (expressiveness (C

_{11}), readability (C

_{12}) usability (C

_{13}), formality (C

_{14}), ease of learning (C

_{15}), application and installation cost (C

_{21}), operating cost (C

_{22}), training cost (C

_{23}), application and installation time (C

_{31}), operating time (C

_{32}), training time (C

_{33}), internal safety (C

_{41}), and external safety (C

_{42})) in a BPM tool choosing issue, causing various ambiguous information. Thus, an optimal assessment method is needed to make effective decisions. Therefore, the BPM tool selection methodology is proposed in this study with the consideration of different BPM tool selection influencing factors.

_{11}) and formality (C

_{14}) criteria can be removed and replaced by the data control or evaluation criteria. Meanwhile, although some criteria do not need to be changed, the contents inside the criteria need to be redefined.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

E1 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E2 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E3 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E4 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E5 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Efficiency (C_{1}) | 0 | 2 | 3 | 3 | C_{1} | 0 | 2 | 2 | 4 | C_{1} | 0 | 3 | 1 | 2 | C_{1} | 0 | 3 | 1 | 3 | C_{1} | 0 | 1 | 1 | 1 |

Economical (C_{2}) | 3 | 0 | 2 | 2 | C_{2} | 2 | 0 | 3 | 1 | C_{2} | 3 | 0 | 2 | 3 | C_{2} | 1 | 0 | 2 | 2 | C_{2} | 2 | 0 | 1 | 2 |

Time (C_{3}) | 1 | 2 | 0 | 1 | C_{3} | 3 | 4 | 0 | 2 | C_{3} | 3 | 3 | 0 | 1 | C_{3} | 1 | 1 | 0 | 2 | C_{3} | 3 | 1 | 0 | 2 |

Safety (C_{4}) | 1 | 4 | 2 | 0 | C_{4} | 1 | 2 | 3 | 0 | C_{4} | 3 | 4 | 2 | 0 | C_{4} | 1 | 3 | 2 | 0 | C_{4} | 2 | 2 | 2 | 0 |

E6 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E7 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E8 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E9 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) | E10 | C_{1}(DFN) | C_{2}(DFN) | C_{3}(DFN) | C_{4}(DFN) |

C_{1} | 0 | 1 | 3 | 4 | C_{1} | 0 | 3 | 2 | 3 | C_{1} | 0 | 3 | 2 | 3 | C_{1} | 0 | 3 | 3 | 3 | C_{1} | 0 | 3 | 4 | 2 |

C_{2} | 2 | 0 | 4 | 3 | C_{2} | 3 | 0 | 1 | 3 | C_{2} | 3 | 0 | 1 | 3 | C_{2} | 3 | 0 | 1 | 3 | C_{2} | 1 | 0 | 2 | 3 |

C_{3} | 3 | 4 | 0 | 1 | C_{3} | 2 | 4 | 0 | 3 | C_{3} | 2 | 3 | 0 | 2 | C_{3} | 2 | 2 | 0 | 1 | C_{3} | 3 | 4 | 0 | 2 |

C_{4} | 4 | 3 | 3 | 0 | C_{4} | 4 | 3 | 4 | 0 | C_{4} | 4 | 3 | 4 | 0 | C_{4} | 4 | 3 | 3 | 0 | C_{4} | 3 | 4 | 2 | 0 |

E1 | C_{11}(DFN) | C_{12}(DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E2 | C_{11}(DFN) | C_{12}(DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E3 | C_{11}(DFN) | C_{12}(DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E4 | C_{11}(DFN) | C_{12}(DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E5 | C_{11}(DFN) | C_{12}(DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

EX (C_{11}) | 0 | 1 | 1 | 1 | 2 | C_{11} | 0 | 2 | 3 | 1 | 3 | C_{11} | 0 | 1 | 2 | 3 | 3 | C_{11} | 0 | 2 | 1 | 2 | 1 | C_{11} | 0 | 0 | 3 | 0 | 2 |

RE (C_{12}) | 2 | 0 | 2 | 4 | 2 | C_{12} | 1 | 0 | 1 | 3 | 1 | C_{12} | 2 | 0 | 1 | 4 | 2 | C_{12} | 3 | 0 | 2 | 3 | 2 | C_{12} | 2 | 0 | 1 | 4 | 2 |

US (C_{13}) | 1 | 1 | 0 | 3 | 3 | C_{13} | 2 | 2 | 0 | 3 | 3 | C_{13} | 1 | 2 | 0 | 3 | 3 | C_{13} | 3 | 1 | 0 | 4 | 3 | C_{13} | 0 | 2 | 0 | 3 | 3 |

FO (C_{14}) | 2 | 3 | 2 | 0 | 4 | C_{14} | 1 | 3 | 2 | 0 | 4 | C_{14} | 1 | 2 | 0 | 0 | 3 | C_{14} | 4 | 3 | 2 | 0 | 3 | C_{14} | 2 | 3 | 1 | 0 | 1 |

EL (C_{15}) | 4 | 1 | 0 | 2 | 0 | C_{15} | 3 | 2 | 2 | 0 | 0 | C_{15} | 3 | 1 | 0 | 3 | 0 | C_{15} | 4 | 1 | 0 | 2 | 0 | C_{15} | 4 | 2 | 2 | 0 | 0 |

E6 | C_{11}(DFN) | C_{12} (DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E7 | C_{11}(DFN) | C_{12} (DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E8 | C_{11}(DFN) | C_{12} (DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E9 | C_{11}(DFN) | C_{12} (DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) | E10 | C_{11}(DFN) | C_{12} (DFN) | C_{13}(DFN) | C_{14}(DFN) | C_{15}(DFN) |

C_{11} | 0 | 2 | 3 | 2 | C_{11} | 0 | 1 | 3 | 1 | 2 | C_{11} | 0 | 2 | 1 | 3 | 2 | C_{11} | 0 | 1 | 2 | 2 | 2 | C_{11} | 0 | 3 | 2 | 1 | 3 | |

C_{12} | 2 | 0 | 2 | 4 | C_{12} | 2 | 0 | 1 | 4 | 2 | C_{12} | 2 | 0 | 2 | 4 | 2 | C_{12} | 1 | 0 | 2 | 4 | 3 | C_{12} | 1 | 0 | 3 | 2 | 3 | |

C_{13} | 1 | 1 | 0 | 3 | C_{13} | 2 | 1 | 0 | 3 | 3 | C_{13} | 1 | 1 | 0 | 3 | 3 | C_{13} | 3 | 3 | 0 | 3 | 3 | C_{13} | 3 | 1 | 0 | 3 | 2 | |

C_{14} | 2 | 3 | 2 | 0 | C_{14} | 3 | 3 | 1 | 0 | 4 | C_{14} | 1 | 2 | 2 | 0 | 3 | C_{14} | 1 | 3 | 2 | 0 | 3 | C_{14} | 3 | 3 | 2 | 0 | 4 | |

C_{15} | 3 | 2 | 2 | 1 | C_{15} | 4 | 1 | 0 | 1 | 0 | C_{15} | 3 | 1 | 0 | 1 | 0 | C_{15} | 3 | 1 | 0 | 2 | 0 | C_{15} | 3 | 2 | 0 | 2 | 0 |

E1 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E2 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E3 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E4 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E5 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Application and Installment cost (C_{21}) | 0 | 2 | 3 | C_{21} | 0 | 1 | 2 | C_{21} | 0 | 2 | 2 | C_{21} | 0 | 1 | 3 | C_{21} | 0 | 3 | 2 |

Operating cost (C_{22}) | 4 | 0 | 4 | C_{22} | 3 | 0 | 1 | C_{22} | 2 | 0 | 3 | C_{22} | 2 | 0 | 3 | C_{22} | 3 | 0 | 3 |

Training cost (C_{23}) | 3 | 3 | 0 | C_{23} | 2 | 2 | 0 | C_{23} | 1 | 2 | 0 | C_{23} | 1 | 3 | 0 | C_{23} | 2 | 3 | 0 |

E6 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E7 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E8 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E9 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) | E10 | C_{21}(DFN) | C_{22}(DFN) | C_{23}(DFN) |

C_{21} | 0 | 1 | 2 | C_{21} | 0 | 3 | 4 | C_{21} | 0 | 3 | 2 | C_{21} | 0 | 3 | 4 | C_{21} | 0 | 3 | 4 |

C_{22} | 3 | 0 | 3 | C_{22} | 3 | 0 | 2 | C_{22} | 3 | 0 | 3 | C_{22} | 3 | 0 | 3 | C_{22} | 3 | 0 | 3 |

C_{23} | 2 | 4 | 0 | C_{23} | 3 | 2 | 0 | C_{23} | 4 | 4 | 0 | C_{23} | 4 | 2 | 0 | C_{23} | 2 | 4 | 0 |

E1 | C_{31}(DFN) | C_{32}(DFN) | C_{33}(DFN) | E2 | C_{31}(DFN) | C_{32}(DFN) | C_{33}(DFN) | E3 | C_{31}(DFN) | C_{32}(DFN) | C_{33}(DFN) | E4 | C_{31}(DFN) | C_{32}(DFN) | C_{33}(DFN) | E5 | C_{31}(DFN) | C_{32}(DFN) | C_{33}(DFN) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Application and Installment time(C_{31}) | 0 | 1 | 2 | C_{31} | 0 | 0 | 1 | C_{31} | 0 | 1 | 1 | C_{31} | 0 | 1 | 2 | C_{31} | 0 | 0 | 2 |

Operating time (C_{32}) | 0 | 0 | 3 | C_{32} | 0 | 0 | 2 | C_{32} | 0 | 0 | 2 | C_{32} | 0 | 0 | 3 | C_{32} | 0 | 0 | 3 |

Training time (C_{33}) | 0 | 2 | 0 | C_{33} | 0 | 3 | 0 | C_{33} | 0 | 1 | 0 | C_{33} | 0 | 2 | 0 | C_{33} | 0 | 2 | 0 |

E6 | C_{31}(DFN) | C_{32} (DFN) | C_{33}(DFN) | E7 | C_{31}(DFN) | C_{32} (DFN) | C_{33}(DFN) | E8 | C_{31}(DFN) | C_{32} (DFN) | C_{33}(DFN) | E9 | C_{31}(DFN) | C_{32} (DFN) | C_{33}(DFN) | E10 | C_{31}(DFN) | C_{32} (DFN) | C_{33}(DFN) |

C_{31} | 0 | 1 | 2 | C_{31} | 0 | 0 | 2 | C_{31} | 0 | 0 | 2 | C_{31} | 0 | 1 | 2 | C_{31} | 0 | 0 | 3 |

C_{32} | 0 | 0 | 4 | C_{32} | 0 | 0 | 3 | C_{32} | 0 | 0 | 2 | C_{32} | 0 | 0 | 3 | C_{32} | 0 | 0 | 3 |

C_{33} | 0 | 2 | 0 | C_{33} | 0 | 2 | 0 | C_{33} | 0 | 3 | 0 | C_{33} | 0 | 3 | 0 | C_{33} | 0 | 3 | 0 |

E1 | C_{41}(DFN) | C_{42}(DFN) | E2 | C_{41}(DFN) | C_{42}(DFN) | E3 | C_{41}(DFN) | C_{42}(DFN) | E4 | C_{41}(DFN) | C_{42}(DFN) | E5 | C_{41}(DFN) | C_{42}(DFN) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Internal safety(C_{41}) | 0 | 1 | C_{41} | 0 | 2 | C_{41} | 0 | 1 | C_{41} | 0 | 2 | C_{41} | 0 | 3 |

External safety (C_{42}) | 3 | 0 | C_{42} | 2 | 0 | C_{42} | 3 | 0 | C_{42} | 3 | 0 | C_{42} | 3 | 0 |

E6 | C_{41}(DFN) | C_{42}(DFN) | E7 | C_{41}(DFN) | C_{42}(DFN) | E8 | C_{41}(DFN) | C_{42}(DFN) | E9 | C_{41}(DFN) | C_{42}(DFN) | E10 | C_{41}(DFN) | C_{42}(DFN) |

C_{41} | 0 | 2 | C_{41} | 0 | 1 | C_{41} | 0 | 2 | C_{41} | 0 | 2 | C_{41} | 0 | 2 |

C_{42} | 2 | 0 | C_{42} | 4 | 0 | C_{42} | 2 | 0 | C_{42} | 3 | 0 | C_{42} | 3 | 0 |

**Table A6.**The corresponding triangle fuzzy numbers in Table A1.

E1 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) | E2 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) |
---|---|---|---|---|---|---|---|---|---|

Efficiency (C_{1}) | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{1} | 0 | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | (0.75, 1, 1) |

Economical (C_{2}) | (0.5, 0.75, 1) | 0 | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | C_{2} | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) | (0, 0.25, 0.5) |

Time (C_{3}) | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | 0 | (0, 0.25, 0.5) | C_{3} | (0.5, 0.75, 1) | (0.75, 1, 1) | 0 | (0.25, 0.5, 0.75) |

Safety (C_{4}) | (0, 0.25, 0.5) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | 0 | C_{4} | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | 0 |

E3 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) | E4 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) |

C_{1} | 0 | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | C_{1} | 0 | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0.5, 0.75, 1) |

C_{2} | (0.5, 0.75, 1) | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | C_{2} | (0, 0.25, 0.5) | 0 | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) |

C_{3} | (0.5, 0.75, 1) | (0.5, 0.75, 1) | 0 | (0, 0.25, 0.5) | C_{3} | (0, 0.25, 0.5) | (0, 0.25, 0.5) | 0 | (0.25, 0.5, 0.75) |

C_{4} | (0.5, 0.75, 1) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | 0 | C_{4} | (0, 0.25, 0.5) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 |

E5 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) | E6 | E6 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) |

C_{1} | 0 | (0, 0.25, 0.5) | (0, 0.25, 0.5) | (0, 0.25, 0.5) | C_{1} | C_{1} | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) |

C_{2} | (0.25, 0.5, 0.75) | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | C_{2} | C_{2} | (0.25, 0.5, 0.75) | 0 | (0.75, 1, 1) |

C_{3} | (0.5, 0.75, 1) | (0, 0.25, 0.5) | 0 | (0.25, 0.5, 0.75) | C_{3} | C_{3} | (0.5, 0.75, 1) | (0.75, 1, 1) | 0 |

C_{4} | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | 0 | E6 | C_{4} | (0.75, 1, 1) | (0.5, 0.75, 1) | (0.5, 0.75, 1) |

E7 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) | E8 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) |

C_{1} | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | C_{1} | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) |

C_{2} | (0.5, 0.75, 1) | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) | C_{2} | (0.5, 0.75, 1) | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) |

C_{3} | (0.25, 0.5, 0.75) | (0.75, 1, 1) | 0 | (0.5, 0.75, 1) | C_{3} | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | 0 | (0.25, 0.5, 0.75) |

C_{4} | (0.75, 1, 1) | (0.5, 0.75, 1) | (0.75, 1, 1) | 0 | C_{4} | (0.75, 1, 1) | (0.5, 0.75, 1) | (0.75, 1, 1) | 0 |

E9 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) | E10 | C_{1} (TFN) | C_{2} (TFN) | C_{3} (TFN) | C_{4} (TFN) |

C_{1} | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{1} | 0 | (0.5, 0.75, 1) | (0.75, 1, 1) | (0.25, 0.5, 0.75) |

C_{2} | (0.5, 0.75, 1) | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) | C_{2} | (0, 0.25, 0.5) | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) |

C_{3} | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | 0 | (0, 0.25, 0.5) | C_{3} | (0.5, 0.75, 1) | (0.75, 1, 1) | 0 | (0.25, 0.5, 0.75) |

C_{4} | (0.75, 1, 1) | (0.5, 0.75, 1) | (0.5, 0.75, 1) | 0 | C_{4} | (0.5, 0.75, 1) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | 0 |

**Table A7.**The corresponding triangle fuzzy numbers in Table A2.

E1 | C11 (TFN) | C12 (TFN) | C13 (TFN) | C14 (TFN) | C15 (TFN) | E2 | C11 (TFN) | C12 (TFN) | C13 (TFN) | C14 (TFN) | C15 (TFN) |
---|---|---|---|---|---|---|---|---|---|---|---|

EX (C_{11}) | 0 | (0, 0.25, 0.5) | (0, 0.25, 0.5) | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | C_{11} | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0.5, 0.75, 1) |

RE (C_{12}) | (0.25, 0.5, 0.75) | 0 | (0.25, 0.5, 0.75) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | C_{12} | (0, 0.25, 0.5) | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) | (0, 0.25, 0.5) |

US (C_{13}) | (0, 0.25, 0.5) | (0, 0.25, 0.5) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{13} | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) |

FO (C_{14}) | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | (0.75, 1, 1) | C_{14} | (0, 0.25, 0.5) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | (0.75, 1, 1) |

EL (C_{15}) | (0.75, 1, 1) | (0, 0.25, 0.5) | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 | C_{15} | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | (0, 0, 0.25) | 0 |

E3 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) | E4 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) |

C_{11} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{11} | 0 | (0.25, 0.5, 0.75) | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | (0, 0.25, 0.5) |

C_{12} | (0.25, 0.5, 0.75) | 0 | (0, 0.25, 0.5) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | C_{12} | (0.5, 0.75, 1) | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) |

C_{13} | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{13} | (0.5, 0.75, 1) | (0, 0.25, 0.5) | 0 | (0.75, 1, 1) | (0.5, 0.75, 1) |

C_{14} | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) | C_{14} | (0.75, 1, 1) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) |

C_{15} | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0, 0, 0.25) | (0.5, 0.75, 1) | 0 | C_{15} | (0.75, 1, 1) | (0, 0.25, 0.5) | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 |

E5 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) | E6 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) |

C_{11} | 0 | (0, 0, 0.25) | (0.5, 0.75, 1) | (0, 0, 0.25) | (0.25, 0.5, 0.75) | C_{11} | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0.75, 1, 1) |

C_{12} | (0.25, 0.5, 0.75) | 0 | (0, 0.25, 0.5) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | C_{12} | (0.25, 0.5, 0.75) | 0 | (0.25, 0.5, 0.75) | (0.75, 1, 1) | (0.5, 0.75, 1) |

C_{13} | 0 | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{13} | (0, 0.25, 0.5) | (0, 0.25, 0.5) | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) |

C_{14} | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0, 0.25, 0.5) | 0 | (0, 0.25, 0.5) | C_{14} | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) |

C_{15} | (0.75, 1, 1) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | (0, 0, 0.25) | 0 | C_{15} | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | (0, 0.25, 0.5) | 0 |

E7 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) | E8 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) |

C_{11} | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | C_{11} | 0 | (0.25, 0.5, 0.75) | (0, 0.25, 0.5) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) |

C_{12} | (0.25, 0.5, 0.75) | 0 | (0, 0.25, 0.5) | (0.75, 1, 1) | (0.25, 0.5, 0.75) | C_{12} | (0.25, 0.5, 0.75) | 0 | (0.25, 0.5, 0.75) | (0.75, 1, 1) | (0.25, 0.5, 0.75) |

C_{13} | (0.25, 0.5, 0.75) | (0, 0.25, 0.5) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{13} | (0, 0.25, 0.5) | (0, 0.25, 0.5) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) |

C_{14} | (0.5, 0.75, 1) | (0.5, 0.75, 1) | (0, 0.25, 0.5) | 0 | (0.75, 1, 1) | C_{14} | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) |

C_{15} | (0.75, 1, 1) | (0, 0.25, 0.5) | 0 | (0, 0.25, 0.5) | 0 | C_{15} | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0, 0, 0.25) | (0, 0.25, 0.5) | 0 |

E9 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) | E10 | C_{11} (TFN) | C_{12} (TFN) | C_{13} (TFN) | C_{14} (TFN) | C_{15} (TFN) |

C_{11} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | C_{11} | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0, 0.25, 0.5) | (0.5, 0.75, 1) |

C_{12} | (0, 0.25, 0.5) | 0 | (0.25, 0.5, 0.75) | (0.75, 1, 1) | (0.5, 0.75, 1) | C_{12} | (0, 0.25, 0.5) | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) |

C_{13} | (0.5, 0.75, 1) | (0.5, 0.75, 1) | 0 | (0.5, 0.75, 1) | (0.5, 0.75, 1) | C_{13} | (0.5, 0.75, 1) | (0, 0.25, 0.5) | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) |

C_{14} | (0, 0.25, 0.5) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) | C_{14} | (0.5, 0.75, 1) | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | (0.75, 1, 1) |

C_{15} | (0.5, 0.75, 1) | (0, 0.25, 0.5) | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 | C_{15} | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 |

**Table A8.**The corresponding triangle fuzzy numbers in Table A3.

E1 | C_{21} (TFN) | C_{22} (TFN) | C_{23} (TFN) | E2 | C_{21} (TFN) | C_{22} (TFN) | C_{23} (TFN) |
---|---|---|---|---|---|---|---|

Application and Installment cost (C_{21}) | 0 | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | C_{21} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) |

Operating cost (C_{22}) | (0.75, 1, 1) | 0 | (0.75, 1, 1) | C_{22} | (0.5, 0.75, 1) | 0 | (0, 0.25, 0.5) |

Training cost (C_{23}) | (0.5, 0.75, 1) | (0.5, 0.75, 1) | 0 | C_{23} | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | 0 |

E3 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) | E4 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) |

C_{21} | 0 | (0.25, 0.5, 0.75) | (0.25, 0.5, 0.75) | C_{21} | 0 | (0, 0.25, 0.5) | (0.5, 0.75, 1) |

C_{22} | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) | C_{22} | (0.25, 0.5, 0.75) | 0 | (0.5, 0.75, 1) |

C_{23} | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | 0 | C_{23} | (0, 0.25, 0.5) | (0.5, 0.75, 1) | 0 |

E5 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) | E6 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) |

C_{21} | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | C_{21} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) |

C_{22} | (0.5, 0.75, 1) | 0 | (0.5, 0.75, 1) | C_{22} | (0.5, 0.75, 1) | 0 | (0.5, 0.75, 1) |

C_{23} | (0.25, 0.5, 0.75) | (0.5, 0.75, 1) | 0 | C_{23} | (0.25, 0.5, 0.75) | (0.75, 1, 1) | 0 |

E7 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) | E8 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) |

C_{21} | 0 | (0.5, 0.75, 1) | (0.75, 1, 1) | C_{21} | 0 | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) |

C_{22} | (0.5, 0.75, 1) | 0 | (0.25, 0.5, 0.75) | C_{22} | (0.5, 0.75, 1) | 0 | (0.5, 0.75, 1) |

C_{23} | (0.5, 0.75, 1) | (0.25, 0.5, 0.75) | 0 | C_{23} | (0.75, 1, 1) | (0.75, 1, 1) | 0 |

E9 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) | E10 | C_{21} (TFN) | C_{22}(TFN) | C_{23} (TFN) |

C_{21} | 0 | (0.5, 0.75, 1) | (0.75, 1, 1) | C_{21} | 0 | (0.5, 0.75, 1) | (0.75, 1, 1) |

C_{22} | (0.5, 0.75, 1) | 0 | (0.5, 0.75, 1) | C_{22} | (0.5, 0.75, 1) | 0 | (0.5, 0.75, 1) |

C_{23} | (0.75, 1, 1) | (0.25, 0.5, 0.75) | 0 | C_{23} | (0.25, 0.5, 0.75) | (0.75, 1, 1) | 0 |

**Table A9.**The corresponding triangle fuzzy numbers in Table A4.

E1 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) | E2 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) |
---|---|---|---|---|---|---|---|

Application and Installment time (C_{31}) | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | C_{31} | 0 | (0, 0, 0.25) | (0, 0.25, 0.5) |

Operating time (C_{32}) | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) | C_{32} | (0, 0, 0.25) | 0 | (0.25, 0.5, 0.75) |

Training time (C_{33}) | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 | C_{33} | (0, 0, 0.25) | (0.5, 0.75, 1) | 0 |

E3 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) | E4 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) |

C_{31} | 0 | (0, 0.25, 0.5) | (0, 0.25, 0.5) | C_{31} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) |

C_{32} | (0, 0, 0.25) | 0 | (0.25, 0.5, 0.75) | C_{32} | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) |

C_{33} | (0, 0, 0.25) | (0, 0.25, 0.5) | 0 | C_{33} | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 |

E5 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) | E6 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) |

C_{31} | 0 | (0, 0, 0.25) | (0.25, 0.5, 0.75) | C_{31} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) |

C_{32} | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) | C_{32} | (0, 0, 0.25) | 0 | (0.75, 1, 1) |

C_{33} | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 | C_{33} | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 |

E7 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) | E8 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) |

C_{31} | 0 | (0, 0, 0.25) | (0.25, 0.5, 0.75) | C_{31} | 0 | (0, 0, 0.25) | (0.25, 0.5, 0.75) |

C_{32} | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) | C_{32} | (0, 0, 0.25) | 0 | (0.25, 0.5, 0.75) |

C_{33} | (0, 0, 0.25) | (0.25, 0.5, 0.75) | 0 | C_{33} | (0, 0, 0.25) | (0.5, 0.75, 1) | 0 |

E9 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) | E10 | C_{31} (TFN) | C_{32} (TFN) | C_{33} (TFN) |

C_{31} | 0 | (0, 0.25, 0.5) | (0.25, 0.5, 0.75) | C_{31} | 0 | (0, 0, 0.25) | (0.5, 0.75, 1) |

C_{32} | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) | C_{32} | (0, 0, 0.25) | 0 | (0.5, 0.75, 1) |

C_{33} | (0, 0, 0.25) | (0.5, 0.75, 1) | 0 | C_{33} | (0, 0, 0.25) | (0.5, 0.75, 1) | 0 |

**Table A10.**The corresponding triangle fuzzy numbers in Table A5.

E1 | C_{41} (TFN) | C_{42} (TFN) | E2 | C_{41} (TFN) | C_{42} (TFN) |
---|---|---|---|---|---|

Internal safety (C_{41}) | 0 | (0, 0.25, 0.5) | C_{41} | 0 | (0.25, 0.5, 0.75) |

External safety (C_{42}) | (0.5, 0.75, 1) | 0 | C_{42} | (0.25, 0.5, 0.75) | 0 |

E3 | C_{41} (TFN) | C_{42} (TFN) | E4 | C_{41} (TFN) | C_{42} (TFN) |

C_{41} | 0 | (0, 0.25, 0.5) | C_{41} | 0 | (0.25, 0.5, 0.75) |

C_{42} | (0.5, 0.75, 1) | 0 | C_{42} | (0.5, 0.75, 1) | 0 |

E5 | C_{41} (TFN) | C_{42} (TFN) | E6 | C_{41} (TFN) | C_{42} (TFN) |

C_{41} | 0 | (0.5, 0.75, 1) | C_{41} | 0 | (0.25, 0.5, 0.75) |

C_{42} | (0.5, 0.75, 1) | 0 | C_{42} | (0.25, 0.5, 0.75) | 0 |

E7 | C_{41} (TFN) | C_{42} (TFN) | E8 | C_{41} (TFN) | C_{42} (TFN) |

C_{41} | 0 | (0, 0.25, 0.5) | C_{41} | 0 | (0.25, 0.5, 0.75) |

C_{42} | (0.75, 1, 1) | 0 | C_{42} | (0.25, 0.5, 0.75) | 0 |

E9 | C_{41} (TFN) | C_{42} (TFN) | E10 | C_{41} (TFN) | C_{42} (TFN) |

C_{41} | 0 | (0.25, 0.5, 0.75) | C_{41} | 0 | (0.25, 0.5, 0.75) |

C_{42} | (0.5, 0.75, 1) | 0 | C_{42} | (0.5, 0.75, 1) | 0 |

**Table A11.**The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A6.

E1 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E2 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E3 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E4 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E5 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Efficiency(C_{1}) | 0.00 | 0.50 | 0.73 | 0.73 | C_{1} | 0.00 | 0.50 | 0.50 | 0.97 | C_{1} | 0.00 | 0.73 | 0.27 | 0.50 | C_{1} | 0.00 | 0.73 | 0.27 | 0.73 | C_{1} | 0.00 | 0.25 | 0.25 | 0.25 |

Economical(C_{2}) | 0.73 | 0.00 | 0.50 | 0.50 | C_{2} | 0.50 | 0.00 | 0.73 | 0.27 | C_{2} | 0.73 | 0.00 | 0.50 | 0.73 | C_{2} | 0.26 | 0.00 | 0.49 | 0.49 | C_{2} | 0.49 | 0.00 | 0.26 | 0.49 |

Time(C_{3}) | 0.26 | 0.49 | 0.00 | 0.26 | C_{3} | 0.73 | 0.97 | 0.00 | 0.50 | C_{3} | 0.73 | 0.73 | 0.00 | 0.27 | C_{3} | 0.26 | 0.26 | 0.00 | 0.49 | C_{3} | 0.73 | 0.27 | 0.00 | 0.50 |

Safety(C_{4}) | 0.27 | 0.97 | 0.50 | 0.00 | C_{4} | 0.27 | 0.50 | 0.73 | 0.00 | C_{4} | 0.73 | 0.97 | 0.50 | 0.00 | C_{4} | 0.27 | 0.73 | 0.50 | 0.00 | C_{4} | 0.49 | 0.49 | 0.49 | 0.00 |

E6 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E7 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E8 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E9 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) | E10 | C_{1}(DCV) | C_{2}(DCV) | C_{3}(DCV) | C_{4}(DCV) |

C_{1} | 0.00 | 0.27 | 0.73 | 0.97 | C_{1} | 0.00 | 0.73 | 0.50 | 0.73 | C_{1} | 0.00 | 0.73 | 0.50 | 0.73 | C_{1} | 0.00 | 0.73 | 0.73 | 0.73 | C_{1} | 0.00 | 0.73 | 0.97 | 0.50 |

C_{2} | 0.50 | 0.00 | 0.97 | 0.73 | C_{2} | 0.73 | 0.00 | 0.27 | 0.73 | C_{2} | 0.73 | 0.00 | 0.27 | 0.73 | C_{2} | 0.73 | 0.00 | 0.27 | 0.73 | C_{2} | 0.27 | 0.00 | 0.50 | 0.73 |

C_{3} | 0.73 | 0.97 | 0.00 | 0.27 | C_{3} | 0.50 | 0.97 | 0.00 | 0.73 | C_{3} | 0.50 | 0.73 | 0.00 | 0.50 | C_{3} | 0.49 | 0.49 | 0.00 | 0.26 | C_{3} | 0.73 | 0.97 | 0.00 | 0.50 |

C_{4} | 0.97 | 0.73 | 0.73 | 0.00 | C_{4} | 0.97 | 0.73 | 0.97 | 0.00 | C_{4} | 0.97 | 0.73 | 0.97 | 0.00 | C_{4} | 0.97 | 0.73 | 0.73 | 0.00 | C_{4} | 0.73 | 0.97 | 0.50 | 0.00 |

**Note. DCV =**De-fuzzy Crisp Number.

**C =**Criteria.

**Table A12.**The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A7.

E1 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E2 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E3 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E4 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E5 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

EX (C_{11}) | 0.00 | 0.40 | 0.40 | 0.40 | 0.92 | C_{11} | 0.00 | 0.60 | 0.95 | 0.28 | 0.95 | C_{11} | 0.00 | 0.28 | 0.60 | 0.95 | 0.95 | C_{11} | 0.00 | 0.92 | 0.40 | 0.92 | 0.40 | C_{11} | 0.00 | 0.05 | 0.95 | 0.05 | 0.60 |

RE (C_{12}) | 0.45 | 0.00 | 0.45 | 0.97 | 0.45 | C_{12} | 0.28 | 0.00 | 0.28 | 0.95 | 0.28 | C_{12} | 0.45 | 0.00 | 0.21 | 0.97 | 0.45 | C_{12} | 0.95 | 0.00 | 0.60 | 0.95 | 0.60 | C_{12} | 0.45 | 0.00 | 0.21 | 0.97 | 0.45 |

US (C_{13}) | 0.28 | 0.28 | 0.00 | 0.95 | 0.95 | C_{13} | 0.60 | 0.60 | 0.00 | 0.95 | 0.95 | C_{13} | 0.28 | 0.60 | 0.00 | 0.95 | 0.95 | C_{13} | 0.70 | 0.21 | 0.00 | 0.97 | 0.70 | C_{13} | 0.05 | 0.60 | 0.00 | 0.95 | 0.95 |

FO (C_{14}) | 0.45 | 0.70 | 0.45 | 0.00 | 0.97 | C_{14} | 0.21 | 0.70 | 0.45 | 0.00 | 0.97 | C_{14} | 0.28 | 0.60 | 0.05 | 0.00 | 0.95 | C_{14} | 0.97 | 0.70 | 0.45 | 0.00 | 0.70 | C_{14} | 0.60 | 0.95 | 0.28 | 0.00 | 0.28 |

EL (C_{15}) | 0.54 | 0.54 | 0.38 | 0.08 | 0.00 | C_{15} | 0.41 | 0.95 | 0.60 | 0.05 | 0.00 | C_{15} | 0.41 | 0.50 | 0.05 | 0.17 | 0.00 | C_{15} | 0.97 | 0.54 | 0.38 | 0.08 | 0.00 | C_{15} | 0.54 | 0.64 | 0.27 | 0.03 | 0.00 |

E6 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E7 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E8 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E9 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) | E10 | C_{11}(DVC) | C_{12}(DVC) | C_{13}(DVC) | C_{14}(DVC) | C_{15}(DVC) |

C_{11} | 0.00 | 0.45 | 0.70 | 0.45 | 0.97 | C_{11} | 0.00 | 0.28 | 0.95 | 0.28 | 0.60 | C_{11} | 0.00 | 0.60 | 0.28 | 0.95 | 0.60 | C_{11} | 0.00 | 0.40 | 0.92 | 0.92 | 0.92 | C_{11} | 0.00 | 0.95 | 0.60 | 0.28 | 0.95 |

C_{12} | 0.45 | 0.00 | 0.45 | 0.97 | 0.70 | C_{12} | 0.45 | 0.00 | 0.21 | 0.97 | 0.45 | C_{12} | 0.45 | 0.00 | 0.45 | 0.97 | 0.45 | C_{12} | 0.21 | 0.00 | 0.45 | 0.97 | 0.70 | C_{12} | 0.28 | 0.00 | 0.95 | 0.60 | 0.95 |

C_{13} | 0.28 | 0.28 | 0.00 | 0.95 | 0.60 | C_{13} | 0.60 | 0.28 | 0.00 | 0.95 | 0.95 | C_{13} | 0.28 | 0.28 | 0.00 | 0.95 | 0.95 | C_{13} | 0.95 | 0.95 | 0.00 | 0.95 | 0.95 | C_{13} | 0.95 | 0.28 | 0.00 | 0.95 | 0.60 |

C_{14} | 0.60 | 0.95 | 0.60 | 0.00 | 0.95 | C_{14} | 0.70 | 0.70 | 0.21 | 0.00 | 0.97 | C_{14} | 0.28 | 0.60 | 0.60 | 0.00 | 0.95 | C_{14} | 0.28 | 0.95 | 0.60 | 0.00 | 0.95 | C_{14} | 0.70 | 0.70 | 0.45 | 0.00 | 0.97 |

C_{15} | 0.65 | 0.95 | 0.60 | 0.05 | 0.00 | C_{15} | 0.73 | 0.54 | 0.21 | 0.03 | 0.00 | C_{15} | 0.41 | 0.50 | 0.50 | 0.05 | 0.00 | C_{15} | 0.41 | 0.95 | 0.50 | 0.11 | 0.00 | C_{15} | 0.95 | 0.95 | 0.50 | 0.11 | 0.00 |

**Table A13.**The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A8.

E1 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E2 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E3 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E4 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E5 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Application and Installment cost (C_{21}) | 0.00 | 0.50 | 0.73 | C_{21} | 0.00 | 0.27 | 0.50 | C_{21} | 0.00 | 0.50 | 0.50 | C_{21} | 0.00 | 0.27 | 0.73 | C_{21} | 0.00 | 0.73 | 0.50 |

Operating cost (C_{22}) | 0.97 | 0.00 | 0.97 | C_{22} | 0.73 | 0.00 | 0.27 | C_{22} | 0.50 | 0.00 | 0.73 | C_{22} | 0.50 | 0.00 | 0.73 | C_{22} | 0.73 | 0.00 | 0.73 |

Training cost (C_{23}) | 0.73 | 0.73 | 0.00 | C_{23} | 0.50 | 0.50 | 0.00 | C_{23} | 0.27 | 0.50 | 0.00 | C_{23} | 0.27 | 0.73 | 0.00 | C_{23} | 0.50 | 0.73 | 0.00 |

E6 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E7 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E8 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E9 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) | E10 | C_{21}(DVC) | C_{22}(DVC) | C_{23}(DVC) |

C_{21} | 0.00 | 0.27 | 0.50 | C_{21} | 0.00 | 0.73 | 0.97 | C_{21} | 0.00 | 0.73 | 0.50 | C_{21} | 0.00 | 0.73 | 0.97 | C_{21} | 0.00 | 0.73 | 0.97 |

C_{22} | 0.73 | 0.00 | 0.73 | C_{22} | 0.73 | 0.00 | 0.50 | C_{22} | 0.73 | 0.00 | 0.73 | C_{22} | 0.73 | 0.00 | 0.73 | C_{22} | 0.73 | 0.00 | 0.73 |

C_{23} | 0.50 | 0.97 | 0.00 | C_{23} | 0.73 | 0.50 | 0.00 | C_{23} | 0.97 | 0.97 | 0.00 | C_{23} | 0.97 | 0.50 | 0.00 | C_{23} | 0.50 | 0.97 | 0.00 |

**Table A14.**The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A9.

E1 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E2 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E3 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E4 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E5 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Application and Installment time(C_{31}) | 0.00 | 0.35 | 0.65 | C_{31} | 0.00 | 0.08 | 0.50 | C_{31} | 0.00 | 0.50 | 0.50 | C_{31} | 0.00 | 0.35 | 0.65 | C_{31} | 0.00 | 0.05 | 0.65 |

Operating time (C_{32}) | 0.03 | 0.00 | 0.73 | C_{32} | 0.05 | 0.00 | 0.65 | C_{32} | 0.05 | 0.00 | 0.65 | C_{32} | 0.03 | 0.00 | 0.73 | C_{32} | 0.03 | 0.00 | 0.73 |

Training time (C_{33}) | 0.05 | 0.65 | 0.00 | C_{33} | 0.03 | 0.73 | 0.00 | C_{33} | 0.08 | 0.50 | 0.00 | C_{33} | 0.05 | 0.65 | 0.00 | C_{33} | 0.05 | 0.65 | 0.00 |

E6 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E7 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E8 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E9 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) | E10 | C_{31}(DVC) | C_{32}(DVC) | C_{33}(DVC) |

C_{31} | 0.00 | 0.35 | 0.65 | C_{31} | 0.00 | 0.05 | 0.65 | C_{31} | 0.00 | 0.05 | 0.65 | C_{31} | 0.00 | 0.35 | 0.65 | C_{31} | 0.00 | 0.03 | 0.73 |

C_{32} | 0.03 | 0.00 | 0.97 | C_{32} | 0.03 | 0.00 | 0.73 | C_{32} | 0.05 | 0.00 | 0.65 | C_{32} | 0.03 | 0.00 | 0.73 | C_{32} | 0.03 | 0.00 | 0.73 |

C_{33} | 0.05 | 0.65 | 0.00 | C_{33} | 0.05 | 0.65 | 0.00 | C_{33} | 0.03 | 0.73 | 0.00 | C_{33} | 0.03 | 0.73 | 0.00 | C_{33} | 0.03 | 0.73 | 0.00 |

**Table A15.**The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A10.

E1 | C_{41}(DVC) | C_{42}(DVC) | E2 | C_{41}(DVC) | C_{42}(DVC) | E3 | C_{41}(DVC) | C_{42}(DVC) | E4 | C_{41}(DVC) | C_{42}(DVC) | E5 | C_{41}(DVC) | C_{42}(DVC) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Internal safety (C_{41}) | 0.00 | 0.27 | C_{41} | 0.00 | 0.50 | C_{41} | 0.00 | 0.27 | C_{41} | 0.00 | 0.50 | C_{41} | 0.00 | 0.73 |

External safety (C_{42}) | 0.73 | 0.00 | C_{42} | 0.50 | 0.00 | C_{42} | 0.73 | 0.00 | C_{42} | 0.73 | 0.00 | C_{42} | 0.73 | 0.00 |

E6 | C_{41}(DVC) | C_{42}(DVC) | E7 | C_{41}(DVC) | C_{42}(DVC) | E8 | C_{41}(DVC) | C_{42}(DVC) | E9 | C_{41}(DVC) | C_{42}(DVC) | E10 | C_{41}(DVC) | C_{42}(DVC) |

C_{41} | 0.00 | 0.50 | C_{41} | 0.00 | 0.27 | C_{41} | 0.00 | 0.50 | C_{41} | 0.00 | 0.50 | C_{41} | 0.00 | 0.50 |

C_{42} | 0.50 | 0.00 | C_{42} | 0.97 | 0.00 | C_{42} | 0.50 | 0.00 | C_{42} | 0.73 | 0.00 | C_{42} | 0.73 | 0.00 |

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Criteria | Sub-Criteria |
---|---|

Efficiency parameters (C_{1}) | Expressiveness (C_{11}) |

Readability (C_{12}) | |

Usability (C_{13}) | |

Formality (C_{14}) | |

Ease of Learning (C_{15}) | |

Economical parameters (C_{2}) | Application and Installment cost (C_{21}) |

Operating cost (C_{22}) | |

Training cost (C_{23}) | |

Time parameters (C_{3}) | Application and Installment time (C_{31}) |

Operating time (C_{32}) | |

Training time (C_{33}) | |

Safety parameters (C_{4}) | Internal safety (C_{41}) |

External safety (C_{42}) |

Rank | Explanation |
---|---|

1 | Very bad |

2 | Good |

3 | Normal |

4 | Bad |

5 | Very good |

Direct Influence Degree | Fuzzy Rank Value | Triangular Fuzzy Scale | ||
---|---|---|---|---|

No impact | 0 | 0 | 0 | 0.25 |

Low impact | 1 | 0 | 0.25 | 0.5 |

Medium impact | 2 | 0.25 | 0.5 | 0.75 |

High impact | 3 | 0.5 | 0.75 | 1 |

Very high impact | 4 | 0.75 | 1 | 1 |

**Table 4.**Evaluation rank value for all the business process modelling tool selection sub-criterions in Table 1.

Alternative | Efficiency (C_{1}) | Economical (C_{2}) | Time (C_{3}) | Safety (C_{4}) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{12} | C_{12} | C_{13} | C_{14} | C_{15} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | |

TL1 | 2 | 3 | 2 | 3 | 5 | 1 | 2 | 2 | 3 | 3 | 3 | 1 | 1 |

TL2 | 3 | 3 | 2 | 3 | 5 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 |

TL3 | 4 | 4 | 3 | 2 | 3 | 3 | 4 | 3 | 3 | 1 | 1 | 4 | 5 |

TL4 | 4 | 5 | 4 | 3 | 4 | 3 | 2 | 3 | 2 | 3 | 3 | 5 | 5 |

TL5 | 4 | 2 | 3 | 4 | 3 | 4 | 3 | 2 | 4 | 4 | 2 | 2 | 2 |

TL6 | 3 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 5 | 5 | 2 | 2 | 1 |

TL7 | 4 | 5 | 4 | 3 | 2 | 1 | 3 | 4 | 1 | 2 | 1 | 4 | 5 |

TL8 | 5 | 4 | 5 | 3 | 3 | 3 | 4 | 4 | 2 | 2 | 5 | 4 | 5 |

T | C_{1} (ADFI) | C_{2} (ADFI) | C_{3} (ADFI) | C_{4} (ADFI) |
---|---|---|---|---|

Efficiency (C_{1}) | 0 | 0.59 | 0.545 | 0.684 |

Economical (C_{2}) | 0.567 | 0 | 0.476 | 0.613 |

Time (C_{3}) | 0.566 | 0.685 | 0 | 0.428 |

Safety (C_{4}) | 0.664 | 0.755 | 0.662 | 0 |

T (C_{1}) | C_{11} (ADFI) | C_{12} (ADFI) | C_{13} (ADFI) | C_{14} (ADFI) | C_{15} (ADFI) |
---|---|---|---|---|---|

Expressiveness (C_{11}) | 0 | 0.493 | 0.675 | 0.548 | 0.786 |

Readability (C_{12}) | 0.442 | 0 | 0.426 | 0.929 | 0.548 |

Usability (C_{13}) | 0.497 | 0.436 | 0 | 0.952 | 0.855 |

Formality (C_{14}) | 0.507 | 0.755 | 0.414 | 0 | 0.866 |

Ease of Learning (C_{15}) | 0.602 | 0.706 | 0.399 | 0.076 | 0 |

T (C_{2}) | C_{21} (ADFI) | C_{22} (ADFI) | C_{23} (ADFI) |
---|---|---|---|

Application and Installment cost (C_{21}) | 0 | 0.546 | 0.687 |

Operating cost (C_{22}) | 0.708 | 0 | 0.685 |

Training cost (C_{23}) | 0.594 | 0.71 | 0 |

T (C_{3}) | C_{31} (ADFI) | C_{32} (ADFI) | C_{33} (ADFI) |
---|---|---|---|

Application and Installment time (C_{31}) | 0 | 0.216 | 0.628 |

Operating time (C_{32}) | 0.036 | 0 | 0.73 |

Training time (C_{33}) | 0.045 | 0.667 | 0 |

T (C_{4}) | C_{41} (ADFI) | C_{42} (ADFI) |
---|---|---|

Internal safety (C_{41}) | 0 | 0.454 |

External safety (C_{42}) | 0.685 | 0 |

G | Efficiency (C_{1}) | Economical (C_{2}) | Time (C_{3}) | Safety (C_{4}) |
---|---|---|---|---|

Efficiency (C_{1}) | 0 | 0.284 | 0.262 | 0.329 |

Economical (C_{2}) | 0.272 | 0 | 0.229 | 0.295 |

Time (C_{3}) | 0.272 | 0.329 | 0 | 0.206 |

Safety (C_{4}) | 0.319 | 0.363 | 0.318 | 0 |

G (C_{1}) | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} |
---|---|---|---|---|---|

Expressiveness (C_{11}) | 0 | 0.18 | 0.25 | 0.2 | 0.29 |

Readability (C_{12}) | 0.16 | 0 | 0.16 | 0.34 | 0.2 |

Usability (C_{13}) | 0.18 | 0.16 | 0 | 0.35 | 0.31 |

Formality (C_{14}) | 0.19 | 0.28 | 0.15 | 0 | 0.32 |

Ease of Learning (C_{15}) | 0.22 | 0.26 | 0.15 | 0.03 | 0 |

G (C_{2}) | C_{21} | C_{22} | C_{23} |
---|---|---|---|

Application and Installment cost (C_{21}) | 0 | 0.39 | 0.49 |

Operating cost (C_{22}) | 0.51 | 0 | 0.49 |

Training cost (C_{23}) | 0.43 | 0.51 | 0 |

G (C_{3}) | C_{31} | C_{32} | C_{33} |
---|---|---|---|

Application and Installment time (C_{31}) | 0 | 0.26 | 0.74 |

Operating time (C_{32}) | 0.04 | 0 | 0.86 |

Training time (C_{33}) | 0.05 | 0.79 | 0 |

G (C_{4}) | C_{41} | C_{42} |
---|---|---|

Internal safety (C_{41}) | 0 | 0.66 |

External safety (C_{42}) | 1 | 0 |

E | Efficiency (C_{1}) | Economical (C_{2}) | Time (C_{3}) | Safety (C_{4}) | ro_{n} | ro_{n} + co_{j} |
---|---|---|---|---|---|---|

Efficiency (C_{1}) | 1.484 | 1.852 | 1.614 | 1.696 | 6.646 | 13.181 |

Economical (C_{2}) | 1.592 | 1.514 | 1.493 | 1.573 | 6.172 | 13.356 |

Time (C_{3}) | 1.585 | 1.756 | 1.299 | 1.513 | 6.153 | 12.347 |

Safety (C_{4}) | 1.874 | 2.062 | 1.788 | 1.593 | 7.317 | 13.692 |

co_{j} | 6.535 | 7.184 | 6.194 | 6.375 | — | — |

_{n}(highlighted in yellow) means the total given both direct and indirect effects from criteria n to the other criteria, and co

_{j}(highlighted in pink) means the total received both direct and indirect effects from other criterions to criterion j. The overall direct and indirect impact matrix E with corresponding ro

_{n}and co

_{j}values, when n equals j, (ro

_{j}+ co

_{j}) (highlighted in lime green) value, represents the centrality of criterion j.

E (C_{1}) | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | ro_{n} | ro_{n} + co_{j} |
---|---|---|---|---|---|---|---|

Expressiveness (C_{11}) | 1.05 | 1.36 | 1.18 | 1.34 | 1.66 | 6.59 | 12.25 |

Readability (C_{12}) | 1.14 | 1.16 | 1.07 | 1.38 | 1.54 | 6.29 | 12.81 |

Usability (C_{13}) | 1.28 | 1.44 | 1.04 | 1.52 | 1.78 | 7.06 | 12.34 |

Formality (C_{14}) | 1.21 | 1.44 | 1.11 | 1.17 | 1.68 | 6.61 | 12.97 |

Ease of Learning (C_{15}) | 0.98 | 1.12 | 0.88 | 0.95 | 1.08 | 5.01 | 12.75 |

co_{j} | 5.66 | 6.52 | 5.28 | 6.36 | 7.74 | — | — |

E (C_{2}) | C_{21} | C_{22} | C_{23} | ro_{n} | ro_{n} + co_{j} |
---|---|---|---|---|---|

Application and Installment cost (C_{21}) | 4.73 | 4.89 | 5.2 | 14.82 | 14.82 |

Operating cost (C_{22}) | 5.51 | 5.03 | 5.65 | 16.19 | 16.19 |

Training cost (C_{23}) | 5.27 | 5.18 | 5.12 | 15.57 | 15.57 |

co_{j} | 15.51 | 15.1 | 15.97 | — | — |

E (C_{3}) | C_{31} | C_{32} | C_{33} | ro_{n} | ro_{n} + co_{j} |
---|---|---|---|---|---|

Application and Installment time (C_{31}) | 0.34 | 3.54 | 4.04 | 7.92 | 7.92 |

Operating time (C_{32}) | 0.35 | 3.04 | 3.73 | 7.12 | 7.12 |

Training time (C_{33}) | 0.34 | 3.36 | 3.15 | 6.85 | 6.85 |

co_{j} | 1.03 | 9.94 | 10.92 | — | — |

E (C_{4}) | C_{41} | C_{42} | ro_{n} | ro_{n} + co_{j} |
---|---|---|---|---|

Internal safety (C_{41}) | 1.94 | 1.94 | 3.88 | 8.76 |

External safety (C_{42}) | 2.94 | 1.94 | 4.88 | 8.76 |

co_{j} | 4.88 | 3.88 | — | — |

**Table 20.**Normalized (ro + co) value (weight) for all the all the business process modelling tool selection criteria and sub-criteria.

Direct Influence Degree | NRC (W) | Sub-Criteria | NRC (W) |
---|---|---|---|

Efficiency parameters (C_{1}) | 0.251 | Expressiveness (C_{11}) | 0.194 |

Readability (C_{12}) | 0.203 | ||

Usability (C_{13}) | 0.196 | ||

Formality (C_{14}) | 0.205 | ||

Ease of Learning (C_{15}) | 0.202 | ||

Economical parameters (C_{2}) | 0.254 | Application and installment cost (C_{21}) | 0.32 |

Operating cost (C_{22}) | 0.35 | ||

Training cost (C_{23}) | 0.33 | ||

Time parameters (C_{3}) | 0.235 | Application and installment time (C_{31}) | 0.36 |

Operating time (C_{32}) | 0.33 | ||

Training time (C_{33}) | 0.31 | ||

Safety parameters(C_{4}) | 0.26 | Internal safety (C_{41}) | 0.5 |

External safety (C_{42}) | 0.5 |

**Table 21.**The weighted normalized value of all the business process modelling tool alternatives in Table 4.

Alternative | Efficiency (C_{1}) | Economical (C_{2}) | Time (C_{3}) | Safety (C_{4}) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{11}Max | C_{12}Max | C_{13}Max | C_{14}Max | C_{15}Max | C_{21}Min | C_{22}Min | C_{23}Min | C_{31}Min | C_{32}Min | C_{33}Min | C_{41}Max | C_{42}Max | |

T1 | 0.009 | 0.015 | 0.011 | 0.019 | 0.025 | 0.012 | 0.022 | 0.021 | 0.03 | 0.027 | 0.029 | 0.014 | 0.013 |

T2 | 0.014 | 0.015 | 0.011 | 0.019 | 0.025 | 0.012 | 0.022 | 0.021 | 0.02 | 0.018 | 0.019 | 0.014 | 0.013 |

T3 | 0.018 | 0.02 | 0.016 | 0.012 | 0.015 | 0.036 | 0.045 | 0.032 | 0.03 | 0.009 | 0.01 | 0.057 | 0.063 |

T4 | 0.018 | 0.025 | 0.021 | 0.019 | 0.02 | 0.036 | 0.022 | 0.032 | 0.02 | 0.027 | 0.029 | 0.057 | 0.063 |

T5 | 0.018 | 0.01 | 0.016 | 0.025 | 0.015 | 0.047 | 0.034 | 0.021 | 0.04 | 0.037 | 0.019 | 0.071 | 0.025 |

T6 | 0.014 | 0.01 | 0.005 | 0.012 | 0.01 | 0.012 | 0.011 | 0.011 | 0.05 | 0.046 | 0.019 | 0.029 | 0.013 |

T7 | 0.018 | 0.025 | 0.021 | 0.019 | 0.01 | 0.012 | 0.034 | 0.042 | 0.01 | 0.018 | 0.01 | 0.029 | 0.063 |

T8 | 0.023 | 0.02 | 0.027 | 0.019 | 0.015 | 0.036 | 0.045 | 0.042 | 0.02 | 0.018 | 0.048 | 0.057 | 0.063 |

Alternative | Efficiency (C_{1}) | Economical (C_{2}) | Time (C_{3}) | Safety (C_{4}) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{12}Max | C_{12}Max | C_{13}Max | C_{14}Max | C_{15}Max | C_{21}Min | C_{22}Min | C_{23}Min | C_{31}Min | C_{32}Min | C_{33}Min | C_{41}Max | C_{42}Max | |

T^{+} | 0.023 | 0.025 | 0.027 | 0.025 | 0.025 | 0.012 | 0.011 | 0.011 | 0.01 | 0.009 | 0.01 | 0.071 | 0.063 |

T^{−} | 0.009 | 0.01 | 0.005 | 0.012 | 0.01 | 0.047 | 0.045 | 0.042 | 0.05 | 0.046 | 0.048 | 0.014 | 0.013 |

Alternative | d_{i}* | d_{i}^{0} | C_{i}* |
---|---|---|---|

T1 | 0.088 | 0.06 | 0.405 |

T2 | 0.082 | 0.071 | 0.464 |

T3 | 0.056 | 0.09 | 0.616 |

T4 | 0.048 | 0.086 | 0.642 |

T5 | 0.074 | 0.073 | 0.497 |

T6 | 0.092 | 0.067 | 0.421 |

T7 | 0.061 | 0.092 | 0.601 |

T8 | 0.069 | 0.083 | 0.546 |

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Jin, G.; Jin, G.; Huo, H.
Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method. *Axioms* **2022**, *11*, 601.
https://doi.org/10.3390/axioms11110601

**AMA Style**

Jin G, Jin G, Huo H.
Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method. *Axioms*. 2022; 11(11):601.
https://doi.org/10.3390/axioms11110601

**Chicago/Turabian Style**

Jin, Guangying, Guangzhe Jin, and Haibo Huo.
2022. "Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method" *Axioms* 11, no. 11: 601.
https://doi.org/10.3390/axioms11110601