# Supplier Selection in the Nuclear Power Industry with an Integrated ANP-TODIM Method under Z-Number Circumstances

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Supplier Selection Methods

#### 2.2. Evaluation Criteria for Nuclear Power Equipment Supplier Selection

- (1)
- Quality assurance (C
_{1}). The NPP usually consists of four parts: nuclear island, convention island, balance of plant, and nuclear fuel assembly [38,43]. All the nuclear power equipment of them must strictly comply with the quality standard set by IAEA, and their failure rate should be controlled at a very low level. Once a quality safety accident occurs, suppliers should have good after-sales service capabilities and provide corrective measures in the first place. - (2)
- Cost control (C
_{2}). In previous studies [44], cost was often regarded as a key factor in supplier selection. Market surveys show that when the quality level of products or services provided by suppliers is the same, nuclear power enterprises tend to choose suppliers with lower prices so as to save the construction and operation costs of NPPs. - (3)
- Technical capacity (C
_{3}). The nuclear power industry is a high-tech industry, and the technological level embodies the comprehensive competitiveness of nuclear power enterprises. In general, technical capacity is positively related to quality assurance capacity, but shows a negative correlation with the cost of production [39]. - (4)
- Enterprise qualification (C
_{4}). The qualification of an enterprise is closely related to its financial status, market condition, and industry influence [39,40,45]. In reality, buyers always look to strengthening cooperation with suppliers with a higher credit reputation and industry ranking. Hence, it is necessary to consider the qualification of candidates in the supplier selection process. - (5)
- Deliver capability (C
_{5}). The delivery cycle and delivery performance are the major concerns in supplier selection [9,46]. Specifically, any delay in delivery will result in cost overruns, or even project failure. In addition, whether it can be successfully delivered in a state of emergency should also be considered in the selection of a nuclear power equipment supplier. - (6)
- Environmental consciousness (C
_{6}). In recent years, the environmental awareness of the company and the government has gradually increased, and many nuclear power equipment suppliers increasingly focus on efficiency, energy saving, environmental protection equipment, and Research and Development (R & D). It has also become an important indicator to measure the corporate social responsibility of suppliers [7,41,47].

## 3. Preliminaries

**Definition**

**1**

**.**Let $\tilde{A}=\left\{\left(x,{\mu}_{\tilde{A}}(x)\right)|x\in X\right\}$ be a fuzzy set on space $X$. The membership function of a Triangular Fuzzy Number (TFN) $\tilde{A}=\left({a}_{1},{a}_{2},{a}_{3}\right)$ is defined as Equation (1). The distance between the two TFNs${\tilde{A}}^{\prime}=\left({{a}_{1}}^{\prime},{{a}_{2}}^{\prime},{{a}_{3}}^{\prime}\right)$ and ${\tilde{A}}^{\u2033}=\left({{a}_{1}}^{\u2033},{{a}_{2}}^{\u2033},{{a}_{3}}^{\u2033}\right)$ can be calculated as Equation (2).

**Definition**

**2**

**.**Assume that ${\tilde{A}}^{\prime}=\left({{a}_{1}}^{\prime},{{a}_{2}}^{\prime},{{a}_{3}}^{\prime}\right)$ and ${\tilde{A}}^{\u2033}=\left({{a}_{1}}^{\u2033},{{a}_{2}}^{\u2033},{{a}_{3}}^{\u2033}\right)$ are TFNs, and the math operations of them are shown below:

**Definition**

**3**

**.**A Z-number $Z=(A,B)$ is an ordered pair of regular fuzzy numbers. A is a restriction on the values, which indicates that the uncertain variable is allowed to take. B is a measure of the reliability of A. Let $\tilde{Z}=(\tilde{A},\tilde{B})$ be a Z-number; the component $\tilde{B}$ can be converted into a crisp number $\alpha $, as in Equation (4). Then, $\tilde{Z}=(\tilde{A},\tilde{B})$ can be translated into a regular fuzzy number, as in Equation (5).

**Definition**

**4**

**Example**

**1.**

## 4. Methodology

#### 4.1. Phase I Determine Criterion Weights with Z-ANP

**Step 1.**Construct a network model and identify the interdependence of criteria.

**Step 2.**Determine the fuzzy aggregated pairwise comparison matrix.

**Step 3.**Calculate the priority weights of criteria.

**Step 3.1.**The value of the fuzzy synthetic extent with respect to each element $i(i=1,2,\cdots ,n)$ is defined as:

**Step 3.2.**The degree of possibility of ${S}_{i}=\left({m}_{i},{n}_{i},{q}_{i}\right)\ge {S}_{j}=\left({m}_{j},{n}_{j},{q}_{j}\right)$ is defined as:

**Step 3.3.**The degree possibility for a convex fuzzy number to be greater than all the other $n-1$ convex fuzzy numbers ${S}_{j}\left(j=1,2,\cdots ,n,j\ne i\right)$ can be defined by:

**Step 3.4.**The normalized weight vector $\overline{W}={\left({\overline{w}}_{1},{\overline{w}}_{2},\cdots ,{\overline{w}}_{n}\right)}^{T}$ is obtained by normalization, as in Equation (11). Then, we construct comparison matrices based on other criteria, so as to obtain the weight matrix $\overline{A}=\left[{\overline{W}}_{1},{\overline{W}}_{2},\cdots ,{\overline{W}}_{n}\right]$ for the main criteria by repeating the above steps.

**Step 4.**Construct the unweighted, weighted, and limit super-matrix.

#### 4.2. Phase II: Rank the Alternatives with Z-TODIM

**Step 5.**Construct the fuzzy aggregated decision matrix.

**Step 6.**Normalize the fuzzy aggregated decision matrix.

**Step 7.**Calculate the relative weight.

**Step 8.**Determine the dominance of each alternative.

**Step 9.**Rank the alternatives.

## 5. Case Study and Result

#### 5.1. Phase I: Determine Criterion Weights with Z-ANP

**Step 1.**Construct a network model and identify the interdependence of the criteria.

**Step 2.**Determine the fuzzy aggregated pairwise comparison matrix.

**Step 3.**Calculate the priority weights of the criteria.

**Step 4.**Construct the unweighted, weighted, and limit super-matrix.

Algorithm 1. Calculate the limit super-matrix |

Input: The weight matrix $\overline{A}=\left[{\overline{W}}_{1},{\overline{W}}_{2},\cdots ,{\overline{W}}_{n}\right]$ and the original Z-linguistic evaluation matrices for the sub-criteria $T=\left[{t}_{ij}\right]$ with respect to each main criterion. |

Output: The limit super-matrix $\tilde{W}$. |

(i) Generated the fuzzy aggregated pairwise comparison matrices ${T}^{\prime}=\left[{{t}_{ij}}^{\prime}\right]$, as in Step 2. |

(ii) Calculate the weight matrix $\overline{a}$ of each block by Step 3. |

(iii) Construct the unweighted super-matrix ${W}^{\prime}=\left[{\overline{a}}_{ij}\right]$. |

(iv) Calculate the weighted super-matrix ${W}_{n}=\overline{A}{W}^{\prime}$. |

(v) Normalize the weighted super-matrix to obtain ${{W}_{n}}^{\prime}$. |

(vi) Generated the limit super-matrix $\tilde{W}=\underset{t\to \infty}{\mathrm{lim}}{W}_{n}^{\prime 2t+1}$. |

#### 5.2. Phase II: Rank the Alternatives with Z-TODIM

**Steps 5 and 6.**Construct the normalized fuzzy decision matrix.

**Steps 7 and 8.**Calculate the relative weight and determine the overall dominance of each alternative.

**Step 9.**Rank the alternatives.

#### 5.3. Results and Discussion

#### 5.3.1. Sensitivity Analysis

#### 5.3.2. Comparison Analysis

#### 5.3.3. Discussion

_{21}) and technical advancement (C

_{31}) are significantly larger than the others. This indicates that advanced technology and an affordable product price are the prerequisites that affect the competitiveness of nuclear-level suppliers, while the impact of enterprise qualifications is relatively weak. In terms of the ranking of alternatives, the extended Z-TODIM is used to obtain the potential priorities for suppliers, and ultimately determines that ${P}_{1}$ is the best choice. Referring to Table 8, the ranking of alternatives has not changed with the variation in parameter $\theta $. This shows that our proposed Z-TODIM is less sensitive to the risk preference of decision-makers, and is more reliable and stable than the traditional TODIM method.

## 6. Conclusions, Limitation, and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Acronym | Definition |
---|---|

ANP | Analytic Network Process |

TODIM | Tomada de Decisão Iterativa Multicritério |

IAEA | International Atomic Energy Agency |

NPPs | Nuclear Power Plants |

SCM | Supply Chain Management |

MCDM | Multi-criteria Decision-making |

QUALIFLEX | Qualitative Flexible Multiple Method |

BWM | Best-worst Method |

VIKOR | VIsekriterijumskao ptimizacija i KOm-promisno Resenje |

AHP | Analytic Hierarchy Process |

DEMATEL | Decision-Making and Evaluation Laboratory |

R & D | Research and Development |

TFN | Triangular Fuzzy Number |

TOPSIS | Technique for Order Performance by Similarity to Ideal Solution |

MAIRCA | Multi-attribute Ideal Real Comparison Analysis |

## Appendix B

C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | C_{51} | C_{52} | C_{53} | C_{61} | C_{62} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{11} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||||||||

C_{12} | √ | √ | √ | √ | √ | √ | √ | √ | ||||||||||

C_{13} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |||||||

C_{14} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||||||||

C_{21} | √ | √ | √ | √ | √ | √ | ||||||||||||

C_{22} | √ | √ | √ | √ | ||||||||||||||

C_{23} | √ | √ | √ | √ | √ | √ | √ | |||||||||||

C_{31} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||||||

C_{32} | √ | √ | √ | √ | √ | √ | √ | √ | √ | |||||||||

C_{33} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |||||

C_{41} | √ | √ | √ | |||||||||||||||

C_{42} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||||||

C_{43} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||||||||

C_{51} | √ | √ | √ | √ | √ | √ | √ | √ | √ | |||||||||

C_{52} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||||||||

C_{53} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |||||||

C_{61} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |||||||

C_{62} | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ |

Expert | Criteria | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} |
---|---|---|---|---|---|---|---|

DM_{1} | C_{1} | Just equal | (MI, N) | (WI, H) | (MI, H) | (WI, H) | (MI, H) |

C_{2} | (MI^{−1}, N) | Just equal | (MI^{−1}, N) | (WI, N) | (EI, H) | (MI, N) | |

C_{3} | (WI^{−1}, H) | (MI, N) | Just equal | (MI, H) | (WI, N) | (MI, H) | |

C_{4} | (MI^{−1}, H) | (WI^{−1}, N) | (MI^{−1}, H) | Just equal | (MI^{−1}, H) | (WI, H) | |

C_{5} | (WI^{−1}, H) | (EI^{−1}, H) | (WI^{−1}, N) | (MI, H) | Just equal | (MI, N) | |

C_{6} | (MI^{−1}, H) | (MI^{−1}, N) | (MI^{−1}, H) | (WI^{−1}, H) | (MI^{−1}, N) | Just equal | |

DM_{2} | C_{1} | Just equal | (EI, H) | (EI, H) | (WI, N) | (WI, H) | (WI, N) |

C_{2} | (EI^{−1}, H) | Just equal | (WI, N) | (WI, VH) | (WI, H) | (MI, N) | |

C_{3} | (EI^{−1}, H) | (WI^{−1}, N) | Just equal | (MI, N) | (WI, VH) | (WI, VH) | |

C_{4} | (WI^{−1}, N) | (WI^{−1}, VH) | (MI^{−1}, N) | Just equal | (WI^{−1}, H) | (EI, H) | |

C_{5} | (WI^{−1}, H) | (WI^{−1}, H) | (WI^{−1}, VH) | (WI, H) | Just equal | (MI, N) | |

C_{6} | (WI^{−1}, N) | (MI^{−1}, N) | (WI^{−1}, VH) | (EI^{−1}, H) | (MI^{−1}, N) | Just equal | |

DM_{3} | C_{1} | Just equal | (WI^{−1}, N) | (EI, H) | (MI, N) | (WI, H) | (MI, N) |

C_{2} | (WI, N) | Just equal | (WI, H) | (MI, N) | (WI, VH) | (MI, H) | |

C_{3} | (EI^{−1}, H) | (WI^{−1}, H) | Just equal | (WI, VH) | (WI, N) | (MI, N) | |

C_{4} | (MI^{−1}, N) | (MI^{−1}, N) | (WI^{−1}, VH) | Just equal | (WI^{−1}, H) | (WI, N) | |

C_{5} | (WI^{−1}, H) | (WI^{−1}, VH) | (WI^{−1}, N) | (WI, H) | Just equal | (WI, VH) | |

C_{6} | (MI^{−1}, N) | (MI^{−1}, H) | (MI^{−1}, N) | (WI^{−1}, N) | (WI^{−1}, VH) | Just equal | |

DM_{4} | C_{1} | Just equal | (EI, N) | (WI, N) | (WI, H) | (WI, N) | (WI, VH) |

C_{2} | (EI^{−1}, N) | Just equal | (WI, N) | (WI, H) | (WI, N) | (WI, H) | |

C_{3} | (WI^{−1}, N) | (WI^{−1}, N) | Just equal | (WI, N) | (WI, H) | (WI, VH) | |

C_{4} | (WI^{−1}, H) | (WI^{−1}, H) | (WI^{−1}, N) | Just equal | (WI^{−1}, N) | (EI, H) | |

C_{5} | (WI^{−1}, N) | (WI^{−1}, N) | (WI^{−1}, H) | (WI, N) | Just equal | (WI, N) | |

C_{6} | (WI^{−1}, VH) | (WI^{−1}, H) | (WI^{−1}, VH) | (EI^{−1}, H) | (WI^{−1}, N) | Just equal |

C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | C_{51} | C_{52} | C_{53} | C_{61} | C_{62} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{11} | 0.000 | 0.667 | 0.375 | 0.636 | 0.217 | 0.000 | 0.000 | 0.000 | 0.000 | 0.393 | 1.000 | 0.324 | 0.374 | 0.334 | 1.000 | 0.000 | 0.000 | 0.000 |

C_{12} | 0.345 | 0.000 | 0.364 | 0.000 | 0.363 | 0.000 | 0.000 | 0.384 | 0.398 | 0.000 | 0.000 | 0.262 | 0.401 | 0.000 | 0.000 | 0.000 | 0.667 | 0.000 |

C_{13} | 0.362 | 0.000 | 0.000 | 0.364 | 0.216 | 0.000 | 0.000 | 0.311 | 0.386 | 0.341 | 0.000 | 0.241 | 0.225 | 0.333 | 0.000 | 0.000 | 0.333 | 0.435 |

C_{14} | 0.293 | 0.333 | 0.261 | 0.000 | 0.204 | 0.000 | 0.000 | 0.305 | 0.216 | 0.266 | 0.000 | 0.173 | 0.000 | 0.333 | 0.000 | 0.000 | 0.000 | 0.565 |

C_{21} | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.622 | 0.487 | 1.000 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 |

C_{22} | 0.000 | 0.000 | 0.000 | 0.433 | 0.500 | 0.000 | 0.513 | 0.000 | 0.000 | 0.000 | 0.000 | 0.500 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

C_{23} | 0.000 | 0.000 | 0.000 | 0.567 | 0.500 | 0.378 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.500 | 0.000 | 1.000 | 1.000 | 1.000 | 0.000 | 0.000 |

C_{31} | 0.000 | 0.361 | 0.614 | 0.500 | 0.343 | 0.000 | 0.000 | 0.000 | 0.717 | 0.691 | 0.000 | 0.334 | 0.464 | 0.000 | 0.502 | 0.575 | 0.321 | 0.520 |

C_{32} | 1.000 | 0.345 | 0.000 | 0.000 | 0.333 | 0.000 | 0.000 | 0.663 | 0.000 | 0.309 | 0.000 | 0.333 | 0.425 | 0.000 | 0.000 | 0.000 | 0.317 | 0.480 |

C_{33} | 0.000 | 0.294 | 0.386 | 0.500 | 0.324 | 0.000 | 0.000 | 0.337 | 0.283 | 0.000 | 1.000 | 0.333 | 0.111 | 1.000 | 0.498 | 0.425 | 0.362 | 0.000 |

C_{41} | 0.312 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.500 | 0.511 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

C_{42} | 0.333 | 0.528 | 0.500 | 0.000 | 0.458 | 0.000 | 0.000 | 0.348 | 1.000 | 0.000 | 0.534 | 0.000 | 0.489 | 0.000 | 0.484 | 1.000 | 0.651 | 1.000 |

C_{43} | 0.355 | 0.472 | 0.500 | 1.000 | 0.542 | 0.000 | 0.000 | 0.652 | 0.000 | 0.000 | 0.466 | 0.500 | 0.000 | 0.000 | 0.516 | 0.000 | 0.349 | 0.000 |

C_{51} | 1.000 | 0.000 | 0.000 | 0.362 | 0.334 | 0.000 | 0.335 | 0.442 | 0.000 | 0.227 | 0.000 | 0.303 | 0.349 | 0.000 | 0.000 | 0.555 | 0.000 | 0.000 |

C_{52} | 0.000 | 0.000 | 1.000 | 0.342 | 0.314 | 0.000 | 0.323 | 0.264 | 0.000 | 0.232 | 0.765 | 0.355 | 0.316 | 0.000 | 0.000 | 0.445 | 0.000 | 0.000 |

C_{53} | 0.000 | 0.000 | 0.000 | 0.296 | 0.352 | 0.000 | 0.342 | 0.294 | 1.000 | 0.541 | 0.235 | 0.342 | 0.335 | 1.000 | 1.000 | 0.000 | 0.000 | 0.000 |

C_{61} | 0.000 | 0.555 | 0.000 | 0.513 | 0.501 | 0.000 | 0.000 | 0.564 | 0.621 | 0.532 | 1.000 | 0.422 | 0.000 | 0.500 | 0.000 | 1.000 | 0.000 | 1.000 |

C_{62} | 0.000 | 0.445 | 1.000 | 0.487 | 0.499 | 0.000 | 0.000 | 0.436 | 0.379 | 0.468 | 0.000 | 0.578 | 1.000 | 0.500 | 0.000 | 0.000 | 1.000 | 0.000 |

C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | C_{51} | C_{52} | C_{53} | C_{61} | C_{62} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{11} | 0.000 | 0.168 | 0.099 | 0.134 | 0.068 | 0.000 | 0.000 | 0.000 | 0.000 | 0.153 | 0.279 | 0.090 | 0.127 | 0.077 | 0.228 | 0.000 | 0.000 | 0.000 |

C_{12} | 0.103 | 0.000 | 0.096 | 0.000 | 0.114 | 0.000 | 0.000 | 0.101 | 0.130 | 0.000 | 0.000 | 0.073 | 0.136 | 0.000 | 0.000 | 0.000 | 0.217 | 0.000 |

C_{13} | 0.108 | 0.000 | 0.000 | 0.076 | 0.068 | 0.000 | 0.000 | 0.082 | 0.126 | 0.133 | 0.000 | 0.067 | 0.076 | 0.076 | 0.000 | 0.000 | 0.108 | 0.116 |

C_{14} | 0.088 | 0.084 | 0.069 | 0.000 | 0.064 | 0.000 | 0.000 | 0.080 | 0.070 | 0.103 | 0.000 | 0.048 | 0.000 | 0.076 | 0.000 | 0.000 | 0.000 | 0.150 |

C_{21} | 0.000 | 0.244 | 0.000 | 0.000 | 0.000 | 0.622 | 0.289 | 0.193 | 0.000 | 0.000 | 0.176 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.183 |

C_{22} | 0.000 | 0.000 | 0.000 | 0.088 | 0.091 | 0.000 | 0.304 | 0.000 | 0.000 | 0.000 | 0.000 | 0.088 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

C_{23} | 0.000 | 0.000 | 0.000 | 0.115 | 0.091 | 0.378 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.088 | 0.000 | 0.201 | 0.200 | 0.225 | 0.000 | 0.000 |

C_{31} | 0.000 | 0.091 | 0.161 | 0.105 | 0.068 | 0.000 | 0.000 | 0.000 | 0.198 | 0.228 | 0.000 | 0.071 | 0.121 | 0.000 | 0.124 | 0.159 | 0.093 | 0.123 |

C_{32} | 0.298 | 0.087 | 0.000 | 0.000 | 0.066 | 0.000 | 0.000 | 0.148 | 0.000 | 0.102 | 0.000 | 0.071 | 0.110 | 0.000 | 0.000 | 0.000 | 0.092 | 0.114 |

C_{33} | 0.000 | 0.074 | 0.101 | 0.105 | 0.064 | 0.000 | 0.000 | 0.075 | 0.078 | 0.000 | 0.214 | 0.071 | 0.029 | 0.247 | 0.123 | 0.117 | 0.105 | 0.000 |

C_{41} | 0.052 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.062 | 0.077 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

C_{42} | 0.055 | 0.074 | 0.073 | 0.000 | 0.048 | 0.000 | 0.000 | 0.046 | 0.162 | 0.000 | 0.066 | 0.000 | 0.074 | 0.000 | 0.060 | 0.139 | 0.129 | 0.162 |

C_{43} | 0.059 | 0.066 | 0.073 | 0.116 | 0.056 | 0.000 | 0.000 | 0.085 | 0.000 | 0.000 | 0.058 | 0.062 | 0.000 | 0.000 | 0.064 | 0.000 | 0.069 | 0.000 |

C_{51} | 0.238 | 0.000 | 0.000 | 0.060 | 0.041 | 0.000 | 0.136 | 0.056 | 0.000 | 0.042 | 0.000 | 0.042 | 0.058 | 0.000 | 0.000 | 0.126 | 0.000 | 0.000 |

C_{52} | 0.000 | 0.000 | 0.210 | 0.057 | 0.039 | 0.000 | 0.131 | 0.033 | 0.000 | 0.043 | 0.105 | 0.049 | 0.053 | 0.000 | 0.000 | 0.101 | 0.000 | 0.000 |

C_{53} | 0.000 | 0.000 | 0.000 | 0.049 | 0.044 | 0.000 | 0.139 | 0.037 | 0.156 | 0.101 | 0.032 | 0.047 | 0.056 | 0.202 | 0.201 | 0.000 | 0.000 | 0.000 |

C_{61} | 0.000 | 0.063 | 0.000 | 0.049 | 0.040 | 0.000 | 0.000 | 0.036 | 0.049 | 0.050 | 0.070 | 0.030 | 0.000 | 0.060 | 0.000 | 0.133 | 0.000 | 0.152 |

C_{62} | 0.000 | 0.051 | 0.119 | 0.046 | 0.039 | 0.000 | 0.000 | 0.028 | 0.030 | 0.044 | 0.000 | 0.040 | 0.085 | 0.060 | 0.000 | 0.000 | 0.186 | 0.000 |

C_{11} | C_{12} | C_{13} | C_{14} | C_{21} | C_{22} | C_{23} | C_{31} | C_{32} | C_{33} | C_{41} | C_{42} | C_{43} | C_{51} | C_{52} | C_{53} | C_{61} | C_{62} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{11} | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 | 0.067 |

C_{12} | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 |

C_{13} | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 |

C_{14} | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 |

C_{21} | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 | 0.083 |

C_{22} | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 | 0.037 |

C_{23} | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 | 0.066 |

C_{31} | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 | 0.089 |

C_{32} | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 | 0.068 |

C_{33} | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 | 0.071 |

C_{41} | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 | 0.010 |

C_{42} | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 | 0.058 |

C_{43} | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 |

C_{51} | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 | 0.052 |

C_{52} | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 |

C_{53} | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 | 0.062 |

C_{61} | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 | 0.039 |

C_{62} | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 |

P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DM_{1} | C_{11} | (G, VH) | (G, VH) | (VG, H) | (G, H) | (G, VH) | DM_{2} | (G, H) | (G, H) | (G, VH) | (G, VH) | (G, H) |

C_{12} | (VG, N) | (G, VH) | (G, VH) | (VG, N) | (VG, N) | (G, VH) | (G, H) | (VG, N) | (G, H) | (G, VH) | ||

C_{13} | (G, H) | (G, H) | (G, H) | (G, VH) | (G, VH) | (G, VH) | (G, H) | (G, VH) | (G, H) | (G, H) | ||

C_{14} | (G, VH) | (G, VH) | (VG, H) | (G, VH) | (VG, N) | (G, VH) | (G, VH) | (G, H) | (VG, H) | (VG, H) | ||

C_{21} | (G, H) | (G, N) | (G, H) | (G, VH) | (G, H) | (G, VH) | (G, H) | (G, VH) | (G, H) | (G, H) | ||

C_{22} | (G, VH) | (G, VH) | (G, H) | (G, H) | (G, VH) | (G, H) | (G, N) | (G, H) | (G, VH) | (VG, N) | ||

C_{23} | (G, H) | (G, N) | (G, H) | (G, VH) | (F, VH) | (G, VH) | (G, H) | (G, VH) | (G, H) | (G, VH) | ||

C_{31} | (G, VH) | (G, H) | (G, VH) | (G, VH) | (G, H) | (G, VH) | (G, H) | (G, VH) | (G, VH) | (VG, H) | ||

C_{32} | (VG, H) | (VG, H) | (G, VH) | (F, VH) | (G, H) | (VG, H) | (VG, H) | (VG, VH) | (G, VH) | (G, VH) | ||

C_{33} | (G, H) | (G, VH) | (G, H) | (G, H) | (G, N) | (G, H) | (G, H) | (G, H) | (G, N) | (G, H) | ||

C_{41} | (G, VH) | (G, VH) | (G, H) | (VG, N) | (G, VH) | (G, VH) | (G, VH) | (VG, N) | (G, VH) | (G, H) | ||

C_{42} | (VG, H) | (VG, H) | (VG, H) | (G, VH) | (G, H) | (VG, H) | (VG, H) | (G, VH) | (VG, N) | (VG, H) | ||

C_{43} | (VG, H) | (G, VH) | (G, VH) | (G, H) | (G, H) | (G, VH) | (G, H) | (G, H) | (G, N) | (G, VH) | ||

C_{51} | (G, H) | (G, VH) | (G, H) | (G, N) | (F, VH) | (G, VH) | (G, H) | (G, VH) | (G, VH) | (G, VH) | ||

C_{52} | (G, VH) | (F, VH) | (VG, N) | (G, H) | (G, N) | (VG, N) | (G, H) | (G, H) | (G, N) | (G, VH) | ||

C_{53} | (G, VH) | (G, H) | (VG, H) | (G, VH) | (VG, N) | (G, VH) | (G, H) | (VG, N) | (F, N) | (G, VH) | ||

C_{61} | (G, H) | (G, H) | (G, VH) | (VG, N) | (G, H) | (G, H) | (G, H) | (G, VH) | (G, N) | (VG, H) | ||

C_{62} | (VG, N) | (G, VH) | (G, H) | (G, H) | (G, H) | (G, VH) | (G, VH) | (G, H) | (F, N) | (G, H) | ||

DM_{3} | C_{11} | (G, H) | (G, H) | (G, H) | (G, H) | (G, N) | DM_{4} | (G, VH) | (G, H) | (G, VH) | (VG, N) | (G, VH) |

C_{12} | (G, H) | (G, N) | (G, N) | (F, VH) | (F, VH) | (G, VH) | (G, VH) | (VG, N) | (G, VH) | (G, H) | ||

C_{13} | (F, VH) | (F, H) | (F, H) | (F, H) | (F, H) | (VG, H) | (VG, N) | (G, VH) | (VG, N) | (G, H) | ||

C_{14} | (G, N) | (G, H) | (G, N) | (G, H) | (G, N) | (G, H) | (G, VH) | (G, H) | (G, VH) | (G, H) | ||

C_{21} | (G, N) | (G, H) | (G, N) | (G, VH) | (G, N) | (G, H) | (F, VH) | (G, VH) | (F, VH) | (G, H) | ||

C_{22} | (G, H) | (G, H) | (G, H) | (G, N) | (G, N) | (G, H) | (G, VH) | (G, H) | (G, VH) | (G, VH) | ||

C_{23} | (F, H) | (F, VH) | (F, H) | (F, H) | (G, H) | (G, N) | (F, VH) | (G, VH) | (G, H) | (G, N) | ||

C_{31} | (G, H) | (G, H) | (G, N) | (G, N) | (G, H) | (G, VH) | (VG, N) | (G, H) | (G, VH) | (VG, N) | ||

C_{32} | (G, N) | (F, VH) | (G, N) | (F, H) | (G, VH) | (G, H) | (G, H) | (G, VH) | (G, VH) | (G, H) | ||

C_{33} | (G, N) | (F, H) | (G, H) | (F, H) | (G, N) | (G, VH) | (VG, N) | (G, H) | (G, VH) | (G, H) | ||

C_{41} | (G, H) | (G, N) | (G, H) | (G, N) | (G, N) | (VG, H) | (G, VH) | (G, VH) | (VG, N) | (G, VH) | ||

C_{42} | (F, VH) | (G, N) | (F, H) | (G, H) | (F, VH) | (G, VH) | (G, VH) | (G, H) | (G, VH) | (G, H) | ||

C_{43} | (G, N) | (G, N) | (G, H) | (G, H) | (G, N) | (G, H) | (VG, N) | (G, VH) | (G, H) | (G, VH) | ||

C_{51} | (F, H) | (G, H) | (F, VH) | (G, N) | (F, VH) | (G, H) | (G, VH) | (G, N) | (G, VH) | (G, H) | ||

C_{52} | (G, N) | (G, H) | (G, VH) | (G, VH) | (G, N) | (G, VH) | (G, H) | (G, VH) | (VG, N) | (G, VH) | ||

C_{53} | (G, N) | (F, VH) | (F, VH) | (F, H) | (F, H) | (G, H) | (F, VH) | (G, H) | (G, VH) | (G, H) | ||

C_{61} | (F, VH) | (F, VH) | (F, H) | (F, H) | (G, N) | (G, VH) | (G, H) | (G, N) | (VG, N) | (G, VH) | ||

C_{62} | (G, H) | (G, N) | (G, H) | (F, VH) | (F, VH) | (G, H) | (G, VH) | (G, H) | (G, H) | (G, H) |

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**Figure 1.**Flowchart of the integrated ANP-TODIM method under Z-information Z-ANP and Z-TODIM method.

Criteria | Sub-Criteria | Category | Related Literature |
---|---|---|---|

Quality assurance | Quality management capacity C_{11} | B | [38,43] |

Quality certification level C_{12} | B | ||

Acceptance rate C_{13} | B | ||

Feedback and improvement C_{14} | B | ||

Cost control | Product price C_{21} | C | [44,48] |

Quantity discount C_{22} | B | ||

Transportation and installation cost C_{23} | C | ||

Technical capacity | Technical advancement C_{31} | B | [4,39,49] |

R&D capability C_{32} | B | ||

Technical equipment level C_{33} | B | ||

Enterprise qualification | Business credit status C_{41} | B | [39,40] |

Market reputation C_{42} | B | ||

Industry ranking C_{43} | B | ||

Deliver capability | Delivery cycle C_{51} | C | [9,46] |

Rate of delivery in time C_{52} | B | ||

Emergency delivery capability C_{53} | B | ||

Environmental consciousness | Energy utilization ratio C_{61} | B | [7,41,50] |

Energy-saving measures C_{62} | B |

Linguistic Term | Triangular Fuzzy Scale | Triangular Fuzzy Reciprocal Scale |
---|---|---|

Just equal | (1,1,1) | (1,1,1) |

Equal importance (EI) | (1/2,1,3/2) | (2/3,1,2) |

Weakly importance (WI) | (1,3/2,2) | (1/2,2/3,1) |

Moderate importance (MI) | (3/2,2,5/2) | (2/5,1/2,2/3) |

Very importance (VI) | (2,5/2,3) | (1/3,2/5,1/2) |

Absolutely importance (AI) | (5/2,3,7/2) | (2/7,1/3,2/5) |

Constraint | Reliability | ||
---|---|---|---|

Linguistic Term | Fuzzy Scale | Linguistic Term | Fuzzy Scale |

Very poor (VP) | (0,0,0.25) | Very low (VL) | (0,0,0.3) |

Poor (P) | (0,0.25,0.5) | Low (L) | (0.1,0.3,0.5) |

Fairly (F) | (0.25,0.5,0.75) | Neutral (N) | (0.3,0.5,0.7) |

Good (G) | (0.5,0.75,1.0) | High (H) | (0.5,0.7,0.9) |

Very good (VG) | (0.75,1.0,1.0) | Very high (VH) | (0.7,1.0,1.0) |

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | |
---|---|---|---|---|---|---|

C_{1} | (1.00,1.00,1.00) | (0.55,0.86,1.20) | (0.60,1.00,1.40) | (0.97,1.35,1.74) | (0.81,1.20,1.61) | (0.99,1.39,1.79) |

C_{2} | (0.51,0.74,1.24) | (1.00,1.00,1.00) | (0.64,0.93,1.24) | (0.89,1.29,1.69) | (0.73,1.14,1.56) | (1.05,1.44,1.83) |

C_{3} | (0.47,0.68,1.22) | (0.55,0.73,1.01) | (1.00,1.00,1.00) | (0.99,1.39,1.79) | (0.80,1.20,1.60) | (1.05,1.48,1.92) |

C_{4} | (0.35,0.45,0.65) | (0.38,0.50,0.74) | (0.36,0.47,0.67) | (1.00,1.00,1.00) | (0.38,0.50,0.74) | (0.60,1.00,1.40) |

C_{5} | (0.40,0.54,0.81) | (0.45,0.63,1.04) | (0.40,0.53,0.80) | (0.91,1.31,1.71) | (1.00,1.00,1.00) | (0.95,1.33,1.71) |

C_{6} | (0.36,0.47,0.67) | (0.33,0.42,0.59) | (0.39,0.51,0.73) | (0.47,0.68,1.22) | (0.35,0.45,0.65) | (1.00,1.00,1.00) |

$V({S}_{1}\ge {S}_{2})=1.000$ | $V({S}_{1}\ge {S}_{3})=1.000$ | $V({S}_{1}\ge {S}_{4})=1.000$ | $V({S}_{1}\ge {S}_{5})=1.000$ | $V({S}_{1}\ge {S}_{6})=1.000$ |

$V({S}_{2}\ge {S}_{1})=0.967$ | $V({S}_{2}\ge {S}_{3})=1.000$ | $V({S}_{2}\ge {S}_{4})=1.000$ | $V({S}_{2}\ge {S}_{5})=1.000$ | $V({S}_{2}\ge {S}_{6})=1.000$ |

$V({S}_{3}\ge {S}_{1})=0.959$ | $V({S}_{3}\ge {S}_{2})=0.992$ | $V({S}_{3}\ge {S}_{4})=1.000$ | $V({S}_{3}\ge {S}_{5})=1.000$ | $V({S}_{3}\ge {S}_{6})=1.000$ |

$V({S}_{4}\ge {S}_{1})=0.522$ | $V({S}_{4}\ge {S}_{2})=0.550$ | $V({S}_{4}\ge {S}_{3})=0.554$ | $V({S}_{4}\ge {S}_{5})=0.723$ | $V({S}_{4}\ge {S}_{6})=1.000$ |

$V({S}_{5}\ge {S}_{1})=0.794$ | $V({S}_{5}\ge {S}_{2})=0.825$ | $V({S}_{5}\ge {S}_{3})=0.833$ | $V({S}_{5}\ge {S}_{4})=1.000$ | $V({S}_{5}\ge {S}_{6})=1.000$ |

$V({S}_{6}\ge {S}_{1})=0.451$ | $V({S}_{6}\ge {S}_{2})=0.478$ | $V({S}_{6}\ge {S}_{3})=0.480$ | $V({S}_{6}\ge {S}_{4})=0.913$ | $V({S}_{6}\ge {S}_{5})=0.643$ |

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

Quality assurance | Quality management capacity C_{11} | 0.067 |

Quality certification level C_{12} | 0.058 | |

Acceptance rate C_{13} | 0.062 | |

Feedback and improvement C_{14} | 0.052 | |

Cost control | Products price C_{21} | 0.083 |

Quantity discount C_{22} | 0.037 | |

Transportation and installation cost C_{23} | 0.066 | |

Technical capacity | Technical advancement C_{31} | 0.089 |

R&D capability C_{32} | 0.068 | |

Technical equipment level C_{33} | 0.071 | |

Enterprise qualification | Business credit status C_{41} | 0.010 |

Market reputation C_{42} | 0.058 | |

Industry ranking C_{43} | 0.040 | |

Deliver capability | Delivery cycle C_{51} | 0.052 |

Rate of delivery in time C_{52} | 0.046 | |

Emergency delivery capability C_{53} | 0.062 | |

Environmental consciousness | Energy utilization ratio C_{61} | 0.039 |

Energy-saving measures C_{62} | 0.040 |

P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | |
---|---|---|---|---|---|

C_{11} | (0.036,0.517,1.000) | (0.010,0.474,0.939) | (0.149,0.629,1.000) | (0.068,0.517,0.870) | (0.000,0.464,0.928) |

C_{12} | (0.128,0.614,1.000) | (0.028,0.513,1.000) | (0.123,0.558,0.792) | (0.000,0.480,0.865) | (0.000,0.480,0.865) |

C_{13} | (0.130,0.610,0.983) | (0.032,0.470,0.810) | (0.027,0.513,1.000) | (0.059,0.513,0.869) | (0.000,0.470,0.940) |

C_{14} | (0.000,0.439,0.879) | (0.060,0.529,1.000) | (0.055,0.464,0.767) | (0.140,0.595,0.943) | (0.110,0.504,0.701) |

C_{21} | (0.055,0.471,0.887) | (0.174,0.590,1.000) | (0.000,0.432,0.862) | (0.000,0.461,0.917) | (0.110,0.511,0.912) |

C_{22} | (0.028,0.513,1.000) | (0.017,0.502,0.988) | (0.000,0.469,0.938) | (0.017,0.502,0.988) | (0.078,0.547,0.915) |

C_{23} | (0.114,0.510,0.907) | (0.190,0.600,1.000) | (0.000,0.425,0.850) | (0.053,0.463,0.874) | (0.076,0.487,0.892) |

C_{31} | (0.060,0.529,1.000) | (0.039,0.448,0.767) | (0.000,0.438,0.878) | (0.024,0.479,0.933) | (0.145,0.555,0.767) |

C_{32} | (0.291,0.661,0.839) | (0.241,0.633,0.839) | (0.277,0.688,0.989) | (0.000,0.419,0.843) | (0.177,0.588,1.000) |

C_{33} | (0.109,0.548,0.987) | (0.093,0.532,0.878) | (0.119,0.559,1.000) | (0.000,0.438,0.878) | (0.046,0.454,0.862) |

C_{41} | (0.149,0.629,1.000) | (0.026,0.506,0.987) | (0.068,0.517,0.870) | (0.090,0.506,0.731) | (0.000,0.464,0.928) |

C_{42} | (0.148,0.660,0.936) | (0.211,0.694,0.936) | (0.010,0.510,0.890) | (0.102,0.602,1.000) | (0.000,0.493,0.872) |

C_{43} | (0.131,0.557,0.876) | (0.080,0.491,0.810) | (0.085,0.542,1.000) | (0.000,0.410,0.820) | (0.050,0.491,0.932) |

C_{51} | (0.145,0.538,0.931) | (0.000,0.407,0.814) | (0.167,0.561,0.949) | (0.118,0.497,0.877) | (0.167,0.588,1.000) |

C_{52} | (0.164,0.594,0.932) | (0.000,0.441,0.886) | (0.200,0.645,1.000) | (0.138,0.553,0.876) | (0.072,0.502,0.932) |

C_{53} | (0.183,0.591,1.000) | (0.000,0.413,0.835) | (0.235,0.624,0.835) | (0.005,0.408,0.816) | (0.145,0.539,0.849) |

C_{61} | (0.027,0.511,1.000) | (0.000,0.467,0.940) | (0.000,0.473,0.946) | (0.065,0.473,0.685) | (0.201,0.658,1.000) |

C_{62} | (0.252,0.617,0.904) | (0.217,0.609,1.000) | (0.183,0.548,0.913) | (0.000,0.357,0.722) | (0.104,0.478,0.857) |

$\mathit{\delta}$ | $\mathit{\theta}=1$ | $\mathit{\theta}=1.5$ | $\mathit{\theta}=2$ | $\mathit{\theta}=2.5$ | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\xi}$ | Ranking | $\mathit{\xi}$ | Ranking | $\mathit{\xi}$ | Ranking | $\mathit{\xi}$ | Ranking | |

${P}_{1}$ | 1.000 | 1 | 1.000 | 1 | 1.000 | 1 | 1.000 | 1 |

${P}_{2}$ | 0.485 | 3 | 0.485 | 3 | 0.485 | 3 | 0.484 | 3 |

${P}_{3}$ | 0.695 | 2 | 0.695 | 2 | 0.696 | 2 | 0.696 | 2 |

${P}_{4}$ | 0.000 | 5 | 0.000 | 5 | 0.000 | 5 | 0.000 | 5 |

${P}_{5}$ | 0.385 | 4 | 0.385 | 4 | 0.385 | 4 | 0.385 | 4 |

Methods | Ranking Orders |
---|---|

Fuzzy AHP-TOPSIS | ${P}_{1}\succ {P}_{5}\succ {P}_{3}\succ {P}_{2}\succ {P}_{4}$ |

Rough BWM-MAIRCA | ${P}_{1}\succ {P}_{3}\succ {P}_{5}\succ {P}_{2}\succ {P}_{4}$ |

The proposed ranking | ${P}_{1}\succ {P}_{3}\succ {P}_{2}\succ {P}_{5}\succ {P}_{4}$ |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.-H.; Peng, H.-M.; Wang, T.-L.; Wang, X.-K.; Wang, J.-Q.
Supplier Selection in the Nuclear Power Industry with an Integrated ANP-TODIM Method under Z-Number Circumstances. *Symmetry* **2020**, *12*, 1357.
https://doi.org/10.3390/sym12081357

**AMA Style**

Liu Y-H, Peng H-M, Wang T-L, Wang X-K, Wang J-Q.
Supplier Selection in the Nuclear Power Industry with an Integrated ANP-TODIM Method under Z-Number Circumstances. *Symmetry*. 2020; 12(8):1357.
https://doi.org/10.3390/sym12081357

**Chicago/Turabian Style**

Liu, Ya-Hua, Heng-Ming Peng, Tie-Li Wang, Xiao-Kang Wang, and Jian-Qiang Wang.
2020. "Supplier Selection in the Nuclear Power Industry with an Integrated ANP-TODIM Method under Z-Number Circumstances" *Symmetry* 12, no. 8: 1357.
https://doi.org/10.3390/sym12081357