# A Supplier Selection Decision-Making Approach for Complex Product Development Based on Hesitant Fuzzy Information

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1**

## 3. Supplier Selection Decision-Making Approach with Hesitant Fuzzy Information

#### 3.1. Bidirectional Projection

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Definition 6.**

- (1)
- The weighted projection of alternative supplier ${S}_{i}$ under the attribute ${G}_{j}$ of the positive ideal direction is described by:

- (2)
- The weighted projection of alternative supplier $S$ under the attribute $G$ of the negative ideal direction is described by:

#### 3.2. Attribute Weight Determination Model

#### 3.3. Decision-Making Procedure

- Step 1
- The hesitant fuzzy decision-making matrix of alternative suppliers is given. After normalized processing, the normalized hesitant fuzzy decision-making matrix $F={\left[{f}_{ij}\right]}_{m\times n}$ is obtained.
- Step 2
- According to Definition 6, in the matrix $F={\left[{f}_{ij}\right]}_{m\times n}$, the positive ideal hesitant fuzzy element and negative ideal hesitant fuzzy element are given.
- Step 3
- By applying Equations (4)–(7), the bidirectional projection value matrices are provided.
- Step 4
- By applying Equation (15), the attribute weights ${\alpha}_{j}=\left\{{\alpha}_{1},{\alpha}_{2},\cdot \cdot \cdot {\alpha}_{n}\right\}$ and ${\beta}_{j}=\left\{{\beta}_{1},{\beta}_{2},\cdot \cdot \cdot {\beta}_{n}\right\}$ can be solved.
- Step 5
- By using Equations (9)–(11), the values of $D\left({S}_{i}\right)$ are calculated, and then the alternative suppliers are ranked according to the values of $D\left({S}_{i}\right)$.

## 4. Illustrative Example

#Regular rectangular word clouds. #Introduction of jieba and wordcloud libraries. import jieba import wordcloud #Open the document in which the extracted comment is located. f = open(“F:\list_attribute.txt”, “r”, encoding=“utf-8”) t = f.read() f.close() #Word. ls = jieba.lcut(t) txt = “ ”.join(ls) #Draw word clouds. w = wordcloud.WordCloud(width = 1000, height = 700, background_color = “white”, font_path = “msyh.ttf”) w.generate(txt) w.to_file(“word cloud.png”) # Statistical word frequency. counts = {} for word in ls: if len(word) == 1: continue else: counts[word] = counts.get(word,0) + 1 items = list(counts.items()) items.sort(key=lambda x:x[1], reverse=True) #Output the words and their corresponding word frequency, meanwhile, the form of “words-> word frequency” is written to the txt document. for i in range(20): word, count = items[i] with open(‘F:// word frequency.txt’,‘a’,encoding=‘utf-8’) as f: f.write(word+‘->’+str(count)+‘\n’) #print (“{0:<10}{1:>5}”.format(word, count)). |

## 5. Discussions

#### 5.1. Sensitivity Analysis

#### 5.2. Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ | ${\mathit{S}}_{5}$ | |
---|---|---|---|---|---|

${G}_{1}$ | $\left\{0.2,0.3,0.7\right\}$ | $\{0.2,0.4,0.5\}$ | $\left\{0.4,0.5,0.6\right\}$ | $\left\{0.3,0.4\right\}$ | $\left\{0.3,0.8\right\}$ |

${G}_{2}$ | $\left\{0.6,0.7\right\}$ | $\left\{0.3,0.7\right\}$ | $\left\{0.3,0.6\right\}$ | $\left\{0.6\right\}$ | $\left\{0.4,0.5,0.8\right\}$ |

${G}_{3}$ | $\left\{0.4,0.6\right\}$ | $\left\{0.3,0.7\right\}$ | $\left\{0.4,0.5\right\}$ | $\left\{0.2,0.8\right\}$ | $\left\{0.6\right\}$ |

${G}_{4}$ | $\{0.4,0.4,0.7\}$ | $\left\{0.4,0.4\right\}$ | $\left\{0.3,0.7\right\}$ | $\left\{0.2,0.6\right\}$ | $\left\{0.1,0.4,0.6\right\}$ |

${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ | ${\mathit{S}}_{5}$ | |
---|---|---|---|---|---|

${G}_{1}$ | $\left\{0.2,0.3,0.7\right\}$ | $\{0.2,0.4,0.5\}$ | $\left\{0.4,0.5,0.6\right\}$ | $\left\{0.3,0.4,0.4\right\}$ | $\left\{0.3,0.8,0.8\right\}$ |

${G}_{2}$ | $\left\{0.6,0.7,0.7\right\}$ | $\left\{0.3,0.7,0.7\right\}$ | $\left\{0.3,0.6,0.6\right\}$ | $\left\{0.6,0.6,0.6\right\}$ | $\left\{0.4,0.5,0.8\right\}$ |

${G}_{3}$ | $\left\{0.4,0.6\right\}$ | $\left\{0.3,0.7\right\}$ | $\left\{0.4,0.5\right\}$ | $\left\{0.2,0.8\right\}$ | $\left\{0.6,0.6\right\}$ |

${G}_{4}$ | $\{0.4,0.4,0.7\}$ | $\left\{0.4,0.4,0.4\right\}$ | $\left\{0.3,0.7,0.7\right\}$ | $\left\{0.2,0.6,0.6\right\}$ | $\left\{0.1,0.4,0.6\right\}$ |

Parameter $\mathit{\theta}$ | $\mathbf{The}\mathbf{Values}\mathbf{of}\mathit{D}\left({\mathit{S}}_{\mathit{i}}\right),\mathit{i}=1,2,3,4,5$. | The Ranking Result of Alternative Suppliers |
---|---|---|

$0.2$ | $D({S}_{1})=0.14$ $D({S}_{2})=0.07$ $D({S}_{3})=0.12$ $D({S}_{4})=0.09$ $D({S}_{5})=0.21$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

$0.4$ | $D({S}_{1})=0.30$ $D({S}_{2})=0.17$ $D({S}_{3})=0.26$ $D({S}_{4})=0.22$ $D({S}_{5})=0.41$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

$0.5$ | $D({S}_{1})=0.39$ $D({S}_{2})=0.24$ $D({S}_{3})=0.35$ $D({S}_{4})=0.29$ $D({S}_{5})=0.51$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

$0.6$ | $D({S}_{1})=0.49$ $D({S}_{2})=0.32$ $D({S}_{3})=0.45$ $D({S}_{4})=0.38$ $D({S}_{5})=0.61$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

$0.8$ | $D({S}_{1})=0.72$ $D({S}_{2})=0.56$ $D({S}_{3})=0.68$ $D({S}_{4})=0.63$ $D({S}_{5})=0.81$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

Methods | Attribute Weights ${\mathit{\alpha}}_{\mathit{i}}$ | Attribute Weights ${\mathit{\beta}}_{\mathit{j}}$ | $\mathbf{The}\mathbf{Values}\mathbf{of}\mathit{D}\left({\mathit{S}}_{\mathit{i}}\right),\mathit{i}=1,2,3,4,5$. | The Ranking Result of Alternative Suppliers |
---|---|---|---|---|

The proposed method of this paper | $\left\{0.26,0.23,0.24,0.27\right\}$ | $\left\{0.36,0.17,0.25,0.22\right\}$ | $D({S}_{1})=0.49$ $D({S}_{2})=0.32$ $D({S}_{3})=0.45$ $D({S}_{4})=0.38$ $D({S}_{5})=0.61$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

The existing model of the literature [56] | $\left\{0.20,0.15,0.20,0.45\right\}$ | $\left\{0.20,0.50,0.20,0.10\right\}$ | $D({S}_{1})=0.56$ $D({S}_{2})=0.36$ $D({S}_{3})=0.51$ $D({S}_{4})=0.46$ $D({S}_{5})=0.57$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

The existing model of the literature [72] | $\left\{0.26,0.24,0.24,0.26\right\}$ | $\left\{0.30,0.21,0.25,0.24\right\}$ | $D({S}_{1})=0.50$ $D({S}_{2})=0.33$ $D({S}_{3})=0.45$ $D({S}_{4})=0.39$ $D({S}_{5})=0.60$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

The existing model of the literature [73] | $\left\{0.29,0.15,0.23,0.33\right\}$ | $\left\{0.55,0.15,0.20,0.10\right\}$ | $D({S}_{1})=0.46$ $D({S}_{2})=0.30$ $D({S}_{3})=0.45$ $D({S}_{4})=0.36$ $D({S}_{5})=0.65$ | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

Attributes | Unit of Measurement | Attribute Types |
---|---|---|

Enterprise reputation | Score | Benefit attribute |

Product quality | Score | Benefit attribute |

Technical ability | Score | Benefit attribute |

Service level | Score | Benefit attribute |

Attributes | $\mathbf{Attribute}\mathbf{Weights}{\mathit{\eta}}_{\mathit{t}}$ |
---|---|

Enterprise reputation | 0.33 |

Product quality | 0.28 |

Technical ability | 0.26 |

Service level | 0.13 |

Attributes | Alternative Supplier 1 | Alternative Supplier 2 | Alternative Supplier 3 | Alternative Supplier 4 | Alternative Supplier 5 |
---|---|---|---|---|---|

Enterprise reputation | 100 | 90 | 100 | 100 | 100 |

Product quality | 75 | 70 | 70 | 60 | 80 |

Technical ability | 75 | 90 | 75 | 90 | 75 |

Service level | 100 | 100 | 95 | 90 | 95 |

Model | The Optimal Ranking Value | The Ranking Result of Alternative Suppliers |
---|---|---|

The developed model of this paper | 0.61 | ${S}_{5}\succ {S}_{1}\succ {S}_{3}\succ {S}_{4}\succ {S}_{2}$ |

The existing model of the literature [74] | 0.54 | ${S}_{5}\succ {S}_{1}\succ {S}_{2}\succ {S}_{4}\succ {S}_{3}$ |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, B.; Su, J.; Yuan, B.; Li, L.; Zhao, Y.; Qin, Z.; Qian, L.
A Supplier Selection Decision-Making Approach for Complex Product Development Based on Hesitant Fuzzy Information. *Axioms* **2023**, *12*, 1006.
https://doi.org/10.3390/axioms12111006

**AMA Style**

Li B, Su J, Yuan B, Li L, Zhao Y, Qin Z, Qian L.
A Supplier Selection Decision-Making Approach for Complex Product Development Based on Hesitant Fuzzy Information. *Axioms*. 2023; 12(11):1006.
https://doi.org/10.3390/axioms12111006

**Chicago/Turabian Style**

Li, Baodong, Jiafu Su, Boqiao Yuan, Lvcheng Li, Yihuan Zhao, Zhidan Qin, and Li Qian.
2023. "A Supplier Selection Decision-Making Approach for Complex Product Development Based on Hesitant Fuzzy Information" *Axioms* 12, no. 11: 1006.
https://doi.org/10.3390/axioms12111006