# Hesitant Fuzzy SWARA-Complex Proportional Assessment Approach for Sustainable Supplier Selection (HF-SWARA-COPRAS)

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## Abstract

**:**

## 1. Introduction

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- A novel HF-SWARA-COPRAS method is introduced.
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- The criteria weights are evaluated by the SWARA method.
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- To illustrate the feasibility and usefulness of the HF-SWARA-COPRAS method, an empirical case study of SSS problem is discussed under HFSs environment.
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- Sensitivity analysis and comparative study are discussed to confirm the stability and validity of the developed methodology.

## 2. Literature Review

#### 2.1. Hesitant Fuzzy Sets

#### 2.2. Step-Wise Weight Assessment Ratio Analysis (SWARA)

#### 2.3. Complex Proportional Assessment (COPRAS) Approach

#### 2.4. Sustainable Supplier Selection (SSS)

## 3. Proposed Methodology

#### 3.1. Prerequisites

**Definition 1**

**.**Given a discourse set Y. A HFS R on Y is defined by the function ${\u0127}_{R}\left(y\right)$ implemented to Y, which maps a finite subset of $[0,1],$ is denoted by

**Definition 2**

**.**Assume that $\u0127,\text{}{\u0127}_{1},\text{}{\u0127}_{2}\in HFEs\left(Y\right),$ then the fundamental laws on HFEs are discussed as below:

**Definition 3.**

**Definition 4**

**.**Consider a set of HFEs $H=\left\{\text{}{\u0127}_{1},\text{}{\u0127}_{2},\text{}\dots ,\text{}{\u0127}_{n}\right\},$ then the Hesitant Fuzzy Weighted Average (HFWA) and geometric (HFWG) operators are mappings ${H}^{n}\to H$ such that

**Definition 5.**

#### 3.2. Hesitant Fuzzy SWARA-COPRAS Method

^{th}DE.

_{j}). The relative significance of each criterion is estimated in relation to preceding criteria.

_{j}) by using

_{j}) by using

## 4. An Empirical Study: Sustainable Supplier Selection (SSS)

_{1}), a production manager (B

_{2}), and a quality control manager (B

_{3}). This team initiates its work with the expectation and description of the assessment criteria. In accordance with the previous operation reviews, the five main suppliers (G

_{1}, G

_{2}, G

_{3}, G

_{4,}and G

_{5}) options are preferred with respect to the various criteria after an initial screening. As the selected supplier options and criteria have different advantages in communal performance, given in Table 1. For this reason, the most favorable supplier cannot be chosen and needs validating during an adequate and logical inspection. Next, the procedure for the execution of the developed HF-SWARA-COPRAS approach is discussed as follows:

#### 4.1. Sensitivity Analysis

_{3}has the highest rank, when $\gamma =0.0$ to 0.2, while G

_{2}has the highest rank when $\gamma =0.3$ to 1.0. Further, the option G

_{5}has the worst rank when $\gamma =0.0$ to 0.7, whereas G

_{1}has the worst rank when $\gamma =0.8$ to 1.0. Accordingly, it can be noticed that the HF-SWARA-COPRAS approach has better stability for the diverse values of the parameter $\gamma $. Furthermore, the subjective weights evaluated by the SWARA technique preserve to enhance the sensitivity of the proposed framework. As per the above discussion, we realize that the use of different values of parameter $\gamma $ will increase the stability of the introduced framework.

#### 4.2. Comparison with Existing Methods

_{2}). On the similar line, from SSS &OA [90] method, the final preference order of the SS option is ${G}_{2}\text{}\succ \text{}{G}_{1}\text{}\succ \text{}{G}_{3}\text{}\succ \text{}{G}_{4}\text{}\succ \text{}{G}_{5}$ and hence, the most suitable SS option is SSS-2 (G

_{2}). Hence, we observe that the optimal SS option, that is, SSS-2(G

_{2}) is same with all the proposed and existing approaches, while the preference order outcomes slightly vary with different methods.

- The HFS can reflect the DE’s hesitancy more objectively than other classical extensions of FS. Therefore, the use of the developed HF-SWARA-COPRAS approach gives a more flexible way to express the uncertainty in the selection of SS.
- The SWARA method is employed to evaluate the criteria weights in the evaluation of the SSS process, which makes the introduced HF-SWARA-COPRAS method more reliable, efficient and sensible tool.
- The proposed HF-SWARA-COPRAS can process the information in a more useful and suitable way and from different perspectives, such as benefit-type and cost-type criteria.

#### 4.3. Discussion and Implications

_{8}) achieves the highest weight (0.1429) and industry reputation (f

_{7}) (0.1359) is attained in second rank, followed by pollution (f

_{6}) (0.1312), cost (f

_{2}) (0.1264), quality (f

_{1}) (0.1204), eco-design (f

_{4}) (0.1203), sustainable materials (f

_{5}) (0.1130) and production capacity (f

_{3}) (0.1099). This weight assessment is instinctively formulated, as we applied to consider that the economic aspect should be the leading factor, even amongst SSs. Several DEs in the case rely on the sustainability of SSSs should initially be revealed in their corporate ethics [95]. Though, criteria containing cost, quality, production capacity, and eco-design are usually supposed to be leading significances in the conventional SSs [11]. It is significant to remark that health and safety, as transparent and assessable indicators of social duty, are observed as the significant factors within eight considered criteria [96]. Nonetheless, the procedure that we have developed for SSS, cannot be static to be more applicable and robust and essential to be dynamic adjusted based on the variations in the real-life circumstances and demand of business organizations.

_{2}(0.412) is the optimal SS in most cases, which spectacles the strength of the introduced framework. Further, to validate the applicability of the introduced framework, we implement the BWM-AQM model, SSS & OA model to compute a SS option-based on the above-mentioned problem. Afterward, we obtained a similar outcome by existing models, although the robustness of the BWM-AQM model, SSS & OA model were weaker than the proposed method. In contrast, the HF-SWARA-COPRAS framework displayed higher robustness when the criteria weights shifted, which facilitated the DEs to offer decision-making orientations to a certain amount. This outcome delivers accurate indication for the comparison of Liu et al. [85] and You et al. [90].

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Dimension | Criteria | Meaning | Type |
---|---|---|---|

Economic [7,13,91,92] | Quality (F_{1}) | The overall quality of products. | Benefit |

Cost (F_{2}) | Price and the share of transaction costs per unit product. | Cost | |

Production Capacity (F_{3}) | Single-shift production per day. | Benefit | |

Environmental [7,13,88,93] | Eco-design (F_{4}) | Refers the design of products for reduced consumption of material/energy, reuse and recycling. | Benefit |

Sustainable Materials (F_{5}) | The level of sustainable materials utilized in manufacturing and packaging per product. | Benefit | |

Pollution (F_{6}) | Refers the average amount of air pollutants, wastes and hazardous materials released per day. | Cost | |

Industry Reputation (F_{7}) | The degree of social recognition of corporate behavior. | Benefit | |

Social [13,88,92,94] | Health and safety (F_{8}) | Including Occupational Safety and Health (OHSAS) 18001, conditions and incidents. | Benefit |

LVs | HFNs | DEs Risk Preference | ||
---|---|---|---|---|

Pessimist | Moderate | Optimist | ||

Very High (VH) | [0.8, 0.9] | 0.80 | 0.85 | 0.90 |

High (H) | [0.70, 0.8] | 0.70 | 0.75 | 0.80 |

Medium (M) | [0.55, 0.70] | 0.55 | 0.625 | 0.70 |

Low (L) | [0.40, 0.55] | 0.40 | 0.475 | 0.55 |

Very Low (VL) | [0.25, 0.40] | 0.25 | 0.325 | 0.40 |

LVs | HFNs | DEs Risk Preference | ||
---|---|---|---|---|

Pessimist | Moderate | Optimist | ||

Extremely Preferable (EP) | [0.9, 1.0] | 0.9 | 0.95 | 1.00 |

Strong Preferable (SP) | [0.75, 0.9] | 0.75 | 0.825 | 0.9 |

Preferable (P) | [0.6, 0.75] | 0.6 | 0.675 | 0.75 |

Moderate (M) | [0.45, 0.6] | 0.45 | 0.525 | 0.6 |

Undesirable (U) | [0.35, 0.45] | 0.35 | 0.4 | 0.45 |

Strong Undesirable (SU) | [0.2, 0.35] | 0.2 | 0.275 | 0.35 |

Extremely Undesirable (EU) | [0.0, 0.15] | 0.00 | 0.075 | 0.15 |

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

F_{1} | {0.2, 0.3, 0.7} | {0.3, 0.4, 0.8} | {0.4, 0.6, 0.7} | {0.5, 0.7, 0.9} | {0.1, 0.4, 0.5} |

F_{2} | {0.5, 0.6,0.7} | {0.5, 0.7,0.8} | {0.4, 0.5, 0.7} | {0.5,0.8, 0.9} | {0.5, 0.7, 0.9} |

F_{3} | {0.4,0.5, 0.7} | {0.7, 0.8,0.9} | {0.5, 0.6,0.9} | {0.2,0.6, 0.7} | {0.3, 0.5, 0.8} |

F_{4} | {0.4, 0.6, 0.8} | {0.4, 0.8, 0.9} | {0.1, 0.2, 0.4} | {0.3, 0.6, 0.8} | {0.2, 0.5, 0.8} |

F_{5} | {0.3, 0.4, 0.6} | {0.4, 0.7, 0.9} | {0.3, 0.6, 0.7} | {0.4, 0.7, 0.8} | {0.2, 0.5, 0.6} |

F_{6} | {0.3, 0.5, 0.7} | {0.1,0.5, 0.8} | {0.4, 0.6, 0.7} | {0.2, 0.5, 0.9} | {0.2, 0.7, 0.9} |

F_{7} | {0.3, 0.5, 0.8} | {0.6, 0.7, 0.8} | {0.3, 0.5, 0.6} | {0.2, 0.4, 0.8} | {0.2, 0.3, 0.7} |

F_{8} | {0.1, 0.3,0.4} | {0.2, 0.5,0.8} | {0.6, 0.7,0.9} | {0.3, 0.4, 0.9} | {0.3, 0.6, 0.8} |

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

F_{1} | 0.435 | 0.548 | 0.577 | 0.744 | 0.346 |

F_{2} | 0.603 | 0.682 | 0.544 | 0.777 | 0.744 |

F_{3} | 0.544 | 0.813 | 0.717 | 0.533 | 0.575 |

F_{4} | 0.627 | 0.762 | 0.237 | 0.607 | 0.555 |

F_{5} | 0.440 | 0.727 | 0.554 | 0.662 | 0.449 |

F_{6} | 0.519 | 0.537 | 0.577 | 0.641 | 0.699 |

F_{7} | 0.575 | 0.706 | 0.474 | 0.527 | 0.435 |

F_{8} | 0.271 | 0.555 | 0.762 | 0.635 | 0.607 |

Criteria | LVs is given by DEs | HFNs Given by DEs | $\mathbb{S}\left({\mathit{\xi}}_{\mathit{k}\mathit{j}}\right)$ | ||||
---|---|---|---|---|---|---|---|

B_{1} | B_{2} | B_{3} | B_{1} | B_{2} | B_{3} | ||

F_{1} | SU | P | P | 0.2 | 0.675 | 0.75 | 0.589 |

F_{2} | M | P | P | 0.45 | 0.675 | 0.75 | 0.639 |

F_{3} | U | M | M | 0.35 | 0.525 | 0.6 | 0.496 |

F_{4} | M | M | P | 0.45 | 0.525 | 0.75 | 0.588 |

F_{5} | M | M | M | 0.45 | 0.525 | 0.6 | 0.524 |

F_{6} | P | P | P | 0.6 | 0.675 | 0.75 | 0.676 |

F_{7} | U | P | EP | 0.35 | 0.675 | 0.9 | 0.712 |

F_{8} | SP | M | SP | 0.75 | 0.525 | 0.9 | 0.763 |

Criteria | Crisp Values | Comparative Significance of Criteria Value $\left({\mathbf{s}}_{\mathbf{j}}\right)$ | Coefficient $\left({\mathbf{k}}_{\mathbf{j}}\right)$ | Recalculated Weight $\left({\mathbf{p}}_{\mathbf{j}}\right)$ | Criteria Weight $\left({\mathbf{w}}_{\mathbf{j}}\right)$ |
---|---|---|---|---|---|

F_{8} | 0.763 | - | 1.000 | 1.000 | 0.1429 |

F_{7} | 0.712 | 0.051 | 1.051 | 0.951 | 0.1359 |

F_{6} | 0.676 | 0.036 | 1.036 | 0.918 | 0.1312 |

F_{2} | 0.639 | 0.037 | 1.037 | 0.885 | 0.1264 |

F_{1} | 0.589 | 0.050 | 1.050 | 0.843 | 0.1204 |

F_{4} | 0.588 | 0.001 | 1.001 | 0.842 | 0.1203 |

F_{5} | 0.524 | 0.064 | 1.064 | 0.791 | 0.1130 |

F_{3} | 0.496 | 0.028 | 1.028 | 0.769 | 0.1099 |

SSS Option | ${\mathbb{S}}^{\ast}\left({\mathit{\sigma}}_{\mathit{i}}\right)$ | ${\mathbb{S}}^{\ast}\left({\mathit{\upsilon}}_{\mathit{i}}\right)$ | ${\mathit{\theta}}_{\mathit{i}}$ | ${\mathbf{\u019b}}_{\mathit{i}}$ | Ranking |
---|---|---|---|---|---|

G_{1} | 0.394 | 0.192 | 0.333 | 80.73% | 4 |

G_{2} | 0.586 | 0.218 | 0.412 | 100.00% | 1 |

G_{3} | 0.483 | 0.191 | 0.378 | 91.58% | 2 |

G_{4} | 0.517 | 0.277 | 0.353 | 85.53% | 3 |

G_{5} | 0.406 | 0.281 | 0.296 | 71.76% | 5 |

$\mathit{\gamma}$ | G_{1} | G_{2} | G_{3} | G_{4} | G_{5} |
---|---|---|---|---|---|

0.0 | 0.272 | 0.239 | 0.273 | 0.188 | 0.186 |

0.1 | 0.284 | 0.274 | 0.294 | 0.221 | 0.208 |

0.2 | 0.296 | 0.308 | 0.315 | 0.254 | 0.230 |

0.3 | 0.309 | 0.343 | 0.336 | 0.287 | 0.252 |

0.4 | 0.321 | 0.378 | 0.357 | 0.320 | 0.274 |

0.5 | 0.333 | 0.412 | 0.378 | 0.353 | 0.296 |

0.6 | 0.345 | 0.447 | 0.399 | 0.386 | 0.318 |

0.7 | 0.357 | 0.482 | 0.420 | 0.418 | 0.340 |

0.8 | 0.369 | 0.516 | 0.441 | 0.451 | 0.362 |

0.9 | 0.382 | 0.551 | 0.462 | 0.484 | 0.384 |

1.0 | 0.394 | 0.586 | 0.483 | 0.517 | 0.406 |

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**MDPI and ACS Style**

Rani, P.; Mishra, A.R.; Krishankumar, R.; Mardani, A.; Cavallaro, F.; Soundarapandian Ravichandran, K.; Balasubramanian, K.
Hesitant Fuzzy SWARA-Complex Proportional Assessment Approach for Sustainable Supplier Selection (HF-SWARA-COPRAS). *Symmetry* **2020**, *12*, 1152.
https://doi.org/10.3390/sym12071152

**AMA Style**

Rani P, Mishra AR, Krishankumar R, Mardani A, Cavallaro F, Soundarapandian Ravichandran K, Balasubramanian K.
Hesitant Fuzzy SWARA-Complex Proportional Assessment Approach for Sustainable Supplier Selection (HF-SWARA-COPRAS). *Symmetry*. 2020; 12(7):1152.
https://doi.org/10.3390/sym12071152

**Chicago/Turabian Style**

Rani, Pratibha, Arunodaya Raj Mishra, Raghunathan Krishankumar, Abbas Mardani, Fausto Cavallaro, Kattur Soundarapandian Ravichandran, and Karthikeyan Balasubramanian.
2020. "Hesitant Fuzzy SWARA-Complex Proportional Assessment Approach for Sustainable Supplier Selection (HF-SWARA-COPRAS)" *Symmetry* 12, no. 7: 1152.
https://doi.org/10.3390/sym12071152