# A Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) and Combined Compromise Solution (CoCoSo) Algorithm in Distribution Center Location Selection: A Case Study in Agricultural Supply Chain

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Research Graph

#### 3.2. Theoretical Basis

#### 3.2.1. Spherical Fuzzy Sets Theory

- Intersection of ${\tilde{A}}_{S}\text{}\mathrm{and}\text{}{\tilde{B}}_{S}$:$$\begin{array}{cc}\hfill {\tilde{A}}_{S}{\displaystyle \cap}{\tilde{B}}_{S}& =\{min\left\{{\mu}_{{\tilde{A}}_{S}},{\mu}_{{\tilde{B}}_{S}}\right\},\text{}max\left\{{v}_{{\tilde{A}}_{S}},{v}_{{\tilde{B}}_{S}}\right\},\text{}max\left\{\right[1\hfill \\ & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-({\left(min\left\{{\mu}_{{\tilde{A}}_{S}},{\mu}_{{\tilde{B}}_{S}}\right\}\right)}^{2}+{\left(max\left\{{v}_{{\tilde{A}}_{S}},{v}_{{\tilde{B}}_{S}}\right\}\right)}^{2}){]}^{0.5},\text{}min\left\{{\pi}_{{\tilde{A}}_{S}},{\pi}_{{\tilde{B}}_{S}}\right\}\}\}\hfill \end{array}$$
- Addition of ${\tilde{A}}_{S}\text{}\mathrm{and}\text{}{\tilde{B}}_{S}$:$${\tilde{A}}_{S}+{\tilde{B}}_{S}=\left\{{\left({\mu}_{{\tilde{A}}_{S}}^{2}+{\mu}_{{\tilde{B}}_{S}}^{2}-{\mu}_{{\tilde{A}}_{S}}^{2}{\mu}_{{\tilde{B}}_{S}}^{2}\right)}^{0.5},{v}_{{\tilde{A}}_{S}}{v}_{{\tilde{B}}_{S}},\text{}{\left(\left(1-{\mu}_{{\tilde{B}}_{S}}^{2}\right){\pi}_{{\tilde{A}}_{S}}^{2}+\left(1-{\mu}_{{\tilde{A}}_{S}}^{2}\right){\pi}_{{\tilde{B}}_{S}}^{2}-{\pi}_{{\tilde{A}}_{S}}^{2}{\pi}_{{\tilde{B}}_{S}}^{2}\right)}^{0.5}\right\}$$
- Multiplication of ${\tilde{A}}_{S}\text{}\mathrm{and}\text{}{\tilde{B}}_{S}$:$${\tilde{A}}_{S}\times {\tilde{B}}_{S}=\left\{{\mu}_{{\tilde{A}}_{S}}{\mu}_{{\tilde{B}}_{S}},\text{}{\left({v}_{{\tilde{A}}_{S}}^{2}+{v}_{{\tilde{B}}_{S}}^{2}-{v}_{{\tilde{A}}_{S}}^{2}{v}_{{\tilde{B}}_{S}}^{2}\right)}^{0.5},\text{}{\left(\left(1-\text{}{v}_{{\tilde{B}}_{S}}^{2}\right){\pi}_{{\tilde{A}}_{S}}^{2}+\left(1-\text{}{v}_{{\tilde{A}}_{S}}^{2}\right){\pi}_{{\tilde{B}}_{S}}^{2}-{\pi}_{{\tilde{A}}_{S}}^{2}{\pi}_{{\tilde{B}}_{S}}^{2}\right)}^{0.5}\right\}$$
- Multiplication of ${\tilde{A}}_{S}\text{}\mathrm{and}\text{}\mathrm{a}\text{}\mathrm{scalar}\text{}\left(\lambda 0\right):$$$\lambda \times {\tilde{A}}_{S}=\left\{{\left(1-{\left(1-{\mu}_{{\tilde{A}}_{S}}^{2}\right)}^{\lambda}\right)}^{0.5},{v}_{{\tilde{A}}_{S}}^{\lambda},\text{}{\left({\left(1-{\mu}_{{\tilde{A}}_{S}}^{2}\right)}^{\lambda}-{\left(1-{\mu}_{{\tilde{A}}_{S}}^{2}-{\pi}_{{\tilde{A}}_{S}}^{2}\right)}^{\lambda}\right)}^{0.5}\right\}$$
- Power of ${\tilde{A}}_{S}$, with $\lambda >0$:$${\tilde{A}}_{S}^{\lambda}=\left\{{\mu}_{{\tilde{A}}_{S}}^{\lambda},\text{}{\left(1-{\left(1-{v}_{{\tilde{A}}_{S}}^{2}\right)}^{\lambda}\right)}^{0.5},\text{}{\left({\left(1-{v}_{{\tilde{A}}_{S}}^{2}\right)}^{\lambda}-{\left(1-{v}_{{\tilde{A}}_{S}}^{2}-{\pi}_{{\tilde{A}}_{S}}^{2}\right)}^{\lambda}\right)}^{0.5}\right\}$$

#### 3.2.2. Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) Model

#### 3.2.3. Combined Compromise Solution (CoCoSo)

## 4. Case Study

#### 4.1. Model Application

_{1}(location A

_{1}) is the optimal location. Findings: the authors described a real case of choosing optimal location for distribution center in Mekong Delta, Vietnam from an agricultural supply chain project.

#### 4.2. Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Mekong Delta Region and South East Region of Vietnam [3].

**Table 1.**Linguistic measures of importance [29].

Definition | $\left(\mathit{\mu},\text{}\mathit{v},\text{}\mathit{\pi}\right)$ | Score Index |
---|---|---|

Absolutely more importance (AM) | (0.9, 0.1, 0.0) | 9 |

Very high importance (VH) | (0.8, 0.2, 0.1) | 7 |

High importance (HI) | (0.7, 0.3, 0.2) | 5 |

Slightly more importance (SM) | (0.6, 0.4, 0.3) | 3 |

Equally importance (EI) | (0.5, 0.4, 0.4) | 1 |

Slightly lower importance (SL) | (0.4, 0.6, 0.3) | 1/3 |

Low importance (LI) | (0.3, 0.7, 0.2) | 1/5 |

Very low importance (VL) | (0.2, 0.8, 0.1) | 1/7 |

Absolutely low importance (AL) | (0.1, 0.9, 0.0) | 1/9 |

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

Cost | A | Land Cost (A1) |

Logistics Cost (A2) | ||

Available Infrastructure | B | Proximity to Airport (B2) |

Proximity to Highway (B3) | ||

Proximity to Railway (B4) | ||

Service Level | C | Transportation Time (C1) |

Distance to Markets (C2) | ||

Distance to Manufacturers(C3) | ||

Sustainability Factors | D | Distance to forest area (D1) |

Distance to surface water (D2) | ||

Ethical Factors (D3) |

Spherical Fuzzy Weights | Crisp Weights | ||
---|---|---|---|

Degree of Membership | Degree of Non-Membership | Degree of Hesitancy | |

0.433 | 0.540 | 0.318 | 0.067 |

0.325 | 0.660 | 0.254 | 0.049 |

0.422 | 0.589 | 0.241 | 0.067 |

0.443 | 0.583 | 0.229 | 0.071 |

0.472 | 0.554 | 0.229 | 0.076 |

0.564 | 0.459 | 0.219 | 0.092 |

0.634 | 0.390 | 0.182 | 0.106 |

0.667 | 0.343 | 0.202 | 0.111 |

0.705 | 0.302 | 0.174 | 0.118 |

0.707 | 0.295 | 0.196 | 0.118 |

0.741 | 0.256 | 0.177 | 0.125 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
---|---|---|---|---|---|---|---|---|---|---|---|

A1 | 0.6667 | 1.0000 | 1.0000 | 0.0000 | 0.3333 | 0.5000 | 0.0000 | 1.0000 | 0.5000 | 0.5000 | 0.8000 |

A2 | 1.0000 | 0.0000 | 0.5000 | 0.5000 | 0.0000 | 0.0000 | 0.3333 | 0.7500 | 0.0000 | 0.0000 | 0.0000 |

A3 | 0.0000 | 0.0000 | 0.5000 | 0.5000 | 1.0000 | 0.5000 | 0.6667 | 0.0000 | 1.0000 | 1.0000 | 0.6000 |

A4 | 0.6667 | 0.5000 | 0.0000 | 1.0000 | 0.3333 | 1.0000 | 0.6667 | 0.2500 | 0.5000 | 0.0000 | 1.0000 |

A5 | 1.0000 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 1.0000 | 1.0000 | 0.2500 | 0.5000 | 0.0000 | 0.8000 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
---|---|---|---|---|---|---|---|---|---|---|---|

A1 | 0.0447 | 0.0490 | 0.0670 | 0.0000 | 0.0253 | 0.0460 | 0.0000 | 0.1110 | 0.0590 | 0.0590 | 0.1000 |

A2 | 0.0670 | 0.0000 | 0.0335 | 0.0355 | 0.0000 | 0.0000 | 0.0353 | 0.0833 | 0.0000 | 0.0000 | 0.0000 |

A3 | 0.0000 | 0.0000 | 0.0335 | 0.0355 | 0.0760 | 0.0460 | 0.0707 | 0.0000 | 0.1180 | 0.1180 | 0.0750 |

A4 | 0.0447 | 0.0245 | 0.0000 | 0.0710 | 0.0253 | 0.0920 | 0.0707 | 0.0278 | 0.0590 | 0.0000 | 0.1250 |

A5 | 0.0670 | 0.0000 | 0.0670 | 0.0710 | 0.0000 | 0.0920 | 0.1060 | 0.0278 | 0.0590 | 0.0000 | 0.1000 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
---|---|---|---|---|---|---|---|---|---|---|---|

A1 | 0.9732 | 1.0000 | 1.0000 | 0.0000 | 0.9199 | 0.9382 | 0.0000 | 1.0000 | 0.9215 | 0.9215 | 0.9725 |

A2 | 1.0000 | 0.0000 | 0.9546 | 0.9520 | 0.0000 | 0.0000 | 0.8901 | 0.9686 | 0.0000 | 0.0000 | 0.0000 |

A3 | 0.0000 | 0.0000 | 0.9546 | 0.9520 | 1.0000 | 0.9382 | 0.9579 | 0.0000 | 1.0000 | 1.0000 | 0.9381 |

A4 | 0.9732 | 0.9666 | 0.0000 | 1.0000 | 0.9199 | 1.0000 | 0.9579 | 0.8574 | 0.9215 | 0.0000 | 1.0000 |

A5 | 1.0000 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 1.0000 | 1.0000 | 0.8574 | 0.9215 | 0.0000 | 0.9725 |

Alternatives | Ka | Ranking | Kb | Ranking | Kc | Ranking | Ki |
---|---|---|---|---|---|---|---|

A_{1} | 0.2301 | 1 | 4.0181 | 1 | 0.9969 | 1 | 2.7215 |

A_{2} | 0.1254 | 5 | 2.0000 | 5 | 0.5435 | 5 | 1.4043 |

A_{3} | 0.2077 | 4 | 3.8739 | 4 | 0.9001 | 4 | 2.5586 |

A_{4} | 0.2283 | 2 | 3.9248 | 3 | 0.9892 | 2 | 2.6747 |

A_{5} | 0.2084 | 3 | 3.9432 | 2 | 0.9031 | 3 | 2.5903 |

Alternative | Final Performance Score (Ki) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

λ = 0.1 | λ = 0.2 | λ = 0.3 | λ = 0.4 | λ = 0.5 | λ = 0.6 | λ = 0.7 | λ = 0.8 | λ = 0.9 | λ = 0.10 | |

A1 | 2.72 | 2.72 | 2.72 | 2.72 | 2.72 | 2.72 | 2.72 | 2.72 | 2.71 | 2.69 |

A2 | 1.41 | 1.41 | 1.41 | 1.41 | 1.40 | 1.40 | 1.40 | 1.39 | 1.38 | 1.33 |

A3 | 2.56 | 2.56 | 2.56 | 2.56 | 2.56 | 2.56 | 2.56 | 2.57 | 2.57 | 2.61 |

A4 | 2.68 | 2.68 | 2.68 | 2.68 | 2.67 | 2.67 | 2.67 | 2.67 | 2.66 | 2.65 |

A5 | 2.59 | 2.59 | 2.59 | 2.59 | 2.59 | 2.59 | 2.60 | 2.60 | 2.61 | 2.63 |

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**MDPI and ACS Style**

Kieu, P.T.; Nguyen, V.T.; Nguyen, V.T.; Ho, T.P.
A Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) and Combined Compromise Solution (CoCoSo) Algorithm in Distribution Center Location Selection: A Case Study in Agricultural Supply Chain. *Axioms* **2021**, *10*, 53.
https://doi.org/10.3390/axioms10020053

**AMA Style**

Kieu PT, Nguyen VT, Nguyen VT, Ho TP.
A Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) and Combined Compromise Solution (CoCoSo) Algorithm in Distribution Center Location Selection: A Case Study in Agricultural Supply Chain. *Axioms*. 2021; 10(2):53.
https://doi.org/10.3390/axioms10020053

**Chicago/Turabian Style**

Kieu, Phan Thuy, Van Thanh Nguyen, Viet Tinh Nguyen, and Thanh Phong Ho.
2021. "A Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) and Combined Compromise Solution (CoCoSo) Algorithm in Distribution Center Location Selection: A Case Study in Agricultural Supply Chain" *Axioms* 10, no. 2: 53.
https://doi.org/10.3390/axioms10020053