# On the Analytic Hierarchy Process Structure in Group Decision-Making Using Incomplete Fuzzy Information with Applications

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Remark**

**1**

**Definition**

**3**

**Definition**

**4.**

**Definition**

**5.**

## 3. Proposed Procedure for the MCGDM Problem

#### 3.1. Estimating Missing Values

#### 3.2. Consistency Measures

- The ${T}_{L}$-consistency Index (${T}_{L}CI$) of each pair of alternatives is computed using the following expression:$${T}_{L}CI({r}_{ij}^{p})=1-d({r}_{ij}^{p},{\tilde{r}}_{ij}^{p}),$$
- ${T}_{L}CI$ values for the alternatives ${a}_{i}$ and $1\le i\le n$ are determined using$${T}_{L}CI\left({a}_{i}\right)=\frac{1}{2(m-1)}\underset{j=1}{{\displaystyle \sum}^{m}}({T}_{L}CI\left({r}_{ij}^{p}\right)+{T}_{L}CI\left({r}_{ji}^{p}\right)).$$
- ${T}_{L}CI$ for an FPR ${R}^{p}$ is therefore evaluated by calculating the mean of ${T}_{L}CI$ against all alternatives ${a}_{i}$:$${T}_{L}CI\left({R}^{p}\right)=\frac{1}{m}\underset{i=1}{{\displaystyle \sum}^{m}}{T}_{L}CI\left({a}_{i}\right).$$
- After evaluating ${T}_{L}CI$ in three stages (9)–(11), higher weights are assigned rationally to the experts with higher consistency degrees. Consistency weights may therefore be allocated to experts in the sense of the following relation:$${w}_{p}=\frac{{T}_{L}CI\left({R}^{p}\right)}{{\displaystyle {\displaystyle \sum}_{p=1}^{l}}{T}_{L}CI\left({R}^{p}\right)}.$$

#### 3.3. Consensus Measures

- First, the degree of consensus for every pair $({a}_{i},{a}_{j})$ of alternatives, referred to as $co{d}_{ij}$, is determined:$$co{d}_{ij}={s}_{ij}.$$
- At level 2, the degree of consensus among the experts on each alternative ${a}_{i}$, referred to as $Co{D}_{i}$ for $1\le i\le m,$ is established as$$Co{D}_{i}=\frac{1}{2(m-1)}\underset{j=1,j\ne i}{{\displaystyle \sum}^{m}}({s}_{ij}+{s}_{ji}).$$
- The third level includes the global consensus degree, symbolized by $CoD$, among all experts on their observations:$$CoD=\frac{1}{m}\underset{i=1}{{\displaystyle \sum}^{m}}Co{D}_{i}.$$

#### 3.4. Final Priority Weights of the Experts

#### 3.5. Ranking of Criteria

#### 3.6. Ranking of Alternatives Regarding Each Criterion

#### 3.7. Final Ranking of Alternatives

## 4. Example

## 5. Comparison

#### Problem Statement

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AHP | Analytical Hierarchy Process |

FPR | Fuzzy preference relation |

IFPR | Incomplete fuzzy preference relation |

MPR | Multiplicative preference relation |

SCM | Supply chain management |

GDM | Group decision-making |

MCDM | Multi-criteria decision-making |

MCGDM | Multi-criteria group decision-making |

## References

- Durmić, E.; Stević, Ž.; Chatterjee, P.; Vasiljević, M.; Tomašević, M. Sustainable supplier selection using combined FUCOM—Rough SAW model. Rep. Mech. Eng.
**2020**, 1, 34–43. [Google Scholar] [CrossRef] - Sałabun, W.; Ziemba, P. Application of the Characteristic Objects Method in Supply Chain Management and Logistics. In Recent Developments in Intelligent Information and Database Systems; Springer: Berlin/Heidelberg, Germany, 2016; pp. 445–453. [Google Scholar]
- Badi, I.; Pamucar, D. Supplier selection for steelmaking company by using combined Grey-MARCOS methods. Decis. Mak. Appl. Manag. Eng.
**2020**, 3, 37–48. [Google Scholar] [CrossRef] - Chakraborty, S.; Chattopadhyay, R.; Chakraborty, S. An integrated D-MARCOS method for supplier selection in an iron and steel industry. Decis. Mak. Appl. Manag. Eng.
**2020**, 3, 49–69. [Google Scholar] - Swaminathan, J.M.; Tayur, S.R. Models for supply chains in e-business. Manage. Sci.
**2003**, 49, 1387–1406. [Google Scholar] [CrossRef] [Green Version] - Chen, C.T.; Lin, C.T.; Huang, S.F. A fuzzy approach for supplier evaluation & selection in supply chain management. Int. J. Prod. Econ.
**2006**, 102, 289–301. [Google Scholar] - De Boer, L.; Labro, E.; Morlacchi, P. A review of methods supporting supplier selection. Eur. J. Purch. Supply Manag.
**2001**, 7, 75–89. [Google Scholar] [CrossRef] - Demirtas, E.; Üstün, Ö. An integrated multiobjective decision making process for supplier selection and order allocation. Omega
**2008**, 36, 76–90. [Google Scholar] [CrossRef] - Hwang, C.L.; Yoon, K. Multiple Attributes Decision Making Methods & Applications; Springer: New York, NY, USA, 1981. [Google Scholar]
- Ng, W.L. An efficient & simple model for multiple criteria supplier selection problem. Eur. J. Oper. Res.
**2008**, 186, 1059–1067. [Google Scholar] - Dickson, G.W. A analysis of vender selection systems & decisions. J. Suppl. Chain Manag.
**1966**, 2, 5–17. [Google Scholar] - Barbarosoglu, G.; Yazgaç, T. An application of the analytic hierarchy process to the supplier selection problem. Prod. Inventory Manag. J.
**1997**, 38, 14–21. [Google Scholar] - Goffin, K.; Szwejczewski, M.; New, C. Managing suppliers when fewer can mean more. Int. J. Phys. Distrib. Logist. Manag.
**1997**, 27, 422–436. [Google Scholar] [CrossRef] - Swift, C.O. Preference for single sourcing & supplier selection criteria. J. Bus. Res.
**1995**, 32, 105–111. [Google Scholar] - Weber, C.A.; Current, J.R.; Benton, W.C. Vender selection criteria & method. Eur. J. Oper. Res.
**1991**, 50, 2–18. [Google Scholar] - Chai, J.; Liu, J.N.K.; Ngai, E.W.T. Application of decision-making techniques in supplier selection: A systematic review of literature. Expert Syst. Appl.
**2013**, 40, 3872–3885. [Google Scholar] [CrossRef] - Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S. Group decision-making for hesitant fuzzy sets based on characteristic objects method. Symmetry
**2017**, 9, 136. [Google Scholar] [CrossRef] - Faizi, S.; Sałabun, W.; Rashid, T.; Zafar, S.; Wątróbski, J. Intuitionistic fuzzy sets in multi-criteria group decision making problems using the characteristic objects method. Symmetry
**2020**, 12, 1382. [Google Scholar] [CrossRef] - Chen, P.C. A fuzzy multiple criteria decision making model in employee recruitment. Int. J. Comput. Sci. Net. Secur.
**2009**, 9, 113–117. [Google Scholar] - Hadi-Vencheh, A.; Mokhtarian, M.N. A new fuzzy MCDM approach based on centroid of fuzzy numbers. Expert Syst. Appl.
**2011**, 38, 5226–5230. [Google Scholar] [CrossRef] - Önüt, S.; Efendigil, T.; Kara, S.S. A combined fuzzy MCDM approach for selecting shopping center site: An example from Istanbul, Turkey. Expert Syst. Appl.
**2010**, 37, 1973–1980. [Google Scholar] [CrossRef] - Srdevic, Z.; Blagojevic, B.; Srdevic, B. AHP based group decision making in ranking loan applicants for purchasing irrigation equipment: A case study. Bulg. J. Agric. Sci.
**2011**, 17, 531–543. [Google Scholar] - Sun, C.C.; Lin, G.T.R.; Tzeng, G.H. The evaluation of cluster policy by fuzzy MCDM: Empirical evidence from Hsin Chu Science Park. Expert Syst. Appl.
**2009**, 36, 11895–11906. [Google Scholar] [CrossRef] - Wu, H.Y.; Tzeng, G.H.; Chen, Y.H. A fuzzy MCDM approach for evaluating banking performance based on balanced scorecard. Expert Syst. Appl.
**2009**, 36, 10135–10147. [Google Scholar] [CrossRef] - Bashir, Z.; Rashid, T.; Wątróbski, J. Sałabun, W.; Malik, A. Hesitant probabilistic multiplicative preference relations in group decision making. Appl. Sci.
**2018**, 8, 398. [Google Scholar] [CrossRef] [Green Version] - Kim, S.H.; Ahn, B.S. Interactive group decision making procedure under incomplete information. Eur. J. Oper. Res.
**1999**, 116, 498–507. [Google Scholar] [CrossRef] - Xu, Z.S. Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. Int. J. Approx. Reason.
**2004**, 36, 261–270. [Google Scholar] [CrossRef] [Green Version] - Gong, Z.W. Least-square method to priority of the fuzzy preference relations with incomplete information. Int. J. Approx. Reason.
**2008**, 47, 258–264. [Google Scholar] [CrossRef] [Green Version] - Herrera-Viedma, E.; Chiclana, F.; Herrera, F.; Alonso, S. Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Trans. Syst. Man Cybern. Part B Cybern.
**2007**, 37, 176–189. [Google Scholar] [CrossRef] - Alonso, S.; Chiclana, F.; Herrera, F.; Herrera-Viedma, E. A consistency based procedure to estimate missing pairwise preference values. Int. J. Intell. Syst.
**2008**, 23, 155–175. [Google Scholar] [CrossRef] - Xu, Y.J. On group decision making with four formats of incomplete preference relations. Comput. Ind. Eng.
**2011**, 61, 48–54. [Google Scholar] [CrossRef] - Lee, L.W. Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency. Expert Syst. Appl.
**2012**, 39, 11666–11676. [Google Scholar] [CrossRef] - Chen, S.M.; Lin, T.E.; Lee, L.W. Group decision making using incomplete fuzzy preference relations based on the additive consistency and the order consistency. Inf. Sci.
**2014**, 259, 1–15. [Google Scholar] [CrossRef] - Rehman, A.; Kerre, E.E.; Ashraf, S. Group decision making by using incomplete fuzzy preference relations based on T-Consistency Order Consistency. Int. J. Intell. Syst.
**2015**, 30, 120–143. [Google Scholar] [CrossRef] [Green Version] - Kerre, E.E.; Rehman, A.U.; Ashraf, S. Group decision making with incomplete reciprocal preference relations based on multiplicative consistency. Int. J. Comput. Intell. Syst.
**2018**, 11, 1030–1040. [Google Scholar] [CrossRef] [Green Version] - Saaty, T.L. The Analytic Hierarchy Process: Planning, Priority Setting & Resource Allocation; ; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Saaty, T.L. How to make a decision: The analytic hierarchy process. Eur. J. Oper. Res.
**1990**, 48, 9–26. [Google Scholar] [CrossRef] - Lin, C.T.; Hsu, P.F. Adopting an analytic hierarchy process to select internet advertising networks. Mark. Intell. Plan.
**2003**, 21, 183–191. [Google Scholar] [CrossRef] - Omasa, T.; Kisshimoto, M.; Kawase, M.; Yagi, K. An attempt at decision making in tissue engineering: Reactor evaluation using the analytic hierarchy process (AHP). Biochem. Eng. J.
**2004**, 20, 173–179. [Google Scholar] [CrossRef] - Wei, C.C.; Chien, C.F.; Wang, M.J. An AHP-based approach to ERP system selection. Int. J. Prod. Econ.
**2005**, 96, 47–62. [Google Scholar] [CrossRef] - Tanino, T. Fuzzy preference relations in group decision making. In Non-Conventional Preference Relations in Decision Making; Lecture Notes in Economics and Mathematical Systems; Springer: Berlin/Heidelberg, Germany, 1988; Volume 301, pp. 54–71. [Google Scholar]
- Venugopalan, P. Fuzzy ordered sets. Fuzzy Sets Syst.
**1992**, 46, 221–226. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Tanino, T. Fuzzy preference orderings in group decision making. Fuzzy Sets Syst.
**1984**, 12, 117–131. [Google Scholar] [CrossRef] - Ureña, R.; Chiclana, F.; Alonso, S.; Morente-Molinera, J.A.; Herrera-Viedma, E. On incomplete fuzzy and multiplicative preference relations in multi-person decision making. Procedia Comput. Sci.
**2014**, 31, 793–801. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Ge, B.; Jiang, J.; Tan, Y. Consensus building in group decision making based on multiplicative consistency with incomplete reciprocal preference relations. Knowl.-Based Syst.
**2016**, 106, 96–104. [Google Scholar] [CrossRef] - Sałabun, W.; Urbaniak, K. A new coefficient of rankings similarity in decision-making problems. In International Conference on Computational Science; Springer: Cham, Switzerland, 2020; pp. 632–645. [Google Scholar]

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ | Priority Weights | |
---|---|---|---|---|---|---|

${\mathbf{c}}_{\mathbf{1}}$ | 0.50000 | 0.4358 | 0.3648 | 0.5642 | 0.4658 | 0.1830 |

${\mathbf{c}}_{\mathbf{2}}$ | 0.5642 | 0.5000 | 0.3679 | 0.5017 | 0.4689 | 0.1903 |

${\mathbf{c}}_{\mathbf{3}}$ | 0.6352 | 0.6321 | 0.5000 | 0.4317 | 0.6300 | 0.2329 |

${\mathbf{c}}_{\mathbf{4}}$ | 0.4358 | 0.4983 | 0.5683 | 0.5000 | 0.2666 | 0.1769 |

${\mathbf{c}}_{\mathbf{5}}$ | 0.5342 | 0.5311 | 0.3700 | 0.7334 | 0.5000 | 0.2169 |

${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | Priority Ratings | |
---|---|---|---|---|---|

${c}_{1}$ | |||||

${a}_{1}$ | 0.5000 | 0.6667 | 0.3667 | 0.4000 | 0.2389 |

${a}_{2}$ | 0.3333 | 0.5000 | 0.3667 | 0.6333 | 0.2222 |

${a}_{3}$ | 0.6333 | 0.6333 | 0.5000 | 0.4333 | 0.2833 |

${a}_{4}$ | 0.6000 | 0.3667 | 0.5667 | 0.5000 | 0.2556 |

${c}_{2}$ | |||||

${a}_{1}$ | 0.5000 | 0.8000 | 0.3000 | 0.4000 | 0.2500 |

${a}_{2}$ | 0.2000 | 0.5000 | 0.3667 | 0.5000 | 0.1778 |

${a}_{3}$ | 0.7000 | 0.6333 | 0.5000 | 0.4667 | 0.3000 |

${a}_{4}$ | 0.6000 | 0.5000 | 0.5333 | 0.5000 | 0.2722 |

${c}_{3}$ | |||||

${a}_{1}$ | 0.5000 | 0.4000 | 0.2667 | 0.6000 | 0.2111 |

${a}_{2}$ | 0.6000 | 0.5000 | 0.1333 | 0.2000 | 0.1556 |

${a}_{3}$ | 0.7333 | 0.8667 | 0.5000 | 0.3333 | 0.3222 |

${a}_{4}$ | 0.4000 | 0.8000 | 0.6667 | 0.5000 | 0.3111 |

${c}_{4}$ | |||||

${a}_{1}$ | 0.5000 | 0.2333 | 0.5000 | 0.4667 | 0.2000 |

${a}_{2}$ | 0.7667 | 0.5000 | 0.4000 | 0.4667 | 0.2722 |

${a}_{3}$ | 0.5000 | 0.6000 | 0.5000 | 0.4000 | 0.2500 |

${a}_{4}$ | 0.5333 | 0.5333 | 0.6000 | 0.5000 | 0.2778 |

${c}_{5}$ | |||||

${a}_{1}$ | 0.5000 | 0.3333 | 0.1667 | 0.3333 | 0.1389 |

${a}_{2}$ | 0.6667 | 0.5000 | 0.5000 | 0.1667 | 0.2222 |

${a}_{3}$ | 0.8333 | 0.5000 | 0.5000 | 0.3667 | 0.2833 |

${a}_{4}$ | 0.6667 | 0.8333 | 0.6333 | 0.5000 | 0.3556 |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ | Priority Weights | |
---|---|---|---|---|---|---|

Criteria weights | 0.1830 | 0.1903 | 0.2329 | 0.1769 | 0.2169 | |

${a}_{1}$ | 0.2389 | 0.2500 | 0.2111 | 0.2000 | 0.1389 | 0.2060 |

${a}_{2}$ | 0.2222 | 0.1778 | 0.1556 | 0.2722 | 0.2222 | 0.2071 |

${a}_{3}$ | 0.2833 | 0.3000 | 0.3222 | 0.2500 | 0.2833 | 0.2896 |

${a}_{4}$ | 0.2556 | 0.2722 | 0.3111 | 0.2778 | 0.3556 | 0.2973 |

**Table 4.**Comparison matrices provided by President of the Fund Council (DM1) [22].

Criteria | Service | LOANH | Insurance | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Service | 2 | 9 | |||||||||||||

LOANH | 5 | ||||||||||||||

Insurance | |||||||||||||||

Service | LOANH | Insurance | |||||||||||||

${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | |

${U}_{1}$ | 3 | 1/2 | 8 | 4 | 1/9 | 1/9 | 1/4 | 1/6 | 2 | 5 | 1/2 | 1/6 | |||

${U}_{2}$ | 1/3 | 7 | 3 | 1 | 3 | 2 | 5 | 1/3 | 1/8 | ||||||

${U}_{3}$ | 9 | 6 | 3 | 2 | 1/7 | 1/9 | |||||||||

${U}_{4}$ | 1/4 | 1/2 | 1/3 | ||||||||||||

${U}_{5}$ |

**Table 5.**Comparison matrices provided by Senior advisor of the Fund (DM2) [22].

Criteria | Service | LOANH | Insurance | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Service | 5 | 9 | |||||||||||||

LOANH | 6 | ||||||||||||||

Insurance | |||||||||||||||

Service | LOANH | Insurance | |||||||||||||

${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | |

${U}_{1}$ | 4 | 1/3 | 6 | 5 | 1/6 | 1/6 | 1/3 | 1/3 | 2 | 4 | 1/2 | 1/4 | |||

${U}_{2}$ | 1/4 | 5 | 3 | 1 | 2 | 2 | 4 | 1/4 | 1/6 | ||||||

${U}_{3}$ | 9 | 8 | 2 | 2 | 1/8 | 1/9 | |||||||||

${U}_{4}$ | 1/3 | 1/2 | 1/2 | ||||||||||||

${U}_{5}$ |

**Table 6.**Comparison matrices provided by Fund manager (DM3) [22].

Criteria | Service | LOANH | Insurance | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Service | 1/5 | 1/3 | |||||||||||||

LOANH | 6 | ||||||||||||||

Insurance | |||||||||||||||

Service | LOANH | Insurance | |||||||||||||

${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | |

${U}_{1}$ | 2 | 1/3 | 7 | 6 | 1/5 | 1/5 | 1/2 | 1/4 | 3 | 6 | 1/2 | 1/7 | |||

${U}_{2}$ | 1/5 | 5 | 3 | 1 | 2 | 2 | 2 | 1/4 | 1/7 | ||||||

${U}_{3}$ | 9 | 7 | 2 | 2 | 1/6 | 1/9 | |||||||||

${U}_{4}$ | 1/4 | 1/2 | 1/5 | ||||||||||||

${U}_{5}$ |

**Table 7.**Comparison matrices provided by External expert advisor (DM4) [22].

Criteria | Service | LOANH | Insurance | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Service | 3 | 7 | |||||||||||||

LOANH | 5 | ||||||||||||||

Insurance | |||||||||||||||

Service | LOANH | Insurance | |||||||||||||

${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | |

${U}_{1}$ | 3 | 1/3 | 8 | 5 | 1/8 | 1/8 | 1/5 | 1/6 | 4 | 6 | 1/3 | 1/5 | |||

${U}_{2}$ | 1/4 | 6 | 3 | 1 | 3 | 2 | 2 | 1/4 | 1/7 | ||||||

${U}_{3}$ | 9 | 5 | 3 | 2 | 1/6 | 1/9 | |||||||||

${U}_{4}$ | 1/4 | 1/3 | 1/4 | ||||||||||||

${U}_{5}$ |

**Table 8.**Comparison matrices provided by Expert representative of the Ministry (DM5) [22].

Criteria | Service | LOANH | Insurance | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Service | 7 | 9 | |||||||||||||

LOANH | 5 | ||||||||||||||

Insurance | |||||||||||||||

Service | LOANH | Insurance | |||||||||||||

${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | ${U}_{1}$ | ${U}_{2}$ | ${U}_{3}$ | ${U}_{4}$ | ${U}_{5}$ | |

${U}_{1}$ | 3 | 1/5 | 7 | 5 | 1/9 | 1/9 | 1/5 | 1/7 | 3 | 6 | 1/3 | 1/5 | |||

${U}_{2}$ | 1/6 | 6 | 5 | 1 | 7 | 5 | 4 | 1/5 | 1/7 | ||||||

${U}_{3}$ | 9 | 7 | 7 | 5 | 1/7 | 1/9 | |||||||||

${U}_{4}$ | 1/3 | 1/4 | 1/4 | ||||||||||||

${U}_{5}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Rehman, A.u.; Shekhovtsov, A.; Rehman, N.; Faizi, S.; Sałabun, W.
On the Analytic Hierarchy Process Structure in Group Decision-Making Using Incomplete Fuzzy Information with Applications. *Symmetry* **2021**, *13*, 609.
https://doi.org/10.3390/sym13040609

**AMA Style**

Rehman Au, Shekhovtsov A, Rehman N, Faizi S, Sałabun W.
On the Analytic Hierarchy Process Structure in Group Decision-Making Using Incomplete Fuzzy Information with Applications. *Symmetry*. 2021; 13(4):609.
https://doi.org/10.3390/sym13040609

**Chicago/Turabian Style**

Rehman, Atiq ur, Andrii Shekhovtsov, Nighat Rehman, Shahzad Faizi, and Wojciech Sałabun.
2021. "On the Analytic Hierarchy Process Structure in Group Decision-Making Using Incomplete Fuzzy Information with Applications" *Symmetry* 13, no. 4: 609.
https://doi.org/10.3390/sym13040609