# Geometric Compatibility Indexes in a Local AHP-Group Decision Making Context: A Framework for Reducing Incompatibility

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. AHP in a Multiactor Decision Making Context

#### 2.2. Compatibility in AHP-GDM

## 3. Compatibility Indexes in AHP

#### 3.1. Geometric Compatibility Indexes. Basic Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**1.**

#### 3.2. Geometric Compatibility Indexes for Families of Matrices and Vectors

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Remark**

**2.**

## 4. A Theoretical Framework for Reducing Geometric Compatibility Measures in a Local Context

#### 4.1. ${\mathrm{GCOMPI}}_{1}$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

#### 4.2. ${\mathrm{GCOMPI}}_{2}$

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

#### 4.3. ${\mathrm{GCOMPI}}_{3}$

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**7.**

**Proof.**

**Corollary**

**8.**

**Proof.**

**Corollary**

**9.**

**Proof.**

#### 4.4. ${\mathrm{GCOMPI}}_{4}$

**Theorem**

**7.**

**Theorem**

**8.**

**Corollary**

**10.**

**Corollary**

**11.**

**Corollary**

**12.**

## 5. Procedure for Incompatibility Reduction and Numerical Example

#### 5.1. Semi-Automatic Procedure for Incompatibility Reduction

**Box 1.**Outline of the algorithm for improving the ${\mathrm{GCOMPI}}_{i}$ in terms of relative changes.

- Step 0.
- Let $J=\left(\right)open="\{"\; close="\}">(r,s),\mathrm{with}rs$.
- Step 1.
- Evaluate ${d}_{rs}={\left(\right)}_{\frac{\partial {\mathrm{GCOMPI}}_{i}}{\partial {t}_{rs}}}{t}_{rs}=1$ for all $(r,s)\in J$.
- Step 2.
- Choose the pair $({r}^{\prime},{s}^{\prime})\in J$ for which ${d}_{{r}^{\prime}{s}^{\prime}}$ has the largest absolute value.
- Step 3.
- If ${p}_{{r}^{\prime}{s}^{\prime}}>1$ then let $(r,s)=({r}^{\prime},{s}^{\prime})$. Otherwise, let $(r,s)=({s}^{\prime},{r}^{\prime})$.
- Step 4.
- Modify ${p}_{rs}$ using expression (39).
- Step 5.
- Update matrix P with new values ${p}_{rs}^{\prime}={p}_{rs}{t}_{rs}$ and ${p}_{sr}^{\prime}=1/{p}_{rs}^{\prime}$.Update $J=J\backslash ({r}^{\prime},{s}^{\prime})$.
- Step 6.
- If J is empty or ${E}_{t}\ge {E}^{*}$, stop and provide ${P}^{\prime}$. Otherwise go to Step 1.

_{rs}to ${a}_{rs}^{G}$ taking into account that the relative variation is limited by permissibility $\rho $. The problem can be expressed as

**Remark**

**3.**

#### 5.2. Numerical Example

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Saaty, T.L. A scaling method for priorities in hierarchical structures. J. Math. Psychol.
**1977**, 15, 234–281. [Google Scholar] [CrossRef] - Saaty, T.L. Multicriteria Decision Making: The Analytic Hierarchy Process; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Altuzarra, A.; Moreno-Jiménez, J.M.; Salvador, M. Consensus Building in AHP-Group Decision Making: A Bayesian Approach. Oper. Res.
**2010**, 58, 1755–1773. [Google Scholar] [CrossRef] - Altuzarra, A.; Gargallo, P.; Moreno-Jiménez, J.M.; Salvador, M. Homogeneous Groups of Actors in an AHP-Local Decision Making Context: A Bayesian Analysis. Mathematics
**2019**, 7, 294. [Google Scholar] [CrossRef] [Green Version] - Aguarón, J.; Escobar, M.T.; Moreno-Jiménez, J.M. Reducing incompatibility in a local AHP-Group Decision Making context. Forthcoming.
- Saaty, T.L. Decision-making with the AHP: Why is the principal eigenvector necessary. Eur. J. Oper. Res.
**2003**, 145, 85–91. [Google Scholar] [CrossRef] - Crawford, G.; Williams, C. A note on the analysis of subjective judgment matrices. J. Math. Psychol.
**1985**, 29, 387–405. [Google Scholar] [CrossRef] - Saaty, T. Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process; AHP Series; RWS Publications: Pittsburgh, PA, USA, 1994. [Google Scholar]
- Aguarón, J.; Moreno-Jiménez, J.M. The Geometric Consistency Index: Approximated Thresholds. Eur. J. Oper. Res.
**2003**, 147, 137–145. [Google Scholar] [CrossRef] - Saaty, T.L. Group Decision Making and the AHP. In The Analytic Hierarchy Process: Applications and Studies; Golden, B.L., Wasil, E.A., Harker, P.T., Eds.; Springer: Berlin/Heidelberg, Germany, 1989; pp. 59–67. [Google Scholar] [CrossRef]
- Ramanathan, R.; Ganesh, L. Group preference aggregation methods employed in AHP: An evaluation and an intrinsic process for deriving members’ weightages. Eur. J. Oper. Res.
**1994**, 79, 249–265. [Google Scholar] [CrossRef] - Forman, E.; Peniwati, K. Aggregating individual judgments and priorities with the analytic hierarchy process. Eur. J. Oper. Res.
**1998**, 108, 165–169. [Google Scholar] [CrossRef] - Aguarón, J.; Escobar, M.T.; Moreno-Jiménez, J.M.; Turón, A. The Triads Geometric Consistency Index in AHP-pairwise comparison matrices. Mathematics
**2020**, 8, 926. [Google Scholar] [CrossRef] - Garuti, C.E. A set theory justification of Garuti’s compatibility index. J. Multi-Criteria Decis. Anal.
**2020**, 27, 50–60. [Google Scholar] [CrossRef] - Lipovetsky, S. Priority vector estimation: Consistency, compatibility, precision. Int. J. Anal. Hierarchy Process
**2020**, 12, 577–591. [Google Scholar] [CrossRef] - Dong, Y.; Zhang, G.; Hong, W.C.; Xu, Y. Consensus models for AHP group decision making under row geometric mean prioritization method. Decis. Support Syst.
**2010**, 49, 281–289. [Google Scholar] [CrossRef] - Escobar, M.T.; Aguarón, J.; Moreno-Jiménez, J.M. Some extensions of the precise consistency consensus matrix. Decis. Support Syst.
**2015**, 74, 67–77. [Google Scholar] [CrossRef] - Garuti, C. Consistency & compatibility (two sides of the same coin). In Proceedings of the ISAHP 2016, London, UK, 4–7 August 2016. [Google Scholar] [CrossRef]
- Saaty, T. The Analyutic Network Process. Decision Making with Dependence and Feedback; AHP Series; RWS Publications: Pittsburgh, PA, USA, 1996. [Google Scholar]
- Garuti, C. Measuring compatibility (closeness) in weighted environments. In Proceedings of the International Symposium on the AHP, Vina del Mar, Chile, 2–6 August 2007. [Google Scholar]
- Lipovetsky, S. Global Priority Estimation in Multiperson Decision Making. J. Optim. Theory Appl.
**2009**, 140, 77–91. [Google Scholar] [CrossRef] - Kahneman, D.; Tversky, A. Prospect Theory: An Analysis of Decision under Risk. Econometrica
**1979**, 47, 263–291. [Google Scholar] [CrossRef] [Green Version] - Barzilai, J.; Golany, B. Ahp Rank Reversal, Normalization Furthermore, Aggregation Rules. INFOR Inf. Syst. Oper. Res.
**1994**, 32, 57–64. [Google Scholar] [CrossRef] - Escobar, M.; Aguarón, J.; Moreno-Jiménez, J. A note on AHP group consistency for the row geometric mean priorization procedure. Eur. J. Oper. Res.
**2004**, 153, 318–322. [Google Scholar] [CrossRef] - Grzybowski, A.Z. New results on inconsistency indices and their relationship with the quality of priority vector estimation. Expert Syst. Appl.
**2016**, 43, 197–212. [Google Scholar] [CrossRef] [Green Version] - Aguarón, J.; Escobar, M.T.; Moreno-Jiménez, J.M. Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process. Eur. J. Oper. Res.
**2021**, 288, 576–583. [Google Scholar] [CrossRef] - Aguarón, J.; Escobar, M.T.; Moreno-Jiménez, J.M.; Turón, A. AHP-Group Decision Making Based on Consistency. Mathematics
**2019**, 7, 242. [Google Scholar] [CrossRef] [Green Version] - Wu, Z.; Jin, B.; Fujita, H.; Xu, J. Consensus analysis for AHP multiplicative preference relations based on consistency control: A heuristic approach. Knowl.-Based Syst.
**2020**, 191, 105317. [Google Scholar] [CrossRef] - Turón, A.; Aguarón, J.; Escobar, M.T.; Moreno-Jiménez, J.M. A Decision Support System and Visualisation Tools for AHP-GDM. Int. J. Decis Support Syst. Technol.
**2019**, 11, 1–19. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Outline of the incompatibility reduction of indicators ${\mathrm{GCOMPI}}_{1}$ and ${\mathrm{GCOMPI}}_{2}$.

**Figure 3.**Outline of the incompatibility reduction of indicators ${\mathrm{GCOMPI}}_{3}$ and ${\mathrm{GCOMPI}}_{4}$.

Measure | ${\mathit{d}}_{\mathit{r}\mathit{s}}={\left(\right)}_{\frac{\mathit{\partial}{\mathbf{GCOMPI}}_{\mathit{i}}}{\mathit{\partial}{\mathit{t}}_{\mathit{r}\mathit{s}}}}{\mathit{t}}_{\mathit{r}\mathit{s}}=1$ | ${\mathit{t}}_{\mathit{r}\mathit{s}}^{*}$ | ${\mathit{\nabla}}^{*}{\mathbf{GCOMPI}}_{\mathit{i}}$ |
---|---|---|---|

${\mathrm{GCOMPI}}_{1}$ | $\frac{4}{(n-1)(n-2)}log\frac{{p}_{rs}}{{a}_{rs}^{G}}$ | $\frac{{a}_{rs}^{G}}{{p}_{rs}}$ | $\frac{2}{(n-1)(n-2)}{log}^{2}\frac{{p}_{rs}}{{a}_{rs}^{G}}$ |

${\mathrm{GCOMPI}}_{2}$ | $\frac{4}{(n-1)(n-2)}log\frac{{v}_{r}/{v}_{s}}{{w}_{r}^{G|J}/{w}_{s}^{G|J}}$ | ${\left(\right)}^{\frac{{w}_{r}^{G|J}/{w}_{s}^{G|J}}{{v}_{r}/{v}_{s}}}$ | $\frac{n}{(n-1)(n-2)}{log}^{2}\frac{{v}_{r}/{v}_{s}}{{w}_{r}^{G|J}/{w}_{s}^{G|J}}$ |

${\mathrm{GCOMPI}}_{3}$ | $\frac{4}{(n-1)(n-2)}log\frac{{p}_{rs}}{{w}_{r}^{G|P}/{w}_{s}^{G|P}}$ | $\frac{{w}_{r}^{G|P}/{w}_{s}^{G|P}}{{p}_{rs}}$ | $\frac{2}{(n-1)(n-2)}{log}^{2}\frac{{p}_{rs}}{{w}_{r}^{G|P}/{w}_{s}^{G|P}}$ |

${\mathrm{GCOMPI}}_{4}$ | $\frac{4}{(n-1)(n-2)}log\frac{{v}_{r}/{v}_{s}}{{w}_{r}^{G|P}/{w}_{s}^{G|P}}$ | ${\left(\right)}^{\frac{{w}_{r}^{G|P}/{w}_{s}^{G|P}}{{v}_{r}/{v}_{s}}}$ | $\frac{n}{(n-1)(n-2)}{log}^{2}\frac{{v}_{r}/{v}_{s}}{{w}_{r}^{G|P}/{w}_{s}^{G|P}}$ |

Iteration t | $(\mathit{r},\mathit{s})$ | ${\mathit{p}}_{\mathit{r}\mathit{s}}$ | ${\mathit{p}}_{\mathit{r}\mathit{s}}^{\prime}$ | ${\mathbf{GCOMPI}}_{1},\mathit{t}$ | ${\mathit{E}}_{\mathit{t}}$ |
---|---|---|---|---|---|

0 | 0.484 | ||||

1 | (1–5) | 3.165 | 3.640 | 0.462 | 15.57% |

2 | (1–2) | 2.049 | 2.356 | 0.443 | 28.87% |

3 | (3–4) | 2.708 | 3.114 | 0.428 | 39.13% |

4 | (4–5) | 2.844 | 3.271 | 0.416 | 47.76% |

5 | (2–3) | 3.000 | 2.608 | 0.410 | 52.01% |

6 | (1–4) | 9.000 | 8.149 | 0.408 | 53.21% |

7 | (3–5) | 1.466 | 1.599 | 0.407 | 54.06% |

8 | (1–3) | 5.509 | 6.007 | 0.406 | 54.98% |

9 | (2–5) | 1.738 | 1.877 | 0.405 | 55.68% |

10 | (2–4) | 6.082 | 5.650 | 0.404 | 56.32% |

${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ${\mathit{v}}_{3}$ | ${\mathit{v}}_{4}$ | ${\mathit{v}}_{5}$ | ${\mathbf{GCOMPI}}_{1}$ | |
---|---|---|---|---|---|---|

P | 0.467 | 0.255 | 0.095 | 0.044 | 0.139 | 0.484 |

${P}^{\prime}$ | 0.486 | 0.238 | 0.096 | 0.042 | 0.138 | 0.404 |

${\mathit{E}}_{10}$ | #Iter | G | |
---|---|---|---|

${\mathrm{GCOMPI}}_{1}$ | 56.32% | 5 | 0.9612 |

${\mathrm{GCOMPI}}_{2}$ | 73.10% | 5 | 0.9380 |

${\mathrm{GCOMPI}}_{3}$ | 44.11% | – | 0.9498 |

${\mathrm{GCOMPI}}_{4}$ | 73.10% | 5 | 0.9380 |

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Aguarón, J.; Escobar, M.T.; Moreno-Jiménez, J.M.; Turón, A.
Geometric Compatibility Indexes in a Local AHP-Group Decision Making Context: A Framework for Reducing Incompatibility. *Mathematics* **2022**, *10*, 278.
https://doi.org/10.3390/math10020278

**AMA Style**

Aguarón J, Escobar MT, Moreno-Jiménez JM, Turón A.
Geometric Compatibility Indexes in a Local AHP-Group Decision Making Context: A Framework for Reducing Incompatibility. *Mathematics*. 2022; 10(2):278.
https://doi.org/10.3390/math10020278

**Chicago/Turabian Style**

Aguarón, Juan, María Teresa Escobar, José María Moreno-Jiménez, and Alberto Turón.
2022. "Geometric Compatibility Indexes in a Local AHP-Group Decision Making Context: A Framework for Reducing Incompatibility" *Mathematics* 10, no. 2: 278.
https://doi.org/10.3390/math10020278