# When Fairness Meets Consistency in AHP Pairwise Comparisons

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*Mathematics*

**2023**,

*11*(3), 604; https://doi.org/10.3390/math11030604 (registering DOI)

## Abstract

**:**

## 1. Introduction

**Hypothesis**

**1.**

## 2. Materials and Methods

#### 2.1. Insight into Related Literature

- There are 35,430 articles published in the period 1980–2021 (from which 15,000 articles are from 2017–2021), according to Madzík and Falát [33] and based on Scopus;
- There are 8441 articles published in the period 1979–2017 (more specifically, there are 86 articles from 1979–1990, 716 articles from 1991–2001, and 7639 articles from 2002–2017), according to Emrouznejad and Marra [34] and based on ISI Web of Science;
- There are 9859 harvested articles published in the period 1982–2018, according to Yu et al. [35] and based on ISI Web of Science. This review also includes an improved Saaty’s version of the AHP, i.e., the analytic network process (ANP) that considers interaction and dependence among hierarchically structured elements.

- Up-down decomposition of decision problem in hierarchy levels, starting from the goal on the top, followed by criteria in the middle, sub-criteria if necessary, in the next/lower level(s), and finally, decision alternatives/options at the bottom of the elaborated hierarchy;
- Comparative judgments, i.e., comparisons of the decomposed elements from the same level in pairs (within PCMs) regarding the above-level goal or criteria (according to the nine-degree scale defined in [10]), to derive principal eigenvectors, i.e., relative priorities;
- Synthesis of the priorities, from local to the global plane, for overall alternatives’ ranking corresponding to the goal.

#### 2.1.1. Common-Mentioned Shortcomings of AHP Pairwise Comparisons

#### 2.1.2. Fairness Notes on AHP Pairwise Comparisons

- Concerning DMs–it is applicable in group decision-making when decision systems should fairly include DMs’ opinions. For example, a decision support system for group MCDM can mitigate or eliminate biased DMs’ opinions [62] or follow a democratic approach in conflict situations by choosing a consensual value for parameter v (related to the VIKOR method, it equals 0.5) [63].
- Concerning decision criteria–criterion weights are essential for two levels of fairness: among criteria and alternatives [64]; while it is generally possible to introduce discrimination based on a single property (e.g., racial discrimination–[65], gender discrimination–[66], age discrimination–[67], etc.), several separate properties (i.e., multiple discrimination–[68]), or one joint property (i.e., intersectional discrimination–[69]). In MCDM, opposite to wash criteria about which alternatives are equally preferred [70], DMs can define criteria (or their weights) that favor/damage some alternatives or groups (the favored group is privileged, and the damaged group is unprivileged).
- Concerning the used algorithms–it can imply the absence of unintentional (coincidentally made) discrimination toward vulnerable groups in society that algorithmic decision-making techniques may amplify. The harmful practice is possible because of, for example, data that reflect historically widespread biases and contain the prejudices of prior DMs [71] or impose discriminatory inferences towards underrepresented groups [23].

#### 2.2. Toward Consistent and Fair AHP Pairwise Methodology

#### 2.2.1. Discrete Optimization Problem

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 2.3. Synthetic Experiment

## 3. Results

**Proof**

**of**

**Hypothesis**

**1.**

## 4. Discussion

## 5. Conclusions

- Expanding and applying the methodology to the whole AHP hierarchy structure;
- Fixing some judgments that DMs do not want to change;
- Setting multiobjective discrete optimizations (such as in [87]) to achieve additional goals (regarding AHP hierarchy structure or the used accuracy/fairness metrics).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AHP | Analytic hierarchy process |

AI | Artificial intelligence |

CR | Consistency ratio |

DI | Disparate impact |

DM(s) | Decision-maker(s) |

GA | Genetic algorithm |

MCDM | Multi-criteria decision-making |

ML | Machine learning |

MM | Mathematical model |

MP | Mathematical programming |

OM | Operation management |

PCM(s) | Pairwise comparison matrice(s) |

RQ | Rayleigh quotient |

UT(s) | Upper triangle(s) |

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**Figure 1.**Average values of ${f}_{d}$ objective function for different PCM sizes observed concerning predefined combinations of CR and DI.

**Figure 2.**Comparison of average values of ${f}_{d}$ objective function for different values of the initial DI and CR set-points.

m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|

$\mathit{RCI}$ | 0.58 | 0.9 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 | 1.49 |

**Table 2.**An overview of AHP modifications dedicated to facilitating pairwise comparisons and improving logical consistency.

Author(s) | Name | Purpose of Transformation | The Used Method | Field of Application |
---|---|---|---|---|

Sangiorgio et al. [42] | Optimized-AHP (O-AHP) | It successfully overcomes the following drawbacks of classical AHP, common in situations when the number of criteria or alternatives is greater than nine:
- a large number of comparisons/judgments and the limited capacity of the human mind,
- consistency issue.
| The weights evaluation procedure based on mathematical programming (MP) redefines the process of forming a judgment matrix in the following ways:
- It uses judgment ranges instead of the exact judgment assignments;
- An MP formulation provides the entries of the judgment matrix to minimize inconsistency.
| Construction |

Ishizaka et al. [43] | AHPSort | It reduces the number of required comparisons and facilitates decision-making. | A new variant sorts alternatives into ordered predefined categories according to DMs’ preferences. | Supplier selection |

Ishizaka and López [44] | Cost-Benefit AHPSort | It provides better and easier comparisons and benchmarks for evaluating alternatives. | A modification of the AHPSort treats cost and benefit criteria separately, i.e., into two distinct hierarchies. | Performance analysis of offshore providers in the aerospace industry |

Li et al. [45] | Improved AHP (IAHP) | It improves consistency in PCMs when the number of elements equals five or more. | The improvement uses the sorting technique (instead of quantification on Saaty’s 9-point scale) to judge between two elements in pairwise comparisons. | Risk identification in construction |

Lin et al. [46] | Adaptive AHP approach (${A}^{3}$) | It helps to improve consistency in pairwise comparisons, reduce costs and timeliness, and improve decision-making quality. | The approach uses a soft computing technique (i.e., a GA) to improve consistency automatically. | Construction |

Xiulin and Dawei [47] | Improved AHP | It overcomes the difficulties of making judgments according to the traditional nine-scaling method and blindness in checking consistency. It is helpful in the determination of target weights. | The improvement uses a 3-scale point method: 1/0.5/0 indicates that the ${i}^{th}$ alternative is more/equally/less important than the ${j}^{th}$ alternative. | Teacher evaluation |

Leal [48] | AHP-express | The simplified version helps make decisions in constrained times. It reduces the number of required comparisons and facilitates calculations. | The simplification requires only $n-1$ comparisons (n–number of alternatives) for each criterion, unlike $({n}^{2}-n)/2$ required comparisons in traditional AHP. | Business application |

Chen [49] | Diversified AHP-tree approach | It allows diverse viewpoints of DMs regarding criteria relative importance. | The approach decomposes an inconsistent judgment matrix into several sub-judgment matrices. It uses the GA for solving nonlinear programming models to maximize the sum of distances between any two sub-judgment matrices. | Supplier selection |

Abastante et al. [50] | New parsimonious AHP methodology | It reduces the number of comparisons and inconsistencies and avoids rank reversal problems compared to the original AHP. | A newly developed proposal implies using reference objects for pairwise comparisons. It avoids comparisons of more relevant objects with less relevant ones. | General application |

**Table 3.**An overview of AHP modifications dedicated to overcoming uncertain and subjective DMs’ judgments.

Author(s) | Name | Purpose of Transformation | The Used Method | Field of Application |
---|---|---|---|---|

Nefeslioglu et al. [51] | Modified AHP (M-AHP) | It compensates for expert subjectivism in pairwise comparisons due to a lack of knowledge or data for the relevant problem. | A computer code is postulated on factors and the decision points, whereby the role of experts in preparing the comparison matrix is limited to defining the maximum scores for factors in the system. | Natural hazards |

Tesfamariam and Sadiq [52] | Fuzzy AHP (F-AHP) | It allows DMs to account for uncertainty (vagueness). | The fuzzy-based technique used fuzzy arithmetic operations. It aggregates the fuzzy global preference weights concerning each alternative. | Environmental risk management |

Banuelas and Antony [53] | Modified AHP (MAHP) | It includes uncertainty and managerial aspects (‘soft’ issues) in judgment comparisons and, therefore, a better understanding of the context of the applied technique. | The method incorporates uncertainty by using probabilistic distributions. It tests the results for statistical significance and analyses rank reversal using ANOVA. | Application in organizations |

Xu et al. [54] | Entropy weight modified AHP hierarchy model (EWMAHPHM) | It improves decision-making efficiency in changing environments where regional information is insufficient. | The method includes the entropy weight method in AHP. The construction of the distributed model precedes the entropy weight correction. | Information-based ecological environment construction |

Sadiq and Tesfamariam [55] | Intuitionistic fuzzy AHP (IF-AHP) | It handles both uncertainty categories–vagueness and ambiguity in human subjective evaluation. | The methodology uses intuitionistic fuzzy sets. | Environmental decision-making |

Lin et al. [56] | Improved AHP (IAHP) | It comprehensively determines the weights of risk indices. | The improvement uses the entropy weight method and integrates objective data variability with subjective judgments. | Flash flood risk assessment |

Deng et al. [57] | D-AHP | It provides a new, effective, and feasible expression of uncertain information. | The method extends the fuzzy preference relation approach by using the so-called D numbers, resulting from the Dempster–Shafer theory extension. | Supplier selection |

PCM Size (m) | Percentage |
---|---|

4 | 100.00% |

5 | 100.00% |

6 | 100.00% |

7 | 99.5% |

8 | 99.25% |

9 | 98.25% |

10 | 99.5% |

Total | 99.5% |

Approach | Successful Optimizations (in%) | $\overline{{\mathit{f}}_{\mathit{d}}}$ | ${\overline{\mathbf{CR}}}_{\mathbf{aft}}$ | ${\overline{\mathbf{DI}}}_{\mathbf{aft}}$ |
---|---|---|---|---|

Iterative–eigenvector | 99.00% | 13.6658 | 0.0939 | 0.8476 |

Approximate–RQ | 86.00% | 12.4885 | 0.0943 | 0.8499 |

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Dodevska, Z.; Radovanović, S.; Petrović, A.; Delibašić, B. When Fairness Meets Consistency in AHP Pairwise Comparisons. *Mathematics* **2023**, *11*, 604.
https://doi.org/10.3390/math11030604

**AMA Style**

Dodevska Z, Radovanović S, Petrović A, Delibašić B. When Fairness Meets Consistency in AHP Pairwise Comparisons. *Mathematics*. 2023; 11(3):604.
https://doi.org/10.3390/math11030604

**Chicago/Turabian Style**

Dodevska, Zorica, Sandro Radovanović, Andrija Petrović, and Boris Delibašić. 2023. "When Fairness Meets Consistency in AHP Pairwise Comparisons" *Mathematics* 11, no. 3: 604.
https://doi.org/10.3390/math11030604