Inconsistent Data Entry and Ignoring the Consistency Index (CI) in the AHP Decision-Making Method

Introduction

 The Analytic Hierarchy Process (AHP) is one of the most powerful and widely used multi-criteria decision-making methods, especially for complex and multi-dimensional problems. This method is based on pairwise comparisons of criteria and alternatives and assigning weights accordingly. However, one of the main challenges in properly implementing AHP is the entry of inconsistent data and neglecting the Consistency Index (CI). These issues can lead to inaccurate results and unreliable decisions. This article explores the problem of inconsistent data entry and the critical importance of considering the Consistency Index in AHP.

What Is Inconsistent Data Entry in AHP?

 In AHP, to determine priorities, a pairwise comparison matrix between criteria or alternatives is constructed. This matrix must follow certain rules to be logical and valid. One key rule is reciprocity; that is, if criterion A compared to B has a value of 3, then criterion B compared to A must be 1/3.

Inconsistency arises when the data or comparisons entered in this matrix contradict each other. For example:

• If A is preferred to B,

• And B is preferred to C,

• Then A should also be preferred to C.

If this transitive relationship does not hold, the data are considered inconsistent.

What Is the Consistency Index (CI)?

 The Consistency Index (CI) quantifies the degree of inconsistency in a pairwise comparison matrix. It is calculated using the eigenvalues of the matrix. The formula for CI is:

CI=λmax⁡−nn−1CI = \frac{\lambda_{\max} – n}{n – 1}

where:

• λmax⁡\lambda_{\max} is the largest eigenvalue of the matrix,

• nn is the number of criteria or alternatives.

A CI close to zero indicates good consistency, while a higher CI shows greater inconsistency.

Consistency Ratio (CR)

 To better assess inconsistency, the CI is compared with the Random Index (RI), which represents the average consistency index of randomly generated matrices of the same size. The Consistency Ratio (CR) is defined as:

CR=CIRICR = \frac{CI}{RI}

If CR is less than 0.1 (10%), the matrix is considered consistent, and the results are acceptable. If CR exceeds 0.1, the data are inconsistent and require correction.

Why Is Ignoring the Consistency Index Dangerous in AHP?

• Leads to incorrect results: Decisions based on inconsistent data can result in poor choices.

• Invalidates the analysis: Without checking CI and CR, the AHP analysis loses scientific credibility.

• Wastes time and resources: Repeated analysis and corrections are required when inconsistency is ignored.

• Reduces stakeholder trust: Illogical results undermine confidence in the decision-making process.

How to Prevent Data Inconsistency?

1. Provide comprehensive user training: Users entering data should fully understand AHP principles and the importance of consistency.
2. Use proper software tools: Tools that calculate and alert users about CI and CR help quickly identify inconsistencies.
3. Review comparisons when CR is high: Revisit and adjust inconsistent pairwise comparisons.
4. Limit the number of criteria: Fewer criteria reduce complexity and the chance of inconsistency.
5. Use the standard Saaty scale: Strict adherence to Saaty’s scale in comparisons helps reduce inconsistency.
   

Practical Example of Inconsistent Data Entry and Its Effect on CI

 Consider three criteria A, B, and C, with these pairwise comparisons:

• A compared to B = 3 (A is moderately preferred to B)

• B compared to C = 4 (B is strongly preferred to C)

• A compared to C = 1/3 (A is less preferred than C) — this creates inconsistency because if A > B and B > C, then A should be preferred to C.

This inconsistency increases CI and CR, signaling the need to correct the data.

Conclusion

 Inconsistent data entry and ignoring the Consistency Index (CI) are among the biggest challenges in applying AHP, which can significantly reduce the accuracy and credibility of the analysis. To achieve reliable results, data consistency must be ensured, and the consistency index must be carefully checked and corrected if needed. Training, using suitable tools, and careful data input are key factors for successful AHP implementation.

Frequently Asked Questions (FAQ)