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<p>Verifying the consistency of the Pairwise Comparison Matrix (PCM) is essential in the Multi-Criteria Decision-Making (MCDM) process, as decision-makers cannot use an inconsistent PCM as a credible reference. To optimize a PCM that is inconsistent, the primary requirement is to minimize the difference between the original and substitute matrices while improving the Consistency Ratio (CR) of the original Matrix. In this article, we employ a novel framework that uses a novel distance formula focused on the Cosine Distance metric to address inconsistencies in the PCM. Additionally, we utilize a swarm intelligence-based Grey Wolf Optimizer (GWO) to address further and repair these inconsistencies in the PCM. GWO leverages the exploration and exploitation strategies of grey wolves to identify the best-optimized value that satisfies the CR threshold while aligning closely with the decision-makers’ (DM) original judgments. Additionally, we have introduced the maximum correction range <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_22310_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon\)</EquationSource> </InlineEquation>. This <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_22310_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon\)</EquationSource> </InlineEquation> is very helpful in achieving consistency based on the preference level of the DM, to what extent the DM wants to change the correction range. The experimental results indicate that we achieve better outcomes for the special case matrix with a CR of 0.546487; we obtain the resultant Matrix with a CR of 0.073397 with a minimal correction range of E = 3 in the 94th iteration. Experimental results also demonstrate that the suggested novel framework successfully generates a consistent matrix with minimal deviation from the original Matrix and outperforms previously used algorithms, such as ANT-based Analytic Hierarchy Process (ANTAHP) and Particle Swarm Optimization (PSO).</p>

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Repairing the inconsistent pairwise comparison matrix using a cosine distance and grey wolf optimiser-based framework in multi-criteria decision-making

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  • 发表日期：

    2025年11月3日

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  • Shalu Kaushik,
  • Sangeeta Pant,
  • Lokesh Kumar Joshi,
  • Anuj Kumar,
  • Ketan Kotecha,
  • Ambarish Kulkarni

摘要

Verifying the consistency of the Pairwise Comparison Matrix (PCM) is essential in the Multi-Criteria Decision-Making (MCDM) process, as decision-makers cannot use an inconsistent PCM as a credible reference. To optimize a PCM that is inconsistent, the primary requirement is to minimize the difference between the original and substitute matrices while improving the Consistency Ratio (CR) of the original Matrix. In this article, we employ a novel framework that uses a novel distance formula focused on the Cosine Distance metric to address inconsistencies in the PCM. Additionally, we utilize a swarm intelligence-based Grey Wolf Optimizer (GWO) to address further and repair these inconsistencies in the PCM. GWO leverages the exploration and exploitation strategies of grey wolves to identify the best-optimized value that satisfies the CR threshold while aligning closely with the decision-makers’ (DM) original judgments. Additionally, we have introduced the maximum correction range \(\epsilon\) . This \(\epsilon\) is very helpful in achieving consistency based on the preference level of the DM, to what extent the DM wants to change the correction range. The experimental results indicate that we achieve better outcomes for the special case matrix with a CR of 0.546487; we obtain the resultant Matrix with a CR of 0.073397 with a minimal correction range of E = 3 in the 94th iteration. Experimental results also demonstrate that the suggested novel framework successfully generates a consistent matrix with minimal deviation from the original Matrix and outperforms previously used algorithms, such as ANT-based Analytic Hierarchy Process (ANTAHP) and Particle Swarm Optimization (PSO).