Abstract
One of the most popular methods of calculating priorities based on the pairwise comparisons matrices (PCM) is the geometric mean method (GMM). It is equivalent to the logarithmic least squares method (LLSM), so some use both names interchangeably, treating it as the same approach. The main difference, however, is in the way the calculations are done. It turns out, however, that a similar relationship holds for incomplete matrices. Based on Harker’s method for the incomplete PCM, and using the same substitution for the missing entries, it is possible to construct the geometric mean solution for the incomplete PCM, which is fully compatible with the existing LLSM for the incomplete PCM. Again, both approaches lead to the same results, but the difference is how the final solution is computed. The aim of this work is to present in a concise form, the computational method behind the geometric mean method (GMM) for an incomplete PCM. The computational method is presented to emphasize the relationship between the original GMM and the proposed solution. Hence, everyone who knows the GMM for a complete PCM should easily understand its proposed extension. Theoretical considerations are accompanied by a numerical example, allowing the reader to follow the calculations step by step.
Highlights
The ability to compare things has accompanied mankind for centuries
His method over time was forgotten and rediscovered in a similar form by Condorcet [2]. Both Llull and Condorcet treated comparisons as binary, i.e., the result of comparisons can be either a win or loss, Thurstone proposed the use of pairwise comparisons (PC) in a more generalized, quantitative way [3]
Since pairwise judgments are very often made by experts, making a large number of paired comparisons can be difficult and expensive. This observation encouraged experts to search for ranking methods using a reduced number of pairwise comparisons. These studies have resulted in several methods, including Harker’s eigenvalue based approach [30], the logarithmic least squares method for incomplete PC matrices (ILLSM) [31,32,33], the spanning-tree approach [34,35,36] that can be used for incomplete PC matrices [37], and missing values estimation and reconstruction [38,39,40]
Summary
The ability to compare things has accompanied mankind for centuries. When comparing products in a convenience store, choosing dishes in a restaurant, and selecting a gas station with the most attractive fuel price, people are trying to make the best choices. During the process of selecting the best option, the available alternatives are compared in pairs The first use of pair comparison as a formal basis for the decision procedure is attributed to the XIII-century mathematician Ramon Llull who proposed a binary electoral system [1] His method over time was forgotten and rediscovered in a similar form by Condorcet [2]. This observation encouraged experts to search for ranking methods using a reduced number of pairwise comparisons These studies have resulted in several methods, including Harker’s eigenvalue based approach [30], the logarithmic least squares method for incomplete PC matrices (ILLSM) [31,32,33], the spanning-tree approach [34,35,36] that can be used for incomplete PC matrices [37], and missing values estimation and reconstruction [38,39,40].
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