scholarly journals A Numerical Comparison of the Sensitivity of the Geometric Mean Method, Eigenvalue Method, and Best–Worst Method

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 554
Author(s):  
Jiří Mazurek ◽  
Radomír Perzina ◽  
Jaroslav Ramík ◽  
David Bartl
Keyword(s):  
Research Method ◽  
Geometric Mean ◽  
Pairwise Comparisons ◽  
Numerical Comparison ◽  
Priority Vector ◽  
Eigenvalue Method ◽  
Fundamental Scale ◽  
Matrix Vector ◽  
Best Worst Method ◽  
Order Violation

In this paper, we compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons—the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best–Worst Method (BWM)—with respect to two features: sensitivity and order violation. As the research method, we apply One-Factor-At-a-Time (OFAT) sensitivity analysis via Monte Carlo simulations; the number of compared objects ranges from 3 to 8, and the comparison scale coincides with Saaty’s fundamental scale from 1 to 9 with reciprocals. Our findings suggest that the BWM is, on average, significantly more sensitive statistically (and thus less robust) and more susceptible to order violation than the GMM and EVM for every examined matrix (vector) size, even after adjustment for the different numbers of pairwise comparisons required by each method. On the other hand, differences in sensitivity and order violation between the GMM and EMM were found to be mostly statistically insignificant.

scholarly journals On the Geometric Mean Method for Incomplete Pairwise Comparisons

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1873
Author(s):  
Konrad Kułakowski
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Geometric Mean ◽  
Computational Method ◽  
Pairwise Comparisons ◽  
Similar Relationship ◽  
Final Solution ◽  
The Difference ◽  
The Relationship ◽  
Theoretical Considerations

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.

scholarly journals On the Statistical Discrepancy and Affinity of Priority Vector Heuristics in Pairwise-Comparison-Based Methods

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1150
Author(s):  
Pawel Tadeusz Kazibudzki
Keyword(s):  
Pairwise Comparison ◽  
Analytic Hierarchy ◽  
Priority Vector ◽  
Approximation Quality ◽  
Statistical Measures ◽  
Eigenvalue Method ◽  
Mathematica Software ◽  
Hierarchy Process ◽  
Alternative Solutions

There are numerous priority deriving methods (PDMs) for pairwise-comparison-based (PCB) problems. They are often examined within the Analytic Hierarchy Process (AHP), which applies the Principal Right Eigenvalue Method (PREV) in the process of prioritizing alternatives. It is known that when decision makers (DMs) are consistent with their preferences when making evaluations concerning various decision options, all available PDMs result in the same priority vector (PV). However, when the evaluations of DMs are inconsistent and their preferences concerning alternative solutions to a particular problem are not transitive (cardinally), the outcomes are often different. This research study examines selected PDMs in relation to their ranking credibility, which is assessed by relevant statistical measures. These measures determine the approximation quality of the selected PDMs. The examined estimates refer to the inconsistency of various Pairwise Comparison Matrices (PCMs)—i.e., W = (wij), wij > 0, where i, j = 1,…, n—which are obtained during the pairwise comparison simulation process examined with the application of Wolfram’s Mathematica Software. Thus, theoretical considerations are accompanied by Monte Carlo simulations that apply various scenarios for the PCM perturbation process and are designed for hypothetical three-level AHP frameworks. The examination results show the similarities and discrepancies among the examined PDMs from the perspective of their quality, which enriches the state of knowledge about the examined PCB prioritization methodology and provides further prospective opportunities.

scholarly journals Cognitive Best Worst Method for Multiattribute Decision-Making

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjun Zhang ◽  
Chengxiang Yin ◽  
Xiuli Qi ◽  
Rui Zhang ◽  
Xingdang Kang
Keyword(s):  
Decision Making ◽  
Rating Scale ◽  
Semantic Representation ◽  
Pairwise Comparison ◽  
Pairwise Comparisons ◽  
Ratio Scale ◽  
Interval Scale ◽  
Real World Application ◽  
The Matrix ◽  
Best Worst Method

Pairwise comparison based multiattribute decision-making (MADM) methods are widely used and studied in recent years. However, the perception and cognition towards the semantic representation for the linguistic rating scale and the way in which the pairwise comparisons are executed are still open to discuss. The commonly used ratio scale is likely to produce misapplications and the matrix based comparison style needs too many comparisons and is not able to guarantee the consistency of the matrix when the number of objects involved is large. This research proposes a new MADM method CBWM (Cognitive Best Worst Method) which adopts interval scale to represent the pairwise difference and only compares each object to the best object and the worst object rather than all the other objects. CBWM is a vector based method which only needs 2n-3 pairwise comparisons and is more likely to generate consistent comparisons and reliable results. The theoretical analysis and a real world application demonstrate the effectiveness of CBWM.

scholarly journals A Model for Determining Weight Coefficients by Forming a Non-Decreasing Series at Criteria Significance Levels (NDSL)

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 745
Author(s):  
Mališa Žižović ◽  
Dragan Pamučar ◽  
Goran Ćirović ◽  
Miodrag M. Žižović ◽  
Boža D. Miljković
Keyword(s):  
Decision Makers ◽  
Pairwise Comparisons ◽  
Small Range ◽  
Analytic Hierarchy ◽  
Original Algorithm ◽  
Real World Problem ◽  
Significance Levels ◽  
Hierarchy Process ◽  
Weight Coefficients ◽  
Best Worst Method

In this paper, a new method for determining weight coefficients by forming a non-decreasing series at criteria significance levels (the NDSL method) is presented. The NDLS method includes the identification of the best criterion (i.e., the most significant and most influential criterion) and the ranking of criteria in a decreasing series from the most significant to the least significant criterion. Criteria are then grouped as per the levels of significance within the framework of which experts express their preferences in compliance with the significance of such criteria. By employing this procedure, fully consistent results are obtained. In this paper, the advantages of the NDSL model are singled out through a comparison with the Best Worst Method (BWM) and Analytic Hierarchy Process (AHP) models. The advantages include the following: (1) the NDSL model requires a significantly smaller number of pairwise comparisons of criteria, only involving an n − 1 comparison, whereas the AHP requires an n(n − 1)/2 comparison and the BWM a 2n − 3 comparison; (2) it enables us to obtain reliable (consistent) results, even in the case of a larger number of criteria (more than nine criteria); (3) the NDSL model applies an original algorithm for grouping criteria according to the levels of significance, through which the deficiencies of the 9-degree scale applied in the BWM and AHP models are eliminated. By doing so, the small range and inconsistency of the 9-degree scale are eliminated; (4) while the BWM includes the defining of one unique best/worst criterion, the NDSL model eliminates this limitation and gives decision-makers the freedom to express the relationships between criteria in accordance with their preferences. In order to demonstrate the performance of the developed model, it was tested on a real-world problem and the results were validated through a comparison with the BWM and AHP models.

WEAK CONSISTENCY AND QUASI-LINEAR MEANS IMPLY THE ACTUAL RANKING

2002 ◽  
Vol 10 (03) ◽  
pp. 227-239 ◽  
Author(s):  
LUCIANO BASILE ◽  
LIVIA D'APUZZO
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Pairwise Comparisons ◽  
Relative Importance ◽  
Analytic Hierarchy ◽  
Left Eigenvector ◽  
The Matrix ◽  
Linear Means ◽  
The Right ◽  
Hierarchy Process

It is known that in the Analytic Hierarchy Process (A.H.P.) a scale of relative importance for alternatives is derived from a pairwise comparisons matrix A = (aij). Priority vectors are basically provided by the following methods: the right eigenvector method, the geometric mean method and the arithmetic mean method. Antipriority vectors can also be considered; they are built by both the left eigenvector method and mean procedures applied to the columns of A. When the matrix A is inconsistent, priority and antipriority vectors do not indicate necessarily the same ranking. We deal with the problem of the reliability of quantitative rankings and we use quasi-linear means for providing a more general approach to get priority and antipriority vectors.

scholarly journals Ranking Alternatives by Pairwise Comparisons Matrix and Priority Vector

2017 ◽  
Vol 64 (s1) ◽  
pp. 85-95 ◽  
Author(s):  
Jaroslav Ramík
Keyword(s):  
Decision Making ◽  
Essential Elements ◽  
Real Interval ◽  
Pairwise Comparisons ◽  
Analytic Hierarchy ◽  
Priority Vector ◽  
Advantages And Disadvantages ◽  
Decision Making Problem ◽  
Ranking Of Alternatives ◽  
Hierarchy Process

Abstract The decision making problem considered here is to rank n alternatives from the best to the worst, using information given by the decision maker(s) in the form of an n×n pairwise comparisons (PC) matrix. We investigate pairwise comparisons matrices with elements from a real interval which is a traditional multiplicative approach used in Analytic hierarchy process (AHP). Here, we deal with two essential elements of AHP: measuring consistency of PC matrix and the method of eliciting the priority vector by which the final ranking of alternatives is derived. Classical approaches introduced by T. Saaty in AHP are compared with later approaches based on the AHP criticism published in the literature. Advantages and disadvantages of both approaches are highlighted and discussed.

scholarly journals A New Model for Determining Weight Coefficients of Criteria in MCDM Models: Full Consistency Method (FUCOM)

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 393 ◽  
Author(s):  
Dragan Pamučar ◽  
Željko Stević ◽  
Siniša Sremac
Keyword(s):  
Pairwise Comparison ◽  
Pairwise Comparisons ◽  
Analytic Hierarchy ◽  
Proposed Model ◽  
Optimal Values ◽  
Definition Of ◽  
Hierarchy Process ◽  
Consistency Method ◽  
Weight Coefficients ◽  
Best Worst Method

In this paper, a new multi-criteria problem solving method—the Full Consistency Method (FUCOM)—is proposed. The model implies the definition of two groups of constraints that need to satisfy the optimal values of weight coefficients. The first group of constraints is the condition that the relations of the weight coefficients of criteria should be equal to the comparative priorities of the criteria. The second group of constraints is defined on the basis of the conditions of mathematical transitivity. After defining the constraints and solving the model, in addition to optimal weight values, a deviation from full consistency (DFC) is obtained. The degree of DFC is the deviation value of the obtained weight coefficients from the estimated comparative priorities of the criteria. In addition, DFC is also the reliability confirmation of the obtained weights of criteria. In order to illustrate the proposed model and evaluate its performance, FUCOM was tested on several numerical examples from the literature. The model validation was performed by comparing it with the other subjective models (the Best Worst Method (BWM) and Analytic Hierarchy Process (AHP)), based on the pairwise comparisons of the criteria and the validation of the results by using DFC. The results show that FUCOM provides better results than the BWM and AHP methods, when the relation between consistency and the required number of the comparisons of the criteria are taken into consideration. The main advantages of FUCOM in relation to the existing multi-criteria decision-making (MCDM) methods are as follows: (1) a significantly smaller number of pairwise comparisons (only n − 1), (2) a consistent pairwise comparison of criteria, and (3) the calculation of the reliable values of criteria weight coefficients, which contribute to rational judgment.

scholarly journals Exploring Multicriteria Elicitation Model Based on Pairwise Comparisons: Building an Interactive Preference Adjustment Algorithm

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Giancarllo Ribeiro Vasconcelos ◽  
Caroline Maria de Miranda Mota
Keyword(s):  
Decision Making ◽  
Decision Maker ◽  
Decision Makers ◽  
Pairwise Comparisons ◽  
Decision Making Process ◽  
Priority Vector ◽  
Effective Decision ◽  
Low Levels ◽  
Effective Decision Making ◽  
Decision Making Tool

Pairwise comparisons have been applied to several real decision making problems. As a result, this method has been recognized as an effective decision making tool by practitioners, experts, and researchers. Although methods based on pairwise comparisons are widespread, decision making problems with many alternatives and criteria may be challenging. This paper presents the results of an experiment used to verify the influence of a high number of preferences comparisons in the inconsistency of the comparisons matrix and identifies the influence of consistencies and inconsistencies in the assessment of the decision-making process. The findings indicate that it is difficult to predict the influence of inconsistencies and that the priority vector may or may not be influenced by low levels of inconsistencies, with a consistency ratio of less than 0.1. Finally, this work presents an interactive preference adjustment algorithm with the aim of reducing the number of pairwise comparisons while capturing effective information from the decision maker to approximate the results of the problem to their preferences. The presented approach ensures the consistency of a comparisons matrix and significantly reduces the time that decision makers need to devote to the pairwise comparisons process. An example application of the interactive preference adjustment algorithm is included.

scholarly journals A Numerical Comparison of Iterative Algorithms for Inconsistency Reduction in Pairwise Comparisons

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Jiri Mazurek ◽  
Radomir Perzina ◽  
Dominik StrzaLka ◽  
Bartosz Kowal ◽  
PaweL Kuras
Keyword(s):  
Iterative Algorithms ◽  
Pairwise Comparisons ◽  
Numerical Comparison
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