scholarly journals The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 926 ◽  
Author(s):  
Juan Aguarón ◽  
María Teresa Escobar ◽  
José María Moreno-Jiménez ◽  
Alberto Turón
Keyword(s):  
Pairwise Comparison ◽  
Consistency Index ◽  
Analytic Hierarchy ◽  
Priority Vector ◽  
Decision Tools ◽  
Geometric Consistency ◽  
The Matrix ◽  
Stability Intervals ◽  
Definition Of ◽  
Hierarchy Process

The paper presents the Triads Geometric Consistency Index ( T - G C I ), a measure for evaluating the inconsistency of the pairwise comparison matrices employed in the Analytic Hierarchy Process (AHP). Based on the Saaty’s definition of consistency for AHP, the new measure works directly with triads of the initial judgements, without having to previously calculate the priority vector, and therefore is valid for any prioritisation procedure used in AHP. The T - G C I is an intuitive indicator defined as the average of the log quadratic deviations from the unit of the intensities of all the cycles of length three. Its value coincides with that of the Geometric Consistency Index ( G C I ) and this allows the utilisation of the inconsistency thresholds as well as the properties of the G C I when using the T - G C I . In addition, the decision tools developed for the G C I can be used when working with triads ( T - G C I ), especially the procedure for improving the inconsistency and the consistency stability intervals of the judgements used in group decision making. The paper further includes a study of the computational complexity of both measures ( T - G C I and G C I ) which allows selecting the most appropriate expression, depending on the size of the matrix. Finally, it is proved that the generalisation of the proposed measure to cycles of any length coincides with the T - G C I . It is not therefore necessary to consider cycles of length greater than three, as they are more complex to obtain and the calculation of their associated measure is more difficult.

JUDGMENT NUMBER REDUCTION: AN ISSUE IN THE ANALYTIC HIERARCHY PROCESS

2010 ◽  
Vol 09 (01) ◽  
pp. 175-189 ◽  
Author(s):  
LONG-TING WU ◽  
XIA CUI ◽  
RU-WEI DAI
Keyword(s):  
Decision Making ◽  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Random Variable ◽  
Analytic Hierarchy ◽  
Priority Vector ◽  
Judgment Error ◽  
Scale Error ◽  
The Analytic Hierarchy Process ◽  
Hierarchy Process

The Analytic Hierarchy Process (AHP) uses pairwise comparison to evaluate alternatives' advantages to a certain criterion. For decision-making problem with many different criteria and alternatives, pairwise comparison causes a prolonged decision-making period and rises fatigue in decision-makers' mentality. A question of practical value is if there exists a way to reduce judgment number and what influence the reduction will have on the overall evaluation of alternative ratings. To answer this question, we introduce scale error and judgment error into AHP judgment matrix. By expanding the scales defined in the AHP, scale error is eliminated. Taking judgment error as random variable, a new estimator to calculate priority vector is presented. In the end, an example is proved to show lowering judgment number will increase the probability of larger errors appearing in priority vector computation.

Intuitionistic Fuzzy AHP and its Application in Evaluation of Ecological Architecture

2012 ◽  
Vol 518-523 ◽  
pp. 4466-4472 ◽  
Author(s):  
Hao Zhang ◽  
Wei Xia Li ◽  
Cheng Yi Zhang
Keyword(s):  
Fuzzy Ahp ◽  
Relative Importance ◽  
Analytic Hierarchy ◽  
Additive Consistency ◽  
Intuitionistic Fuzzy ◽  
Judgement Matrix ◽  
The Matrix ◽  
Addition And Subtraction ◽  
Definition Of ◽  
Hierarchy Process

In this paper, the definition of additive consistent intuitionistic fuzzy complementary judgement matrix (ACIFCJM) was given; The addition and subtraction algorithms of intuitionistic fuzzy value representing the relative importance degree in the matrix were given, then the definition of the scale transition matrix of intuitionistic fuzzy complementary judgement matrix (IFCJM) was given; The additive consistency recursive iterative adjustment algorithm about the IFCJM was given, then priority vectors formula of IFCJM was introduced; At last, the steps of intuitionistic fuzzy analytic hierarchy process (IFAHP) were introduced, then the method was applied in actual examples, and its effectiveness was verified.

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 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.

Purchasing strategies for relief items in humanitarian operations

2019 ◽  
Vol 9 (2) ◽  
pp. 151-171 ◽  
Author(s):  
Arthur Abreu da Silva Lamenza ◽  
Tharcisio Cotta Fontainha ◽  
Adriana Leiras
Keyword(s):  
Empirical Test ◽  
Analytic Hierarchy ◽  
Content Type ◽  
Humanitarian Operations ◽  
Humanitarian Organizations ◽  
The Matrix ◽  
Relief Operations ◽  
Definition Of ◽  
Hierarchy Process ◽  
Practical Implications

Purpose The purpose of this paper is to develop a Humanitarian Purchasing Matrix to guide purchasing strategies for relief items in humanitarian operations. Design/methodology/approach The research synthesizes the structures of a Purchasing Portfolio Model and the characteristics of purchasing in humanitarian operations, validating them with academics and practitioners to develop a Humanitarian Purchasing Matrix. Then, based on the Analytic Hierarchy Process to classify the relief items in the matrix, an illustrative example is used as an empirical test for the proposed Humanitarian Purchasing Matrix. Findings The academic literature on purchasing in general and purchasing in humanitarian operations share some similarities in terms of “Importance of Purchasing” and “Complexity of Supply Market” dimensions. Moreover, the analysis of such criteria supports the definition of purchasing strategies for different relief items in humanitarian operations. Practical implications The Humanitarian Purchasing Matrix can be considered a tool/guide for professionals of humanitarian organizations in the adoption of purchasing strategies for the different relief items purchased for humanitarian operations. Originality/value Considering a scenario of a constant increase in the variety of relief items, the high purchasing volume and the pressure to more efficient relief operations, the research discusses the intersectionality of business purchasing models and the purchasing characteristics of humanitarian operations. Moreover, the research deliveries a tool/guide to the adoption of purchasing strategies that are composed of criteria observed in the literature and suggested by both humanitarian logistic academics and practitioners.

scholarly journals Characterization of Consistent Completion of Reciprocal Comparison Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Julio Benítez ◽  
Laura Carrión ◽  
Joaquín Izquierdo ◽  
Rafael Pérez-García
Keyword(s):  
Pairwise Comparison ◽  
Explicit Calculation ◽  
Frobenius Norm ◽  
Analytic Hierarchy ◽  
Linear System Of Equations ◽  
Decision Aiding ◽  
The Matrix ◽  
Decision Making Processes ◽  
Reciprocal Matrix ◽  
Hierarchy Process

Analytic hierarchy process (AHP) is a leading multi-attribute decision-aiding model that is designed to help make better choices when faced with complex decisions involving several dimensions. AHP, which enables qualitative analysis using a combination of subjective and objective information, is a multiple criteria decision analysis approach that uses hierarchical structured pairwise comparisons. One of the drawbacks of AHP is that a pairwise comparison cannot be completed by an actor or stakeholder not fully familiar with all the aspects of the problem. The authors have developed a completion based on a process of linearization that minimizes the matrix distance defined in terms of the Frobenius norm (a strictly convex minimization problem). In this paper, we characterize when an incomplete, positive, and reciprocal matrix can be completed to become a consistent matrix. We show that this characterization reduces the problem to the solution of a linear system of equations—a straightforward procedure. Various properties of such a completion are also developed using graph theory, including explicit calculation formulas. In real decision-making processes, facilitators conducting the study could use these characterizations to accept an incomplete comparison body given by an actor or to encourage the actor to further develop the comparison for the sake of consistency.

scholarly journals Morphometric parameters based prioritization of a Mid-Himalayan watershed using fuzzy analytic hierarchy process

2021 ◽  
Vol 280 ◽  
pp. 10004
Author(s):  
Mallika Joshi ◽  
Pankaj Kumar ◽  
Purabi Sarkar
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Decision Support Tool ◽  
Morphometric Parameters ◽  
Fuzzy Analytic Hierarchy Process ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Support Tool ◽  
The Matrix ◽  
Hierarchy Process

Watershed prioritization has become increasingly crucial for managing natural resources, especially the watersheds. A useful decision support tool to provide appropriate weights to different morphological attributes with lineage with soil erosion is required to identify environmentally stressed areas for the watershed resources. This study examines the Western Nayar watershed delineation and further examination of the watershed’s morphometric parameters. The morphometric parameters were quantified under the linear, areal, and relief heads for the watershed. The prioritization of sub-watersheds was done by the fuzzy analytic hierarchy process (FAHP). The study included nine morphometric parameters for forming a pairwise comparison matrix. The fuzzy analytic hierarchy process was employed for assigning the suitable weights to morphometric parameters, and further, these weights are normalized to assign the final ranks to the sub-watershed. In Western Nayar, SW9 got the highest priority, and SW1 was categorized as the least priority. The results were validated by the consistency ratio index, which depends on the matrix consistency index’s size that should be less than 10%. The consistency index of the present study was found to be 2%.

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