scholarly journals The extention rank ordering criteria weighting methods in fuzzy enviroment

2020 ◽  
Vol 30 (2) ◽  
Author(s):  
Ewa Roszkowska
Keyword(s):  
Geometric Mean ◽  
Pairwise Comparison ◽  
Consistency Index ◽  
Priority Vector ◽  
Order Preservation ◽  
Rank Ordering ◽  
Inconsistency Index ◽  
A Priority ◽  
Weighting Methods ◽  
Different Levels

We examine the satisfaction of the condition of order preservation (COP) concerning different levels of inconsistency for randomly generated multiplicative pairwise comparison matrices (MPCMs) of the order from 3 to 9, where a priority vector is derived both by the eigenvalue (eigenvector) method (EV) and the geometric mean (GM) method. Our results suggest that the GM method and the EV method preserve the COP almost identically, both for the less inconsistent matrices (with Saaty’s consistency index below 0.10), and the more inconsistent matrices (Saaty’s consistency index equal to or greater than 0.10). Further, we find that the frequency of the COP violations grows (almost linearly) with the increasing inconsistency of MPCMs measured by Koczkodaj’s inconsistency index and Saaty’s consistency index, respectively, and we provide graphs to illustrate these relationships.

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.

Reducing inconsistency in AHP by combining Delphi and Nudge theory and network analysis of the judgements: an application to future scenarios

2021 ◽  
pp. 87-92
Author(s):  
Simone Di Zio
Keyword(s):  
Network Analysis ◽  
Geometric Mean ◽  
Pairwise Comparison ◽  
Future Scenarios ◽  
Analytic Hierarchy ◽  
Direct Graph ◽  
Decision Factors ◽  
Inconsistency Index ◽  
Textual Material ◽  
Hierarchy Process

The Analytic Hierarchy Process (AHP) is a Multi-Criteria method in which a number of decision factors (typically criteria and alternatives) are compared pairwise by one or more experts, using the Saaty scale, with the goal of sorting the alternatives (Saaty, 1977; 1980). For group AHP the Delphi method can be used in parallel with the AHP (Di Zio and Maretti, 2014), and this allows the search for a consensus on each pairwise judgement. A big issue of the AHP regards the inconsistency of the pairwise comparison matrices and here we propose a new method to reduce the inconsistency. As a solution we exploit the Nudge theory (Thaler and Sunstein, 2008) and from the second round of the Delphi survey, we calculate and circulate a Nudge to “gentle push” the experts towards more consistent evaluations. Furthermore, we propose the representation of the AHP matrices through graphs. In a direct graph two nodes are linked with two direct and weighted edges (or one edge with the direction based on the weights), where the weights indicate the evaluation given by an expert or, for a group, the geometric mean of the judgements. This type of visualization facilitates the reading of the results and could also be used as real-time feedback in the Delphi process, by displaying on the edges also a measure of variability. An application is proposed, on the evaluation of four future scenarios on the regulation of genetic modification experiments, assessed by a panel of 27 experts according to different criteria (plausibility, consistency and simplicity). The application demonstrated that it is possible to: a) reduce the inconsistency; b) collect useful textual material which enrich the AHP itself; c) use the inconsistency index as a stopping criterion for the Delphi rounds; d) display the pairwise comparison matrices with graphs.

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.

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.

scholarly journals AHP PRIORITIES AND MARKOV-CHAPMAN-KOLMOGOROV STEADY-STATES PROBABILITIES

2015 ◽  
Vol 7 (2) ◽  
Author(s):  
Stan Lipovetsky
Keyword(s):  
Steady State ◽  
General Solution ◽  
Stochastic Modeling ◽  
Pairwise Comparison ◽  
System Of Equations ◽  
Pairwise Comparison Matrix ◽  
Pair Comparison ◽  
Priority Vector ◽  
Discrete States ◽  
Steady State Probabilities

<div class="MsoTitle" style="margin: 12pt 0in 15pt;"><p>An AHP matrix of the quotients of the pair comparison priorities is transformed to a matrix of shares of the preferences which can be used in Markov stochastic modeling via the Chapman-Kolmogorov system of equations for the discrete states. It yields a general solution and the steady-state probabilities. The AHP priority vector can be interpreted as these probabilities belonging to the discrete states corresponding to the compared items. The results of stochastic modeling correspond to robust estimations of priority vectors not prone to influence of possible errors among the elements of a pairwise comparison matrix.</p></div><div class="MsoTitle" style="margin: 12pt 0in 15pt;"> </div>

Studies of the Biting-habits of African Mosquitos. An Appraisal of Methods employed, with special Reference to the twenty-four-hour Catch

1954 ◽  
Vol 45 (1) ◽  
pp. 199-242 ◽  
Author(s):  
A. J. Haddow
Keyword(s):  
Vertical Distribution ◽  
Special Reference ◽  
Geometric Mean ◽  
Seasonal Abundance ◽  
Central Tendency ◽  
Valuable Method ◽  
Different Seasons ◽  
Working Out ◽  
Different Levels ◽  
Biting Behaviour

It is felt that the 24-hour catch presents a valuable method of studying the biting-behaviour of mosquitos in the field, and further that conclusions concerning seasonal abundance, vertical distribution and times of biting-activity must be based on catches of this type if serious errors of interpretation are to be avoided.The time-divisions of the catch should not exceed one hour, and even shorter intervals may be desirable. Timing should be related to the actual times of sunrise and sunset. Where possible, a series of consecutive catches should be carried out, and a shift system must be carefully considered in relation to the particular series projected.Usually the results from different levels above ground should be treated separately in working out biting-cycles, and this may also apply to results from different seasons or different localities. Further, while in some instances the summation of long series of catches seems permissible, in others it is not, and in these detailed analysis of the figures may be necessary before consistent behaviour-patterns become apparent.In cases where a measure of the central tendency is to be employed, the geometric mean as modified by C. B. Williams appears to be the most suitable.

scholarly journals A Multi-Criteria Decision Framework Considering Different Levels of Decision-Maker Involvement to Reconfigure Manufacturing Systems

Machines ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 8
Author(s):  
Mohammed M. Mabkhot ◽  
Saber Darmoul ◽  
Ali M. Al-Samhan ◽  
Ahmed Badwelan
Keyword(s):  
Manufacturing Systems ◽  
Evaluation Criteria ◽  
System Capacity ◽  
Reconfigurable Manufacturing ◽  
New Production ◽  
Industrial Case Study ◽  
Set Up ◽  
Quantitative Indicators ◽  
Weighting Methods ◽  
Different Levels

Reconfigurable Manufacturing Systems (RMSs) rely on a set of technologies to quickly adapt the manufacturing system capacity and/or functionality to meet unexpected disturbances, such as fluctuation/uncertainty of demand and/or unavailability/unreliability of resources. At the operational stage, such disturbances raise new production requirements and risks, which call upon Decision-Makers (DMs) to analyze the opportunity to move from a running configuration to another more competitive one. Such a decision is generally based on an evaluation of a multitude of criteria, and several multi-criteria decision-making (MCDM) approaches have been suggested to help DMs with the reconfiguration process. Most existing MCDM approaches require some assignment of weights to the criteria, which is not a trivial task. Unfortunately, existing studies on MCDM for an RMS have not provided guidelines to weigh the evaluation criteria. This article fills in this gap by offering a framework to set up such weights. We provide a comprehensive set of quantitative indicators to evaluate the reconfiguration decisions during the operation of the RMS. We suggest three weighting methods that are convenient to different levels of DM expertise and desired degree of involvement in the reconfiguration process. These weighting methods are based on (1) intuitive weighting, (2) revised Simos procedural weighting combined with the Technique for Order of Preferences by Similarity to Ideal Solution (TOPSIS), and (3) DM independent weighting using ELECTRE IV. The implementation of the suggested framework and a comparison of the suggested methods carried out on an industrial case study are described herein.

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