A Note on the Sampling Distribution of the Information Content of the Priority Vector of a Consistent Pairwise Comparison Judgment Matrix of AHP

2000 ◽  
Vol 51 (2) ◽  
pp. 237
Author(s):  
V. M. Rao Tummala ◽  
H. Ling
Keyword(s):  
Information Content ◽  
Pairwise Comparison ◽  
Sampling Distribution ◽  
Priority Vector ◽  
Comparison Judgment ◽  
Judgment Matrix

scholarly journals AN INTERPRETATION OF THE AHP EIGENVECTOR SOLUTION FOR THE LAY PERSON

2010 ◽  
Vol 2 (2) ◽  
Author(s):  
Stan Lipovetsky
Keyword(s):  
Decision Making ◽  
Linear Algebra ◽  
Iterative Process ◽  
Pairwise Comparison ◽  
Forthcoming Paper ◽  
Pairwise Comparison Matrix ◽  
Priority Vector ◽  
Comparison Matrix ◽  
Judgment Matrix

<p class="MsoBodyText" style="margin: 0in 0in 0pt;">An AHP priority vector represents the importance, preference, or likelihood of its elements with respect to a certain property or criterion and here we examine how that priority vector can be derived through an iterative process applied to the pairwise comparison matrix. Further, we show that the vector obtained in this way satisfies the definition for an eigenvector of the original judgment matrix. Practical managers using AHP in decision making would most likely be better able to appreciate this approach than they would a process phrased in the language of linear algebra. The overall priority vector for the alternatives in a hierarchy and, further, in a network, can be obtained in the same way by applying the iterative process to the supermatrix of the ANP. This claim is examined in depth in a forthcoming paper that will appear in this journal.</p><p class="MsoBodyText" style="margin: 0in 0in 0pt;">http://dx.doi.org/10.13033/ijahp.v2i2.42</p>

scholarly journals Words from the Managing Editor

2010 ◽  
Vol 2 (2) ◽  
Author(s):  
Rozann Saaty
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Analytic Hierarchy ◽  
Comparison Judgment ◽  
The Subject ◽  
Judgment Matrix ◽  
The Analytic Hierarchy Process ◽  
Hierarchy Process

<p><span>At the ISAHP 2009 meeting in Pittsburgh, Thomas L. Saaty offered a challenge to the attendees to explain the eigenvector solution of the pairwise comparison judgment matrix that is the basis of the Analytic Hierarchy Process so that it would be understandable to any intelligent layman. He wrote an essay himself on it during the meeting, and submitted it as his offering. A second essay on the subject, also during the meeting, was written by Stan Lipovetsky. These two papers appear in this issue of the journal. </span></p><p><span>Rozann Saaty</span></p><p><span>http://dx.doi.org/10.13033/ijahp.v2i2.86<br /></span></p>

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>

FARSJUM, A FUZZY SYSTEM FOR RANKING SPARSE JUDGMENT MATRICES: A CASE STUDY IN SOCCER TOURNAMENTS

2007 ◽  
Vol 15 (01) ◽  
pp. 115-129 ◽  
Author(s):  
M. R. GHOLAMIAN ◽  
S. M. T. FATEMI GHOMI ◽  
M. GHAZANFARI
Keyword(s):  
Pairwise Comparison ◽  
Major Constituent ◽  
Pairwise Comparisons ◽  
Ahp Method ◽  
World Cup ◽  
Analytic Hierarchy ◽  
Rank Reversal ◽  
Judgment Matrix ◽  
Hierarchy Process

The establishment of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). However, the number of pairwise comparisons necessary in real problems often becomes overwhelming. In such cases, generally the experts are not able to answer all questions and consequently sparse judgment matrix is generated which caused "equal ranks" and "rank reversal" based on AHP method. In this paper, a new ranking system (FARSJUM) is developed for such sparse judgment matrix. The system is constructed on fuzzy rules and fuzzy reasoning methods. The numerical example of world cup soccer tournament is brought to clarify the performance of the developed system comparing with AHP method in ranking the sparse judgment matrices.

Study on the Difficulty Sorting of Experimental Projects Based on the AHP

2011 ◽  
Vol 271-273 ◽  
pp. 363-367
Author(s):  
Lian Hui Liu ◽  
Juan He
Keyword(s):  
Analytic Hierarchy Process ◽  
Ahp Method ◽  
Analytic Hierarchy ◽  
Experimental Instruction ◽  
Comparison Judgment ◽  
Judgment Matrix ◽  
The Analytic Hierarchy Process ◽  
Set Up ◽  
Hierarchy Process

To classify the difficulty of the experiments is the basis of the effective mutlit-level experimental instruction. In this paper, we employ the analytic hierarchy process (AHP) method to compare and analyse the difficult of the experiments, and set up a corresponding comparison judgment matrix to the difficulty of the projects. We obtain some sortings of difficulty of experiments, which coincide with the fact. Our studies offer an effective way to the rational sorting of difficulty of experiments.

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 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.

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