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A Pairwise Comparison Matrix Framework for Large-Scale Decision MakingJanuary 2013 (has links)
abstract: A Pairwise Comparison Matrix (PCM) is used to compute for relative priorities of criteria or alternatives and are integral components of widely applied decision making tools: the Analytic Hierarchy Process (AHP) and its generalized form, the Analytic Network Process (ANP). However, a PCM suffers from several issues limiting its application to large-scale decision problems, specifically: (1) to the curse of dimensionality, that is, a large number of pairwise comparisons need to be elicited from a decision maker (DM), (2) inconsistent and (3) imprecise preferences maybe obtained due to the limited cognitive power of DMs. This dissertation proposes a PCM Framework for Large-Scale Decisions to address these limitations in three phases as follows. The first phase proposes a binary integer program (BIP) to intelligently decompose a PCM into several mutually exclusive subsets using interdependence scores. As a result, the number of pairwise comparisons is reduced and the consistency of the PCM is improved. Since the subsets are disjoint, the most independent pivot element is identified to connect all subsets. This is done to derive the global weights of the elements from the original PCM. The proposed BIP is applied to both AHP and ANP methodologies. However, it is noted that the optimal number of subsets is provided subjectively by the DM and hence is subject to biases and judgement errors. The second phase proposes a trade-off PCM decomposition methodology to decompose a PCM into a number of optimally identified subsets. A BIP is proposed to balance the: (1) time savings by reducing pairwise comparisons, the level of PCM inconsistency, and (2) the accuracy of the weights. The proposed methodology is applied to the AHP to demonstrate its advantages and is compared to established methodologies. In the third phase, a beta distribution is proposed to generalize a wide variety of imprecise pairwise comparison distributions via a method of moments methodology. A Non-Linear Programming model is then developed that calculates PCM element weights which maximizes the preferences of the DM as well as minimizes the inconsistency simultaneously. Comparison experiments are conducted using datasets collected from literature to validate the proposed methodology. / Dissertation/Thesis / Ph.D. Industrial Engineering 2013
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INCOMPLETE PAIRWISE COMPARISON MATRICES AND OPTIMIZATION TECHNIQUESTekile, Hailemariam Abebe 08 May 2023 (has links)
Pairwise comparison matrices (PCMs) play a key role in multi-criteria decision making, especially in the analytic hierarchy process. It could be necessary for an expert to compare alternatives based on various criteria. However, for a variety of reasons, such as lack of time or insufficient knowledge, it may happen that the expert cannot provide judgments on all pairs of alternatives. In this case, an incomplete pairwise comparison matrix is formed. In the first research part, an optimization algorithm is proposed for the optimal completion of an incomplete PCM. It is intended to numerically minimize a constrained eigenvalue problem, in which the objective function is difficult to write explicitly in terms of variables. Numerical simulations are carried out to examine the performance of the algorithm. The simulation results show that the proposed algorithm is capable of solving the minimization of the constrained eigenvalue problem. In the second part, a comparative analysis of eleven completion methods is studied. The similarity of the eleven completion methods is analyzed on the basis of numerical simulations and hierarchical clustering. Numerical simulations are performed for PCMs of different orders considering various numbers of missing
comparisons. The results suggest the existence of a cluster of five extremely similar methods, and a method significantly dissimilar from all the others. In the third part, the filling in patterns (arrangements of known comparisons) of incomplete PCMs based on their graph representation are investigated under given conditions: regularity, diameter and number of vertices, but without prior information. Regular and quasi-regular graphs with minimal diameter are proposed. Finally, the simulation results indicate that the proposed graphs indeed provide better weight vectors than alternative graphs with the same number of comparisons. This research problem’s contributions include a list of (quasi-)regular graphs with diameters of 2 and 3, and vertices from 5 up to 24.
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[en] THE AHP - CONCEPTUAL REVIEW AND PROPOSAL OF SIMPLIFICATION / [pt] O MÉTODO AHP - REVISÃO CONCEITUAL E PROPOSTA DE SIMPLIFICAÇÃOCRISTINA SANTOS WOLFF 27 October 2008 (has links)
[pt] Muitos problemas de transportes, assim como de outras áreas
do
conhecimento, envolvem tomada de decisão. Em decisões
complexas, a escolha
da melhor alternativa ou plano de ação pode envolver mais
de um critério e é
necessário estudar como cada ação afeta cada critério. O
método AHP, Analytic
Hierarchy Process, proposto por Thomas L. Saaty, é um
método de decisão
multicriterial que funciona para os mais diversos tipos de
decisões, solucionando
problemas com fatores quantitativos e qualitativos. Ele
reúne a opinião dos
tomadores de decisão em matrizes de comparação. Este
trabalho faz uma revisão
geral de conceitos básicos do método, mostrando diferentes
maneiras de cálculo
da solução. A primeira explorada é o cálculo exato através
dos autovalores e
autovetores das matrizes. Para esse cálculo, foi utilizado
o software francês
Scilab, semelhante ao mais conhecido Matlab, mas distibuído
gratuitamente na
internet. É discutida a questão da consistência dos
julgamentos, com maneiras de
medi-la e melhorá-la. Finalmente, é feita uma proposta de
solução aproximada,
que questiona a idéia original de que um certo nível de
inconsistência é desejável.
É uma solução simplificada que, supondo consistência
absoluta, facilita não só os
cálculos como o trabalho inicial dos tomadores de decisão.
Em vez de comparar
todas as alternativas com as outras, duas a duas, passa a
ser necessário comparar
apenas uma alternativa com as outras. A nova solução
aproximada é comparada
com a solução exata em três casos retirados da literatura. / [en] Several transportation problems, as well as problems in
other knowledge
areas, request decision making. In complex decisions, the
choice of best
alternative or course of action can contain more than one
criterion and it is
necessary to study how each alternative affects each
criterion. The AHP, Analytic
Hierarchy Process, proposed by Thomas L. Saaty, is a
multicriteria decision
method that works well for very diverse decision types,
solving problems with
tangible and intangible factors. It gathers the opinion of
decision makers in
comparison matrices. This study makes a general review of
basic concepts of the
method, showing different manners of calculating the
solution. The first one to be
displayed is the exact solution using the eigenvalues and
eigenvectors of the
matrices. For this solution the French software Scilab was
used, which is similar
to the well-known Matlab, but free and distributed on the
web. The issue of
judgment consistency is discussed, including ways of
measuring and improving it.
Finally, a proposal of approximated solution is made,
questioning the original idea
which says that a certain level of inconsistency is
desirable. It is a simplification
that, considering absolute consistency, facilitates not
only the calculations but also
the early work of decision makers when judging the
alternatives. Instead of
making pair wise comparisons of all alternatives with each
other, it becomes
necessary to compare only one alternative with the others.
The new approximated
solution is compared to the real solution in three cases
taken from the literature.
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