New results on the quality of recently introduced index for a consistency control of pairwise judgments (original) (raw)
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Journal of Applied Mathematics and Computational Mechanics, 2016
There is a theory which meets a prescription of the efficient and effective multicriteria decision making support system called the Analytic Hierarchy Process (AHP). It seems to be the most widely used approach in the world today, as well as the most validated methodology for decision making. The consistency measurement of human judgments appears to be the crucial problem in this concept. This research paper redefines the idea of the triad's consistency within the pairwise comparison matrix (PCM) and proposes a few seminal indices for PCM consistency measurement. The quality of new propositions is then studied with application of computer simulations coded and run in Wolfram Mathematica 9.0.
Annals of Operations Research, 2016
A review of contemporary literature devoted to decision making support systems draws attention to the Analytic Hierarchy Process (AHP). At the core of the AHP are various prioritization procedures which elicit priorities for alternative solutions for complex decisional problems. Certainly, the procedures coincide when decision makers' preferences of alternative solutions are cardinally transitive, otherwise the results differ. This is why consistency measurement of human judgments is so important. It has been scientifically proven that a high inconsistency of decision makers' preferences concerning alternative solutions of decisional problems may lead to fallacious choices. Research verifies the thesis that consistency measures derived from different prioritization procedures are interrelated. It turns out that one of the independent consistency measures is extremely closely related to the consistency index embedded in original approach of the AHP. The main objective of this study is realized through the novel and sophisticated simulation algorithm designed for the AHP and executed within its exemplary decisional framework for three levels. The outcome of the research proves that consistency can be measured in various ways, but recently devised concepts can indicate better solutions as a result of significantly improved methodology.
Consistency Indices in Analytic Hierarchy Process: A Review
Mathematics
A well-regarded as well as powerful method named the ‘analytic hierarchy process’ (AHP) uses mathematics and psychology for making and analysing complex decisions. This article aims to present a brief review of the consistency measure of the judgments in AHP. Judgments should not be random or illogical. Several researchers have developed different consistency measures to identify the rationality of judgments. This article summarises the consistency measures which have been proposed so far in the literature. Moreover, this paper describes briefly the functional relationships established in the literature among the well-known consistency indices. At last, some thoughtful research directions that can be helpful in further research to develop and improve the performance of AHP are provided as well.
Acceptable consistency of aggregated comparison matrices in analytic hierarchy process
European Journal of Operational Research, 2012
The analytic hierarchy process (AHP) is a method for solving multiple criteria decision problems, as well as group decision making. The weighted geometric mean method (WGMM) is appropriate when aggregation of individual judgements (AIJ) is used. This paper presents a new proof which confirms the theorem that if the comparison matrices of all decision makers are of acceptable consistency, then the weighted geometric mean complex judgement matrix (WGMCJM) also is of acceptable consistency. This theorem was presented and first proved by Xu (2000), but Lin et al. (2008) rejected the proof. We also discuss under what conditions the WGMCJM is of acceptable consistency when not all comparison matrices of decision makers are of acceptable consistency. For this case we determine the upper bound for the consistency ratio of WGMCJM and provide numerical examples.
Improving Consistency of Comparison Matrices in Analytical Hierarchy Process
2013
In the field of decision-making, the concept of priority is archetypal and how priorities are derived influence the choices one makes. Priorities should not only be unique but should also reflect the dominance of the order expressed in the judgments of pair wise comparison matrix. In addition, judgments are much more sensitive and responsive to small perturbations. They are highly related to the notion of consistency of a pair wise comparison matrix si mply because when dealing with intangibles, if one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision. This paper endeavors to accomplish nearly consistent matrices in pair wise comparisons by subsiding the effects of hypothetical decisions made by the decision makers. The proposed methodology efficiently improves group decisions by incorporating corrective measures for inconsistent judgments.
Judgment Scales and Consistency Measure in AHP
Procedia Economics and Finance, 2014
The Analytic Hierarchy Process (AHP) is widely used method in multiple-attribute decision making. In the recent literature many authors used different judgment scales which influenced the results and decisions. In this paper the author reviews and discusses effects of utilization of various judgment scales on priority estimation in AHP. There has been studies that have been concerned with the comparison of judgment scales but there were no studies concerned with consistency measures that are needed. The goal of this paper is to compare and discuss the application of various judgment scales on the results in particular practical example that has been used in previous paper by Saaty (2003). Thus the focus of the paper is to analyze the impact of using different judgment scales on the resulting priorities and consistency to default scale as proposed by Saaty. Results suggest that judgment scales have a profound impact on criteria priorities but not on ranking of criteria. However, the consistency varies among applied judgment scales. Authors calculated the values of random index needed for calculation of the consistency index in AHP for all concerned scales. Based on them the consistency index was computed and compared. Both consistent and inconsistent Saaty matrices were used for comparison.
Decision making with the analytic hierarchy process
Decisions involve many intangibles that need to be traded off. To do that, they have to be measured along side tangibles whose measurements must also be evaluated as to, how well, they serve the objectives of the decision maker. The Analytic Hierarchy Process (AHP) is a theory of measurement through pairwise comparisons and relies on the judgements of experts to derive priority scales. It is these scales that measure intangibles in relative terms. The comparisons are made using a scale of absolute judgements that represents, how much more, one element dominates another with respect to a given attribute. The judgements may be inconsistent, and how to measure inconsistency and improve the judgements, when possible to obtain better consistency is a concern of the AHP. The derived priority scales are synthesised by multiplying them by the priority of their parent nodes and adding for all such nodes. An illustration is included.
Incorporating the uncertainty of decision judgements in the analytic hierarchy process
European Journal of Operational Research, 1991
The uncertainty in the relative weights of a pairwise comparison matrix in the Analytic Hierarchy Process (AHP) is caused by the uncertainty in our decision judgements and in many cases can not be avoided. In this paper, it is explicitly shown how such uncertainties can be incorporated within the framework of AHP and how the resulting uncertainties in the relative priorities of the decision alternatives can be computed. The required algorithm and the computational procedures are also developed and illustrated with examples. Uncertainty is introduced as a fundamental concept independent of the concept of consistency with a view to extend the AHP as a decision analysis procedure.
arXiv (Cornell University), 2017
An overview of current debates and contemporary research devoted to the modeling of decision making processes and their facilitation directs attention to the Analytic Hierarchy Process (AHP). At the core of the AHP are various prioritization procedures (PPs) and consistency measures (CMs) for a Pairwise Comparison Matrix (PCM) which, in a sense, reflects preferences of decision makers. Certainly, when judgments about these preferences are perfectly consistent (cardinally transitive), all PPs coincide and the quality of the priority ratios (PRs) estimation is exemplary. However, human judgments are very rarely consistent, thus the quality of PRs estimation may significantly vary. The scale of these variations depends on the applied PP and utilized CM for a PCM. This is why it is important to find out which PPs and which CMs for a PCM lead directly to an improvement of the PRs estimation accuracy. The main goal of this research is realized through the properly designed, coded and executed seminal and sophisticated simulation algorithms in Wolfram Mathematica 8.0. These research results convince that the embedded in the AHP and commonly applied, both genuine PP and CM for PCM may significantly deteriorate the quality of PRs estimation; however, solutions proposed in this paper can significantly improve the methodology.