On Inconsistency of a Pairwise Comparison Matrix (original) (raw)
Related papers
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.
A New Approach to Improve Inconsistency in the Analytical Hierarchy Process
pvamu.edu
In this paper, a new approach based on the generalized Purcell method for solving a system of homogenous linear equations is applied to improve near consistent judgment matrices. The proposed method relies on altering the components of the pairwise comparison matrix in such a way that the resulting sequences of improved matrices approach a consistent matrix. The complexity of the proposed method, together with examples, shows less cost and better results in computation than the methods in practice.
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.
Removing Inconsistency in Pairwise Comparisons Matrix in the AHP
Multiple Criteria Decision Making
The Analytic Hierarchy Process (AHP) allows to create a final ranking for a discrete set of decision variants on the basis of an earlier pairwise comparison of all the criteria and all the decision variants within each criterion. The properties of the obtained ranking depend on the quality of pairwise comparisons; this quality can be evaluated on the basis of consistency measured by means of certain measures. The paper discusses a mathematical model which is the foundation of the AHP and a starting point for a new method which allows to significantly reduce-and even eliminate-the inconsistency of pairwise comparisons measured by the consistency index. The proposed method allows to reduce the consistency index well below the threshold of 0.1.
Journal of Intelligent & Fuzzy Systems
Pairwise comparison matrix (PCM) with crisp or fuzzy elements should satisfy consistency requirements when it is used in analytic hierarchy process (AHP) or in fuzzy AHP methodologies. An algorithm has been presented to obtain a new modified consistent PCM for the corresponding inconsistent original one. The algorithm sets a linear programming problem based on all of the constraints. To obtain the optimum eigenvector of the middle value of the new PCM, segment tree is used to gradually approach the greatest lower bound of distance with the original PCM. As to obtain the lower value and upper value of the new PCM, a theory is proposed to reduce adding uncertainty factors and could maximum maintain the similarity with original PCM. The experiments for crisp elements show that the proposed approach can preserve more the original information than references. The experiments for fuzzy elements show that our method can effectively reduce inconsistency and obtain suitable modified fuzzy PCMs.
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.
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.
Applied Mathematics & Information Sciences, 2014
The paper discusses the problem of performing the prioritization of decision elements within the multicriteria optimization method, analytic hierarchy process (AHP), with incomplete information. An approach is proposed on how to fill in the gap in the pair-wise comparison matrix generated within an AHP standard procedure; that is, to reproduce one missing judgment of the decision maker while assuring the reproduced judgment belongs to the same ratio scale used while other judgments are elicited. The first-level transitivity rule (FLTR) approach is proposed based on screening matrix entries in the neighborhood of a missing one. Scaling (where necessary) and geometric averaging of screened entries allows filling of the gap in the matrix and later prioritization of involved decision elements by the eigenvector, or any other known method. Illustrative examples are provided to compare the proposed method with the other two known methods also aimed to fill-in gaps in AHP matrices. The results indicate some similarities in attaining consistency. However, unlike other methods, the FLTR assures coherency of the generating process in a sense that all numeric values in a matrix (original entries, plus one generated) come from the same ratio scale and have correct element-wise semantic equivalents.
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.