In decision-making methods, it is common to assume that the experts are honest and professional. However, this is not the case when one or more experts in the pairwise-based group decision-making framework, such as the group analytic hierarchy process, try to manipulate results in their favor. This paper aims to introduce two heuristics enabling detection of manipulators and minimizing their effect on the group consensus by diminishing their weights. The first heuristic is based on the assumption that manipulators will provide judgments that can be considered outliers with respect to those of the other experts in the group. The second heuristic assumes that dishonest judgments are less consistent than the average consistency of the group. Both approaches are illustrated with numerical examples and simulations.

Abstract The role of an expert in the decision-making process is crucial. If we ask an expert to help us to make a decision we assume their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is for an expert to provide manipulated data in a given case of decision-making process becomes essential. In the presented work, we consider manipulation of a ranking obtained by the Geometric Mean Method applied to a pairwise comparisons matrix. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives in a ranking. We also define a new index which measures how difficult such manipulation is in a given case.

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced. The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.

We define a new inconsistency index for tournament matrices and two tournaments-based indices for multiplicative pairwise comparison matrices. We compare them with the index defined by Kulakowski and illustrate the fundamental differences on examples.

This article analyzes the question of reliability of entries collected in the pairwise comparison matrices used for prioritization of alternatives. The inappropriate input data may result in an invalid output hierarchy. Basing on the Geometric Mean Method introduced by Crawford and Williams and the geometric consistency index defined by Aguarón and Moreno-Jiménez we introduce an algorithm which allows to search for the least reliable pairwise comparison matrix elements and improve them by means of their estimations.

The method of pairwise comparisons is widely applied in the decision making process. The inconsistency of data may significantly affect the final result. Since the notion of consistency is based on triads or cycles, there is a great need for defining the measure of a triad or cycle inconsistency. In the paper a set of properties of a good cycle inconsistency index is proposed. Two construction methods of a cycle-based inconsistency index for a pairwise comparisons matrix are introduced. All is supported by the examples.

This study has been inspired by numerous requests for clarification from researchers who often confuse Saaty’s Analytic Hierarchy Process (AHP) with the pairwise comparisons (PC) method, taking AHP as the only representation of PC. This study should be regarded as an interpretation and clarificatio n of past investigations of PC. In addition, this article is a reflection on general PC research at a higher level of abstraction: the philosophy of science. It delves into the foundations and implications of pairwise comparisons. Some results of this study are based on a recently published work by Koczkodaj and Szwarc. Finally, open problems have also been reported for future research.

This study presents theoretical proof and empirical evidence of the reduction algorithm convergence for the distance-based inconsistency in pairwise comparisons. Our empirical research shows that the convergence very quick. It usually takes less than 10 reductions to bring the inconsistency of the pairwise comparisons matrix below the assumed threshold of 1/3 (sufficient for most applications). We believe that this is the first Monte Carlo study demonstrating such results for the convergence speed of inconsistency reduction in pairwise comparisons.

Pairwise comparisons are widely recognized method supporting decision making process based on the subjective judgments. The key to this method is the notion of inconsistency that has a significant impact on the reliability of results. Inconsistency is expressed by means of inconsistency indices. Depending on their construction, such indices may pay attention to different aspects of the set of pairwise comparisons. The family of indices proposed in this article tries to combine the advantages coming from different indices, thereby increases the expressiveness of the family elements. The newly introduced notion of equivalence can help in comparing the indices and identifying their common properties.

Organization of a tender is not easy. Preparation of the relevant specification, taking into account the non-price criteria, implementation of the objective and fair assessment procedure, and last but not least, selecting a satisfactory offer are in practice a considerable challenge. In meeting this challenge appropriate multi-criteria assessment models can help. Models that can cope with different kinds of tangible and intangible criteria. The paper presents the hierarchical bid assessment (HBA) model of making decision in a tender procedure based on the pairwise comparisons method. It combines structural elements known from AHP with the Heuristic Rating Estimation approach. Two different schemes of rating tangible and intangible attributes are proposed. The notion of the success of the customer is defined and the practical method for its use is proposed. Theoretical considerations are illustrated in the relevant example.