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.

This paper examines the differences in ordinal rankings obtained from a pairwise comparison matrix using the eigenvalue method and the geometric mean method. First, we introduce several propositions on the (dis)similarity of both rankings concerning the matrix size and its inconsistency expressed by the Koczkodaj's inconsistency index. Further on, we examine the relationship between differences in both rankings and Kendall's rank correlation coefficient τ and Spearman's rank coefficient ρ. Apart from theoretical results, intuitive numerical examples and Monte Carlo simulations are also provided.

We propose parametrized families of algorithms for approximating inconsistent matrices with consistent ones. We present experimental results, and optimize distance versus performance.

Weights or weighted attributes are a part of most measurement, indexing and classification techniques.However, when judgments are subjective; weight assignment, and especially weight consistency, is almost always problematic.A ranking or preference is usually defined as a weakly ordered relationship between a set of items such that, for any two items, the first is either "less preferred", "more preferred" or "indifferent" to the second one.While most existing methods involve numbers, in many cases using only qualitative assessments might be more trustworthy.Formulas and rules involving numbers are considered more scientific and credible than those that involve qualitative values only.This is obviously true when the notions of interest can be measured directly or indirectly, as for instance velocity, height, voltage, pressure etc.However, when it comes to subjective notions as love, importance, taste, beauty, etc., we have to be very careful when numbers are used.One of the ways to deal with such intangible concepts is the pairwise comparisons method.This method is based on the observation that it is much easier to judge the mutual relationship (preference, importance, intensity, etc.) of two objects than to do this for several objects at once.This special issue of Fundamenta Informaticae is devoted to different aspects of the pairwise comparisons method.It is comprised of fourteen excellent articles that present the phenomenon of pairwise comparisons from various perspectives.The work, "Continuous Pairwise Comparisons" written by Thomas Saaty definitely goes far beyond currently ongoing discussions and opens up new horizons for researchers.In the article he proposes changing perspective from a discrete to a continuous one.The suggested solution is to determine the rankings for continuous pairwise comparisons based on solving Fredholm's integral equation of the second kind.In "Complex Ranking Procedures" the authors Barbara Sandrasagra and Michael Soltys investigate pairwise ranking problems where relatively few items are to be ranked with a complex procedure and according to a large number of criteria.They discuss their solutions in the context of tender procedures.Andrew Schumann and Jan Woleński enrich the discussion on pairwise comparisons methods by presenting their logical approach enclosed in the article "Two Squares of Oppositions and Their Applications in Pairwise Comparisons Analysis".

The pairwise comparisons method can be used when the relative order of preferences among different concepts (alternatives) needs to be determined. There are several popular implementations of this method, including the Eigenvector Method, the Least Squares Method, the Chi Squares Method and others. Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision. This article explores the relationship between inconsistency of input and error of output. An error describes to what extent the obtained results correspond to the single expert’s assessments. On the basis of the inconsistency and the error, two properties of the weight deriving procedure are formulated. These properties are proven for eigenvector method and Koczkodaj’s inconsistency index. Several estimates using Koczkodaj’s inconsistency index for a principal eigenvalue, Saaty’s inconsistency index and the Condition of Order Preservation are also provided.

Heuristic rating estimation is a newly proposed method that supports decisions analysis based on the use of pairwise comparisons. It allows the ranking values of some alternatives (herein referred to as concepts) to be initially known, whilst ranks for other concepts have yet to be estimated. To calculate the missing ranks it is assumed that the priority of every single concept can be determined as the weighted arithmetic mean of the priorities of all the other concepts. It has been shown that the problem has an admissible solution if the inconsistency of the pairwise comparisons is not too high. The proposed approach adopts heuristics according to which a weighted geometric mean is used to determine the missing priorities. In this approach, despite increased complexity, a solution always exists and its existence does not depend on the inconsistency or reciprocity of the input matrix. Thus, the presented approach might be appropriate for a larger number of problems than previous methods. Moreover, it turns out that the geometric approach, as proposed in the article, can be optimal. The optimality condition is presented in the form of a corresponding theorem. A formal definition of the proposed geometric heuristics is accompanied by two numerical examples.

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.

SUMMARY Increasing demand for computationally efficient algorithms and processors has turned the attention of researchers toward parallel and concurrent solutions. Because the frequency of contemporary processors cannot be tweaked infinitely, the only hopes for squeezing more performance from computers are parallel processing and parallel computation. The important part of every parallel solution is concurrent data structures, which allow multithread programming environments to be taken advantage of. In this article, a new concurrent dynamic set structure is proposed. It is based on the van Emde Boas trees concept, where on every node of a tree, an array of the node's children is stored. The structure is equipped with a simple but effective locking algorithm, which allows it to be used concurrently by any number of threads. The presented algorithm idea is accompanied by an experimental implementation written in JAVA 6. Preliminary tests prove that, especially for moderately larger data sets with a predominance of read operations, the concurrent van Emde Boas array proposed in this article may be a viable alternative for other structures providing a similar functionality. Copyright © 2013 John Wiley & Sons, Ltd.

The Heuristic Ratio Estimation (HRE) approach proposes a new way of using the pairwise comparisons matrix. It allows the assumption that the weights of some alternatives (herein referred to as concepts) are known and fixed, hence the weight vector needs to be estimated only for the other unknown values. The main purpose of this paper is to extend the previously proposed iterative HRE algorithm and present all the heuristics that create a generalized approach. Theoretical considerations are accompanied by a few numerical examples demonstrating how the selected heuristics can be used in practice.