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Source author record

Konrad Kułakowski

Konrad Kułakowski appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Discrete MathematicsArtificial IntelligenceData Structures and AlgorithmsDistributed, Parallel, and Cluster Computingecon.EMInformation RetrievalLogic in Computer Sciencemath.STStatistics Theory

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Published work

11 published item(s)

preprint2026arXiv

Sufficient conditions for a Heuristic Rating Estimation Method application

A series of papers has introduced the Heuristic Rating Estimation method, which evaluates a set of alternatives based on pairwise comparisons and the weights of reference alternatives. We formulate the conditions under which the HRE method can be applied correctly. The research considers both arithmetic and geometric algorithms for complete and incomplete pairwise comparison methods. The illustrative examples show that the estimations of inconsistency in the arithmetic variant are optimal.

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preprint2022arXiv

Multiple-criteria Heuristic Rating Estimation

One of the most widespread multi-criteria decision-making methods is the Analytic Hierarchy Process (AHP). AHP successfully combines the pairwise comparisons method and the hierarchical approach. It allows the decision-maker to set priorities for all ranked alternatives. But what if, for some of them, their ranking value is known (e.g., it can be determined differently)? The Heuristic Rating Estimation (HRE) method proposed in 2014 tried to bring the answer to this question. However, the considerations were limited to a model that did not consider many criteria. In this work, we go a step further and analyze how HRE can be used as part of the AHP hierarchical framework. The theoretical considerations are accompanied by illustrative examples showing HRE as a multiple-criteria decision-making method.

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preprint2022arXiv

Some Notes on the Similarity of Priority Vectors Derived by the Eigenvalue Method and the Geometric Mean Method

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.

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preprint2019arXiv

Inconsistency indices for incomplete pairwise comparisons matrices

Comparing alternatives in pairs is a very well known technique of ranking creation. The answer to how reliable and trustworthy ranking is depends on the inconsistency of the data from which it was created. There are many indices used for determining the level of inconsistency among compared alternatives. Unfortunately, most of them assume that the set of comparisons is complete, i.e. every single alternative is compared to each other. This is not true and the ranking must sometimes be made based on incomplete data. In order to fill this gap, this work aims to adapt the selected twelve existing inconsistency indices for the purpose of analyzing incomplete data sets. The modified indices are subjected to Monte Carlo experiments. Those of them that achieved the best results in the experiments carried out are recommended for use in practice.

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preprint2016arXiv

Axioms of the Analytic Hierarchy Process (AHP) and its Generalization to Dependence and Feedback: The Analytic Network Process (ANP)

The AHP/ANP are multicriteria decision-making theories that deal with both hierarchic structures when the criteria are independent of the alternatives and with networks when there is any dependence within and between elements of the decision. Both of them have been repeatedly used in practice by various researchers and practitioners. From the perspective of almost 40 years of practice in solving problems using both theories, some of their properties seem to be more important than others. The article indicates four of them as fundamental for understanding AHP/ANP. These are the axioms related to structure, computation, and expectation. The mathematical formulation of the axioms is preceded by an introduction explaining the motivation behind the introduced concepts. The article is expository and it is an improved and refined version of the work [1].

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preprint2015arXiv

Dynamic concurrent van Emde Boas array

The growing popularity of shared-memory multiprocessor machines has caused significant changes in the design of concurrent software. In this approach, the concurrently running threads communicate and synchronize with each other through data structures in shared memory. Hence, the efficiency of these structures is essential for the performance of concurrent applications. The need to find new concurrent data structures prompted the author some time ago to propose the cvEB array modeled on the van Emde Boas Tree structure as a dynamic set alternative. This paper describes an improved version of that structure - the dcvEB array (Dynamic Concurrent van Emde Boas Array). One of the improvements involves memory usage optimization. This enhancement required the design of a tree which grows and shrinks at both: the top (root) and the bottom (leaves) level. Another enhancement concerns the successor (and predecessor) search strategy. The tests performed seem to confirm the high performance of the dcvEB array. They are especially visible when the range of keys is significantly larger than the number of elements in the collection.

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preprint2014arXiv

Concurrent bisimulation algorithm

The coarsest bisimulation-finding problem plays an important role in the formal analysis of concurrent systems. For example, solving this problem allows the behavior of different processes to be compared or specifications to be verified. Hence, in this paper an efficient concurrent bisimulation algorithm is presented. It is based on the sequential Paige and Tarjan algorithm and the concept of the state signatures. The original solution follows Hopcroft's principle "process the smaller half". The presented algorithm uses its generalized version "process all but the largest one" better suited for concurrent and parallel applications. The running time achieved is comparable with the best known sequential and concurrent solutions. At the end of the work, the results of tests carried out are presented. The question of the lower bound for the running time of the optimal algorithm is also discussed.

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preprint2014arXiv

Heuristic rating estimation - geometric approach

Heuristic Rating Estimation (HRE) is a newly proposed method supporting decisions analysis based on the use of pairwise comparisons. It allows that the ranking values of some alternatives (herein referred to as concepts) are initially known, whilst the ranks for the 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 priorities of all the other concepts. It has been shown that the problem has admissible solution if the inconsistency of pairwise comparisons is not too high. The proposed approach adopts the heuristics according to which to determine the missing priorities a weighted geometric mean is used. In this approach, despite an increased complexity, the solution always exists and their existence does not depend on the inconsistency of the input matrix. Thus, the presented approach might be appropriate for a larger number of problems than the previous method. The formal definition of the proposed geometric heuristics is accompanied by two numerical examples.

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preprint2014arXiv

Heuristic Rating Estimation Approach to The Pairwise Comparisons Method

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.

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preprint2014arXiv

Notes on discrepancy in the pairwise comparisons method

The pairwise comparisons method is a convenient tool used when the relative order among different concepts (alternatives) needs to be determined. One popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix. In such cases the ranking result the principal eigenvector of the pairwise comparison matrix is adopted, whilst the eigenvalue is used to determine the index of inconsistency. A lot of research has been devoted to the critical analysis of the eigenvalue based approach. One of them is the work (Bana e Costa and Vansnick, 2008). In their work authors define the conditions of order preservation (COP) and show that even for a sufficiently consistent pairwise comparisons matrices, this condition can not be met. The present work defines a more precise criteria for determining when the COP is met. To formulate the criteria a discrepancy factor is used describing how far the input to the ranking procedure is from the ranking result.

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preprint2014arXiv

Notes on the existence of solutions in the pairwise comparisons method using the Heuristic Rating Estimation approach

Pairwise comparisons are a well-known method for modelling of the subjective preferences of a decision maker. A popular implementation of the method is based on solving an eigenvalue problem for M - the matrix of pairwise comparisons. This does not take into account the actual values of preference. The Heuristic Rating Estimation (HRE) approach is a modification of this method in which allows modelling of the reference values. To determine the relative order of preferences is to solve a certain linear equation system defined by the matrix A and the constant term vector b (both derived from M). The article explores the properties of these equation systems. In particular, it is proven that for some small data inconsistency the A matrix is an M-matrix, hence the equation proposed by the HRE approach has a unique strictly positive solution.

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