arXiv subjects

Konrad Kułakowski

Publications and source records attributed to Konrad Kułakowski.

27 records · Page 2Linked to original sources

arXiv2017

Inconsistency in the ordinal pairwise comparisons method with and without ties

Comparing alternatives in pairs is a well-known method of ranking creation. Experts are asked to perform a series of binary comparisons and then, using mathematical methods, the final ranking is prepared. As experts conduct the individual assessments, they may not always be consistent. The level of inconsistency among individual assessments is widely accepted as a measure of the ranking quality. The higher the ranking quality, the greater its credibility. One way to determine the level of inconsistency among the paired comparisons is to calculate the value of the inconsistency index. One of the earliest and most widespread inconsistency indexes is the consistency coefficient defined by Kendall and Babington Smith. In their work, the authors consider binary pairwise comparisons, i.e., those where the result of an individual comparison can only be: better or worse. The presented work extends the Kendall and Babington Smith index to sets of paired comparisons with ties. Hence, this extension allows the decision makers to determine the inconsistency for sets of paired comparisons, where the result may also be "equal." The article contains a definition and analysis of the most inconsistent set of pairwise comparisons with and without ties. It is also shown that the most inconsistent set of pairwise comparisons with ties represents a special case of the more general set cover problem.

cs.DM↗

arXiv2016

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].

cs.DM↗

arXiv2015

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.

cs.DS↗

arXiv2014

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.

cs.DM↗

arXiv2014

On the Properties of the Priority Deriving Procedure in the Pairwise Comparisons Method

The pairwise comparisons method is a convenient tool 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 discrepancy of output. A global ranking discrepancy describes to what extent the obtained results correspond to the single expert's assessments. On the basis of the inconsistency and discrepancy indices, 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.

cs.DM↗

arXiv2014

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.

cs.DM↗

arXiv2014

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.

cs.DM↗

arXiv2014

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.

cs.DM↗

arXiv2014

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.

cs.LO↗