arXiv subjects

Sándor Bozóki

Publications and source records attributed to Sándor Bozóki.

25 records · Page 2Linked to original sources

arXiv2019

Eccentric pie charts and an unusual pie cutting

The eccentric pie chart, a generalization of the traditional pie chart is introduced. An arbitrary point is fixed within the circle and rays are drawn from it. A sector is bounded by a pair of neighboring rays and the arc between them, The sector's area, aimed to be equal to a given proportion, is calculated from some well known equations in coordinate geometry. The resulting system of polynomial and trigonometric equations can be approximated by a fully polynomial system, once the non-polynomial functions are approximated by their Taylor series written up to the first few terms. The roots of the polynomial system have been found by the homotopy continuation method, then used as starting points of a Newton iteration for the original (non-polynomial) system. The method is illustrated on a special pie cutting problem, and is applicable to a wide class of nonlinear, non-polynomial systems.

math.MG↗

arXiv2018

Efficiency analysis of double perturbed pairwise comparison matrices

Efficiency is a core concept of multi-objective optimization problems and multi-attribute decision making. In the case of pairwise comparison matrices a weight vector is called efficient if the approximations of the elements of the pairwise comparison matrix made by the ratios of the weights cannot be improved in any position without making it worse in some other position. A pairwise comparison matrix is called double perturbed if it can be made consistent by altering two elements and their reciprocals. The most frequently used weighting method, the eigenvector method is analyzed in the paper, and it is shown that it produces an efficient weight vector for double perturbed pairwise comparison matrices.

math.OC↗

arXiv2017

Efficient weight vectors from pairwise comparison matrices

Pairwise comparison matrices are frequently applied in multi-criteria decision making. A weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix, and strictly better in at least one position. A weight vector is weakly efficient if the pairwise ratios cannot be improved in all non-diagonal positions. We show that the principal eigenvector is always weakly efficient, but numerical examples show that it can be inefficient. The linear programs proposed test whether a given weight vector is (weakly) efficient, and in case of (strong) inefficiency, an efficient (strongly) dominating weight vector is calculated. The proposed algorithms are implemented in Pairwise Comparison Matrix Calculator, available at pcmc.online.

math.OC↗

arXiv2017

Seven mutually touching infinite cylinders

We solve a problem of Littlewood: there exist seven infinite circular cylinders of unit radius which mutually touch each other. In fact, we exhibit two such sets of cylinders. Our approach is algebraic and uses symbolic and numerical computational techniques. We consider a system of polynomial equations describing the position of the axes of the cylinders in the 3 dimensional space. To have the same number of equations (namely 20) as the number of variables, the angle of the first two cylinders is fixed to 90 degrees, and a small family of direction vectors is left out of consideration. Homotopy continuation method has been applied to solve the system. The number of paths is about 121 billion, it is hopeless to follow them all. However, after checking 80 million paths, two solutions are found. Their validity, i.e., the existence of exact real solutions close to the approximate solutions at hand, was verified with the alphaCertified method as well as by the interval Krawczyk method.

math.MG↗

arXiv2016

An application of incomplete pairwise comparison matrices for ranking top tennis players

Pairwise comparison is an important tool in multi-attribute decision making. Pairwise comparison matrices (PCM) have been applied for ranking criteria and for scoring alternatives according to a given criterion. Our paper presents a special application of incomplete PCMs: ranking of professional tennis players based on their results against each other. The selected 25 players have been on the top of the ATP rankings for a shorter or longer period in the last 40 years. Some of them have never met on the court. One of the aims of the paper is to provide ranking of the selected players, however, the analysis of incomplete pairwise comparison matrices is also in the focus. The eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete PCMs. In our results the top three players of four decades were Nadal, Federer and Sampras. Some questions have been raised on the properties of incomplete PCMs and remains open for further investigation.

cs.AI↗

arXiv2016

Efficiency analysis of simple perturbed pairwise comparison matrices

Efficiency, the basic concept of multi-objective optimization is investigated for the class of pairwise comparison matrices. A weight vector is called efficient if no alternative weight vector exists such that every pairwise ratio of the latter's components is at least as close to the corresponding element of the pairwise comparison matrix as the one of the former's components is, and the latter's approximation is strictly better in at least one position. A pairwise comparison matrix is called simple perturbed if it differs from a consistent pairwise comparison matrix in one element and its reciprocal. One of the classical weighting methods, the eigenvector method is analyzed. It is shown in the paper that the principal right eigenvector of a simple perturbed pairwise comparison matrix is efficient.

math.OC↗

arXiv2013

On reducing inconsistency of pairwise comparison matrices below an acceptance threshold

A recent work of the authors on the analysis of pairwise comparison matrices that can be made consistent by the modification of a few elements is continued and extended. Inconsistency indices are defined for indicating the overall quality of a pairwise comparison matrix. It is expected that serious contradictions in the matrix imply high inconsistency and vice versa. However, in the 35-year history of the applications of pairwise comparison matrices, only one of the indices, namely CR proposed by Saaty, has been associated to a general level of acceptance, by the well known ten percent rule. In the paper, we consider a wide class of inconsistency indices, including CR, CM proposed by Koczkodaj and Duszak and CI by Peláez and Lamata. Assume that a threshold of acceptable inconsistency is given (for CR it can be 0.1). The aim is to find the minimal number of matrix elements, the appropriate modification of which makes the matrix acceptable. On the other hand, given the maximal number of modifiable matrix elements, the aim is to find the minimal level of inconsistency that can be achieved. In both cases the solution is derived from a nonlinear mixed-integer optimization problem. Results are applicable in decision support systems that allow real time interaction with the decision maker in order to review pairwise comparison matrices.

math.OC↗