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1

Note on traces of matrix products involving inverses of positive definite ones

Grzybowski A. Z.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 1

29--36

EN

This short note is devoted to the analysis of the trace of a product of two matrices in the case where one of them is the inverse of a given positive definite matrix while the other is nonnegative definite. In particular, a relation between the trace of A-1 H and the values of diagonal elements of the original matrix A is analysed.

2

New optimization based method for estimating priority weights

Grzybowski A. Z.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 1

33--44

EN

The estimation of priority vectors from pairwise comparison matrices is a core of the Analytic Hierarchy Process. Perhaps the most popular approach for deriving the priority weights is the right eigenvalue method (EM). Despite its popularity, various shortcomings of the EM have been described in literature. In this paper a new method for deriving priority vectors is proposed. This method makes use of the idea underlying the EM but in difference to the latter, the new one is optimization based. Important features of this new technique are studied via computer simulations and illustrated by some numerical examples.

3

Optimization methods in modeling the mechanical properties of heavy steel plates

Grzybowski A. Z.

Archives of Metallurgy and Materials

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2012

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Vol. 57, iss. 4

971-979

EN

The paper is devoted to an optimization approach to a problem of statistical modeling of mechanical properties of heavy steel plates during a real industrial manufacturing process. The approach enables the manufacturer to attain a specific set of the final product properties by optimizing the alloying composition within the grade specifications. Because this composition has to stay in the agreement with earlier indicated specifications, it leads to the large system of linear constraints, and the problem itself can be expressed in the form of linear programming (LP) task. It turns out however, that certain of the constraints contain the coefficients which have to be estimated on the base of the data gathered in the production process and as such they are uncertain. Consequently, the initial optimization task should be modeled as so-called Chance Constrained Programming problem (CCP), which is a special class within the stochastic programming problems. The paper presents mathematical models of the optimization problem that result from both approaches and indicates differences which are important for the decision makers in the production practice. Some examples illustrating the differences in solutions resulting from LP and CCP models are presented as well. Although the statistical analysis presented in this paper is based on the data gathered in the ISD Czestochowa Steelworks, the proposed approach can be adopted in any other process of steel production.

PL

Praca poświęcona jest optymalizacyjnemu podejściu do modelowania wybranych własności mechanicznych blach grubych w trakcie rzeczywistego procesu produkcyjnego. Proponowane podejście umozliwia producentowi uzyskanie ustalonego zbioru własności blach stalowych z jednoczesna optymalizacja ich składu chemicznego w zakresie określonym poprzez normy definiujące dany gatunek. Poniewaz ów skład musi pozostawać w zgodzie z wczesniej wskazanymi w zamówieniu specyfikacjami prowadzi to do dużego układu liniowych ograniczeń, a sam problem można wyrazic w postaci zadania programowani liniowego. W praktyce okazuje się, że pewne z tych ograniczen zawierają współczynniki, które musza być oszacowane na podstawie danych zebranych w procesie produkcyjnym i jako takie ich wartości sa niepewne. W konsekwencji wyjściowy problem powinien zostać sformułowany jako zadanie programowania stochastycznego. Wpracy przedstawia sie modele optymalizacyjne wynikajace z obu podejsc i wskazuje na różnice wazne z punktu widzenia podejmowania decyzji w praktyce produkcyjnej. Prezentowane są równiez przykłady ilustrujace różnice w rozwiązaniach uzyskiwanych z pomoca tych dwóch modeli. Chociaż analiza statystyczna przedstawiona w pracy oparta jest na danych zebranych w ISD Huta Częstochowa, to proponowane podejście może być zaadaptowane w dowolnym innym procesie produkcji stali.

4

Simulation approach to optimal stopping in some blackjack type problems

Grzybowski A. Z.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2011

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Vol. 10, nr 2

75-86

EN

In the paper, an unbounded blackjack type optimal stopping problem is considered. A decision maker (DM) observes sequentially the values of an infinite sequence of nonnegative random variables. After each observation, the DM decides whether to stop or to continue. If the DM decides to stop at a given moment, the obtains a payoff dependent on the sum of already observed values. The greater the sum, the more the DM gains, unless the sum exceeds a given positive number. If so, the decision maker loses all or part of the payoff. It turns out that under some elementary assumptions the optimal stopping rule (OSR) for such a problem has a very simple, so-called threshold form. However, even in very simple cases, the value of the problem has no closed analytical form. Therefore, it is very hard to evaluate the value directly. Thus, in order to find the relationship between the problem design parameters and the value of the problem, is proposed studying the relation via Monte Carlo simulations combined with regression analysis The same approach is adopted to examine the OSR risk characteristics.

5

Goal programming approach for deriving priority vectors : some new ideas

Grzybowski A. Z.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2010

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Vol. 9, nr 1

17-27

EN

The generation of priority vectors from pairwise comparison matrices is an essential part of the Analytic Hierarchy Process. Apart from the well-known Saaty's right eigenvalue method various other procedures have been proposed for priority modelling. Two most important alternative approaches are the statistical estimation techniques and methods based on constrained optimization models. In the paper a new goal programming model for deriving priority vectors and for measurement of consistency is proposed. In this approach the idea of goal programming is combined with the idea of Saaty's eigenvalue method. Some features of the method are studied via computer simulations.

6

On some new method of incorporation prior information along with its uncertainty in regression estimation

Grzybowski A. Z.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2006

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Vol. 5, nr 1

23-31

EN

The paper is devoted to the problem of incorporating prior information in the regression estimation. In series of papers, see [3-6], we have proposed and analyzed some model of uncertainty which allow incorporating prior information along with its uncertainty via some Bayes estimators. We also introduced the notion of an Index of Uncertainty (IU) which indicate how useful the information and consequently the proposed estimators are. The results and methodology are summarized in [7]. Here, assuming different than in the mentioned papers prior knowledge about the regression problems, we propose a new description of uncertainty along with an index of uncertainty which was developed on the base of computer simulation.