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1

Opportunities for Making Production-Related Decisions on the basis of Shadow Prices

Fuksa Dariusz

Inżynieria Mineralna

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2019

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R. 21, nr 1

295--300

EN

This article describes how shadow prices can be used as active constraints (in this case constraints of mine production capacity) to address and support production-related decision-making. This is an algorithm from a post-optimal analysis developed by the author as part of a method for rationalising production decisions for a formal group (PGG, a company) of hard coal mines. Opportunities for using shadow prices are presented using examples of actual mines. The developed algorithm provides a quick way of obtaining information, with no need to solve the problem again, about possible gains or losses resulting from an increase or a decrease in a selected production limit, to determine how changes to such constraints will affect the profits and production and sales structures for specific coal sizes.

PL

W artykule zaprezentowano sposób wykorzystania cen dualnych aktywnych ograniczeń (w tym przypadku ograniczeń zdolności wydobywczych kopalń) do rozwiązywania i wspomagania decyzji produkcyjnych. Jest to jeden z algorytmów opracowanej przez autora analizy postoptymalnej opracowanej w ramach metody racjonalizacji decyzji produkcyjnych dla sformalizowanej grupy (spółki, PGG) kopalń węgla kamiennego. Możliwości wykorzystania cen dualnych przedstawiono na rzeczywistym przykładzie kopalń. Dzięki opracowanemu algorytmowi można szybko uzyskać informację, bez konieczności ponownego rozwiązywania zadania, co uzyskalibyśmy lub stracili zwiększając lub zmniejszając wybrany limit produkcji, i określić wpływ zmian tych ograniczeń na osiągany zysk i strukturę wielkości produkcji i sprzedaży poszczególnych sortymentów węgla.

2

A general iterative solver for unbalanced inconsistent transportation problems

Carp D., Popa C., Serban C.

Archives of Transport

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2016

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Vol. 37, iss. 1

7--13

EN

The transportation problem, as a particular case of a linear programme, has probably the highest relative frequency with which appears in applications. At least in its classical formulation, it involves demands and supplies. When, for practical reasons, the total demand cannot satisfy the total supply, the problem becomes unbalanced and inconsistent, and must be reformulated as e.g. finding a least squares solution of an inconsistent system of linear inequalities. A general iterative solver for this class of problems has been proposed by S. P. Han in his 1980 original paper. The drawback of Han’s algorithm consists in the fact that it uses in each iteration the computation of the Moore-Penrose pseudoinverse numerical solution of a subsystem of the initial one, which for bigger dimensions can cause serious computational troubles. In order to overcome these difficulties we propose in this paper a general projection-based minimal norm solution approximant to be used within Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Numerical experiments and comparisons on some inconsistent transport model problems are presented.

3

Optymalizacja procesu produkcyjnego z wykorzystaniem Octave 3.4.3

Szkutnik J., Ziółkowski J.

Logistyka

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2015

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nr 4

6025--6032, CD 2

PL

W artykule opisano przykład praktycznego zastosowania programowania liniowego przy wykorzystaniu Notepad ++ oraz Octave 3.4.3. W oparciu o dane liczbowe przedstawiono zalety wykorzystywania języka programowania dla optymalizacji procesu produkcyjnego. Cel pracy, jakim było zaprezentowanie efektywności zastosowania nowoczesnego oprogramowania komputerowego oraz zagadnienia programowania liniowego z matematycznego punku widzenia został osiągnięty.

EN

The article describes practical application of linear programming with using Notepad ++ and Octave 3.4.3. The paper shows advantages of programming language on basis of the figures for optimization of production process. The main aim of this article, which was to present efficiency of modern computer software and linear programming from the mathematical point of view, has been achieved. The calculations shows that the lowest computational complexity can be achieved using Octave 3.4.3 environment, however traditional calculation method is more time – consuming.

4

Investigation of effectiveness of α-constrained simplex method applied to design of optimal induction motors

Dąbrowski M., Rudeński A.

Computer Applications in Electrical Engineering

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2010

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Vol. 8

61--71

EN

The paper presents a modified nonlinear simplex algorithm with lexicographic order comparison of solutions and its application to the design of optimal induction motors. In the comparison of the solutions generated in the optimization process, both the objective function value and the additional parameter, called the satisfaction level of constraints, have been taken into account. The comparison method assigns some advantage degree to feasible solutions, thus allows for the control of this advantage degree during the optimization process. Special attention has been paid to the choice of the algorithm parameters and to the kind of the mutation operator. The presented algorithm has been implemented in the object-oriented software. Calculation results of the selected double-cage induction motors have been compared with the results obtained with the evolution strategy (μ+λ)-ES and with the hybrid algorithm assembled with the modified Price algorithm. An additional calculation experiment allows for the comparison of exploitation properties between the α-Constrained Simplex Method and the Modified Price Algorithm. As the investigations showed, the presented algorithm can be successively used for the optimization of the induction motors, however, with constraints, which are not very restrictive concerning respective functional parameters.

5

A linear programming based analysis of the CP-rank of completely positive matrices

Li Y., Kummert A., Frommer A.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 1

25-31

EN

A real matrix A is said to be completely positive (CP) if it can be decomposed as A= B BT, where the real matrix B has exclusively non-negative entries. Let k be the rank of A and Phik the least possible number of columns of the matrix B, the so-called completely positive rank (cp-rank) of A. The present work is devoted to a study of a general upper bound for the cp-rank of an arbitrary completely positive matrix A and its dependence on the ordinary rank k. This general upper bound of the cp-rank has been proved to be at most k(k + 1)/2. In a recent pioneering work of Barioli and Berman it was slightly reduced by one, which means that Phik \leq k(k + 1)/2-1 holds for k \geq 2. An alternative constructive proof of the same result is given in the present paper based on the properties of the simplex algorithm known from linear programming. Our proof illuminates complete positivity from a different point of view. Discussions concerning dual cones are not needed here. In addition to that, the proof is of constructive nature, i.e. starting from an arbitrary decomposition A= B1 B1T (B1\geq 0) a new decomposition A= B2 B2T (B2\geq 0) can be generated in a constructive manner, where the number of column vectors of B2 does not exceed k(k + 1)/2-1. This algorithm is based mainly on the well-known techniques stemming from linear programming, where the pivot step of the simplex algorithm plays a key role.

6

Obliczanie namiaru wsadu metalowego z wykorzystaniem algorytmu simpleks

Piątkowski J., Szymszal J., Binczyk F.

Archiwum Odlewnictwa

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2002

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R. 2, nr 6

179-188

PL

Uniwersalną metodą rozwiązywania zagadnień optymalizacyjnych z zastosowaniem programowania liniowego jest algorytm simpleks. Jest to nowoczesny sposób badań operacyjnych, polegający na poszukiwaniu kolejnych rozwiązań bazowych programowania liniowego w postaci kanonicznej. W pracy przedstawiono metodę wyznaczania namiaru wsadu metalowego opartą o optymalizację simpleks, pozwalającą na określenie położenia ekstremum funkcji wieIu zmiennych. Obliczenia prowadzono na przykladzie nadeutektycznego siluminu AISi17 z dodatkiem Cu, Ni i Mg.

EN

Simplex algorithm is universal method of dissolving of questions optimisation with utilization of linear programmes. It is this modern way of operating investigations, depending on search of next base solutions of linear programme in canonical form. In work metal leaning of marking of bearing of batch method was introduced about optimisation simplex, permitting onto qualification of position of extreme variable function. Calculation hypereutectic AISi17 alloy with addition Cu, Ni and Mg alloys was led on example.

7

Zastosowanie symulowanego wyżarzania do optymalizacji aparatów i procesów

Jeżowski J., Poplewski G., Słoma R.

Inżynieria Chemiczna i Procesowa

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2001

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T. 22, z. 3C

573--578

PL

Opracowano metodę optymalizacji według strategii symulowanego wyżarzania (SW) opartą na propozycji z [1, 2]. Polega ona na wbudowaniu algorytmu simpleksów w ramy podstawowych koncepcji SW. Opracowana metoda umożliwia rozwiązywanie problemów nieliniowych z ograniczeniami równościowymi i nierównościowymi. W tej pracy przedstawiono przykłady zastosowania do rozwiązania dwóch zagadnień z zakresu inżynierii chemicznej.

EN

Simulated annealing (SA) optimisation method has developed based on the proposition from [1,2]. The approach relies on embedding simplex algorithm into basic concepts of SA. The optimisation method can be applied for solving non-linear problems with equality and inequality constraints. Examples of applications are presented for two chemical engineering problems.