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Algorithms solving the Internet shopping optimization problem with price discounts

Musial J., Pecero J. E., Lopez-Loces M. C., Fraire-Huacuja H. J., Bouvry P., Blazewicz J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 3

505--516

EN

The Internet shopping optimization problem arises when a customer aims to purchase a list of goods from a set of web-stores with a minimum total cost. This problem is NP-hard in the strong sense. We are interested in solving the Internet shopping optimization problem with additional delivery costs associated to the web-stores where the goods are bought. It is of interest to extend the model including price discounts of goods. The aim of this paper is to present a set of optimization algorithms to solve the problem. Our purpose is to find a compromise solution between computational time and results close to the optimum value. The performance of the set of algorithms is evaluated through simulations using real world data collected from 32 web-stores. The quality of the results provided by the set of algorithms is compared to the optimal solutions for small-size instances of the problem. The optimization algorithms are also evaluated regarding scalability when the size of the instances increases. The set of results revealed that the algorithms are able to compute good quality solutions close to the optimum in a reasonable time with very good scalability demonstrating their practicability.

2

Generalized method of Lie-algebraic discrete approximations for solving Cauchy problems with evolution equation

Kindybaliuk A.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 2

51--62

EN

We consider solving the Cauchy problem with an abstract linear evolution equation by means of the Generalized Method of Lie-algebraic discrete approximations. Discretization of the equation is performed by all variables in equation and leads to a factorial rate of convergence if Lagrange interpolation is used for building quasi representation of differential operator. The rank of a finite dimensional operator and approximation properties have been determined. Error estimations and the factorial rate of convergence have been proved.

3

Aproksymacja jednostajna odwzorowań kartograficznych

Pędzich P.

Roczniki Geomatyki

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2009

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T. 7, z. 4

67-76

PL

W artykule zaprezentowane zostaną własności wielomianów Czebyszewa oraz ich zastosowanie do aproksymacji odwzorowań kartograficznych. Ponadto przedstawione zostanie porównanie wyników aproksymacji jednostajnej oraz aproksymacji średniokwadratowej odwzorowań kartograficznych.

EN

Usually the least square method is used for approximation of map projection. Determination of polynomial coefficients requires solution of a complicated system of equations. It is possible to avoid such a problem using orthogonal Chebyshev polynomials. This is a completely different method of approximation, where the maximum difference between the value of the function and the value calculated from polynomial is minimized. In the paper, properties of Chebyshev polynomials approximations are presented as wall as their application to map projection approximation and comparison with other methods of map projection.

4

On the Kantorovich variant of generalized Bernstein type rational functions

Gupta V., Ispir N.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

117-130

EN

In the present paper we define Kantorovich variant of generalized Bernstein type rational functions. We establish the order of approximation for continuous functions in different normed spaces and also estimate the rate of convergence for functions of bounded variation.

5

Properties of transient states in electrical circuits : transformation of F(s) in to the Maclaurin series f(t)

Piwowarczyk T.

Czasopismo Techniczne. Elektrotechnika

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2005

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R. 102, z. 5-E

45--52

EN

Transients in linear electrical circuits can be expressed as functions f(t), or as their Laplace transform F(s). The relation between them takes the form of a well known integral transform. The Laplace transform F(s) can also be computed from some finite subset of derivatives of f(t). In this approach the appropriate formulas take the form of the matrix transformation. It contains four arithmetic operations exclusively. Thus it is easy to be implemented in the assembly language. It also delivers a new interpretation of transients. A set of interesting theorems has been proved in the article. One of theorems states that some approximation of f(t) can contain all the information about a transient which is coded in a special way. This unexpected fact provides a basis for a new method of finding transients in linear electrical circuits.

PL

Stany nieustalone w liniowych obwodach elektrycznych mogą być wyrażone jako transformaty Laplace'a F(s) oryginału f(t). Jeżeli funkcja F(s) jest dana, to oryginał f(t) jest obliczany z zastosowaniem odwrotnej transformacji Laplace'a. W tym celu należy znać wszystkie zera mianownika wy.miernej funkcji F(s). Dla wyższych stopni funkcji F(s) rozkład na ułamki proste staje się poważnym problemem ze względu na twierdzenie Abela. W szczególności jest to zasadniczy problem, gdy elementy obwodu elektrycznego są podane w formie parametrycznej, tak jak to jest w przypadku syntezy obwodu. Możemy ominąć ten problem używając uogólnionych ciągów Fibonacciego. Co więcej, można do obliczeń użyć pewnego skończonego podciągu będącego częścią uogólnionego ciągu Fibonacciego bez straty informacji potrzebnej do odtworzenia dokładnego rozwiązania. Takie skończone podciągi tworzą podstawę nowej teorii rozwiązywania i analizowania stanów nieustalonych w obwodach elektrycznych.