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Optimisation of the stock structure of a single stock item taking into account stock quantity constraints, using a Lagrange multiplier

Krzyżaniak Stanisław

LogForum

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2022

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Vol. 18, no. 2

261--269

EN

Background: Optimisation in the area of stock management is most often performed in relation to cycle stock. The classic example here is the Harris-Wilson formula for calculating the economic order quantity. Often these models are not subject to any constraints imposed on the optimised quantities. However, in practice, taking such constraints into account is important. The application of the so-called Lagrange multiplier is helpful here, but the examples of its application usually refer to the multi-position sets of stock items (e.g. the search for the optimum structure of stock of material groups in the case of capital constraints). This paper attempts to optimise the structure of the stock (cycle stock vs. safety stock) for a single stock item. Methods: To achieve the objective of determining the optimum stock structure for the various conditions under which stock replenishment is implemented, a general model has been built, a component of which is a Lagrange function containing the constraint conditions for the solution. Next, this model has been implemented in the form of an EXCEL spreadsheet application. Results: The result of solving the optimization task based on the proposed model is a system of equations, the solution of which (with the help of the EXCEL application) allows to determine the optimum value of the Lagrange multiplier, on the basis of which the components of the inventory structure and other related quantities (service level indicators and costs, such as stock replenishment, stock maintenance and stock deficit costs) are calculated. This has been illustrated using a fictitious example, which at the same time made it possible to observe certain general relationships between the adopted constraints and the recorded quantities. Conclusions: Two types of conclusions can be presented. The first type concerns the approach itself. The possibility of determining the optimum structure of the stock (cycle stock vs. safety stock) depending on various values characterising the adopted stock replenishment system as well as the adopted limitations has been demonstrated. The second type of conclusions results from the presented example of application of the method for the assumed ranges of changes of selected quantities.

2

Tropical cyclone detection in South Pacifc and Atlantic coastal area using optical fow estimation and RESNET deep learning model

Malothu Nagu, Prasad Varre Venkata Kanaka Durga Vara, Krishna Battula Tirumala

Acta Geophysica

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2022

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Vol. 70, no 6

2855--2871

EN

Tropical cyclones (TC) are among the worst natural disasters, that cause massive damage to property and lives. The meteorologists track these natural phenomena using Satellite imagery. The spiral rain bands appear in a cyclic pattern with an eye as a center in the satellite image. Automatic identification of the cyclic pattern is a challenging task due to the clouds present around the structure. Conventional approaches use only image data to detect the cyclic structure using deep learning algorithms. The training and testing data consist of positive and negative samples of TC. But the cyclic structure’s texture pattern makes it difficult for the deep learning algorithms to extract useful features. This paper presents an automatic TC detection algorithm using optical flow estimation and deep learning algorithms to overcome this draw-back. The optical flow vectors are estimated using the Horn-Schunck estimator, the Liu-Shen estimator, and the Lagrange multiplier. The deep learning algorithms take the optical flow vectors as input during the training stage and extract the features to identify the cyclone’s circular pattern. The software used for experimental analysis is MATLAB 2021a. The proposed method increases the accuracy of detecting the cyclone pattern through optical flow vectors compared to using the pixel intensity values. By using proposed method 98% of accuracy will be achieved when compared with the existing methods.

3

Distance between two Keplerian orbits

Mohamed Ayman Homda, Dwidar Hany Ramdan, Adham Inal, Bakry Abd-Elazaz, El-Raffie Ahmed

Artificial Satellites : Journal of Planetary Geodesy

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2021

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Vol. 56, No. 3

57--77

EN

In this paper, constrained minimization for the point of closest approach of two conic sections is developed. For this development, we considered the nine cases of possible conics, namely, (elliptic-elliptic), (elliptic-parabolic), (elliptic–hyperbolic), (paraboli-elliptic), (paraboli-parabolic), (parabolic–hyperbolic), (hyperbolic-elliptic), (hyperbolic-parabolic), and (hyperbolic–hyperbolic). The developments are considered from two points of view, namely, analytical and computational. For the analytical developments, the literal expression of the minimum distance equation (S) and the constraint equation (G), including the first and second derivatives for each case, are established. For the computational developments, we construct an efficient algorithm for calculating the minimum distance by using the Lagrange multiplier method under the constraint on time. Finally, we compute the closest distance S between two conics for some orbits. The accuracy of the solutions was checked under the conditions that L|solution ≤ε1G||solution ≤ε2, where ε1,2 < 10 -10. For the cases of (paraboli-parabolic), (paraboli-hyperbolic), and (hyperbolic-hyperbolic), we studied thousands of comets, but the condition of the closest approach was not met.

4

The Lagrange multiplier and the stationary Stokes equations

Ożański W. S.

Journal of Applied Analysis

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2017

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Vol. 23, nr 2

137--140

EN

We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure p appearing in the stationary Stokes equations is the Lagrange multiplier of the constraint div u = 0.

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Zastosowanie formalizmu mnożników Lagrange'a w analizie drgań poprzecznych belek o zmiennym przekroju z elementami dyskretnymi

Posiadała B., Wilczak R.

Górnictwo Odkrywkowe

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2006

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R. 48, nr 7-8

99-102

PL

Artykuł dotyczy zagadnienia swobodnych drgań poprzecznych belek o skokowo-zmiennym polu przekroju poprzecznego w połączeniu z dodatkowymi elementami dyskretnymi typu: sprężyna transakcyjna i rotacyjna, oscylator harmoniczny, masa skupiona oraz dodatkowe zamocowania. Do sformułowania i rozwiązania zagadnienia wykorzystano formalizm mnożników Lagrange'a.

EN

The modeling and solution of the free lateral vibration problem of stepped beam with additional discrete like: translational and rotational spring, harmonic oscillator, mass and fix elements, has been presented. The procedure has been based on the Lagrange multiplier formalism. Additionally, the presented paper contains the numerical example illustrating the use of the derived expressions.

6

Free longitudinal vibrations of step like rods with additional discrete elements

Posiadała B., Wilczak R.

Vibrations in Physical Systems

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2004

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Vol. 21

311--314