Supply Chain Networking Models Under Fuzzy Uncertainty

In decision-making model, the techniques of numerical analysis have been widely adopted. It is rare for someone to solve a linear program by hand — except perhaps in a class-room. Large-scale simulations would be all but impossible without the aid of a computer. For many people, numerical techniques have superseded analytic techniques as a tool for solving mathematical problems. This paper proposed Generalized LUExponential Trapezoidal Fuzzy Number and their ranking based on numerical integration. In this ranking method, the values are calculated with left and right spreads at some 𝜶 −level of generalized LU-exponential trapezoidal fuzzy numbers and Weddle‘s rule for numerical integration. To illustrate the proposed methods, a fuzzy four dimensional transportation problem (FDTP) is proposed and solved. This ranking approach is very simple and useful for the real life inequality based decision making problems.

Investigation of error in evaluation of spectral statistical characteristics of numerical solution of stochastic dynamic system

In this paper, an algorithm for numerical simulations is developed for calculating a discrete dynamic system with a stochastic perturbation and an analysis of the quality of numerical solutions is carried out. For this, an algorithm for the numerical solution of a second-order differential equation with a stochastic right-hand side was developed and this algorithm was implemented as a program. The next step was to carry out a set of computational studies by varying the parameters of numerical integration with the subsequent assessment of their impact on the error and accuracy of simulations. To estimate the spectral density, the Welch periodogram method was used. To check the quality of simulations and assess the accuracy of solutions, it is proposed to compare the results of numerical integration and subsequent digital processing with analytical solutions that are known for the linear problem, given by the equation. As a result of the work, a comparative analysis of the dispersion of displacements relative to the lengths of signals from a different number of blocks was carried out, into which the signal is divided for the Welch method; the confidence interval of the error at different signal lengths and the confidence interval of the error with a different number of blocks at a certain signal length. Comparison of the variance with a different number of blocks showed that with a signal length of 30 s and from 90 s, there is a slight scatter of the variance values within an error of ± 5%.

Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction. Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

Accurate Quaternion Polar Harmonic Transform for Color Image Analysis

Polar harmonic transforms (PHTs) have been applied in pattern recognition and image analysis. But the current computational framework of PHTs has two main demerits. First, some significant color information may be lost during color image processing in conventional methods because they are based on RGB decomposition or graying. Second, PHTs are influenced by geometric errors and numerical integration errors, which can be seen from image reconstruction errors. This paper presents a novel computational framework of quaternion polar harmonic transforms (QPHTs), namely, accurate QPHTs (AQPHTs). First, to holistically handle color images, quaternion-based PHTs are introduced by using the algebra of quaternions. Second, the Gaussian numerical integration is adopted for geometric and numerical error reduction. When compared with CNNs (convolutional neural networks)-based methods (i.e., VGG16) on the Oxford5K dataset, our AQPHT achieves better performance of scaling invariant representation. Moreover, when evaluated on standard image retrieval benchmarks, our AQPHT using smaller dimension of feature vector achieves comparable results with CNNs-based methods and outperforms the hand craft-based methods by 9.6% w.r.t mAP on the Holidays dataset.