Analysis of an Inventory System for Items with Stochastic Demand and Time Dependent Three-Parameter Weibull Deterioration Function

In recent times, mathematical models have been developed to describe various scenarios obtainable in the management of inventories. These models usually have as objective the minimizing of inventory costs. In this research work we propose a mathematical model of an inventory system with time-dependent three-parameter Weibull deterioration and a stochastic type demand in the form of a negative exponential distribution. Explicit expressions for the optimal values of the decision variables are obtained. Numerical examples are provided to illustrate the theoretical development.

Analysis of an Inventory System for Items with Price-Dependent Demand and Time Dependent Three-Parameter Weibull Deterioration Function

In this research work we propose a mathematical model of an inventory system with time dependent three-parameter Weibull deterioration and price-dependent demand rate. The model incorporates shortages and deteriorating items are considered in which inventory is depleted not only by demand but also by decay, such as, direct spoilage as in fruits, vegetables and food products, or deterioration as in obsolete electronic components. Furthermore, the rate of deterioration is taken to be time-proportional, and a power law form of the price dependence of demand is considered. This price-dependence of the demand function is nonlinear, and is such that when price of a commodity increases, demand decreases and when price of a commodity decreases, demand increases. The objective of the model is to minimize the total inventory costs. From the numerical example presented to illustrate the solution procedure of the model, we obtain meaningful results. We then proceed to perform sensitivity analysis of our model. The sensitivity analysis illustrates the extent to which the optimal solution of the model is affected by slight changes or errors in its input parameter values.

Solving Riccati-Type Nonlinear Differential Equations with Novel Artificial Neural Networks

In this study we investigate neural network solutions to nonlinear differential equations of Ricatti-type. We employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but avoid the standard back-propagation algorithm for updating the intrinsic weights. Our objective is to minimize an error, which is a function of the network parameters i.e., the weights and biases. Once the weights of the neural network are obtained by our systematic procedure, we need not adjust all the parameters in the network, as postulated by many researchers before us, in order to achieve convergence. We only need to fine-tune our biases which are fixed to lie in a certain given range, and convergence to a solution with an acceptable minimum error is achieved. This greatly reduces the computational complexity of the given problem. We provide two important ODE examples, the first is a Ricatti type differential equation to which the procedure is applied, and this gave us perfect agreement with the exact solution. The second example however provided us with only an acceptable approximation to the exact solution. Our novel artificial neural networks procedure has demonstrated quite clearly the function approximation capabilities of ANN in the solution of nonlinear differential equations of Ricatti type.

Application of <i>q</i>-Calculus to the Solution of Partial <i>q</i>-Differential Equations

We introduce the concept of q-calculus in quantum geometry. This involves the q-differential and q-integral operators. With these, we study the basic rules governing q-calculus as compared with the classical Newton-Leibnitz calculus, and obtain some important results. We introduce the reduced q-differential transform method (RqDTM) for solving partial q-differential equations. The solution is computed in the form of a convergent power series with easily computable coefficients. With the help of some test examples, we discover the effectiveness and performance of the proposed method and employing MathCAD 14 software for computation. It turns out that when q = 1, the solution coincides with that for the classical version of the given initial value problem. The results demonstrate that the RqDTM approach is quite efficient and convenient.

A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.