Interval Differential Equations Research Articles

Comments on “An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand”

Comments on “An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand”

Impact of Carbon Emissions and Advance Payment on Optimal Decisions for Perishable Products via Parametric Approach of Interval

Under the heading of sustainability challenges, this study incorporates payment procedures and inventory selections. The article’s goal is to acquire insight into how the methods of payment affect perishable product inventory selections under the widely utilised carbon tax regime in view of policies to reduce emissions. Uncertainty arises as a natural consequence of the unpredictable behaviour of customers. Keeping these impacts in mind, an interval-valued inventory model is introduced, where all the related inventory parameters are considered interval-valued. In addition, the vendor offers an interval-valued discount rate to customers against advance payment. Due to the inventory parameters being chosen as interval-valued, the objective function is changed into an interval-valued form, and the corresponding differential equation is changed into an interval differential equation. To solve the interval-valued differential equation, a parametric approach to interval is introduced, and the corresponding interval-valued objective function is constructed. Interval order relations and the MATHEMATICA software are used to solve the objective function with interval values. To assess the validity of the proposed model, one numerical example is solved, and a sensitivity analysis of the ideal course of action is performed.

Study of dynamical behaviors of harvested stage-structured predator–prey fishery model with fear effect on prey under interval uncertainty

The lack of clarity in many parameters used in ecological models is due to the constant changes in nature. The fundamental predator–prey model discussed in this article takes into account several ecological parameters as parametric-functional interval numbers. It discusses the dynamic characteristics of the model within the context of an uncertain environment. To account for this uncertainty, the model incorporates the concepts of bionomic equilibria and maximum sustainable yield (MSY). The article explores the optimal harvesting policy by applying Pontryagin’s maximum principle in an imprecise and uncertain environment. The proposed imprecise model formulation includes two additional variables: the influence of fear on prey population growth rate and the harvesting of prey and juvenile predator populations. The system of interval differential equations governs the interactions between these imprecise species, allowing for mathematical analysis of the proposed system. The aim of this study is to evaluate the robustness of an imprecise prey–predator model within an unpredictable environment. The article concludes by presenting rigorous numerical simulations that support all the analytical findings.

Interval valued inventory model for deterioration, carbon emissions and selling price dependent demand considering buy now and pay later facility

Uncertainty refers to a lack of precise knowledge or information about a particular event, situation, or outcome. It is an inherent characteristic of many real-world phenomena and is often associated with the presence of risk or ambiguity. Uncertainty can arise due to various factors, such as incomplete information, unpredictability, complexity, or variability of a system or process. In many cases, uncertainty can lead to difficulties in decision-making and pose challenges in planning, forecasting, and managing risks. There are different types of uncertainty, including epistemic uncertainty, which arises from the inherent randomness or variability of a system or process. This study encompasses the inclusion of all inventory parameters as uncertainty parameters, which are expressed in the form of intervals. Furthermore, demand for the product is taken into account in interval form. The conversion of the differential equation into an interval differential equation is carried out in order to account for the variability in demand within a certain interval. The utilisation of the centre-radius technique leads to the derivation of the related optimisation problem under a ‘buy now, pay later’ (BNPL) payment scheme. In order to demonstrate the concept, a numerical case is selected and solved with the MATHEMATICA software and interval order relations. In the end, a comprehensive sensitivity analysis has been conducted to derive insightful conclusions pertaining to this study.

New quantum integral inequalities for left and right log-ℏ-convex interval-valued functions

Abstract We introduce the concept of quantum integration for interval-valued functions and establish new q-Hermite–Hadamard and q-Hermite–Hadamard–Fejér inequalities for left and right log ⁢ - ⁢ h {\mathrm{log}\text{-}h} -convex interval-valued functions. Our results generalize the known ones in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and interval differential equations. We illustrate our results with examples.

Inventory model for perishable items for interval-valued price dependent demand and advance payment policy with parametric approach via particle swarm optimization

ABSTRACT Advance payment for replenishment and the lifetime of an item are the most vital issues in the business sector. With the consideration of these issues, the formulation of a non-deterministic inventory model in the interval environment is of great importance. In this work, two different non-deterministic inventory models are formulated along with a variable holding cost, variable demand, and advertisement frequency under an advance payment policy in the interval environment. In the first inventory model, a no-shortage situation is discussed, whereas in the second model, a stock-out scenario is adopted by partially backordering during modelling. The modelling framework for both problems is accomplished based on interval differential equations. The interval optimization problems related to both models are obtained utilizing parametric approach and interval mathematics. Then, using interval order relations and different types of quantum particle swarm optimization, both interval optimization problems are solved. By means of two examples, the authentication of both models is examined and the benefits of the proposed approach are demonstrated. The impact of deviations of various parameters on optimality is studied graphically through the performance of sensitivity analyses, and the work is concluded by providing potential management insights and future scopes.

Transformation of Mathematical Model for Complex Object in Form of Interval Difference Equations to a Differential Equation

Mathematical models of complex objects in the form of interval difference equations are built on the basis of the obtained experimental interval data within the limits of the inductive approach. At the same time, interpretation of physical properties of the object on the base of such model is complex enough. A method of transformation of a mathematical model in the form of interval differential equations was proposed in the article. The proposed method is based on the formulas for representing the values of the function at the node of the difference grid in the Taylor series in the neighborhood of the base node, as well as the differential representation of the derivatives in the same neighborhood. The developed approach creates opportunities for the identification of interval models of complex objects based on the analysis of interval data with further interpretation of the physical properties of the modeled object according to the classical scheme.

The stability of interval two-dimensional semi-linear differential equations based on constrained interval arithmetic

In this study, the constrained interval arithmetic (CIA) is used as an effective mathematical tool for solving the stability analysis for interval two-dimensional semi-linear differential equations. Under certain assumptions, the origin is a focus of the interval semi-linear differential equations if it is a focus of the interval linear ones. Meanwhile, the origin can be a center, a center-focus or a focus of interval semi-linear differential equations if it is a center of the interval linear ones. On the other word, the types of equilibrium point are still determined by the linear part when a nonlinear disturbance is added to the interval linear differential equations. Based on CIA, the stability results of interval differential equations are the same as those of the real differential equations. At last, three illustrative examples validate the stability results of the origin for interval two-dimensional semi-linear differential equations.

Manifestation of interval uncertainties for fractional differential equations under conformable derivative

We propose a generalization of conformable calculus for Type-2 interval-valued functions. We investigated the differentiability and integrability properties of such functions. The conformable generalized Hukuhara (gH) differentiability of fractional order is introduced in this study. We prove a number of essential theorems on the conformable differentiability of the sum, gH difference, and product in a Type 2 interval setting. Furthermore, we define conformable Laplace transformation of Type-2 interval-valued functions. We interpret uncertain linear differential equations by using proposed theories. Several examples are given in detail to illustrate and clarify these rules and theorems. Applications to solving Type-2 interval differential equations with conformable derivatives are shown. Type-2 interval generalizes the interval uncertainty. On the other hand, conformable calculus extends the notion of integer calculus. This paper contributes a generalized theory that includes several existing results of classical integral and differential calculus and their conformable extensions in crisp and interval environments.

Modeling of a carbon emitted production inventory system with interval uncertainty via meta-heuristic algorithms

Modeling of a production inventory system under uncertain economic situation is a difficult task. Tackling this difficulty, the proposed work attempts to formulate a production inventory model with time dependent defective rate, greenness of the product dependent demand and production rates under interval uncertainty. Consideration of carbon emission during the production with interval valued time dependent emission rate is also an important feature of the proposed model. Furthermore, reduction technology is taken into the account for reducing the large scale of carbon emission. The proposed model is formulated mathematically with the help of interval differential equations and interval parametric approach. Thereafter, various interval valued cost components as well as average profit related to the model are obtained using interval mathematics. To handle the interval valued average profit, center-radius technique of interval optimization is applied. To verify the validity of the developed model, four numerical examples are considered and solved by meta-heuristic algorithms. Also, sensitivity analyses are carried out on the best found policy with respect to different system parameters of this model.

Interval valued demand and prepayment-based inventory model for perishable items via parametric approach of interval and meta-heuristic algorithms

This work mainly focuses on the mathematical formulation of an imprecise inventory model with partial prepayment policy in interval environment. In the proposed model, two different cases (without shortage and with partially backlogged shortages) are taken into consideration. Here, the demand and deterioration rates are considered as interval valued. Also, all the inventory costs, viz. ordering cost, purchase cost, holding cost, shortage cost and lost sale cost are considered as interval valued. Then, the continuous change of inventory level with time of the proposed model is defined mathematically by the interval differential equations. For both cases of the proposed model, the interval valued average profits are obtained by using parametric approach of interval. Here, the selling price and business period are taken as decision variables of the corresponding interval maximization problem. Finally, two numerical examples are considered and solved by using different meta-heuristic algorithms for checking the validity of the proposed model.

A new approach based on inventory control using interval differential equation with application to manufacturing system

<p style='text-indent:20px;'>Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued average profit of the proposed model has been obtained in parametric form and it is maximized by centre-radius optimization technique. Then to validate the proposed model, two numerical examples have been solved using MATHEMATICA software. In addition, we have shown the concavity of the objective function graphically using the code of 3D plot in MATHEMATICA. Finally, the post optimality analyses are carried out with respect to different system parameters.</p>

Исследование робастности наноэлектронных структур на базе резонансно-туннельных элементов

Multi-input logic gates based on two-level logic cells MOBILE have short (picosecond) switching times and higher functionality due to the ability to implement logic functions with fewer gates. This creates good prospects for the development of ultra-high-speed FPGAs with a high degree of integration, which are required for organizing high-performance computing. However, the extremely high sensitivity of resonant tunneling elements to changes in the energies of quantum states requires an assessment of the stability of such structures to external influences in real operation. In this work, the problem of assessing the stability of nanoelectronic structures that include resonant tunneling elements is considered. The method for studying the robustness of logic cells MOBILE based on a resonant tunneling diode and an НВТ transistor was proposed, making it possible to find an external interval estimate of the output voltage of the device under study for given interval models of the initial components. The technique is based on the use of systems of topological and parametric equations written in finite increments. It was shown that the proposed decomposition principle for the initial interval model ensures the algorithmic solvability of the problem posed. A computational algorithm for calculating processes in a two-level logical cell MOBILE has been developed. The algorithm provides for step-by-step integration of interval differential equations and solution of interval nonlinear algebraic equations at each step of integration using Kaucher interval arithmetic. The obtained results of the study of processes in a two-level logic cell MOBILE create prerequisites for expanding the field of application of resonant tunneling devices in high-speed monolithic integrated circuits.

Studies of different types of bifurcations analyses of an imprecise two species food chain model with fear effect and non-linear harvesting

Study of a food chain model under uncertainty is quite difficult. Because, in an uncertain food chain model, the biological parameters can’t be determined accurately. The aim of this work is to study the stability and local bifurcations (Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation) of an imprecise prey–predator system in an uncertain environment. The proposed imprecise model is formulated by considering two more realistic factors: the effect of a fear factor on the growth rate of prey population and non-linear harvesting of predator population. To study the proposed imprecise system mathematically, the dynamical interactions between the imprecise species are presented by the system of governing interval differential equations. And to study the dynamics of the proposed imprecise system theoretically, it is modelled in a precise way by the linear parametric representation of the interval. Then all the theoretical analyses, including Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens (BT) bifurcation of the interior equilibrium point of the proposed imprecise model are discussed in parametric form. To verify all the theoretical analyses of the proposed imprecise model, numerical simulations with interval-valued hypothetical data of the imprecise parameters are performed graphically. Finally, the work is concluded with some biological consequences.

An application of interval differential equation on a production inventory model with interval‐valued demand via center‐radius optimization technique and particle swarm optimization

Due to the fluctuation of market economy and uncertainty of customers' demand, it is quite difficult to develop an appropriate inventory model under uncertain situations. To overcome this difficulty, a food production model with preservation technology and credit-linked demand under default risk of capital in uncertain environment is developed with the help of parametric approach and interval mathematics. In this proposed model, all the inventory parameters, including demand rate, production rate and deterioration rate are considered as interval-valued. Because of the consideration of demand rate, production rate as well as deterioration rate as interval-valued, to represent the proposed model mathematically, the interval differential equations have been used. Solving these differential equations by using parametric approach, all the cost components and the corresponding average profit are obtained in the form of intervals. Therefore, the optimization problem of this model becomes interval-optimization problem. Then, to solve the interval-optimization problem, the center-radius optimization technique is established with the help of interval order relations. With the help of this technique, the interval-optimization problem is converted into crisp problem and then it is solved numerically by using different variants (Gaussian Quantum-behaved Particle Swarm Optimization, Weighted Quantum-behaved Particle Swarm Optimization, and Adaptive Quantum-behaved Particle Swarm Optimization) of Quantum-behaved Particle Swarm Optimization technique. Some real-life problems are considered and solved to justify the validity of the proposed model. The same model is also analyzed in crisp environment to verify the result of interval environment. Finally, the sensitivity analyses of both crisp and interval environments are performed separately with respect to the different system parameters.

Non-instantaneous impulses interval-valued fractional differential equations with Caputo-Katugampola fractional derivative concept

In this paper, by using the Caputo-Katugampola fractional derivative concept for the interval functions, a non-instantaneous impulsive value problem of interval differential equations is investigated. The first purpose is to study the existence and uniqueness results of the solution of the given problem, and the second purpose is to present the Ulam-Hyers-Mittag-Leffler's stability results of the above problem. Finally, some examples are given to illustrate our main results.

An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand

Due to the uncertainty of market economy as well as fluctuation of customer’s demand, appropriate modelling of an inventory problem is a challenging task to the researchers/OR practitioners. In order to overcome this type of challenging situation, in this article, using the parametric approach of interval a non-deterministic inventory model for deteriorating items with partially backlogged shortages and interval-valued deterioration rate has been developed. Here, the demand rate is considered to be interval-valued and dependent upon the selling price which is also interval-valued. Also, all the parameters (like ordering cost, holding cost, etc.) except the backlogging parameter have been considered as interval-valued. The corresponding problem has been formulated as a maximization problem with an interval-valued objective. Then, to solve the said problem, we have used different variants of the QPSO algorithm with interval fitness using interval ranking. Then, the computational results have been illustrated with the help of three numerical examples. Finally, the sensitivity of the solution has been analysed with the changes in the values of different parameters associated with the model and a fruitful conclusion has been drawn.

Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative.

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

Optimal Control for Partially Observed Nonlinear Interval Systems.

In this paper, we consider the optimal control problem for a class of systems governed by nonlinear time-varying partially observed interval differential equations. The control process is assumed to be governed by linear time varying interval differential equation driven by the observed process. Using the fact that the state, observation, and control processes possess lower and upper bounds, we have developed sets of (ordinary) differential equations that describe the behavior of the bounds of these processes. Using these differential equations, the interval control problem can be transformed into an equivalent ordinary control problem in which interval mathematics and extension principle of Moore are not required. Using variational arguments, we have developed the necessary conditions of optimality for the equivalent (ordinary) control problem. Finally, we present some numerical simulations to illustrate the effectiveness of the proposed control scheme.

A new interval differential equation for edge detection and determining breast cancer regions in mammography images

ABSTRACTBreast cancer is the most common form of cancer in women. The importance of diagnosing breast cancer is one of the important issues in medical science. Diagnosis of benign or malignant cancer is of great importance in addition to reducing costs in the direction of treatment. A non-destructive test method for early detection of breast cancer is image processing. Image processing has various uncertainties which are generated by different reasons such as sampling to noise, initial digitalization, intensity, and special domain. In this study, a strong image segmentation method based on interval uncertainties is proposed for the breast cancer images diagnosis. The main purpose of this paper is to improve the ordinary Sobel filter based on interval analysis by considering the intensity uncertainties. Simulation results have been implemented on MIAS which is an applicable database for breast cancer detection. The results of the proposed method have been compared with some state of the art methods such as LoG, Prewitt and canny filters. Final results showed that using the proposed method gives better achievement than the others by considering some kinds of uncertainties like Gaussian noise and salt and pepper noise.