Modeling Of Real-world Problems Research Articles

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>

The role of generating questions in mathematical modeling

This study aims to clarify the role of generating questions in mathematical modeling and construct principles for teaching and learning mathematical modeling with an emphasis on generating questions. To achieve this purpose, the role of generating questions was determined by solving a sprinkler problem and considering the investigation process. Generating questions requires noticing the unknown and cognitive steps to elucidate the unknown and solve the problem. The process and outcome of generating questions were explored using an open-ended problem. This study demonstrates (1) the role in advancing mathematical modeling from various points of view, (2) the role of fostering mathematical modeling competencies, and (3) the role in facilitating the extension of the applicability of the mathematical model. Having established these roles, the principle of teaching and learning was constructed, which is expected to foster the ability to generate questions. This study demonstrates that, in mathematical modeling, generating questions is crucial for developing a clear sense of direction for modeling real-world problems. Furthermore, generating questions reflects on the structures of mathematical models, allowing the execution of the mathematical modeling process.

Mathematical analysis of an eco-epidemiological model with different competition factors in its fractional-stochastic form

In this paper, we considered a complex real-world problem with existing mathematical equations. These equations were converted to fractional-stochastic differential and integral equations. Analysis of existence and uniqueness was presented for each case. Numerical simulations have been performed for different values of fractional orders and the density of randomness. We discuss in detail the results for the Caputo and Atangana-Baleanu case. The algorithm is given to obtain the numerical solution of each model. The numerical results are shown for the integer case, arbitrary derivative, and stochastic case. The results obtained indicated that this concept will open new doors of investigation toward modeling real-world problems.

A Study on Greedy Technique in Container Loading Problem and Knapsack Problem

The main aim of the paper is to use application of greedy algorithm in container loading problem and Knapsack problem. Greedy method gives an optimal solution to the problem by considering the inputs one at a time, checking to see if it can be included in the set of values which give an optimal solution and then check if it is the feasible solution. The Greedy algorithm could be understood very well with a well-known problem referred to as container loading problem and Knapsack problem. The basic Container Loading Problem can be defined as the problem of placing a set of boxes into the container respecting the geometric constraints: the boxes cannot overlap and cannot exceed the dimensions of the container. The knapsack problem is in combinatorial optimization problem. It appears as a sub problem in many, more complex mathematical models of real world problems.

Solving Generalized DEA/AR Model With Fuzzy Data and Its Application to Evaluate the Performance of Manufacturing Enterprises

The use of conventional data envelopment analysis (DEA) models in real-world problems are limited because of some restrictions that must be considered in the model such as imprecise or vague data in inputs and outputs as well as additional information or assumptions. One way to handle this problem is by using fuzzy DEA with assurance regions (FDEA/AR) models. There is a common approach in almost all the suggested methods for solving FDEA/AR models. However, in this paper, we show that in some DEA/AR models, applying this approach can be led to inappropriate results. Four theorems are given to provide some sufficient conditions for a DMU to be the DEA/AR efficient. These theorems can be used to check the accuracy of the presented methods for solving FDEA/AR models, too. Moreover, a new method for solving a generalized FDEA/AR model that includes established DEA models such as CCR model (Charnes et al., 1978), BCC model (Banker et al., 1984), FG model (Fare & Grosskopf, 1985), and ST model (Seiford & Thrall, 1990) is proposed. These models are constant, variable, non-decreasing, and non-increasing returns to scale models, respectively. The proposed method is applied to evaluate the performance of manufacturing enterprises.

New concept in calculus: Piecewise differential and integral operators

• Piecewise derivative with classical and global derivative is presented. • Piecewise derivative with singular and non-singular kernel is presented. • Some properties of piecewise derivative is introduced. • Numerical algorithm based on the Newton polynomial is constructed for Cauch problem with such derivative. • Applications of this derivative to some chaotic and epidemiological models are given and numerical simulations are depicted for different values of fractional orders and fractal dimensions. In the last decades, many methodologies have been suggested to depict behaviors of some complex world’s problems arising in many academic fields. One of these problems is the multi-steps behavior displayed by some problems. A concept of piecewise derivative is introduced in this paper with the aim to model real world problems following multiples processes. We have presented some important properties of these definitions. We considered different scenarios and presented numerical schemes that could be used to solve such problems. Illustrative examples, including chaotic and epidemiological models are presented to see the effectiveness of the suggested concept.

Stochastic fractional model of Alzheimer disease

Differential operators based on convolution definitions have been recognized as powerful mathematics tools to help model real world problems due to the properties associated to their different kernels. In particular the power law kernel helps include into mathematical formulation the effect of long range, while the exponential decay helps with fading memory, also with Poisson distribution properties that lead to a transitive behavior from Gaussian to non-Gaussian phases respectively. In this paper, we presented the well-poseness of the models for different differential operators that were presented in detail.

Retraction Note to: Derivative with two fractional orders: A new avenue of investigation toward revolution in fractional calculus

In order to describe more complex problems using the concept of fractional derivatives, we introduce in this paper the concept of fractional derivatives with orders. The new definitions are based upon the concept of power law together with the generalized Mittag-Leffler function. The first order is included in the power law function and the second one is in the generalized Mittag-Leffler function. Each order therefore plays an important role while modeling, for instance, problems with two layers with different properties. This is the case, for instance, in thermal science for a reaction diffusion within a media with two different layers with different properties. Another case is that of groundwater flowing within an aquifer where geological formation is formed with two layers with different properties. The paper presents new fractional operators that will open new doors for research and investigations in modeling real world problems. Some useful properties of the new operators are presented, in particular their relationship with existing integral transforms, namely the Laplace, Sumudu, Mellin and Fourier transforms. The numerical approximation of the new fractional operators are presented. We apply the new fractional operators on the model of groundwater plume with degradation and limited sorption and solve the new model numerically with some numerical simulations. The numerical simulation leaves no doubt in believing that the new fractional operators are powerfull mathematical tools able to portray complexes real world problems.

A study of fractional order Ambartsumian equation involving exponential decay kernel

<abstract> Recently, non-singular fractional operators have a significant role in the modeling of real-world problems. Specifically, the Caputo-Fabrizio operators are used to study better dynamics of memory processes. In this paper, under the non-singular fractional operator with exponential decay kernel, we analyze the Ambartsumian equation qualitatively and computationally. We deduce the result of the existence of at least one solution to the proposed equation through Krasnoselskii's fixed point theorem. Also, we utilize the Banach fixed point theorem to derive the result concerned with unique solution. We use the concept of functional analysis to show that the proposed equation is Ulam-Hyers and Ulam-Hyers-Rassias stable. We use an efficient analytical approach to compute a semi-analytical solution to the proposed problem. The convergence of the series solution to an exact solution is proved through non-linear analysis. Lastly, we present the solution for different fractional orders. </abstract>

A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980’s but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.

A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment

Abstract A tumor is most dangerous disease of medical science which is a mass or lumps of tissue that’s formed by an accumulation of abnormal cells. A famous fractional tumor-immune model is interpreting the dynamics of tumor and effector cells. In this work, we provide a comparative and chaotic study of tumor and effector cells through fractional tumor-immune dynamical model. A new arbitrary operator based on the Mittag-Leffler law is assumed for this study. Again, we examine the interactions among distinct tumor cell inhabitants and immune structure through a model of real world problem of medical science. We First investigate the dynamical effect of the activation of the effector immune and tumor cells by using Adams-Bashforth-Moulton and Toufik-Atangana methods. Furthermore, this paper analyses the existence and uniqueness of given tumor-immune model of arbitrary order. Further, we have examined the dynamical behaviors of the fractional tumor-immunne model and obtained results are compared with exiting results by other methods. Numerical simulations are executed by Adams-Bashforth-Moulton and Toufik-Atangana methods using popular Atangana-Baleanu fractional derivative. Our obtained results will be useful for biologists to the treatment of cancer disease.

Mathematical model of Ebola and Covid-19 with fractional differential operators: Non-Markovian process and class for virus pathogen in the environment.

Differential operators based on convolution definitions have been recognized as powerful mathematics tools to help model real world problems due to the properties associated to their different kernels. In particular the power law kernel helps include into mathematical formulation the effect of long range, while the exponential decay helps with fading memory, also with Poisson distribution properties that lead to a transitive behavior from Gaussian to non-Gaussian phases respectively, however, with steady state in time and finally the generalized Mittag-Leffler helps with many features including the queen properties, transitive behaviors, random walk for earlier time and power law for later time. Very recently both Ebola and Covid-19 have been a great worry around the globe, thus scholars have focused their energies in modeling the behavior of such fatal diseases. In this paper, we used new trend of fractional differential and integral operators to model the spread of Ebola and Covid-19.

A Review on Some Arithmetic Concepts of Z-Number and Its Application to Real-World Problems

Zadeh introduced the concept of Z-numbers in 2011 to deal with imprecise information. In this regard, many research works have been published in an attempt to introduce some basic theoretical concepts of Z-numbers to model real-world problems. To understand the current challenges when dealing with Z-numbers and the feasibility of using Z-number in solving real-world problems, a comprehensive review of the existing work on Z-number is paramount. This paper consists of an overview of existing literature on Z-number and identifies some of the key areas that are required for further improvement.

A multi-objective open set orienteering problem

We have constructed a multi-objective set orienteering problem to model real-world problems more fittingly than the existing models. It attaches (i) a predefined profit associated with each cluster of customers, and (ii) a preset maximum service time associated with each customer of all the clusters. When a customer from cluster visits, it allows the earning of a profit score. Our purpose is primarily to search for a route that (i) on the one hand allows us to service each cluster for maximizing customer satisfaction and (ii) on the other hand allows us to maximize our profits, too. The model assumes that the more time we spend on servicing, the more customer satisfaction it yields. It tries to cover as many clusters as possible within a specified time budget. In this paper, we also consider third-party logistics to allow the flexibility of ending our journey at any cluster of our choice. The proposed model is solved using the nondominated sorting genetic algorithm and the strength Pareto evolutionary algorithm. Here, we also generate the dataset to test the proposed model by using instances from the literature of the generalized traveling salesman problem. Finally, we present a comparative result analysis with the help of some statistical tools and discuss the results.

An efficient numerical approach for fractional diffusion partial differential equations

The main purpose of this manuscript is to present a new approximation method to find the numerical solution of partial differential equations with Caputo fractional derivatives. We have also included the stability analysis of the proposed technique. To illustrate the effectiveness and reliability of the method, we have examined the method with several numerical examples. The high computational efficiency of the proposed method can be seen by observing the accurate approximate solutions obtained from the process. The suggested procedure is a potent tool and it can be used to solve other similar problems in modeling real-world problems.

Neuro-Fuzzy Networks and Their Applications in Medical Fields.

The fusion of artificial neural networks and fuzzy logic systems allows researchers to model real-world problems through the development of intelligent and adaptive systems. Artificial neural networks are able to adapt and learn by adjusting the interconnections between layers, while fuzzy logic inference systems provide a computing framework based on the concept of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. The combined use of those adaptive structures is known as "neuro-fuzzy" systems. In this paper, the basic elements of both approaches are analyzed, noticing that this blending could be applied for pattern recognition in medical applications.

A comparative study of online consumer behavior: a tale of two research methods

PurposeThe use of linear regression analysis is common in the social sciences. The purpose of this paper is to show the advantage of a qualitative research method, namely, structured qualitative analysis (SQA), over the linear regression method by using different characteristics of data.Design/methodology/approachData were gathered from a study of online consumer behavior in Taiwan. The authors changed the content of the data to have different sets of data. These data sets were used to demonstrate how SQA and linear regression works individually, and to contrast the empirical analyses and empirical results from linear regression and SQA.FindingsThe linear regression method uses one equation to model different characteristics of data. When facing a data set containing a big and a small size of different characteristics, linear regression tends to provide an equation by modeling the characteristics of the big size data and subsuming those of the small size. When facing a data set containing similar sizes of data with different characteristics, linear regression tends to provide an equation by averaging these data. The major concern is that the one equation may not be able to reflect the data of various characteristics (different values of independent variables) that result in the same outcome (the same value of dependent variable). In contrast, SQA can identify various variable combinations (multiple relationships) leading to the same outcome. SQA provided multiple relationships to represent different sizes of data with different characteristics so it created consistent empirical results.Research limitations/implicationsTwo research methods work differently. The popular linear regression tends to use one equation to model different sizes and characteristics of data. The single equation may not be able to cover different behaviors but may lead to the same outcome. Instead, SQA provides multiple relationships for different sizes of data with different characteristics. The analyses are more consistent and the results are more appropriate. The academics may re-think the existing literature using linear regression. It would be interesting to see if there are new findings for similar problems by using SQA. The practitioners have a new method to model real world problems and to understand different possible combinations of variables leading to the same outcome. Even the relationship obtained from a small data set may be very valuable to practitioners.Originality/valueThis paper compared online consumer behavior by using two research methods to analyze different data sets. The paper offered the manipulation of real data sets to create different data sizes of different characteristics. The variations in empirical results from both methods due to the various data sets facilitate the comparison of both methods. Hence, this paper can serve as a complement to the existing literature, focusing on the justification of research methods and on limitations of linear regression.

Strange chaotic attractors under fractal-fractional operators using newly proposed numerical methods

Two classes of differentiation and integral operators have been successfully used in modeling real world problems. On the one hand, researchers used the concept of non-local differentiation to capture the heterogeneity of nature linked to heavy-tailed, and many other, nonlocalities’ dependency. On the other hand, they used a local operator with power law setting to capture processes with similarities. Both operators were used for different purposes; however, we shall note that there exist in nature chaotic processes that exhibit both kinds of behavior. Thus, neither the nonlocal operator nor the differential operator with power law setting can capture such processes. In connection to this, a new class of differential operators has been recently introduced. The differential operator has two orders: the first is the fractional order and the second is the fractal dimension. To open new doors to capturing more chaotic behaviors, in this paper, the Thomas cyclically symmetric attractor, the Shilnikov attractor, the Rossler attractor, the Langford attractor and the King Cobra attractor under the domain of newly proposed fractal-fractional derivative operators, are investigated. Three different types of fractal-fractional derivative operators called the Caputo, Caputo-Fabrizio-Caputo and the Atangana-Baleanu-Caputo operators are employed to comprehend the dynamics of the attractors with the help of varying fractional-order $ \Omega$ and fractal-dimension $ \Delta$ parameters, where $ \Omega,\Delta\in ]0, 1]$. The new strange behaviour of the attractors obtained in this study was not possible earlier in the literature for either classical or fractional cases. Various new three-dimensional graphs are presented for the numerical simulations, which have been carried out with the application of well-convergent methods devised to find approximate solutions for the fractal-fractional order systems.

New insight kinetic modeling: Models above classical chemical mechanic

Abstract New trends of differential operators have been suggest very recently and have been proven to be accurate in modeling real world problems in many fields of science. We present in this paper some kinetic reaction model where the process does not follow the classical law of chemical mechanic. We present kinetic reactions where the processes follow the power law, exponential decay law and crossover behavior from exponential to power law. Analytical technique were used where the model is linear and new numerical where the model is non-linear. Results obtained here are very closer to reality than those models designed with classical chemical mechanic.

Differential and integral operators with constant fractional order and variable fractional dimension

Abstract The complexities of nature have pushed humankind to construct complex mathematical formula that can be used to capture such natural occurrence. Very recently the concept differential and integral operators with fractional order and fractal dimension were introduced. The concept has opened new doors for investigations. In this paper, we present a step forward, where the constant fractal dimension is replaced by variable dimension. We present in detail some properties of this new operators, we suggested a new numerical approach that can be used to solve differential and integral equations associate to this operators. We presented some examples and simulations are presented to underpin the strength of the new operators. We strongly believe that this paper will open many new doors of investigation toward modeling real world problems.