Classical Integer Research Articles

A method for demand-accurate one-dimensional cutting problems with pattern reduction.

The main objective in the one-dimensional cutting stock problem (1D-CSP) is to minimize material costs. In practice, it is useful to focus on auxiliary objectives, one of which is to reduce the number of different cutting patterns. This paper discusses the classical integer IDCSP, where only one type of stock object is included. Meanwhile, the demands of various items must be precisely satisfied in the constraints. In other words, no overproduction or underproduction is allowed. Therefore, to solve this issue, a variable-to-constant method based on a new mathematical model is proposed. In addition, we integrate the approach with two other representative methods to demonstrate its effectiveness. Both benchmark instances and real instances are used in the experiments, and the results show that the methodology is effective in reducing patterns. In particular, in terms of the solutions to the real-life instances, the proposed approach presents a 31.93 to 37.6% pattern reduction compared to other similar methods (including commercial software).

Novel Robust Control Using a Fractional Adaptive PID Regulator for an unstable system

Recent advances in fractional order calculus led to the improvement of control theory and resulted in potential use of fractional adaptive PID controller in advanced academic and industrial applications as compared to the conventional adaptive PID controller. Basically, a fractional order adaptive PID controller is an improved version of classical integer order adaptive PID controller that outperformed its classical counterpart. In case of a closed loop system, a minute change would result in instability of the overall system. An efficient PID controller can be used to control the response of such system. Among various parameters of an instable system, speed of the system is an important parameter to be controlled efficiently. The current research work presents the speed controlling mechanism for an uncertain instable system by using fractional order adaptive PID controller.To validate the arguments, effectiveness and robustness of the proposed fractional order adaptive PID controller have been studied in comparison to the classical adaptive PID controller using The Criterion of quadratic error. Simulation findings and comparisons demonstrated that the proposed controller has superior control performance and outstanding robustness in terms of percentage overshoot, settling time, rising time, and disturbance rejection.

Investigation of Novel Piecewise Fractional Mathematical Model for COVID-19

The outbreak of coronavirus (COVID-19) began in Wuhan, China, and spread all around the globe. For analysis of the said outbreak, mathematical formulations are important techniques that are used for the stability and predictions of infectious diseases. In the given article, a novel mathematical system of differential equations is considered under the piecewise fractional operator of Caputo and Atangana–Baleanu. The system is composed of six ordinary differential equations (ODEs) for different agents. The given model investigated the transferring chain by taking non-constant rates of transmission to satisfy the feasibility assumption of the biological environment. There are many mathematical models proposed by many scientists. The existence of a solution along with the uniqueness of a solution in the format of a piecewise Caputo operator is also developed. The numerical technique of the Newton interpolation method is developed for the piecewise subinterval approximate solution for each quantity in the sense of Caputo and Atangana-Baleanu-Caputo (ABC) fractional derivatives. The numerical simulation is drawn against the available data of Pakistan on three different time intervals, and fractional orders converge to the classical integer orders, which again converge to their equilibrium points. The piecewise fractional format in the form of a mathematical model is investigated for the novel COVID-19 model, showing the crossover dynamics. Stability and convergence are achieved on small fractional orders in less time as compared to classical orders.

Integrated Backup Rolling Stock Allocation and Timetable Rescheduling with Uncertain Time-Variant Passenger Demand Under Disruptive Events

Railway traffic management focuses on regulating train movements and delivering improved service quality to passengers; however, such efforts are subject to many uncertainties in terms of disruptions and passenger demand on a rail transit line. In contrast to most existing studies, which focus on the rescheduling of passenger timetables in a deterministic framework, this study proposes a two-stage stochastic optimization model for allocating backup rolling stocks (BRS) to storage lines to reschedule the timetable and serve passengers delayed by disruptions. The first stage is an assignment problem to determine the optimal plan for the allocation of BRS to storage lines to achieve a good trade-off between the investment cost for the BRS and the expected travel time of delayed passengers across different stochastic scenarios. The second stage is explicitly formulated as a network flow model to optimize the timetable of the delayed trains on the tracks and the BRS from the storage lines such that the passenger travel time is minimized under each stochastic scenario. To improve the efficiency of convergence, we develop an improved L-shaped method with several accelerating techniques. Among these, we show that the classical integer L-shaped cut can be tightened given the property of the second-stage problem, which can also be generalized to other two-stage integer stochastic programs. Real-world case studies based on historical data from the Beijing metro verify the effectiveness of the proposed approach in reducing the travel time for passengers. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This research was supported by the National Natural Science Foundation of China [Grants 71621001, 71825004, and 71901016]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1233 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.6892548 ].

Transient and passage to steady state in fluid flow and heat transfer within fractional models

PurposeThe classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a quiescent Newtonian fluid within a circular pipe are initially generalized by introducing fractional derivatives. The purpose of this paper is to represent solutions as steady and transient parts. Afterward, making use of separation of variables, a fractional Sturm–Liouville eigenvalue task is posed whose eigenvalues and eigenfunctions enable us to write down the transient solution in the Fourier series involving also Mittag–Leffler function. An alternative solution based on the Laplace transform method is also provided.Design/methodology/approachIn this work, an analytical formulation is presented concerning the transient and passage to steady state in fluid flow and heat transfer within the diffusion fractional models.FindingsFrom the closed-form solutions, it is clear to visualize the start-up process of physical diffusion phenomena in fractional order models. In particular, impacts of fractional derivative in different time regimes are clarified, namely, the early time zone of acceleration, the transition zone and the late time regime of deceleration.Originality/valueWith the newly developing field of fractional calculus, the classical heat and mass transfer analysis has been modified to account for the fractional order derivative concept.

Every Choice Function Is Pro-Con Rationalizable

Dogan and Yildiz introduce and analyze the pro-con model that is inspired by Franklin’s prudential algebra. Consider an agent who is endowed with two sets of orderings: pro- and con-orderings. For each choice set, if an alternative is the top-ranked by a pro-ordering (con-ordering), then this is a pro (con) for choosing that alternative. The alternative with more pros than cons is chosen from each choice set. Each ordering may have a weight reflecting its salience. In this case, the probability that an alternative is chosen equals the difference between the total weights of its pros and cons. Although, this is an additive model similar to the random utility model with structurally invariant primitives, authors show that every (random) choice function is (random) pro-con rational. Their technique requires a generalization of Ford-Fulkerson theorem. The connection between the random model and its deterministic counterpart demonstrates a fruitful use of classical integer programming techniques in choice theory.

Investigation of a time-fractional COVID-19 mathematical model with singular kernel

We investigate the fractional dynamics of a coronavirus mathematical model under a Caputo derivative. The Laplace–Adomian decomposition and Homotopy perturbation techniques are applied to attain the approximate series solutions of the considered system. The existence and uniqueness solution of the system are presented by using the Banach fixed-point theorem. Ulam–Hyers-type stability is investigated for the proposed model. The obtained approximations are compared with numerical simulations of the proposed model as well as associated real data for numerous fractional-orders. The results reveal a good comparison between the numerical simulations versus approximations of the considered model. Further, one can see good agreements are obtained as compared to the classical integer order.

A noninteger order SEITR dynamical model for TB

This research paper designs the noninteger order SEITR dynamical model in the Caputo sense for tuberculosis. The authors of the article have classified the infection compartment into four different compartments such as newly infected unrecognized individuals, diagnosed patients, highly infected patients, and patients with delays in treatment which provide better detail of the TB infection dynamic. We estimate the model parameters using the least square curve fitting and demonstrate that the proposed model provides a good fit to tuberculosis confirmed cases of India from the year 2000 to 2020. Further, we compute the basic reproduction number as Re _{0} approx 1.73 of the model using the next-generation matrix method and the model equilibria. The existence and uniqueness of the approximate solution for the SEITR model is validated using the generalized Adams–Bashforth–Moulton method. The graphical representation of the fractional order model is given to validate the result using the numerical simulation. We conclude that the fractional order model is more realistic than the classical integer order model and provide more detailed information about the real data of the TB disease dynamics.

On the optimality of pseudo-polynomial algorithms for integer programming

In the classic Integer Programming Feasibility (IPF) problem, the objective is to decide whether, for a given \(m \times n\) matrix A and an m-vector \(b=(b_1,\dots , b_m)\), there is a non-negative integer n-vector x such that \(Ax=b\). Solving (IPF) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IPF) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ITCS 2019]. Jansen and Rohwedder designed an algorithm for (IPF) with running time \({\mathcal {O}}(m \varDelta )^m\) \(\log (\varDelta ) \log (\varDelta +\Vert b\Vert _{\infty })+{\mathcal {O}}(mn)\). Here, \(\varDelta \) is an upper bound on the absolute values of the entries of A. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal, by proving a lower bound of \(n^{o(\frac{m}{\log m})} \cdot \Vert b\Vert _{\infty }^{o(m)}\). We also prove that assuming ETH, (IPF) cannot be solved in time \(f(m)\cdot (n \cdot \Vert b\Vert _{\infty })^{o(\frac{m}{\log m})}\) for any computable function f. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IPF) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IPF) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IPF) when the path-width of the corresponding column-matroid is a constant .

A scaleable projection‐based branch‐and‐cut algorithm for the p‐center problem

The p-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose p of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds.We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation.Finally, we also present a computational study on instances from the literature with up to more than 700,000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters.

Modeling the dynamics of coronavirus with super-spreader class: A fractal-fractional approach

Super-spreaders of the novel coronavirus disease (or COVID-19) are those with greater potential for disease transmission to infect other people. Understanding and isolating the super-spreaders are important for controlling the COVID-19 incidence as well as future infectious disease outbreaks. Many scientific evidences can be found in the literature on reporting and impact of super-spreaders and super-spreading events on the COVID-19 dynamics. This paper deals with the formulation and simulation of a new epidemic model addressing the dynamics of COVID-19 with the presence of super-spreader individuals. In the first step, we formulate the model using classical integer order nonlinear differential system composed of six equations. The individuals responsible for the disease transmission are further categorized into three sub-classes, i.e., the symptomatic, super-spreader and asymptomatic. The model is parameterized using the actual infected cases reported in the kingdom of Saudi Arabia in order to enhance the biological suitability of the study. Moreover, to analyze the impact of memory index, we extend the model to fractional case using the well-known Caputo–Fabrizio derivative. By making use of the Picard-Lindelöf theorem and fixed point approach, we establish the existence and uniqueness criteria for the fractional-order model. Furthermore, we applied the novel fractal-fractional operator in Caputo–Fabrizio sense to obtain a more generalized model. Finally, to simulate the models in both fractional and fractal-fractional cases, efficient iterative schemes are utilized in order to present the impact of the fractional and fractal orders coupled with the key parameters (including transmission rate due to super-spreaders) on the pandemic peaks.

Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

<p style='text-indent:20px;'>The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.</p>

Fractional order PID controlled quasi Z-source inverter with front-line semiconductor material made of power electronics switches fed PMSM motor for EV applications

In the area of control theory, conventional PID controller is most commonly used for various applications to bring out the output of the system at higher efficacy. Now due various technological advancements, PID controller is derived into various advanced controllers. In the advancement of PID controller, there is a controller called Fractional Order PID controller which used widely in industrial applications. The advancement o9f classical integer order PID controller is the FOPID controller. This paper deals with the control of PMSM by FOPID controller. In this work, different control strategies such as Proportional Integral (PI) controller, and Fractional Order Proportional Integral Derivative (FOPID) controller is exploited to sustain the constant yield motor speed of the quasi z-source with 3-phase inverter fed PMSM drive. Also, the source disturbance is introduced to analyze the performance of quasi z-source with 3-phase inverter fed PMSM driven by different controllers. The implementation of FOPID controller will gives out the enhanced response in Quazi Z source inverter fed PMSM. The closed loop system with the FOPID controller has reduced steady state error, improved rise time, settling time and the peak time, when compared with the closed loop system with PI controller. The simulation has been done utilizing MATLAB Simulink software. As discussed, outcomes reveal that the quasi z-source with 3-phase inverter fed PMSM driven by the FOPID-FOPID controller has good-performance such as transient response etc., when compared to the PI-PI controller. Material properties affect the performance and efficiency of power electronic switching devices in three phase inverter circuits. These are generally silicon switches. To increase material technology and performance, switching devices employ wide band-gap semiconductors. This paper also examines the optimal semiconductor material utilized in the three-phase inverter circuit's power electronic switches.

Structure Preserving Algorithm for Fractional Order Mathematical Model of COVID-19

In this article, a brief biological structure and some basic properties of COVID-19 are described. A classical integer order model is modified and converted into a fractional order model with as order of the fractional derivative. Moreover, a valued structure preserving the numerical design, coined as Grunwald–Letnikov non-standard finite difference scheme, is developed for the fractional COVID-19 model. Taking into account the importance of the positivity and boundedness of the state variables, some productive results have been proved to ensure these essential features. Stability of the model at a corona free and a corona existing equilibrium points is investigated on the basis of Eigen values. The Routh–Hurwitz criterion is applied for the local stability analysis. An appropriate example with fitted and estimated set of parametric values is presented for the simulations. Graphical solutions are displayed for the chosen values of (fractional order of the derivatives). The role of quarantined policy is also determined gradually to highlight its significance and relevancy in controlling infectious diseases. In the end, outcomes of the study are presented.

Mathematical assessment of the dynamics of the tobacco smoking model: An application of fractional theory

<abstract><p>In this paper we consider fractional-order mathematical model describing the spread of the smoking model in the sense of Caputo operator with tobacco in the form of snuffing. The threshold quantity $ \mathcal{R}_0 $ and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of of the solution of the considered model. The new version of numerical approximation's framework for the approximation of Caputo operator is used. Finally, the numerical results are presented to justify the significance of the arbitrary fractional order derivative. The analysis shows fractional-order model of tobacco smoking in Caputo sense gives useful information as compared to the classical integer order tobacco smoking model.</p></abstract>

Modeling the Virus Infection at the Population Level.

As pointed out by many researchers in the last few decades, differential equations with fractional (non-integer) order differential operators, in comparison with classical integer order ones, have apparent advantages in modeling. A Caputo fractional order system of ordinary differential equations is introduced to model the virus infection at the population level in this chapter. As well known, there are two main methods to study the dynamics of a model: qualitative analysis and numerical modeling. Here the qualitative analysis, including uniqueness, invariant set, and stability, is first presented with intuitive derivation. Then the famous genetic algorithm is introduced to numerically model the dynamics of virus infection, i.e. to adjust the parameters of the Caputo fractional model such that its solution can properly fit real data and predict future.

A simulation approach for closed-loop control of coupled four tank system

The coupled tank system is the most widely used sub-component in chemical process industries. Fluid mixing is a major step in chemical processes that alters the material properties and cost. Fluid flow and its level regulation between several tanks are important control problems. As the first step, this paper addresses the level regulation problem using classical integer order proportional, derivative, integral (PID), fractional order PID controllers. As a second step, model-based robust fractional order controllers are derived using sliding mode approach in order to achieve the desired response, parameters of the proposed controllers are tuned using genetic algorithm. Finally, system performance under all variants of control schemes has been tested using numerical simulations.

Solving binary programming problems using homotopy theory ideas

PurposeThe aim of the study is to show that merging two areas of mathematics – topology and discrete optimization – could result in a viable option to solve classical or specialized integer problems.Design/methodology/approachIn the paper, discrete topology concepts are applied to propose a metaheuristic algorithm that is capable to solve binary programming problems. Particularly, some of the homotopy for paths principles are used to explore the solution space associated with four well-known NP-hard problems herein considered as follows: knapsack, set covering, bi-level single plant location with order and one-max.FindingsComputational experimentation confirms that the proposed algorithm performs in an effective manner, and it is able to efficiently solve the sets of instances used for the benchmark. Moreover, the performance of the proposed algorithm is compared with a standard genetic algorithm (GA), a scatter search (SS) method and a memetic algorithm (MA). Acceptable results are obtained for all four implemented metaheuristics, but the path homotopy algorithm stands out.Originality/valueA novel metaheuristic is proposed for the first time. It uses topology concepts to design an algorithmic framework to solve binary programming problems in an effective and efficient manner.

Numerical stability of Grünwald–Letnikov method for time fractional delay differential equations

This paper is concerned with the numerical stability of time fractional delay differential equations (F-DDEs) based on Grünwald–Letnikov (GL) approximation for the Caputo fractional derivative. In particular, we focus on the numerical stability region and the Mittag–Leffler stability. Using the boundary locus technique, we first derive the exact expression of the numerical stability region in the parameter plane, and show that the fractional backward Euler scheme is not \(\tau (0)\)-stable, which is different from the backward Euler scheme for integer DDE models. Secondly, we prove the numerical Mittag–Leffler stability for the numerical solutions provided that the parameters fall into the numerical stability region, by employing the singularity analysis of generating function. Our results show that the numerical solutions of F-DDEs are completely different from the classical integer order DDEs, both in terms of \(\tau (0)\)-stability and the long-time decay rate. Numerical examples are given to confirm the theoretical results.

Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission

The coronavirus infectious disease (COVID-19) is a novel respiratory disease reported in 2019 in China. The infection is very destructive to human lives and caused millions of deaths. Various approaches have been made recently to understand the complex dynamics of COVID-19. The mathematical modeling approach is one of the considerable tools to study the disease spreading pattern. In this article, we develop a fractional order epidemic model for COVID-19 in the sense of Caputo operator. The model is based on the effective contacts among the population and environmental impact to analyze the disease dynamics. The fractional models are comparatively better in understanding the disease outbreak and providing deeper insights into the infectious disease dynamics. We first consider the classical integer model studied in recent literature and then we generalize it by introducing the Caputo fractional derivative. Furthermore, we explore some fundamental mathematical analysis of the fractional model, including the basic reproductive number R0 and equilibria stability utilizing the Routh-Hurwitz and the Lyapunov function approaches. Besides theoretical analysis, we also focused on the numerical solution. To simulate the model, we use the well-known generalized Adams–Bashforth Moulton Scheme. Finally, the influence of some of the model essential parameters on the dynamics of the disease is demonstrated graphically.