International Journal of Mathematical Modelling and Numerical Optimisation (16 papers in press)
A New Approximate Solution of the Fuzzy Delay Differential Equations
by A.L.I. JAMEEL, Nidal Anakira, AbdulKareem Alomari, Azizan Saaban, Mohammad Al Mahameed
Abstract: In this paper, an approximate analytical algorithm namely Homotopy Analysis Method (HAM) is presented for the ﬁrst time to obtain approximate analytical solutions of first order fuzzy delay differential equations (FDDEs). This method allows for the solution of the FDDEs to be calculated in the form of an infinite series with the components that can be easily calculated. The HAM utilizes a convergence control parameter the convergence region of the infinite series solution. Numerical examples are tested to highlight the important features of the HAM algorithm.
Keywords: Fuzzy Number; Fuzzy differential equation; Delay differential equation; Homotopy Analysis Method.
The Caputo-Fabrizio fractional derivative applied to a singular perturbation problem
by Abdon Atangana, Emile Franc Doungmo Goufo
Abstract: The Garden equation is a nonlinear partial differential equation that has application in more than two different fields. In this paper, we use the Caputo-Fabrizio derivative with fractional order to extend this model to the concept of fractional calculus. In the process, we prove that the new derivative satisfies the equality of mixed partial and in the extended equation, we present the analysis of existence and uniqueness of the exact solution. We propose a special solution using the Laplace iterative methods. Some numerical simulations are preformed for different values of alpha and also the perturbed parameter.
Keywords: Non linear Garden equation; Caputo-Fabrizio fractional derivative; equality of mixed partial; existence and uniqueness; numerical solutions.
Permutation flow shop scheduling problem under non-renewable resources constraints
by Imane Laribi, Farouk Yalaoui, Zaki Sari
Abstract: The majority of flow shop scheduling problems consider machines as the only resource. However, in most real-life manufacturing environments, jobs for their processing on machines may require additional non-renewable resources. Considering such resources, the scheduling problem is more realistic and much harder to solve. In this paper, we investigate the permutation flow shop scheduling problem subject to non-renewable resources constraints. The objective is to find a schedule that minimizes the maximum completion time. An integer linear programming model is developed. Because of the computation time constraint, we propose an approximate resolution method based on Genetic Algorithm.
To obtain better and more robust solutions, the Taguchi method is performed for tuning the parameters and operators of the algorithm. Furthermore, a local search is proposed to enhance the searching ability. Finally, computational experiments are conducted to evaluate the performance of both mathematical model and algorithm on different configurations of non-renewable resources availability.
Keywords: Scheduling; Permutation flow shop; Non-renewable resources; Optimisation; Mathematical programming; Genetic Algorithm; Local search.
Performance analysis and design optimisation of 3-Ø Packed U Cell inverter for industrial drive applications
by Rajanand Patnaik Narasipuram, Ravindranath Tagore Yadlapalli
Abstract: Nowadays, the importance for multilevel inverters are getting more and more in field of medium and high power applications. This paper discusses about Packed U Cell (PUC) multilevel inverter which is the most advanced topology. The figure of merit of this topology is reduction in the number of switches as level increases. Hence, it reduces the cost implementation besides topology complexity compared to other existing topologies such as neutral-point clamping (NPC), flying capacitor (FC), cascaded H-bridge (CHB) and hybrid cascaded H-bridge (HCHB). Furthermore, there is no need of transformers in this whole concept. Hence, it avoids the bulky installations. The 3-Ø induction motor is fed with 7, 15 and 31 levels Packed U Cell (PUC) inverter topologies individually. The stator current, speed response and electromagnetic torque is shown for each level. The performance of induction motor is analysed in terms of % torque ripples and % total harmonic distortion (%THD). The whole simulations are carried out by using MATLAB/Simulink version R2012b.
Keywords: torque ripples; packed u cell (PUC); total harmonicrndistortion; (THD); load torque (TL); multi-level inverter (MLI); neutralrnpoint clamping (NPC); flying capacitor (FC); cascaded H-Bridgern(CHB) hybrid cascaded H-Bridge (HCHB); induction motor (IM).
Mathematical modelling of human cardiovascular-respiratory system responses to exercise in Rwanda
by Jean Marie Ntaganda, Japhet Niyobuhungiro, Wellars Banzi, Lydie Mpinganzima, Froduald Minani, Jean Bosco Gahutu, Vincent Dusabejambo, Immaculate Kambutse
Abstract: In this paper, we present a nonlinear dynamic model for the interactive mechanism of cardiovascular and respiratory system. The model is designed and analyzed for human during physical exercises. In order to verify the adequacy of the designed model, data collected in Rwanda are used for validation. We have simulated the impact of heart rate and alveolar ventilation as controls of cardiovascular and respiratory system respectively to steady state response of the main cardiovascular hemodynamic quantities i.e., systemic arterial and venous blood pressures, arterial oxygen partial pressure and arterial carbon dioxide partial pressure, to the stabilised values of controls. We used data collected in Rwanda for both male and female during physical activities. We obtained a good agreement with physiological data in the literature. The model may represent an important tool to improve the understanding of exercise physiology.
Keywords: Hemodynamic quantities; cardiovascular/respiratory system,
Numerical simulation; Exercise; Physical activity; Sportsmen in Rwanda.
Temperature and Rainfall Dependent Mathematical Modeling for Progression of Zika Virus Infection
by Narender Kumar, Md Imam Faizan, Shama Parveen, Ravins Dohare
Abstract: Zika virus is a arthropod mediated infection that has re-emerged recently leading to global pandemic of 2015-16. Prevention of ZIKV infection is a challenge in absence of antiviral agents and an effective vaccine. We formulated a susceptible-exposed-infected-recovered (SEIR) mathematical model for mosquito-borne transmission of ZIKV. The model was used to estimate different parameters using the data obtained from Puerto Rico during 2015-16 that were responsible for the outbreak of ZIKV in this region. Further, the inclusion of different climatic factors like rainfall and temperature in the model assisted in more realistic predictions of the transmission dynamics of ZIKV. The value basic reproduction number $R_0 = 3.2869$ calculated at estimated parameters suggested outbreak of viral infection in this region. In addition, the sensitivity analysis revealed that $R_0$ was highly influenced by three different parameters i.e death rate, mosquito biting rate and maturation rate of immature mosquitoes. The $R_0$ further implied that around $70%$ of the individuals should be immunized so that they develop herd immunity to prevent propagation of the virus in this population. The simulation of controlled reproduction number revealed the values of different isolation coefficients ($epsilon_1>0.4$, $epsilon_2>0.14$, $tau_1<0.1$). These coefficients might be utilized by policy makers in future planning of the control strategies against this viral infection. The present investigation on mathematical modeling thus suggested some crucial parameters and climatic factors that affected the progression dynamics of transmission of ZIKV. These simulations are likely to contribute towards formulation of control measures for prevention of future outbreaks.
Keywords: Zika virus infection; basic reproduction number; disease free equilibrium; simulation.
Dynamic Indicator of Intimate Partner Violence: Self-regulatory Perpetrator
by Erik Leal-Enríquez, Aime Renata Gutiérrez-Antúnez
Abstract: A dynamic indicator to measure the level of violence between a perpetrator and victim of intimate partner violence is developed in this paper. Herein it is considered that violence can be measured through the rates of change over time of the violent interactions, considering the weight of severity of the violence committed. These rates of change are modeled by a logistic differential equation with a probabilistic parameter, the principle being the self-regulation of a perpetrator. This parameter of self-regulation is modeled by two changes of states: one of self-control and the other being a loss thereof. The transition of each state is modeled by a discrete Markov chain. With the proposed dynamic indicator, computational simulations are created to generate probable scenarios of partner violence with varying levels of severity. This dynamic indicator is then applied to a specific study of physical violence, using the prevalence of violence to approximate a state of loss of control. With the results obtained from the simulation, a qualitative analysis is made of the probable behaviors, controllable (semi-stable) and uncontrollable (potentially deadly), that can occur in a time interval of three and twelve months.
Keywords: Dynamic indicator; Mathematical model; Violent Scenarios; Selfregulation;
Intimate Partner Violence.
A novel Particle Swarm Optimization with search space tuning parameter to avoid premature convergence
by Raja Chandrasekaran, Agilesh Saravanan, Ashok Kumar, Gangatharan Narayanan
Abstract: Abstract: Particle Swarm Optimization is a trendy optimization technique that is inhaled from the space navigational intelligence of birds. The optimization technique is popular among the researchers for several decades because of the fact that it is inspired by the zonal and Universal best members in all the generations. Though the results excel other optimization techniques, the more-than-modest orientation style of the algorithm leads the population to premature convergence often. Inertia weight parameter is used to tune the explorability of the population. In this paper, a zonal monitor (based on success in the recent iterations) based Inertia weight tuning is redressed by including Universal monitors (based on thernsuccess with a Universal fitness perspective). The proposed algorithm excels the conventional PSO, the PSO with zonal monitors alone. The inertia weight of the PSO with zonal monitor is also not dynamic whereas the proposed PSOs inertia weight are found to be more dynamic with tuning the explore ability with regard to zonal and Universal context of fitness.
Keywords: PSO; Inertia weight.
OPTIMIZATION OF EOQ MODEL WITH WEIBULL DETERIORATION UNDER CRISP AND FUZZY ENVIRONMENT
by A.N.U. SAYAL, A.P. SINGH, DEEPAK AGGARWAL
Abstract: Deterioration is an unavoidable condition prevalent in all spheres of the inventory system involving perishable goods. Though other factors are also responsible for the depletion of the level of the inventory but the effect of deterioration in this regards is quite high. In the present paper we have considered the deterioration rate of the form of weibull distribution and the demand rate is taken as a ramp type function of time, when the inventory system starts without any kind of shortage. We have developed a mechanism for the optimization of the total cost of the EOQ inventory system in both crisp and fuzzy environment. An appropriate numerical example has been proposed in order to validate the model in both crisp and fuzzy system.
Keywords: Optimization; Weibull distribution; deterioration; ramp type demand; EOQ model; fuzzy inventory system.
Natural convection analysis of water near its density extremum between finite vertical plates: a differential transform approach
by Ryoichi Chiba
Abstract: Using a two-dimensional differential transform method, we solve the steady natural convection problem of cold water between vertical isothermal plates of finite length. The cold water exhibits a density variation approximated as a quadratic function of temperature. Given the temperature-dependent viscosity, we present approximate analytical solutions in the form of power series for temperature, vertical flow velocity, and pressure defect. Numerical calculations are carried out for two cases of water temperature in which the following occur with respect to increases in temperature: (i) the density decreases monotonously, and (ii) it increases and subsequently decreases. The numerical results reveal how the temperature-dependent properties affect the developing temperature and velocity profiles and pressure defect distribution along the streamwise direction.
Keywords: cold water; DTM; differential transform method; semi-analytical method; natural convection; temperature-dependent property; vertical channel; series solution; mathematical modelling; heat transfer; convective flow; density extremum.
Numerical analysis on thermal performance of a trapezoidal micro-channel heat sink using an improved version of the augmented -constraint method
by Lagouge Tartibu
Abstract: This work proposes the use of an improved version of the augmented ɛ-constraint method (AUGMENCON2) for the analysis of the thermal performance of a micro-channel heat sink. In order to highlight the strength and the effectiveness of this new approach, a trapezoidal micro-channel heat sink has been considered. The geometrical configuration namely the micro-channel widths and depth are the main variables considered in this study. Surrogate models based on the Response Surface methodology have been adopted to approximate the thermal resistance and the pumping power which provide an indication of the thermal performance of the micro-channel heat sink. A two-objective Non-Linear Problem have been formulated and implemented within the General Algebraic Modelling System. Global Pareto optimal solutions has been computed using the proposed method. Despite being a relatively straightforward method, the AUGMENCON2 provides a reasonable level of accuracy and shortens the required computational time in comparison to two existing approaches.
Keywords: Multi-objective optimization; ε-constraint method; heat sink; AUGMENCON.
A surrogate model-based method to obtain optimal design in spiral casing of Francis turbine
by Parameswara Rao Nakkina, K. Arul Prakash, G. Saravana Kumar
Abstract: Numerical simulations of fluid flow through various spiral casings like accelerated, free vortex and decelerated type with different aspect ratios (AR) are carried out to construct surrogates. These surrogates are utilised for analysing design sensitivity of spiral casing to obtain its optimal design. Responses like spiral velocity coefficient, total pressure loss and average radial velocities obtained from numerical computations are used for surrogates' construction. Different surrogate models considered are Kriging, polynomial response surface, support vector regression and weighted average surrogate. Surrogates are validated using average error analysis for the selection of best surrogate. Weighted average surrogate performs well in most of the cases among all responses. Near optimal solutions obtained from the best surrogates are proposed.
Keywords: spiral casing; spiral velocity coefficient; finite element method; surrogates; design sensitivity; aspect ratio; total pressure loss; average radial velocity; Kriging; optimisation.
A robust computational method for singularly perturbed system of 2D parabolic convection-diffusion problems
by Maneesh Kumar Singh, Srinivasan Natesan
Abstract: This article presents a numerical scheme to solve singularly perturbed system of 2D parabolic convection-diffusion problem exhibiting exponential boundary layers. The numerical scheme consists of a fractional implicit-Euler scheme on uniform mesh for time discretisation and the classical upwind scheme on a piecewise uniform Shishkin mesh for spatial discretisation. For the proposed scheme, the stability analysis is presented and parameter-uniform error estimates are derived. It is shown that the numerical scheme is uniformly convergent with respect to the singular perturbation parameter. The proposed method is applied to a test problem to verify theoretical results numerically.
Keywords: singularly perturbed system; 2D parabolic problems; boundary layers; Shishkin meshes; finite difference scheme; fractional implicit-Euler method; uniform convergence.
Kernel function-based interior-point algorithms for linear optimisation
by Bachir Bounibane, El Amir Djeffal
Abstract: We propose a primal-dual interior-point algorithm for linear optimisation based on a class of kernel functions which is eligible. New search directions and proximity measures are defined based on these functions. We derive the complexity bounds for large and small-update methods respectively. These are currently the best known complexity results for such methods.
Keywords: kernel function; linear optimisation; primal-dual interior-point methods; large-update methods.
Environmental assessment and dealing with undesirable outputs: DEA-based approach
by M. Zoriehhabib, M. Rostamy-Malkhalifeh, F. Hosseinzadeh-Lotfi
Abstract: This paper discusses an indirect portion oriented approach based on a traditional data envelopment analysis (DEA), requiring a proportional reduction of undesirable outputs. The proposed technique prepares an accommodation of undesirable factors in their primary form. In particular, this study is interested in the concept of weak disposability axioms. Given a decreased output vector, each decision making unit attempts to decrease a vector of desirable outputs proportionally with respect to the least wastage. These issues lead to the development of a model based on the weak disposable technology which can analyse the changeable production with reduction of both the desirable and undesirable outputs. Real-life data of 22 organisation for economic cooperation and development (OECD) countries is used in the empirical analysis. This analysis demonstrates that the proposed model is suitable for any assessment involving the proportional reduction of output indicators when undesirable outputs are presented. To achieve this aim the proposed epsilon-based model is applied to evaluate the efficiency of data for 27 Japanese electric power companies.
Keywords: data envelopment analysis; efficiency; undesirable output; weak disposability; abatement factor; portion oriented reduction.
Mathematical analysis and numerical simulation of a fractional reaction-diffusion system with Holling-type III functional response
by Kolade M. Owolabi
Abstract: In recent years, many investigators have questioned the use of convectional diffusion equation to model many physical or real life situations. As a result, fractional space derivatives have been proposed to model anomalous diffusion or related processes, where a particle plume spreads at inconsistent rate with the classical Brownian motion model. By replacing the second derivative in the classical diffusion model with fractional derivative, results to enhance a process known as superdiffusion. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. Analysis of the main equation guides in the correct choice of parameter values. We established the condition for local and global stabilities. We also show that the system undergoes a Hopf bifurcation subject to a small perturbation of the steady-state solution. The complexity of fractional derivative at some instances of order α for the superdiffusive scenario is demonstrated with some numerical experiments in one, two and three dimensions. The effectiveness of the numerical method is demonstrated through numerical simulations to confirm the theoretical results.
Keywords: Fourier spectral method; Runge-Kutta method; fractional reaction-diffusion; Hopf bifurcation; oscillations; Holling-type III; predator-prey; stability analysis.