International Journal of Mathematical Modelling and Numerical Optimisation (13 papers in press)

Regular Issues

• 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.
  
  
• Frequency regulation of a power system integrated with renewables using a novel DE-DA optimized controller.  
  by Sayantan Sinha, Ranjan Kumar Mallick, Srikanta Patnaik 
  Abstract: The proposed research paper mainly focuses on the automatic generation control of an interconnected power system integrated with solar power. The work has taken into consideration a two area power system consisting of a conventional thermal power source and a solar power plant in one area. The penetration of renewable sources give rise to deviations of frequency from their scheduled values. The AGC plays a very important role in minimizing the frequency deviations of the system by reducing the Area Control error to zero with the help of suitable controllers. A maiden attempt has been made to incorporate two degree of freedom proportional integral and derivative controller for AGC purposed. A novel attempt has been made to design a hybrid optimized Dragonfly algorithm- Differential Evolution technique for tuning the controller gains. The performance of the controller is also compared with the conventional PID controller under a system disturbance of 0.01 p.u when applied to area 1. The system performances are further analyzed with varying system parameters and different loading conditions. Comparisons with traditional PID controllers have also been done in terms of dynamic system parameters like settling time, maximum overshoot and minimum undershoot
  Keywords: AGC; Deregulated; benchmark; hybrid; PID; 2 DOF PID; renewables; two area.
  
  
• Numerical Analysis of the European and American Options with the SPH method.  
  by Abdelmjid Qadi El Idrissi, Boujemaa Achchab, Abdellahi Cheikh Maloum 
  Abstract: In this paper, we propose a numerical method to solve the European and the American options by using the SPH method. Because its robustness and efficacy, this numerical method has been widely applied in the computation of partial differential equations particularly in fluid dynamic. To model these financial options, we use the Black Scholes equation. It is a mathematical model consisting of a set of partial differential equation supplemented by some boundary conditions. We evaluate the accuracy of our numerical method by giving some comparisons between the analytic solution and the numerical simulation.
  Keywords: Black Scholes equation; European option; American option; SPH Method.