International Journal of Mathematical Modelling and Numerical Optimisation (20 papers in press)
Regular Issues
 New SemiAnalytical Method for Solving Two Point Nth Order
Fuzzy Boundary Value Problem
by A.L.I. JAMEEL, Nidal Ratib Anakira, ABEDELKARREM Alomari, Doaa Alsharo, Azizan Saaban Abstract: In this paper, a semi analytical algorithm, namely Homotopy Analysis Method (HAM) is presented for the first time to obtain approximate analytical solutions of nth order two point fuzzy boundary value problems (TPFBVP) involving ordinary differential equations. This method allows for the solution of the TPFBVP 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 to control the convergence region of the infinite series solution. Numerical experiment is tested to highlight the important features of the HAM algorithm. Comparison of the HAM results and available exact solution reveal that this algorithm is effective and simple and for solving TPFBVP. Keywords: Fuzzy numbers; Two point fuzzy boundary value problems; Homotopy Analysis Method.
 BOUNDARY ELEMENT APPROACH OF SOLVING FREDHOLM AND VOLTERRA INTEGRAL EQUATIONS
by Sudeshna Banerjea, Rumpa Chakraborty, Anushree Samanta Abstract: A simple numerical technique namely boundary element method is employed here tornsolve Fredholm and Volterra integral equations of second kind. In this method the integral equation is converted in to a system of linear algebraic equations by discretizing the range of the integration and interval of definition into finite number of line elements. By solving the system of linear equations by standard technique the solution of the integral equation is obtained for points in each line element. The method is computationally very simple and gives quite accurate results. Keywords: Fredholm and Volterra integral equation. Boundary element method. Linernelements; System of linear equations; Approximate solution; Relative error.
 New Hybrid Algorithm based on Nonmonotone spectral gradient and Simultaneous Perturbation
by Zineb Tabbakh, Rachid Ellaia, Abderrahmane Habbal Abstract: In this paper, we introduce a new hybrid method called Nonmonotone Spectral Gradient and Simultaneous Perturbation (NSGSP).
It combines the advantages of nonmonotone spectral gradient (NSG), and simultaneous perturbation (SP) methods.
The main idea of our approach is to use the simultaneous perturbation (SP) method in order to get a non expensive estimate of the gradient, and exploit the good properties of the nonmonotone spectral gradient (NSG) method in order to compute an efficient line search.
Several numerical experiments are provided. The results indicate that the new method is effective and outperforms most of other popular methods. Keywords: Nonmonotone line search; spectral gradient method; Simultaneous Perturbation.
 Hierarchical Multiobjective Approach for Optimizing RFID Reader Deployment
by Abdelkader Raghib, Badr Abou El Majd Abstract: Radio frequency identification (RFID) systems have received increased attention from academia and practitioners. RFID facilitates data acquisition and storage; it can provide accurate and realtime data without human intervention. This work proposes a new approach based on multilevel strategy with the aim to solve the deployment of RFID readers problem by focusing on the coverage, the interference and the number of deployed readers as the main objectives to optimize. In this way, Speed Constrained Multiobjective Particle Swarm Optimization algorithm (SMPSO) is adopted in order to minimize the total number of readers required to identify all tags in a given area. The proposed multilevel approach based on SMPSO Algorithm is more efficient and converge rapidly which makes it ideal for our research involving optimal deployment of RFID readers. The simulation results show the effectiveness of the proposed approach. Keywords: SMPSO Algorithm; RFID Network Planning; Deployment; MultiLevel Algorithm; MultiObjective Problem; Optimization.
 Mathematical Modelling for Low Temperature Batch Drying of Paddy using Fluidized Bed Technology
by Binayak Pattanayak, SibaShankar Mohapatra, Harish Chandra Das Abstract: Rice is considered as the major staple food throughout India. Drying is one of the major alternatives available for proper storage of rice. The drying process can be suitably represented by many mathematical models. In this study ten different mathematical models are chosen for describing the mathematical behaviour of drying. The experimental data obtained from the drying of raw rice at different temperatures, drying air velocity and relative humidity are fitted with the chosen mathematical models. The best suitable model to describe the drying behaviour of raw rice is determined by nonlinear regression analysis. The statistical indicators Correlation coefficient (R2), Root Mean Square Error (RMSE) and Square of Squares due to Error (SSE) showed that Midilli et al. model is suitable model for all operating conditions. The coefficients of Midilli et al. model is also expressed as a function of drying air temperature, velocity and inventory of paddy. Keywords: Paddy; Mathematical Modelling; Drying; Correlation coefficient (R2); Root Mean Square Error (RMSE) and Sum of Squares due to Error (SSE).
 Optimal Orbital Elements of Binary Stars Based on Flower Pollination Algorithm
by Ibrahim Elhenawy, Mohamed AbdelBasset, Nabil Diab Abstract: In the field of eclipsing binary stars, the relative coordinate positions of each star with respect to the other stars in the system are adapted. Therefore the orbital parameters of the system will be adapted by the criterion of dynamical and physical (photometric) properties. Orbital elements have been determined for binaries from a combination of velocities in the literature. Formulas that are obtained for computing the orbital elements are determined from the directions of the bases of space data. So the present work is a new presented method for the determination of optimal orbital elements of binary stars based on flower pollination algorithm. The proposed algorithm seek to solve the problem of detecting global minimizers as a function instead of problem of finding periodic orbits, to get a best fit of keplerian and phase curves. The experimental results implemented on star η Bootis of MK type G0 IV. Demonstrate that the Flower Pollination Algorithm (FPA) reaches good results compared to the results of the standard particle swarm optimization and other optimization algorithms. Keywords: Flower pollination algorithm; Metaheuristics; Optimization; orbital element; binary stars.
 A surrogate model based method to obtain optimal design in spiral casing of Francis turbine
by Parameswara Rao Nakkina, Arul Prakash Karaiyan, Saravana Kumar Gurunathan Abstract: Numerical simulations of fluid flow through various spiral casings like accelerated, free vortex and decelerated type with various aspect ratios (AR) are carried out to construct surrogates. These surrogates are utilized for analyzing 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 model. 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 ratios; Total pressure loss; Average radial velocity; Kriging; Optimization.
 A robust computational method for singularly perturbed system of 2D parabolic convectiondiffusion problems
by Manees Kumar Singh, Srinivasan Natesan Abstract: This article presents a numerical scheme to solve singularly perturbed system of 2D parabolic convectiondiffusion problem exhibiting exponential boundary layers. The numerical scheme consists of a fractional implicitEuler scheme on uniform mesh for time discretization and the classical upwind scheme on a piecewise uniform Shishkin mesh for spatial discretization. For the proposed scheme, the stability analysis is presented, and parameteruniform 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 of 2D parabolic problems; Boundary layers; Shishkin meshes; Finite difference scheme; Fractional implicitEuler method; Uniform convergence.
 Kernel Function Based InteriorPoint Algorithms for Linear Optimization
by El Amir Djeffal, Bounibane Bachir Abstract: We propose a primaldual interiorpoint algorithm for linear optimization (LO) based on a class of kernel functions which is eligible. New search directions and proximity measures are defined based on these functions. We show that the algorithm has Opnp+12p logn"andOppn logn"; p > 0 complexity results for small and largeupdate methods, respectively. These are the best known complexity results for such methods. Keywords: Kernel function ; Linear optimization ; Primaldual interiorpoint method ; Largeupdate methods.
 Environmental assessment and dealing with undesirable outputs: DEA based approach
by Monireh Zoriehhabib, Mohsen Rostamy Malkhalifeh, Farhad 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 analyze the changeable production with reduction of both the
desirable and undesirable outputs. Real life data of 22 OECD (Organization
for Economic Cooperation and Development) 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 epsilonbased 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 reactiondiffusion system with Hollingtype III functional response
by Kolade 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 classicalrn 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 highdimensional predatorprey reactiondiffusion system with Hollingtype III functional response, where the usual secondorder derivatives give place to a fractional derivative of order $alpha$ with $1 Keywords: Fourier spectral method; RungeKutta method; Fractional reactiondiffusion; Hopf bifurcation; Oscillations; Hollingtype III; Predatorprey; Stability analysis.
 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 CaputoFabrizio 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 CaputoFabrizio 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; CaputoFabrizio fractional derivative; equality of mixed partial; existence and uniqueness; numerical solutions.
 Permutation flow shop scheduling problem under nonrenewable 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 reallife manufacturing environments, jobs for their processing on machines may require additional nonrenewable 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 nonrenewable 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 nonrenewable resources availability. Keywords: Scheduling; Permutation flow shop; Nonrenewable resources; Optimisation; Mathematical programming; Genetic Algorithm; Local search.
 Performance analysis and design optimisation of 3Ø 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 neutralpoint clamping (NPC), flying capacitor (FC), cascaded Hbridge (CHB) and hybrid cascaded Hbridge (HCHB). Furthermore, there is no need of transformers in this whole concept. Hence, it avoids the bulky installations. The 3Ø 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); multilevel inverter (MLI); neutralrnpoint clamping (NPC); flying capacitor (FC); cascaded HBridgern(CHB) hybrid cascaded HBridge (HCHB); induction motor (IM).
 Mathematical modelling of human cardiovascularrespiratory 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 reemerged recently leading to global pandemic of 201516. Prevention of ZIKV infection is a challenge in absence of antiviral agents and an effective vaccine. We formulated a susceptibleexposedinfectedrecovered (SEIR) mathematical model for mosquitoborne transmission of ZIKV. The model was used to estimate different parameters using the data obtained from Puerto Rico during 201516 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: Selfregulatory Perpetrator
by Erik LealEnríquez, Aime Renata GutiérrezAntúnez 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 selfregulation of a perpetrator. This parameter of selfregulation is modeled by two changes of states: one of selfcontrol 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 (semistable) 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 morethanmodest 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.
