International Journal of Mathematical Modelling and Numerical Optimisation (20 papers in press)

Regular Issues

• Uniformly convergent numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Bakhvalov-Shishkin mesh  
  by Srinivasan Natesan, Abhishek Das 
  Abstract: In this article, we consider a class of singularly perturbed 2D delay parabolic convection-diffusion initial-boundary-value problems. To solve the problem numerically, we consider upwind finite difference scheme on a modified Shishkin mesh (Bakhvalov-Shishkin mesh) to discretize the domain in spatial directions and we apply implicit-Euler scheme for the time derivative on uniform mesh in the temporal direction. We derive some conditions on the mesh-generating functions which are useful for the convergence of the method, uniformly with respect to the perturbation parameter. We prove that the applied scheme on the Bakhvalov-Shishkin mesh is first-order convergent in the discrete supremum norm, which is optimal and does not require any extra computational effort compare to the standard Shishkin mesh. Numerical experiments verify the theoretical results.
  Keywords: Singularly perturbed 2D delay parabolic convection-diffusion problems; boundary layers; finite difference scheme; Bakhvalov-Shishkin mesh; uniform convergence.
  
  
• Information technology value model and its optimal application in IT-based firms  
  by Lukman Abdurrahman, Suhardi Suhardi, Armein Z.R. Langi, Togar M. Simatupang 
  Abstract: This paper proposes the developed business performance based on the IT value model, which offers superior performance with profit maximization due to increased revenue and reduced costs. The model stands on the resource-based view theory, the partial adjustment valuation theory, and the systems engineering approach. Therefore, the model consists of firm performance, firm core competence, firm capability, and information technology resource subsystems. Each subsystem has inputs, i.e. the regular capital (K), the regular labour expense (L), and IT capital (I) where the output is y (the gross revenue). In addition, the synthesis result states that the model with the structure of the composition of each subsystem with the addition operation is more acceptable. The data used came from Telkom, Indosat, and XL, which are telco firms in Indonesia. Similarly, the IT value model optimization testing displays that the model is able to provide superior performance with increased revenue, but at a lower cost. Thus, the proposed model can address the mentioned main problems. However, it needs further studies to enhance this model.
  Keywords: capital;systems engineering;information technology;model;partial adjustment;performance;revenue;cost;value.
  
  
• An inventory System with Two Heterogeneous Servers and Orbital Search for Feedback Customers  
  by Suganya C, Amirthakodi M, Sivakumar B 
  Abstract: This model deals a continuous review (s, S) inventory system two heterogeneous servers, say, server - 1, server - 2. Server - 1 serve for primary customers and server - 2 is served for both primary and feedback customers. The primary customers arrival according to a Markovian arrival process (MAP) and service time for both servers has exponential distribution. The lead time is assumed to be exponential distribution. The primary customers, who finds either two servers are busy or no item is in the stock, waits in the finite waiting hall. If the waiting hall is full, then arriving, customer considered to be lost. After the completion of service the primary customer will decide either to join the orbit, which is an infinite size, for additional service or leaves the system according to a Bernoulli trail. These orbiting (feedback) customers compete for their service according to constant retrial policy, and the service times for these feedback customers are assumed to be exponential distribution. After completing the service for feedback customers server - 2 becomes idle or busy for primary customer. If the server - 2 completes the service for a primary customer, then he searches the customers in the orbit or idle according to a Bernoulli trail. The results are illustrated numerically.
  Keywords: Markovian arrival process; Feedback customers; Heterogeneous serves; Service facility; Orbital search.
  
  
• The impact of hygiene care and maternal immunity on stability behavior of rotavirus infection model for children under the age of five in Thailand  
  by RATCHADA VIRIYAPONG, Gunyarat Koompawan 
  Abstract: Rotavirus infection is one of the significant causes of death in infants and children under five years of age in Thailand and worldwide. In this paper, a deterministic model for rotavirus transmission of children under five years old in Thailand incorporating with hygiene care control and maternal immunity is developed. The basic reproduction number (Rv) is derived and becomes a threshold indicating the stability of both disease-free and endemic equilibrium points. Through the analysis, the results show that the disease-free equilibrium point is locally asymptotically stable when Rv < 1 and the endemic equilibrium point is locally stable when Rv > 1 and satisfies the Routh-Hurwitz criteria. Further, numerical simulation demonstrates that both hygiene care and maternal immunity to the infection due to breastfeeding should be encouraged as strategy to control rotavirus infection.
  Keywords: diarrhea; rotavirus; mathematical modeling; basic reproduction number; hygiene care; breastfeeding.
  
  
• A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy  
  by P. Rajadurai 
  Abstract: This paper deals with a single server retrial queueing system with working vacations and vacation interruption. There are three different types of customers are considered, which are priority customers, ordinary customers and negative customers. The priority customers do not form any queue and have an exclusive preemptive priority to receive their services over ordinary customers. The negative customer is arriving during the service time of any positive customer, will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he will leave the system. As soon as the orbit becomes empty at the time of service completion, the server goes for a working vacation. The server works at a lower speed during a working vacation period. Using the supplementary variable technique, the steady state probability generating function of the system and its orbit are found. Some numerical examples are presented.
  Keywords: retrial queues; priority queues; negative customers; working vacations; supplementary variable technique;.
  
  
• New Semi-Analytical Method for Solving Two Point Nth Order Fuzzy Boundary Value Problem  
  by A.L.I. JAMEEL, Nidal Ratib Anakira, ABEDEL-KARREM 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.
  
  
• Portfolio Optimization with Nonlinear Transaction Costs  
  by Jimbo Henri Claver 
  Abstract: In this paper, we consider the portfolio selection problem, with nonlinear transaction costs, basic constraints and probabilistic constraints. Such a problem cannot be handled by the usual quadratic or convex optimization methods. We develop a heuristic method which yields to computation of efficient (suboptimal) solution of the problem. We describe our heuristic method for finding optimal portfolio based on solving many small optimization problems over large generation number, thus we obtain a good suboptimal solution. Experimental results are demonstrated with various widely used indexes: stocks in US market: U.S. three months treasury bills, U.S. long term government bonds, S&P 500, Wilshire 5000, NASDAQ, Lehman Brothers corporate bond index, EAFE foreign stock index, and Gold recorded from (Jan 2000 Jan 2008) with enhanced performance. Finally, our results suggest that nonlinear transactions costs improve considerably the value of optimal portfolio over investment period, especially for portfolio with smaller among of assets.
  Keywords: Portfolio selection; exponential transaction cost; efficient frontier; optimization; expected return; portfolio risk; portfolio gain.
  
  
• 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 real-time data without human intervention. This work proposes a new approach based on multi-level 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 Multi-objective 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 multi-level 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; Multi-Level Algorithm; Multi-Objective 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 non-linear 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 Abdel-Basset, 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; Meta-heuristics; 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 convection-diffusion problems  
  by Manees 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 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 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 of 2D parabolic problems; Boundary layers; Shishkin meshes; Finite difference scheme; Fractional implicit-Euler method; Uniform convergence.
  
  
• Kernel Function Based Interior-Point Algorithms for Linear Optimization  
  by El Amir Djeffal, Bounibane Bachir 
  Abstract: We propose a primal-dual interior-point 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 large-update methods, respectively. These are the best known complexity results for such methods.
  Keywords: Kernel function ; Linear optimization ; Primal-dual interior-point method ; Large-update 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 Co-operation 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 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 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 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 $alpha$ with $1 Keywords: Fourier spectral method; Runge-Kutta method; Fractional reaction-diffusion; Hopf bifurcation; Oscillations; Holling-type III; Predator-prey; 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 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.