## Forthcoming articles

### International Journal of Mathematical Modelling and Numerical Optimisation

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 International Journal of Mathematical Modelling and Numerical Optimisation (18 papers in press) Regular Issues The Integral-Differential and Integral Approach for the Estimation of the Classical Lennard-Jones and Biswas-Hamann Potentials.   by Samuel Surulere, Michael Shatalov, Andrew Mkolesia, Julius Ehigie Abstract: Many well-known semiempirical potential energy functions have been used to construct potential energy curves from the physical or chemical properties of atoms. In this study, we identify the Lennard-Jones and Biswas-Hamann potential parameters and use these to calculate and reconstruct potential energy curves using experimental data sets of gold atom. Two different approaches are studied in detail. The Lennard-Jones potential yielded complex conjugate eigenvalues for both approaches. Numerical estimates proved the considered approaches gives better approximations as constructed and reconstructed potential energy curves were almost graphically indistinguishable. Keywords: Potential parameters; Objective functions; Potential energy curves; Objective Least Squares function. Method of solving optimal design problems based on flexible tolerance strategy   by Larysa Korotka, Dmitriy Zelentsov Abstract: On solving problems of optimal design, structures working in aggressive external environments, a modified method is proposed, which is based on the flexible tolerance method. Based on the information about the degree of closeness of the point of the solution space to the local extremum point, which is received by the neural network controller, its parameters change. For this purpose, various matrices of neural network synapses, trained for different precisions of calculating the functions of constraints, are used. This strategy is used to modify the flexible tolerance method, based on the use of a neural network controller. As a criterion of the flexible tolerance, the error of calculating the constraint functions is used. Keywords: neural network regulator; flexible tolerance; optimal design; fuzzy variables. Mathematical Modelling of Piezoelectric Elastic Materials   by Boudjour Allaoua, Mohamed Dalah Abstract: We consider a quasistatic contact modelled with the regularized friction law for electro-elastic and the foundation is assumed to be electrically conductive. This regularization is obtained by replacing the function $j(.)$ by the function $j_{\rho}(.)$, where $\rho$ is a strictly positive parameter. The classical formulation for the antiplane problem is formulated as a time dependent of corresponding variational formulation and is solved by the Banach fixed-point theorem and classical results for variational inequalities. We provide a weak formulation of the contact problem in the form variational system in which the unknowns are the displacement and the stress fields, then we establish the existence of a unique weak solution to the model. Finally, we have given a convergence criterion of the solution as the paramater of regularization $\rho$ converges to zero. Keywords: Quasistatic contact; electro-elastic material; antiplane; regularised friction law; weak solution; variational formulation. Backtracking Immune Algorithm for Continuous Multi-objective Optimization   by Ahmed TCHVAGHA ZEINE Abstract: In this paper, a new Backtracking Immune Algorithm for Continuous Multi-objective Optimization (BIAMO) is proposed. It uses the update archive to sort the non-dominated solutions of the Pareto front and the mutation and crossover operators of the Backtracking Search Algorithm (BSA). Experimental results are produced for various benchmark problems and for a variety of engineering design problems. They show that, compared to the recent multi-objective optimization evolutionary algorithms, the proposed algorithm improves not only the convergence capacity but also preserves the diversity of the population. eleven benchmark problems and two engineering design problems have been solved and the obtained results were compared with other well-known optimizers. The obtained\r\nresults demonstrate that, the proposed algorithm requires less number of function evaluations and in most cases gives better results compared to other considered algorithms.\r\n Keywords: Multi-objective optimization; Evolutionary Algorithms; Backtracking Search; Hybrid recombination; Hybrid mutation. Study of Effects of Toxicants and Acidity on Oxygen-Dependent Aquatic Population: A Mathematical Model   by Preety Kalra, Shreya Tangri Abstract: The rapidly elevating contamination of water bodies is one of the present-day critical problems occurring worldwide. It is a well-determined fact that the direct discharge of household wastes, industrial effluents, chemicals, acids and other pollutants into water bodies along with eutrophication are degrading the water quality. Washing-off of land pollutants in water bodies coupled with acid rain further increase the toxicity and acidity of water. These phenomena are leading to decrease of dissolved oxygen in water which is endangering the growth and survival of populations residing in water. To investigate the stress of increasing toxicity and acidity of water bodies like ponds, lakes and rivers on the aquatic population due to decrease in dissolved oxygen, a non-linear mathematical model is proposed. The model is analysed and the stability conditions are derived in terms of the parameters. To substantiate the analytical results, numerical simulations are carried out using Matlab. Keywords: Aquatic Population; Toxicity; Acidity; Dissolved Oxygen; Model; Stability. A new fuzzy transportation algorithm for finding fuzzy optimal solution   by Farikhin Farikhin, Muhammad Sam'an Abstract: Based on the literature, many algorithms have been proposed to solve fuzzy transportation problems in real life. Among the existing algorithms, some do not use the ranking method to convert fuzzy numbers into crisp numbers, whereas some use ranking methods that fail to rank non-normal fuzzy numbers correctly and compensate with the use of areas. Therefore, in this paper, a new fuzzy transportation algorithm, i.e., the NNWC, NLC and NVA, is used to solve a fuzzy transportation problem. The ranking methods involving non-normal trapezoidal fuzzy numbers and triangular fuzzy numbers as well as a new ranking method using total integral value are used to solve case studies 1 and 2, and the results are compared with the results from existing methods. Because of the proposed method is a direct extension of a classical method, it is reasonable to apply it to real-life transportation problems. Keywords: fuzzy transportation problem; new fuzzy transportation algorithm;new total integral value; non-normal trapezoidal fuzzy numbers. A Metropolis within Gibbs algorithm for knowledge Discovery in Language Assessments   by Mengta Chung Abstract: The Reduced Reparameterized Unified Model (RRUM) has been frequently used in language assessment studies. The objective of this research is to advance an MCMC algorithm for the Bayesian RRUM. The algorithm starts with estimating correlated attributes. Using a saturated model and a binary decimal conversion, the algorithm transforms possible attribute patterns to a Multinomial distribution. Along with the likelihood of an attribute pattern, a Dirichlet distribution is used as the prior to sample from the posterior. The Dirichlet distribution is constructed using Gamma distributions. Correlated attributes of examinees are generated using the inverse transform sampling. Model parameters are estimated using the Metropolis within Gibbs sampler sequentially. Two simulation studies are conducted to evaluate the performance of the algorithm, and an empirical study uses the ECPE data obtained from the CDM R package are performed. Keywords: CDM; RUM; RRUM; Q-matrix; Bayesian; MCMC. A new efficient hybrid conjugate gradient method based on LS-DY-HS conjugate gradient parameter   by Nirmalya Ku Mohanty, Rupaj K. Nayak Abstract: A nonlinear conjugate gradient method solves unconstrained noptimisation problem based on an efficient line search technique and maintains a decent direction search (in case of a minimisation problem) with the help of conjugate gradient parameter. In this paper, a new hybrid conjugate gradient method based on a hybrid conjugate gradient parameter k is proposed. The proposed k combines linearly the conjugate gradient parameters of LS, DY and HS method. The present work also discusses the global convergence of the modified algorithm with inexact line search. Moreover, the proposed method is tested on the unconstrained problems from the library CUTEr [GOT15] and the results have been compared with the other state of the art algorithms. The results in the numerical experiment show that the proposed hybrid algorithm is efficient. Keywords: Nonlinear conjugate gradient method; Unconstrained optimisation;rnLine search; Hybrid Conjugate gradient parameter; Descent property. A Collection of Bivariate Nonlinear Optimization Test Problems with Graphical Representations   by Petru A. Simionescu Abstract: Plotting objective functions, whenever possible, allows constraint activity, modality and convexity, as well as landscape features like basins, plateaus, valleys and saddle points, to be intuitively identified. Knowledge of these properties allow researchers to best select a set of functions for rating existing optimization algorithms, when developing new search algorithms or for their control parameter tune-up. This paper offers a selection of 30 bivariate constrained objective functions accompanied by graphical representations. It is the first time a significant number of constrained-functions with their plots are gathered in one place. Also provided are the minima, maxima (global and local) and saddle point information of these functions, very useful in all optima studies. Keywords: nonlinear optimization; inequality constrains; objective functions; all optima; visualization. Discrete Time Portfolio Optimization managing Value at Risk under heavy tail return distribution   by Subhojit Biswas, Diganta Mukherjee Abstract: We consider an investor, whose portfolio consists of a single risky asset and a risk free asset, who wants to maximize his expected utility of the portfolio subject to the Value at Risk assuming a heavy tail distribution of the stock price return. We use Markov Decision Process and dynamic programming principle to get the optimal strategies and the value function which maximize the expected utility for parametric as well as non parametric distributions. Due to lack of explicit solution in the non-parametric case, we use numerical integration for optimization. Keywords: Portfolio Optimization; Markov Decision Process; Parametric distribution; Non parametric distribution. MINLP models for dimensional synthesis of planar four-bar linkages   by Andrea Piga Carboni, Laio Oriel Seman, Eduardo Camponogara Abstract: Most works on mechanism synthesis consider models that employ trigonometric functions and rely on derivative-free optimization, such as bio-inspired and hybrid heuristics. Derivative-free strategies have the advantage of being applied directly to explore the design space. However, such strategies require tuning to achieve satisfactory results and do not provide certificates of solution quality. To this end, this work presents two alternative models: the first one consists of a trigonometric functions based model; while the second relies solely on natural coordinates, in which the nonlinear terms are bilinear. Both models result in Mixed-Integer Nonlinear Programming (MINLP) formulations. Mechanism constraints are conveniently expressed in this formulation, including Grashof's condition, prescribed timing, and increasing and decreasing angle sequence. In standard numerical cases of four-bar linkage path synthesis, global and local MINLP solvers yielded provably optimal designs in some cases and in others they outperformed works from the literature. Keywords: Dimensional Synthesis; Mathematical Programming; MINLP; Bilinear Terms; Four-bar Linkage. Marine reserve and its consequences in a predator-prey system for ecotourism and fishing   by Debabrata Das, T.K. Kar Abstract: This article investigates the effects of introducing density regulation in terms of theta logistic, over a fishery with the implementation of marine reserve. For various density regulations, we discuss the inter-relationship between fishing effort and marine reserve with fish biomass, catches and total rent. It also depicts the impacts of exploitation up to maximum sustainable yield (MSY) and maximum economic yield (MEY) both for single species and predator prey system. We also illustrate some economic consequences of introducing marine reserve both on fishery and ecotourism including the effect of density dependence on these issues. It is interpreted that introducing reserve area, extinction due to over exploitation could be prevented but when there is a predator prey interaction, the benefit of reserve is diminished to some extent. It is found that the safe minimum biomass level (SMBL) also increases with increasing reserve size. With the increase of density regulation, fish biomass increases both in single species and predator-prey model, though in the absence of reserve catches vanishes as effort exceeds a certain level, irrespective of given importance to density regulation. Finally, it is observed that the optimal reserve size corresponding to fisheries management is less than the reserve size when both the fishery rent and ecotourism rent are considered jointly. Keywords: Predator-prey; theta logistic; marine reserve; MSY; ecotourism. Modelling Biological Control Dynamics of the Banana Weevil Cosmopolites sordidus (Germar) by Generalist Predators in a Banana Agroecosystem   by Eliab Horub Kweyunga, Julius Tumwiine, Eldad B. Karamura Abstract: The biological control dynamics of the banana weevil Cosmopolites sordidus (Germar) by the generalist predator, P. javanus are studied by adopting a logistic model incorporating ratiodependent functional response. The steady states are obtained and their stability established. The expression for the banana weevil intrinsic growth rate, a key threshold that governs the control or persistence of the banana weevil is derived and its implications studied. Analysis of the model reveals that the model can have at least one and at most two equilibria depending on the relative sizes of the intrinsic growth rate vis-vis the carrying capacity and the predation rate. The banana weevil intrinsic growth rate and the predation rate are identified as the target parameters in managing infestation of the banana weevil. Numerical simulations are carried out to validate the theoretical findings. The study highlights the need for implementation of a complete integrated pest management (IPM) package against the banana weevil. Keywords: banana weevil; biological control; logistic growth; integrated pest management; ratio-dependent functional response; transcritical bifurcation. Global stability of a fractional order eco-epidemiological system with infected prey   by Mahmoud Moustafa, Mohd Mohd, Ahmad Ismail, Farah Abdullah Abstract: A fractional order eco-epidemiological system with infected prey is proposed and studied. Threshold parameters are obtained to determine the existence and stability conditions of equilibrium points of the proposed fractional order system. In order to clarify the characteristics of the proposed fractional order system, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions, as well as the stability conditions of the system are examined. The global properties of all biologically feasible equilibrium points are studied by constructing suitable Lyapunov functions. It is proved that all equilibrium points of the proposed fractional order system are globally stable whenever they are locally stable. Some numerical simulations are conducted to verify the theoretical results. Keywords: Eco-epidemiological system; Predator-prey; Fractional order system; Global stability. Guiding iterative optimisation methods to a predefined kind of optima for unconstrained optimisation problems   by Christina Nikolakakou, Athanasia Papanikolaou, Eirine Nikolopoulou, Theodoula Grapsa, George Androulakis Abstract: One of the most fundamental issues in the field of mathematical optimisation is the convergence of an iterative optimization method and by this we are referring to two things. First, will the method find an optimum and second, will this optimum be a local one or a global one? A recently proposed technique [NGNA15] that is used in order to lessen the dependance a locally convergent iterative optimization method has on the initial guess, is exploited in this paper. A way so that such a method may be guided to a predefined kind of minimum (local or global) is presented. Well known test functions were used for experimentation. Statistical analysis was conducted by applying a logistic regression classification model on data arisen from the numerical results which confirmed that iterative optimisation methods can be guided to a predefined kind of optimum. Keywords: Unconstrained Optimisation; Priorities; Global minimum; Local minimum; Prediction; Classification; Hypothesis Testing; Logistic Regression. A bioeconomic model applied to a small pelagic species along the Moroccan coast   by Imane AGMOUR, Meriem BENTOUNSI, Naceur ACHTAICH, Youssef EL FOUTAYENI Abstract: This paper proposes a case study of five small pelagic species: sardina pilchrdus, sardinella, engraulis encrasicolus, scomber scombrus and trachurus exploited by seiners in the maritime zones of Morocco. A comparison is made between the fishing effort calculated using the generalized Nash equilibrium problem and the actual fishing effort given by the National Institute of Fisheries Research. In this work, fishing effort represents the number of fishing trips that must be made by seiners and that allows them to maximize their gain taking into account the preservation of marine species. As a result, we find that master data of the National Institute of Fisheries Research support the outcomes shown in the end of this paper. Keywords: Bio-economic model; Small pelagic species dynamics;\r\nFishing effort; Linear Complementarity Problems; Nash equilibrium. Numerical investigation for solutions and derivatives of singularly perturbed initial value problems   by Subal Ranjan Sahu, Jugal Mohapatra Abstract: This article proposes a hybrid scheme on layer-adapted meshes for solving singularly perturbed initialrnvalue problem depending on a parameter. Layer-adapted meshes namely standard Shishkin mesh and modified Shishkin mesh (Bakhvalov-Shishkinrnmesh and Vulanovi'{c} mesh) are considered. The hybrid scheme is a combinationrnof second order central difference scheme on the fine mesh and a modified midpoint upwindrnscheme on the coarse mesh. The error analysis is carried out. We establish a second order parameter uniformrnconvergence rate for the numerical solution and also for the scaled numerical derivative.rnIt is also shown that the modified Shishkin mesh and graded mesh likernGartland Shishkin mesh and Duran Shishkin mesh give better results than the standardrnShishkin mesh. In order to illustrate the efficiency of thernproposed method, some numerical experiments are shown which support the theoretical findings. Keywords: Singular perturbation; Initial value problem; Hybrid scheme; Layer adapted meshes; Uniform convergence. A Hybrid Backtracking Search Algorithm for Energy Management in a Microgrid   by Zineb Garroussi, Rachid Ellaia, El-Ghazali Talbi, Jean-yves Lucas Abstract: Recently, due to the high penetration of distributed generators and storages, and the rapid growth of environmental concerns, energy optimization plays a vital role in the operation of microgrids. In this paper, we propose a hybrid backtracking search matheuristic algorithm with a multiobjective indirect encoding (H-MOBSA) for the multiobjective mixed-integer non-linear formulation of economic-emission dispatch in a grid-connected microgrid. In this approach, each partial discrete solution represented by the backtracking search algorithm is associated with a set of Pareto solutions of the related continuous subproblem where the fitness and diversity assignments are adapted to take into account all associated decoded solutions. The proposed algorithm aims to provide generation planning and the battery size that satisfy the trade-off between the total operating cost and pollutant emissions under equality and inequality constraints. To show the effectiveness of our approach, the proposed H-MOBSA is tested on typical MG and compared to other algorithms. Keywords: Backtracking Search Algorithm; battery storage sizing; distributed generation; matheuristics; microgrid.