Mathematical Modelling and Optimization of Engineering Problems

In this research, we consider a real-time order acceptance and scheduling (OAS) problem in metal additive manufacturing (MAM) production environment, where the manufacturer with multiple machines makes decisions on the acceptance and scheduling of dynamic arriving part orders simultaneously. The objective is to maximize profit per unit time within the planning horizon. An MAM machine is a kind of batch processing machine (BPM) in which a batch of non-identical parts can be processed simultaneously as a production job according to its capacity, and the process time of the job is a function of the properties of all parts assigned to this job as well as the specifications of the MAM machine to conduct this job. This is the first time that a real-time OAS problem is considered in MAM production environment with capacity and due date constraints. We define the problem and propose a mathematical formulation. As this problem is shown to be strongly NP-hard, meta-heuristic procedures based on various selection rules are proposed for the generation of feasible schedule results. The difference of bad schedule results from those good ones is investigated first according to the results obtained with the stochastic selection. Afterwards, the performance of non-random selection rules is evaluated by comparing with the best and the worst results from the stochastic selection. Experimental tests indicate that the proposed non-random selection rules are able to provide promising schedule results without iteration.

This chapter covers the fundamentals of energy economics by providing details of commonly used several optimization models such as economic dispatch, unit commitment, and optimal power flow. Later, various methods of energy storage modeling are defined. Finally, a two-stage optimization problem is introduced to support the development of nationwide energy storage policy. A case study with a complete data set for the power grid at Turkey is evaluated and the results are discussed in terms of effectiveness of the model.

In this chapter, we aim to apply the approach of weak Pontryagin’s principles to a class of discrete time infinite horizon optimal growth problems. The idea of this approach is to transform the optimal growth problem into a dynamical system which is governed by a difference equation or a difference inequation. We establish necessary and sufficient conditions of optimality in terms of weak Pontryagin’s principles.

The aim of this paper is to predict different types of pathological subjects from a population through the physical observational variables (HB, RBC, MCH, WBC, MCV, HCT, MCHC, PLT), sex, and age. To achieve this, we have used a multilinear regression model. The obtained results in terms of the proposed model are seen to be accurate enough. Thus, the regression model is expected to provide important preliminary information to the clinicians in order to plan appropriate treatment programs for their patients. This work is carried out in terms of the dataset consisting of 539 subjects provided from blood laboratories in Iraq. The resulting model is the regression model consisting of observations (the blood variables, age, and sex) and the type of anemia. Therefore, the model is developed for predicting the type of anemia to help for diagnosis of these diseases.

A two-dimensional laminar flow driven by fluid injection through porous surface which represents an incompressible fluid inside a filtration chamber during extraction of particles from the fluid is investigated. The study constructs a mathematical model that represents internal flow field during filtration process proficiently by using basic conservation laws of mass, momentum and energy. For better understanding of dynamics of the case study, solutions that lead to stable filtration process (balanced dynamical system) are obtained, hence provide an insight of the important dynamics that lead to an optimal filtration process. To solve a system of partial differential equations representing the internal flow field similarity transformation based on Lie group method is used to reduce the system to ordinary differential equations. Thereafter, double perturbation method is used to find semi-analytical solutions of the reduced system. Effects of various parameters that arise from the configuration (design) of the filter are presented graphically and analysed to show the connection between the case study and findings.

The purpose of micrositing is to find an optimal layout of a group of wind turbines in order to extract maximum power production from a wind farm. In the case of wind farm design, the wake interactions between wind turbines are one of the most critical subjects that should be considered. Because, not only they cause a decrease in wind speed which causes less energy production but also they lead to blade damages on wind turbines and high maintenance costs. Offering high quality layout solutions that needs to be decided before the design of a wind farm will lead to high profits for wind farm investors. Providing options to the investors regarding the quantity and optimal locations of wind turbines is the main concern of this paper, since erecting more turbines in certain locations sometimes may cause energy losses. In this study, a series of latitude-longitude data was generated by scanning the digital map of the wind farm site. The determination of locations where turbines can be placed is presented as a new approach in terms of wind farm area characterization. By doing so, a continuous search space is generated that brings more flexibility to mobilize wind turbines. The solution starts with a heuristic approach, and then a genetic algorithm is followed to find optimal placements of wind turbines considering minimizing the wake loss. At last, the optimum locations of the wind turbines are obtained, and the maximum number of turbines is recommended for the given wind farm.

A wind turbine system with a self-excited induction generator (SEIG) is one of the best options as power supplier in rural areas because of its low cost, wide speed operation range, brushless structure and low maintenance. Beside its advantages, it has poor voltage and frequency regulation which depend on the generator speed, load impedance, excitation capacitance and magnetizing reactance. This restriction leads the researchers to select the best value of excitation capacitor to maintain the terminal voltage within the upper and lower acceptable limits. The determination of generator speed is another point to be focused to remain frequency at desired level.

In this paper, the response surface method (RSM) is applied in order to determine the optimal steady state performance for the SEIG instead of the commonly used nodal admittance method or the loop impedance technique. Proposed method does not need knowledge of induction machine parameters which makes it superior against classical methods. The main objective of the proposed approach is to determine the excitation capacitance and shaft speed to maintain a constant terminal voltage magnitude and frequency of the SEIG. Consequently, a response surface model is established in which the capacitance value and the shaft speed are considered the inputs, whereas the voltage magnitude and frequency are assumed to be the outputs. The simulation results show the effectiveness of the method proposed in this paper since the regression value (R ²) obtained was 99.98%. In particular, for a 4 kW squirrel cage induction generator with a 950 Ω resistive load per phase, the excitation capacitance and shaft speed were found to be 6.897 μF and 1504 rpm respectively. Moreover, the output voltage magnitude and frequency obtained were 230.2 V and 50 Hz, respectively.

Type 1 diabetes (T1D) is an autoimmune disease characterized by the destruction of β-cells, which are responsible for the production of insulin. T1D develops from an abnormal immune response, where specific clones of cytotoxic T-cells invade the pancreatic islets of Langerhans. Other immune cells, such as macrophages and dendritic cells, are also involved in the onset of T1D. In this paper, we generalize an integer-order model for T1D to include a non-integer order (also known as, fractional order (FO)) derivative. We study the local and the global stabilities of the disease-free equilibrium. Then, we discuss the results of the simulations of the FO model and investigate the role of macrophages from non-obese diabetic (NOD) mice and from control (Balb/c) mice in triggering autoimmune T1D. We observe that, for a value of the order of the fractional derivative equal to 1 (α = 1), an apoptotic wave can trigger T1D in NOD but not in Balb/c mice. The apoptotic wave is cleared efficiently in Balb/c mice preventing the onset of T1D. For smaller values of α, the inflammation persists for NOD and control mice. This alludes to a specific role of the order of the fractional derivative α in disease progression.

IPMS Gyroid was discovered by Alan Schoen in 1970. At that time, he was studying super-strong, super light structures. The mathematical equation of the IPMS Gyroid is complicated because it consists of elliptic integrals. However, a trigonometric equation gives an approximation to the IPMS Gyroid surface looks like the actual Gyroid. This trigonometric equation is considered to create a mathematical model of the IPMS Gyroid by using mathematical software called K3DSurf v0.6.2. Once mathematical model is created, then it is exported as “.obj” data format in order to print it by a 3D fused deposition modelling (FDM) printer. To print the mathematical model of the IPMS Gyroid a low-cost 3D FDM printer is used.

The industry of 3D printing is considered a part of 4th Industrial Revolution and it is the latest piece in a chain of visualisation techniques. Production of complex mathematical model such as an IPMS Gyroid is impossible with traditional chip removal methods. But using 3D printing technology allows us to fabricate such models quite easily. The objective of this study is to fabricate a complex mathematical model of an IPMS Gyroid in order to use it for active teaching and learning of mathematics and to prove that the equations are not only a mathematical expressions but also they are tangible solid objects.