Thesis (Ph.D. (Statistics)) -- University of Limpopo, 2017 / Operations Research (OR) is a scientific method for developing quantitatively well-grounded recommendations for decision making. While it is true that it uses a variety of mathematical techniques, OR has a much broader scope. It is in fact a systematic approach to solving problems, which uses one or more analytical tools in the process of analysis. Over the years, OR has evolved through different stages. This study is motivated by new real-world challenges needed for efficiency and innovation in line with the aims and objectives of OR – the science of better, as classified by the OR Society of the United Kingdom. New real-world challenges are encountered on a daily basis from problems arising in the fields of water, energy, agriculture, mining, tourism, IT development, natural phenomena, transport, climate change, economic and other societal requirements. To counter all these challenges, new techniques ought to be developed. The growth of global markets and the resulting increase in competition have highlighted the need for OR techniques to be improved. These developments, among other reasons, are an indication that new techniques are needed to improve the day-to-day running of organisations, regardless of size, type and location. The principal aim of this study is to modify and develop new OR techniques that can be used to solve emerging problems encountered in the areas of linear programming, integer programming, mixed integer programming, network routing and travelling salesman problems. Distribution models, resource allocation models, travelling salesman problem, general linear mixed integer ii programming and other network problems that occur in real life, have been modelled mathematically in this thesis. Most of these models belong to the NP-hard (non-deterministic polynomial) class of difficult problems. In other words, these types of problems cannot be solved in polynomial time (P). No general purpose algorithm for these problems is known. The thesis is divided into two major areas namely: (1) network models and (2) resource allocation and distribution models. Under network models, five new techniques have been developed: the minimum weight algorithm for a non-directed network, maximum reliability route in both non-directed and directed acyclic network, minimum spanning tree with index less than two, routing through 0k0 specified nodes, and a new heuristic to the travelling salesman problem. Under the resource allocation and distribution models section, four new models have been developed, and these are: a unified approach to solve transportation and assignment problems, a transportation branch and bound algorithm for the generalised assignment problem, a new hybrid search method over the extreme points for solving a large-scale LP model with non-negative coefficients, and a heuristic for a mixed integer program using the characteristic equation approach. In most of the nine approaches developed in the thesis, efforts were done to compare the effectiveness of the new approaches to existing techniques. Improvements in the new techniques in solving problems were noted. However, it was difficult to compare some of the new techniques to the existing ones because computational packages of the new techniques need to be developed first. This aspect will be subject matter of future research on developing these techniques further. It was concluded with strong evidence, that development of new OR techniques is a must if we are to encounter the emerging problems faced by the world today. Key words: NP-hard problem, Network models, Reliability, Heuristic, Largescale LP, Characteristic equation, Algorithm.

In the first part of the thesis, the molecular structure and electronic properties of phospholipids at the single molecule level and also for a monolayer structure are investigated via ab initio calculations under different degrees of hydration. The focus of the study is on phosphatidylcholines, in particular dipalmitoylphosphatidylcholine (DPPC), which are the most abundant phospholipids in biological membranes. Upon hydration, the phospholipid shape into a sickle-like structure. The hydration dramatically alters the surface potential, dipole and quadrupole moments of the lipids, and probably guides the interactions of the lipids with other molecules and the communication between cells. The vibrational spectrum of DPPC and DPPC-water complexes are completely assigned and it is shown that water hydrating the lipid head groups enables efficient energy transfer across membrane leaflets on sub-picosecond time scales. Moreover, the vibrational modes and lifetimes of pure and hydrated DPPC lipids, at human body temperature, are estimated by performing ab initio molecular dynamics simulations. The vibrational modes of the water molecules close to the head group of DPPC are active in the frequency range between 0.5 - 55 THz, with a peak at 2.80 THz in the energy spectrum. The computed lifetimes for the high-frequency modes agree well with recent data measured at room temperature, where high-order phonon scattering is not negligible. The structure and auto-ionization of water at the water-phospholipid interface are investigated by ab initio molecular dynamics and ab initio Monte Carlo simulations using local density approximation and generalized gradient approximation for the exchange-correlation energy functional. Depending on the lipid head group, strongly enhanced ionization is observed, leading to dissociation of several water molecules into H+ and OH- per lipid. The results can shed light on the phenomena of the high proton conductivity along membranes that has been reported experimentally. In the second part of the thesis, Monte Carlo simulations of the lipid bilayer, on the basis of a coarse grained model, are performed to gain insight into the mechanical properties of planar lipid bilayers. By using a rescaling method, the Poisson's ratio is calculated for different phases. Additional information on the bending rigidity, determined from height fluctuations on the basis of the Helfrich Hamiltonian, allows for calculation of the Young's modulus for each phase. In addition, the free energy barrier for lipid flip-flop process in the fluid and gel phases are estimated. The main rate-limiting step to complete a flip-flop process is related to a free energy barrier that has to be crossed in order to reach the center of the bilayer. The free energy cost for performing a lipid flip-flop in the gel phase is found to be five times greater than in the fluid phase, demonstrating the rarity of such events in the gel phase. Moreover, an energy barrier is estimated for formation of transient water pores that often precedes lipid translocation events and accounts for the rate-limiting step of these pore-associated lipid translocation processes.