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In order to understand how bunchgrasses achieve dominance over other plant growth forms and how they achieve dominance over one another in different environments, it is first necessary to develop a detailed understanding of how their growth strategy interacts with the resource limits of their environment. Two properties which have been studied separately in limited detail are architecture and disproportionate resource allocation. Architecture is the structural layout of organs and objects at different hierarchical levels. Disproportionate resource allocation is the manner in which resources are allocated across objects at each level of hierarchy. Clonal architecture and disproportionate resource allocation may interact significantly to determine the growth ability of clonal plants. These interactions have not been researched in bunchgrasses. This thesis employs a novel simulation technique, functional-structural plant modelling, to investigate how bunchgrasses interact with the resource constraints imposed in humid grasslands. An appropriate functional-structural plant model, the TILLERTREE model, is developed that integrates the architectural growth of bunchgrasses with environmental resource capture and disproportionate resource allocation. Simulations are conducted using a chosen model species Themeda triandra, and the environment is parameterised using characteristics of the Southern Tall Grassveld, a humid grassland type found in South Africa. Behaviour is considered at two levels, namely growth of single ramets and growth of multiple ramets on single bunchgrass clones. In environments with distinct growing and non-growing seasons, bunchgrasses are subjected to severe light depletion during regrowth at the start of each growing season because of the accumulation of dead material in canopy caused by the upright, densely packed manner in which they grow. Simulations conducted here indicate that bunchgrass tillers overcome this resource bottleneck through structural adaptations (etiolation, nonlinear blade mass accretion, residual live photosynthetic surface) and disproportionate resource allocation between roots and shoots of individual ramets that together increase the temporal resource efficiency of ramets by directing more resources to shoot growth and promoting extension of new leaves through the overlying dead canopy. The architectural arrangement of bunchgrasses as collections of tillers and ramets directly leads to consideration of a critical property of clonal bunchgrasses: tiller recruitment. Tiller recruitment is a fundamental discrete process limiting the vegetative growth of bunchgrass clones. Tiller recruitment occurs when lateral buds on parent tillers are activated to grow. The mechanism that controls bud outgrowth has not been elucidated. Based on a literature review, it is here proposed that lateral bud outgrowth requires suitable signals for both carbohydrate and nitrogen sufficiency. Subsequent simulations with the model provide corroborative evidence, in that greatest clonal productivity is achieved when both signals are present. Resource allocation between live structures on clones may be distributed proportionately in response to sink demand or disproportionately in response to relative photosynthetic productivity. Model simulations indicate that there is a trade-off between total clonal growth and individual tiller growth as the level of disproportionate allocation between ramets on ramet groups and between tillers on ramets increases, because disproportionate allocation reduces tiller population size and clonal biomass, but increases individual tiller performance. Consequently it is proposed that different life strategies employed by bunchgrasses, especially annual versus perennial life strategies, may follow more proportionate and less proportionate allocation strategies respectively, because the former favours maximal resource capture and seed production while the latter favours individual competitive ability. Structural disintegration of clones into smaller physiologically integrated units (here termed ramet groups) that compete with one another for resources is a documented property of bunchgrasses. Model simulations in which complete clonal integration is enforced are unable to survive for long periods because resource bottlenecks compromise all structures equally, preventing them from effectively overcoming resource deficits during periods when light is restrictive to growth. Productivity during the period of survival is also reduced on bunchgrass clones with full integration relative to clones that disintegrate because of the inefficient allocation of resources that arises from clonal integration. This evidence indicates that clonal disintegration allows bunchgrass clones both to increase growth efficiency and pre-empt potential death, by promoting the survival of larger ramet groups and removing smaller ramet groups from the system. The discrete nature of growth in bunchgrasses and the complex population dynamics that arise from the architectural growth and the temporal resource dynamics of the environment, may explain why different bunchgrass species dominate under different environments. In the final section this idea is explored by manipulating two species tiller traits that have been shown to be associated with species distributions across non-selective in defoliation regimes, namely leaf organ growth rate and tiller size (mass or height). Simulations with these properties indicate that organ growth rate affects daily nutrient demands and therefore the rate at which tillers are terminated, but had only a small effect on seasonal resource capture. Tiller mass size affects the size of the live tiller population where smaller tiller clones maintain greater numbers of live tillers, which allows them to them to sustain greater biomass over winter and therefore to store more reserves for spring regrowth, suggesting that size may affect seasonal nitrogen capture. The greatest differences in clonal behaviour are caused by tiller height, where clones with shorter tillers accumulate substantially more resources than clones with taller tillers. This provides strong evidence there is trade-off for bunchgrasses between the ability to compete for light and the ability to compete for nitrogen, which arises from their growth architecture. Using this evidence it is proposed that bunchgrass species will be distributed across environments in response to the nitrogen productivity. Shorter species will dominate at low nitrogen productivity, while taller species dominate at high nitrogen productivity. Empirical evidence is provided in support of this proposal. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2005.

Algorithms that use relational data are increasingly used to allocate resources within society. As researchers and decision-makers have adapted the role of algorithms from a descriptive one (describing patterns in data) to a prescriptive one (making decisions in predictive systems), there is an increasing concern that algorithms may replicate and even amplify societal bias, allocating worse or less resources to minorities and underrepresented groups. This dissertation proposes methodology for diagnosing when and how algorithms amplify inequality on networks as well as designing interventions for mitigating algorithmic bias. We leverage methods from network modeling, algorithmic game theory, and fair machine learning to uncover the root driver of bias in network data and to leverage this knowledge in order to design fair algorithms. In this thesis, we mostly focus on unsupervised learning problems, which present unique challenges that require a multi-faceted approach. We propose a unifying formulation for unifying different problems in unsupervised learning on networks and use it to propose methods to find the root cause of bias through modeling patterns of connections and embeddings. We leverage this knowledge to design fairer algorithms as well as to define diagnoses metrics for evaluating inequality before and after an algorithm is introduced. Furthermore, we argue for the need to bridge optimization-based learning and utility-based learning in creating stable, efficient, and useful systems. We use network models and mathematical formulations of distributional inequality in diagnosing the algorithmic amplification of bias in social recommendations and ranking algorithms. We find that the most common and neutral algorithms may further underrepresent minority groups in creating new connections or achieving high levels of visibility in networks that exhibit competition in increasing social capital and homophily (the tendency of people to connect with those similar to them). We uncover the role of homophily in helping a minority group overcome their initial disadvantage and we leverage it to design fairer information campaigns that equitable distribute messages across a population. Akin to this goal, we incorporate notions of utility and welfare in our algorithmic design, re-designing heuristics for grouping and clustering that improve the diversity of groups while preserving their usefulness, with applications in political and educational districting. Overall, this set of results aims to investigate the impact of algorithms on the outcomes of different populations and to open new avenues for inter-disciplinary research methods that can alleviate algorithmic bias. We close by discussing connections between different fields and methods as well as directions for future research.

Location-allocation problems can be described generically as follows: Given the location or distribution (perhaps, probabilistic) of a set of customers and their associated demands for a given product or service, determine the optimum location of a number of service facilities and the allocation of products or services from facilities to customers, so as to minimize total (expected) location and transportation costs. This study is concerned with a particular subclass of location-allocation problems involving capacitated facilities and a continuum of demand. Specifically, two minisum, network-based location-allocation problems are analyzed in which facilities having known finite capacities are to be located so as to optimally supply/serve a known continuum of demand. The first problem considered herein, is an absolute p-median problem in which p > l capacitated facilities are to be located on a chain graph having both nodal and link demands, the latter of which are defined by nonnegative, integrable demand functions. In addition, the problem is balanced, in that it is assumed the total demand equals the total supply. An exact solution procedure is developed, wherein the optimality of a certain location-allocation scheme (for any given ordering of the facilities) is used to effect a branch and bound approach by which one can identify an optimal solution to the problem. Results from the chain graph analysis are then used to develop an algorithm with which one can solve a dynamic, sequential location-allocation problem in which a single facility per period is required to be located on the chain. Finally, an exact solution procedure is developed for locating a capacitated, absolute 2-median on a tree graph having both nodal and link demands and for which the total demand is again equal to the total supply. This procedure utilizes an algorithm to construct two subtrees, each of whose ends constitute a set of candidate optimal locations for one of the two elements of an absolute 2-median. Additional localization results are used to further reduce the number of candidate pairs (of ends) that need to be considered, and then a post-localization analysis provides efficient methods of comparing the relative costs of the remaining pairs. / Ph. D.

The IEEE 802.16 standard popularly known as WiMAX is at the forefront of the technological drive. Achieving high system throughput in these networks is challenging due to interference which limits concurrent transmissions. In this thesis, we study routing and link scheduling inWiMAX mesh networks. We present simple joint routing and link scheduling algorithms that have outperformed most of the existing proposals in our experiments. Our session based routing and links scheduling produced results approximately 90% of a trivial lower bound. We also study the problem of quality of service (QoS) provisioning in WiMAX mesh networks. QoS has become an attractive area of study driven by the increasing demand for multimedia content delivered wirelessly. To accommodate the different applications, the IEEE 802.16 standard defines four classes of service. In this dissertation, we propose a comprehensive scheme consisting of routing, link scheduling, call admission control (CAC) and channel assignment that considers all classes of service. Much of the work in the literature considers each of these problems in isolation. Our routing schemes use a metric that combines interference and traffic load to compute routes for requests while our link scheduling ensures that the QoS requirements of admitted requests are strictly met. Results from our simulation indicate that our routing and link scheduling schemes significantly improve network performance when the network is congested. / ix, 77 leaves : ill. ; 29 cm

Fairness emerges as a vital concern to decision makers as crucial as efficiency, if not more important. Fair operations decisions are aimed at distributive justice in various scenarios. In this dissertation, we study two examples of distributively fair decision making in operations research, a dynamic fair allocation problem and a subpopulational robustness assessment problem for machine learning models. We first study a dynamic allocation problem in which 𝑇 sequentially arriving divisible resources are to be allocated to a number of agents with concave utilities. The joint utility functions of each resource to the agents are drawn stochastically from a known joint distribution, independently and identically across time, and the central planner makes immediate and irrevocable allocation decisions. Most works on dynamic resource allocation aim to maximize the utilitarian welfare, i.e., the efficiency of the allocation, which may result in unfair concentration of resources on certain high-utility agents while leaving others' demands under-fulfilled. In this work, aiming at balancing efficiency and fairness, we instead consider a broad collection of welfare metrics, the Hölder means, which includes the Nash social welfare and the egalitarian welfare. To this end, we first study a fluid-based policy derived from a deterministic surrogate to the underlying problem and show that for all smooth Hölder mean welfare metrics it attains an 𝑂 (1) regret over the time horizon length 𝑇 against the hindsight optimum, i.e., the optimal welfare if all utilities were known in advance of deciding on allocations. However, when evaluated under the non-smooth egalitarian welfare, the fluid-based policy attains a regret of order 𝛩 (√𝑇). We then propose a new policy built thereupon, called Backward Infrequent Re-solving (𝖡𝖨𝖱), which consists of re-solving the deterministic surrogate problem at most 𝑂 (log 𝑇) times. We show under a mild regularity condition that it attains a regret against the hindsight optimal egalitarian welfare of order 𝑂 (1) when all agents have linear utilities and 𝑂 (log 𝑇) otherwise. We further propose the Backward Infrequent Re-solving with Thresholding (𝖡𝖨𝖱𝖳) policy, which enhances the (𝖡𝖨𝖱𝖳) policy by thresholding adjustments and performs similarly without any assumption whatsoever. More specifically, we prove the (𝖡𝖨𝖱𝖳) policy attains an 𝑂 (1) regret independently of the horizon length 𝑇 when all agents have linear utilities and 𝑂 (log²⁺^𝜀) otherwise. We conclude by presenting numerical experiments to corroborate our theoretical claims and to illustrate the significant performance improvement against several benchmark policies. The performance of ML models degrades when the training population is different from that seen under operation. Towards assessing distributional robustness, we study the worst-case performance of a model over 𝒂𝒍𝒍 subpopulations of a given size, defined with respect to core attributes 𝑍. This notion of robustness can consider arbitrary (continuous) attributes 𝑍, and automatically accounts for complex intersectionality in disadvantaged groups. We develop a scalable yet principled two-stage estimation procedure that can evaluate the robustness of state-of-the-art models. We prove that our procedure enjoys several finite-sample convergence guarantees, including 𝒅𝒊𝒎𝒆𝒏𝒔𝒊𝒐𝒏-𝒇𝒓𝒆𝒆 convergence. Instead of overly conservative notions based on Rademacher complexities, our evaluation error depends on the dimension of 𝑍 only through the out-of-sample error in estimating the performance conditional on 𝑍. On real datasets, we demonstrate that our method certifies the robustness of a model and prevents deployment of unreliable models.

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.