The general public considers Fixed-Route Transit (FRT) to be inconvenient while Demand-Responsive Transit (DRT) provides much of the desired flexibility with a door-to-door type of service. However, FRT is typically more cost efficient than DRT to deploy. Therefore, there is an increased interest in flexible transit services including all types of hybrid services that combine FRT and pure DRT. The demand-responsive feeder transit, also known as Demand-Responsive Connector (DRC), is a flexible transit service because it operates in a demand-responsive fashion within a service area and moves customers to/from a transfer point that connects to a FRT network. In this research we develop analytical models, validated by simulation, to design the DRC system. Feeder transit services are generally operated with a DRC policy which might be converted to a traditional FRT policy for higher demand. By using continuous approximations, we provide an analytical modeling framework to help planners and operators in their choice of the two policies. We compare utility functions of the two policies to derive rigorous analytical and approximate closed-form expressions of critical demand densities. They represent the switching conditions, that are functions of the parameters of each considered scenario, such as the geometry of the service area, the vehicle speed and also the weights assigned to each term contributing to the utility function: walking time, waiting time and riding time. We address the problem faced by planners in determining the optimal number of zones for dividing a service area. We develop analytical models representing the total cost functions balancing customer service quality and vehicle operating cost. We obtain close-form expressions for the FRT and approximation formulas for the DRC to determine the optimal number of zones. Finally we develop a real-case application with collected customer demand data and road network data of El Cenizo, Texas. With our analytical formulas, we obtain the optimal number of zones, and the times for switching FRT and DRC policies during a day. Simulation results considering the road network of El Cenizo demonstrate that our analytical formulas provide good estimates for practical use.

In this thesis, we use multitype Galton-Watson branching processes in random environments as individual-based models for the evolution of structured populations with both demographic stochasticity and environmental stochasticity, and investigate the phenotype allocation problem. We explore a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. When the population under consideration is large and the time scale is fast, we deduce the continuous approximation for multitype Markov branching processes in random environments. Many problems in evolutionary biology involve the allocation of some limited resource among several investments. It is often of interest to know whether, and how, allocation strategies can be optimized for the evolution of a structured population with randomness. In our work, the investments represent different types of offspring, or alternative strategies for allocations to offspring. As payoffs we consider the long-term growth rate, the expected number of descendants with some future discount factor, the extinction probability of the lineage, or the expected survival time. Two different kinds of population randomness are considered: demographic stochasticity and environmental stochasticity. In chapter 2, we solve the allocation problem w.r.t. the above payoff functions in three stochastic population models depending on different kinds of population randomness. Evolution is often understood as an optimization problem, and there is a long tradition to look at evolutionary models from a variational perspective. In chapter 3, we deduce a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process. Continuous approximation of branching processes is of both practical and theoretical interest. However, to our knowledge, there is no literature on approximation of multitype branching processes in random environments. In chapter 4, we firstly construct a multitype Markov branching process in a random environment. When conditioned on the random environment, we deduce the Kolmogorov equations and the mean matrix for the conditioned branching process. Then we introduce a parallel mutation-selection Markov branching process in a random environment and analyze its instability property. Finally, we deduce a weak convergence result for a sequence of the parallel Markov branching processes in random environments and give examples for applications.

This thesis focuses on the developing of statistical methodology in reliability and quality engineering, and to assist the decision-makings at enterprise level, process level, and product level. In Chapter II, we propose a multi-level statistical modeling strategy to characterize data from spatial logistics systems. The model can support business decisions at different levels. The information available from higher hierarchies is incorporated into the multi-level model as constraint functions for lower hierarchies. The key contributions include proposing the top-down multi-level spatial models which improve the estimation accuracy at lower levels; applying the spatial smoothing techniques to solve facility location problems in logistics. In Chapter III, we propose methods for modeling system service reliability in a supply chain, which may be disrupted by uncertain contingent events. This chapter applies an approximation technique for developing first-cut reliability analysis models. The approximation relies on multi-level spatial models to characterize patterns of store locations and demands. The key contributions in this chapter are to bring statistical spatial modeling techniques to approximate store location and demand data, and to build system reliability models entertaining various scenarios of DC location designs and DC capacity constraints. Chapter IV investigates the power law process, which has proved to be a useful tool in characterizing the failure process of repairable systems. This chapter presents a procedure for detecting and estimating a mixture of conforming and nonconforming systems. The key contributions in this chapter are to investigate the property of parameter estimation in mixture repair processes, and to propose an effective way to screen out nonconforming products. The key contributions in Chapter V are to propose a new method to analyze heavily censored accelerated life testing data, and to study the asymptotic properties. This approach flexibly and rigorously incorporates distribution assumptions and regression structures into estimating equations in a nonparametric estimation framework. Derivations of asymptotic properties of the proposed method provide an opportunity to compare its estimation quality to commonly used parametric MLE methods in the situation of mis-specified regression models.

Manufacturing outsourcing in the U.S. has never been stronger than it is today. Increased outsourcing has led to significant changes in the design of the retail distribution network. While the traditional distribution network had the manufacturing plants supplying goods to the retail stores directly, the off-shore manufacturing has increased the network's demand for transportation and warehousing to deliver the goods. Thus, most companies have a complex distribution network with several import and regional distribution centers (RDC). In this thesis, we study an integrated facility location and inventory allocation problem for designing a distribution network with multiple national (import) distribution centers (NDC) and retailers. The key decisions are where to locate the RDCs and how much inventory to hold at the different locations such that the total network cost is minimized under a pre-defined operational rule for the distribution of goods. In particular, the inventory cost analysis is based on the continuous review batch ordering policy and the base-stock policy. Both Type-I (probability of stock-outs) and Type-II (fill-rate) service level measures are used in the analysis. Two different models are presented in this thesis for solving the integrated facility location-inventory allocation problem. The first model, continuous approximation (CA), assumes the distribution network to be located in a continuous region and replaces the discrete store locations with a store density function. The second model is a discrete representation of the problem as a mixed integer programming problem. Both the models take a nonlinear form and solution techniques are developed using the theory of nonlinear programming and linear reformulation of nonlinear problems. The goal of the first part of the thesis is to model the problem using a modified CA approach and an iterative solution scheme is presented to solve it. The main contribution of this work lies in developing a refined CA modeling technique when the discrete data cannot be modeled by a continuous function. In addition, the numerical analysis suggests that the total network cost is significantly lower in the case of the integrated model as compared with the non-integrated model. It is also shown that the regular CA approach leads to a solution which is inferior to the solution obtained by the modified CA approach. Our analysis shows that the type of service measure used affects the network design. In the second part of the thesis, the problem is modeled as a nonlinear mixed integer program and a linear reformulation solution technique is proposed to obtain a lower bound on the original problem. Computational results are presented for small problem instances. We conclude this part of the thesis by presenting an integrated model when a base stock inventory policy is used. A drop-decomposition heuristic is proposed to solve this problem.

This dissertation contains two topics in operations research. The first topic is to design a distribution network to facilitate the repeated movement of shipments from many origins to many destinations. A method is developed to estimate transportation costs as a function of the number of terminals and moreover to determine the best number of terminals. The second topic is to study dynamics of a buyer's behavior when the buyer can buy goods through both option contracts and a spot market and the buyer attempts to learn the probability distribution of the spot price. The buyer estimates the spot price distribution as though it is exogenous. However, the spot price distribution is not exogenous but is endogenous because it is affected by the buyer's decision regarding option purchases.

In this thesis, we use multitype Galton-Watson branching processes in random environments as individual-based models for the evolution of structured populations with both demographic stochasticity and environmental stochasticity, and investigate the phenotype allocation problem. We explore a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. When the population under consideration is large and the time scale is fast, we deduce the continuous approximation for multitype Markov branching processes in random environments. Many problems in evolutionary biology involve the allocation of some limited resource among several investments. It is often of interest to know whether, and how, allocation strategies can be optimized for the evolution of a structured population with randomness. In our work, the investments represent different types of offspring, or alternative strategies for allocations to offspring. As payoffs we consider the long-term growth rate, the expected number of descendants with some future discount factor, the extinction probability of the lineage, or the expected survival time. Two different kinds of population randomness are considered: demographic stochasticity and environmental stochasticity. In chapter 2, we solve the allocation problem w.r.t. the above payoff functions in three stochastic population models depending on different kinds of population randomness. Evolution is often understood as an optimization problem, and there is a long tradition to look at evolutionary models from a variational perspective. In chapter 3, we deduce a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process. Continuous approximation of branching processes is of both practical and theoretical interest. However, to our knowledge, there is no literature on approximation of multitype branching processes in random environments. In chapter 4, we firstly construct a multitype Markov branching process in a random environment. When conditioned on the random environment, we deduce the Kolmogorov equations and the mean matrix for the conditioned branching process. Then we introduce a parallel mutation-selection Markov branching process in a random environment and analyze its instability property. Finally, we deduce a weak convergence result for a sequence of the parallel Markov branching processes in random environments and give examples for applications.

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In this dissertation, we study two important problems related to additive manufacturing (AM). In the first part, we investigate the economic feasibility of using AM to fabricate biomedical implants at the sites of hospitals AM versus traditional manufacturing (TM). We propose a cost model to quantify the supply-chain level costs associated with the production of biomedical implants using AM technology, and formulate the problem as a two-stage stochastic programming model, which determines the number of AM facilities to be established and volume of product flow between manufacturing facilities and hospitals at a minimum cost. We use the sample average approximation (SAA) approach to obtain solutions to the problem for a real-world case study of hospitals in the state of Mississippi. We find that the ratio between the unit production costs of AM and TM (ATR), demand and product lead time are key cost parameters that determine the economic feasibility of AM. In the second part, we investigate the AM facility deployment approaches which affect both the supply chain network cost and the extent of benefits derived from AM. We formulate the supply chain network cost as a continuous approximation model and use optimization algorithms to determine how centralized or distributed the AM facilities should be and how much raw materials these facilities should order so that the total network cost is minimized. We apply the cost model to a real-world case study of hospitals in 12 states of southeastern USA. We find that the demand for biomedical implants in the region, fixed investment cost of AM machines, personnel cost of operating the machines and transportation cost are the major factors that determine the optimal AM facility deployment configuration. In the last part, we propose an enhanced sample average approximation (eSAA) technique that improves the basic SAA method. The eSAA technique uses clustering and statistical techniques to overcome the sample size issue inherent in basic SAA. Our results from extensive numerical experiments indicate that the eSAA can perform up to 699% faster than the basic SAA, thereby making it a competitive solution approach of choice in large scale stochastic optimization problems.