Network Routing problems focus on exploiting the network-based struc- ture of a mathematical optimization problem to establish e cient solutions that are tailored to the problem at hand. The topic of this dissertation relates to a speci c class of network routing problems, those in which the properties of the nodes and/or links in the network can be represented as instances of a particular network-state realization, where the set possible network-state can be represented by a discrete probability distribution. The main contribution of this research is to formalize the de nition of such families of network-states, a construct we de ne as Stochastic-State Networks (SSN), and show that certain properties of such networks can allow for the systematic development of exact and heuristic solution procedures for a speciric class of network routing problems. The class of network problems considered are those in which dynamic routing decisions are seeked, and where information about the network can only be gathered through direct observation of the instantiation of the stochastic elements of the network. Two speci c instances of routing problems are considered: a dynamic instance of a Traveling Salesman Problem, and a routing problem in the presence of stochastic link failures. Exact methods and heuristics are developed by exploiting the underlaying stochastic-state network formulation and numerical results are presented. / text

published_or_final_version / Mathematics / Master / Master of Philosophy

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The main goal is to develop decision policies for individual forest stand management. It addresses three major areas of interest in the optimal management of individual forest stands: incorporating a two-species growth and yield model into a single stand management model, incorporating a comprehensive list of management options into a single stand management model, and incorporating uncertainty into a single stand management model. Dynamic programming (DP) is a natural framework to study forest management with uncertainty. The forest stand management problem, as modelled in this thesis, has a large dimensional state space with a mix of discrete and continuous state variables. The DP model used to study this problem is solved by value iteration with the objective of understanding infinite horizon policies. However, since some of the state variables are continuous, all states can’t be examined in an attempt to create the cost-to-go function. Therefore, the cost-to-go function value is calculated at a given stage of the algorithm at a finite set of state points and then the cost-to-go values are approximated on the continuous portion of the state space using a continuous function. All of this is done with random processes impacting state transitions. With the mixed-species growth model developed in this thesis, a comprehensive list of management options can be incorporated into the DP model and, with the addition of uncertainty from sources such as market prices and natural disasters, near optimal stand management policies are developed. Solving the DP model with the required level of detail lead to the development of insight into function fitting on continuous state spaces and to the development of cost-to-go function approximation bounds. Studying the policies shows that the addition of uncertainty to the model captures the dynamics between market prices and stand definitions, and leads to policies that are better suited to decision making in a stochastic environment, when compared with policies that are developed with a deterministic model. Enough precision is built into the DP model to give answers to typical questions forest managers would ask.

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