Stochastic Networks: Tractable Approaches for Identifying Strategic Paths

Abstract

In Chapter 1, we present a stochastic shortest path problem that we refer to as the Most Likely Path Problem (MLPP). We prove that optimal solutions to the MLPP are composed of optimal subpaths, a property which is essential for solving the classical deterministic shortest path problem. On series-parallel networks, we produce analytical bounds for the probability of the Most Likely Path (MLP), which we compute efficiently via dynamic programming and ordinal optimization.In Chapter 2, we present an algorithm for preprocessing a class of stochastic shortest path problems on directed acyclic networks. Our method significantly increases the utility of many existing frameworks. Given random costs with finite lower and upper bounds on each edge, our algorithm removes edges that cannot be in any optimal solution to the deterministic shortest path problem for any realization of the random costs. Although this problem is NP-complete, our algorithm efficiently preprocesses many edges in a given network and our computational results show that on average only .2% of the edges in the test problems remain unclassified after preprocessing.In Chapter 3, we introduce a methodology for generating scenario trees from independent samples via k-means clustering. The trees are then input into a well-known stochastic optimization model used for operation planning on the Brazilian power system. The dual prices from this model are by contract used in Brazil to determine prices paid to operators of hydroelectric energy plants. Our computational results provide key insights regarding the variability introduced into these dual prices by both the choice of scenario tree topology and by k-means clustering. We show that for sufficiently large scenario trees, the dual prices are quite stable.