A mean-risk mixed integer nonlinear program for transportation network protection

This paper focuses on transportation network protection to hedge against extreme events such as earthquakes. Traditional two-stage stochastic programming has been widely adopted to obtain solutions under a risk-neutral preference through the use of expectations in the recourse function. In reality, decision makers hold different risk preferences. We develop a mean-risk two-stage stochastic programming model that allows for greater flexibility in handling risk preferences when allocating limited resources. In particular, the first stage minimizes the retrofitting cost by making strategic retrofit decisions whereas the second stage minimizes the travel cost. The conditional value-at-risk (CVaR) is included as the risk measure for the total system cost. The two-stage model is equivalent to a nonconvex mixed integer nonlinear program (MINLP). To solve this model using the Generalized Benders Decomposition (GBD) method, we derive a convex reformulation of the second-stage problem to overcome algorithmic challenges embedded in the non-convexity, nonlinearity, and non-separability of first- and second-stage variables. The model is used for developing retrofit strategies for networked highway bridges, which is one of the research areas that can significantly benefit from mean-risk models. We first justify the model using a hypothetical nine-node network. Then we evaluate our decomposition algorithm by applying the model to the Sioux Falls network, which is a large-scale benchmark network in the transportation research community. The effects of the chosen risk measure and critical parameters on optimal solutions are empirically explored

A Mean-Risk Mixed Integer Nonlinear Program for Network Protection

Many of the infrastructure sectors that are considered to be crucial by the Department of Homeland Security include networked systems (physical and temporal) that function to move some commodity like electricity, people, or even communication from one location of importance to another. The costs associated with these flows make up the price of the network\u27s normal functionality. These networks have limited capacities, which cause the marginal cost of a unit of flow across an edge to increase as congestion builds. In order to limit the expense of a network\u27s normal demand we aim to increase the resilience of the system and specifically the resilience of the arc capacities. Divisions of critical infrastructure have faced difficulties in recent years as inadequate resources have been available for needed upgrades and repairs. Without being able to determine future factors that cause damage both minor and extreme to the networks, officials must decide how to best allocate the limited funds now so that these essential systems can withstand the heavy weight of society\u27s reliance. We model these resource allocation decisions using a two-stage stochastic program (SP) for the purpose of network protection. Starting with a general form for a basic two-stage SP, we enforce assumptions that specify characteristics key to this type of decision model. The second stage objective---which represents the price of the network\u27s routine functionality---is nonlinear, as it reflects the increasing marginal cost per unit of additional flow across an arc. After the model has been designed properly to reflect the network protection problem, we are left with a nonconvex, nonlinear, nonseparable risk-neutral program. This research focuses on key reformulation techniques that transform the problematic model into one that is convex, separable, and much more solvable. Our approach focuses on using perspective functions to convexify the feasibility set of the second stage and second order conic constraints to represent nonlinear constraints in a form that better allows the use of computational solvers. Once these methods have been applied to the risk-neutral model we introduce a risk measure into the first stage that allows us to control the balance between an efficient, solvable model and the need to hedge against extreme events. Using Benders cuts that exploit linear separability, we give a decomposition and solution algorithm for the general network model. The innovations included in this formulation are then implemented on a transportation network with given flow demand

Robust Modeling Framework for Transportation Infrastructure System Protection Under Uncertainty

This dissertation presents a modelling framework that will be useful for decision makers at federal and state levels to establish efficient resource allocation schemes to transportation infrastructures on both strategic and tactical levels. In particular, at the upper level, the highway road network carries traffic flows that rely on the performance of individual bridge infrastructure which is optimized through robust design at lower level. A system optimization model is developed to allocate resources to infrastructure systems considering traffic impact, which aims to reduce infrastructure rehabilitation cost, long term economic cost including travel delays due to realization of future natural disasters such as earthquakes. At the lower level, robust design for each individual bridge is confined by the resources allocated from upper level network optimization model, where optimal rehabilitation strategies are selected to improve its resiliency to hedge against potential disasters. The above two decision making processes are interdependent, thus should not be treated separately. Thus, the resultant modeling framework will be a step forward in the disaster management for transportation infrastructure network. This dissertation first presents a novel formulation and a solution algorithm of network level resource allocation problem. A mean-risk two-stage stochastic programming model is developed with the first-stage considering resources allocation and second-stages shows the response from system travel delays, where the conditional value-at-risk (CVaR) is specified as the risk measure. A decomposition method based on generalized Benders decomposition is developed to solve the model, with a concerted effort on overcoming the algorithmic challenges imbedded in non-convexity, nonlinearity and non-separability of first- and second- stage variables. The network level model focusing on traffic optimization is further integrated into a bi-level modeling framework. For lower level, a method using finite element analysis to generate a nonlinear relationship between structural performances of bridges and retrofit levels. This relationship was converted to traffic capacity-cost relationship and used as an input for the upper-level model. Results from the Sioux Falls transportation network demonstrated that the integration of both network and FE modeling for individual structure enhanced the effectiveness of retrofit strategies, compared to linear traffic capacity-cost estimation and conventional engineering practice which prioritizes bridges according to the severity of expected damages of bridges. This dissertation also presents a minimax regret formulation of network protection problem that is integrated with earthquake simulations. The lower level model incorporates a seismic analysis component into the framework such that bridge columns are subject to a set of ground motions. Results of seismic response of bridge structures are used to develop a Pareto front of cost-safety-robustness relationship from which bridge damage scenarios are generated as an input of the network level model

Models, Theoretical Properties, and Solution Approaches for Stochastic Programming with Endogenous Uncertainty

In a typical optimization problem, uncertainty does not depend on the decisions being made in the optimization routine. But, in many application areas, decisions affect underlying uncertainty (endogenous uncertainty), either altering the probability distributions or the timing at which the uncertainty is resolved. Stochastic programming is a widely used method in optimization under uncertainty. Though plenty of research exists on stochastic programming where decisions affect the timing at which uncertainty is resolved, much less work has been done on stochastic programming where decisions alter probability distributions of uncertain parameters. Therefore, we propose methodologies for the latter category of optimization under endogenous uncertainty and demonstrate their benefits in some application areas. First, we develop a data-driven stochastic program (integrates a supervised machine learning algorithm to estimate probability distributions of uncertain parameters) for a wildfire risk reduction problem, where resource allocation decisions probabilistically affect uncertain human behavior. The nonconvex model is linearized using a reformulation approach. To solve a realistic-sized problem, we introduce a simulation program to efficiently compute the recourse objective value for a large number of scenarios. We present managerial insights derived from the results obtained based on Santa Fe National Forest data. Second, we develop a data-driven stochastic program with both endogenous and exogenous uncertainties with an application to combined infrastructure protection and network design problem. In the proposed model, some first-stage decision variables affect probability distributions, whereas others do not. We propose an exact reformulation for linearizing the nonconvex model and provide a theoretical justification of it. We designed an accelerated L-shaped decomposition algorithm to solve the linearized model. Results obtained using transportation networks created based on the southeastern U.S. provide several key insights for practitioners in using this proposed methodology. Finally, we study submodular optimization under endogenous uncertainty with an application to complex system reliability. Specifically, we prove that our stochastic program\u27s reliability maximization objective function is submodular under some probability distributions commonly used in reliability literature. Utilizing the submodularity, we implement a continuous approximation algorithm capable of solving large-scale problems. We conduct a case study demonstrating the computational efficiency of the algorithm and providing insights

Market integration in network industries

What is the effect of product market integration on the market equilibrium in the presence of international network externalities in consumption? To address this question, we set up a spatial two-country model and we find that the economic forces at work may have an ambiguous effect on prices.compatibility, horizontal differentiation, network effect

Market integration and network industries

What is the effect of product market integration on the market equilibrium in the presence of international externalities in consumption ? To address this question, we set up a spatial two-country model and we find that the economic forces at work may have an ambiguous effect on prices.compatibility; horizontal differentiation; network effect