## Network modeling and design : a distributed problem solving approach

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This dissertation is concerned with developing new solution algorithms for network modeling and design
problems using a distributed problem solving approach. Network modeling and design are fundamental problems in the field of transportation science, and numerous transportation applications such as urban travel demand forecasting, congestion pricing, defining optimal toll values, and scheduling traffic lights all involve some form of network modeling or network design.
The first part of this dissertation focuses on developing a distributed scheme for the static traffic assignment problem, based on a spatial decomposition. The objective of the traffic assignment problem is to estimate traffic flows on a network and the resulting congestion considering the mutual interactions between travelers. A traffic assignment model takes as input the network topology, link performance functions, and a demand matrix indicating the traffic volume between each pair of origin-destination nodes. There are efficient algorithms to solve the traffic assignment problem, but, as computational hardware and algorithms advance, attention shifts to more demanding applications of the traffic assignment problem (bilevel programs whose solution often requires the solution of many traffic assignment problem instances
as subproblems, accounting for forecasting errors with Monte Carlo simulation of input parameters, and
broadening the geographic scope of models to the statewide or national levels.)
In Chapter 2, we propose a network contraction technique based on the theory of equilibrium sensitivity analysis. In the proposed algorithm, we replace the routes between each origin-destination (OD) pair with a single artificial link. These artificial links model the travel time between the origin and destination nodes of each OD pair as a function of network demands. The network contraction method can be advantageous in network design applications where many equilibrium problems must be solved for different design scenarios. The network contraction procedure can also be used to increase the accuracy
of subnetwork analysis. The accuracy and complexity of the proposed methodology are evaluated using
the network of Barcelona, Spain. Further, numerical experiments on the Austin, Texas regional network validate its performance for subnetwork analysis applications.
Using this network contraction technique, we then develop a decentralized (distributed) algorithm
for static traffic assignment in Chapter 3. In this scheme, which we term a decentralized approach to the static traffic assignment problem (DSTAP), the complete network is divided into smaller networks, and the algorithm alternates between equilibrating these networks as subproblems, and master iterations using a simplified version of the full network. The simplified network used for the master iterations is based
on linearizations to the equilibrium solution for each subnetwork obtained using sensitivity analysis techniques. We prove that the DSTAP method converges to the equilibrium solution on the complete network, and demonstrate computational savings of 35-70% on the Austin network. Natural applications of this method are statewide or national assignment problems, or cities with rivers or other geographic features where subnetworks can be easily defined.
The second part of this dissertation, found in Chapter 4, deals with network design problems. In a network design problem, the goal is to optimize an objective function (minimize the travel time, pollution, maximize safety, social welfare, etc.) by making investment decisions subject to budget and feasibility constraints. Network design is a bi-level problem where the leader chooses the design parameters, and travelers, as followers, react to the leader’s decision by changing their route. These problems are hard to solve, and distributed problem solving approach can be used to develop an efficient framework for scaling these
problems.
In the proposed distributed algorithm for network design problems, different planning agencies may have different objective functions and priorities, while a regional agent (state or federal officials) allocates the finding between the urban cities. In this model, the urban planning agencies do their own planning and design independently while capturing the system-level effects of their local decisions and plans. The
regional agent has limited and indirect authorities over the subnetworks through budget allocation. In
addition to computational advantages for traditional bi-level network design problems, the proposed algorithm can be used to model the linkage between different entities for multi-resolution applications. We develop a solution algorithm based on a sensitivity-analysis heuristic, and test our algorithm on two case studies: a hypothetical network composed of two copies of Sioux Falls network, and the Austin regional network. We evaluate the correctness of the decentralized algorithm, and discuss the benefits of the algorithm in modeling the global impacts of local decisions. Furthermore, the implementation of distributed algorithm on Austin regional network demonstrates a computational saving of 22%.