A Smoothing Approach For Solving Equilibrium Transportation Problems

Traffic congestion has become a central problem for road transport and brings a challenge in most populated cities. In traditional Stackelberg game theory, transportation network design problem involving two players (leader-follower) can be described as a bilevel programming problem. The leader is the central authority dedicated to regulating traffic plan, and the follower represents the road users who minimize the travel cost/time based on Wardrop's user equilibrium conditions. These problems require the transport planners to tackle piecewise smooth decision variables such as prices, accessibility, equity and toll necessary for controlling transportation systems. The focus of this study is to propose a new numerical method for solving various road network design problems.In this dissertation, three types of user equilibrium transportation problems in practices are reformulated as bilevel programming problems. We propose smoothing approach based on smoothing F-B function for solving the MPCCs, which are equivalent to the bilevel pro-gramming problem. We demonstrate that the smoothing approach has the global convergence property when the parameter ??0. And we adopt Newton method to solve the smoothing subproblems. Transportation network design models concerned with adding or removing road links of a particular transportation network are network capacity expansion problems which aim to determine the set of link capacity expansion and its corresponding equilibrium flow that im-proves the system performance. A bilevel programming problem which involves two players can be applied to formulate this problem. In this study, we formulate the capacity expansion bi-level problem as a mathematical program with complementarity constraints (MPCC). We apply the perturbation based approach proposed in Chapter 3 in solving the developed problem and demonstrate the convergence properties by using the variational analysis techniques. Applied to different numerical experiments that appear in the various literature of transportation problems, the perturbation method generate better solutions than those produced by previously presented methods.Most importantly, this dissertation investigates the role of combining capacity network ex-pansion and road toll pricing problem as the decision variables, which have significant benefits in transportation systems. We formulate this problem as a mathematical program with complemen-tarity constraints (MPCC), which can be transformed to single-level continuously differentiable problem using a smoothing function based approach. The proposed approach is illustrated by simple numerical examples.The consequences associated with urban transportation problems, especially air pollution is the primary concerns requiring traffic planners/researchers to optimize more than one objective simultaneously. For that reason, various types of environmental functions have been incorporat-ed into the objective function of urban network design problem. In this thesis, the continuous network design problem (CNDP) considering emission is formulated as bilevel optimization problem. We model the bi-level CNDP considering emission optimization problem as math-ematical program with complementarity constraints (MPCC). The MPCC problem is approxi-mated by a single level nonlinear problem (NLP) using the perturbation based approach. The sequential quadratic program is used to solve the resulting single-level differentiable optimiza-tion problem. We present a simple numerical example to illustrate the ability of the proposed methodology and the numerical results are acceptable.