This paper proposes a bi-level risk averse network design model for transportation networks with heterogeneous link travel time distributions. The objective of the network design is to minimize the total system travel time budget, which consists of the mean total system travel time and a safety margin. The design is achieved by selecting optimal link capacity expansions subject to a fixed expansion budget. Users' selfish behavior and risk attitude are captured in the lower-level traffic assignment constraints, in which travelers select routes to minimize their own path travel time budget. The properties of the design problem are analyzed analytically and numerically. The analysis shows that despite the lack of knowledge of travel time distributions, the probabilities that the actual total system travel time and the actual path travel time are respectively within the optimal total system travel time budget and the minimum path travel time budget under optimal design have lower bounds. The lower bounds are related to the system manager's and travelers' risk aversion. The optimal total system travel time budget is proven to be bounded below even when the link expansion budget is unlimited.

This paper proposes reliability-based system-optimal traffic assignment under supply uncertainty based on the concept of the total system travel time budget, and defines the price of anarchy for the corresponding user equilibrium traffic assignment. An analytical formula for a set of the upper bounds of the price of anarchy for the equilibrium assignment to the networks with polynomial link travel time functions is derived. These bounds are proven to be independent of the network topology and demands. The formula for the minimum upper bound is also derived and can be reduced to the upper bound formula for traditional user equilibrium traffic assignment as a special case. The price of anarchy for the traditional user equilibrium network design problem with polynomial link travel time functions is also proven to be bounded by the upper bound of that for traditional user equilibrium traffic assignment of the same instance, before any link expansion. The price of anarchy for the reliability-based user equilibrium network design problem with polynomial link travel time functions is also proven to be bounded by the set of upper bounds of that for reliability-based user equilibrium traffic assignment of the same instance, before any link expansion

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