A probit-based stochastic user equilibrium assignment model

Stochastic methods of traffic assignment have received much less attention in the literature than those based on deterministic user equilibrium (UE). The two best known methods for stochastic assignment are those of Burrell and Dial, both of which have certain weaknesses which have limited their usefulness. Burrell’s is a Monte Carlo method, whilst Dial’s logit method takes no account of the correlation, or overlap, between alternative routes. This paper describes, firstly, a probit stochastic method (SAM) which does not suffer from these weaknesses and which does not require path enumeration. While SAM has a different route-finding methodology to Burrell, it is shown that assigned flows are similar. The paper then goes on to show how, by incorporating capacity restraint (in the form of link-based cost-flow functions) into this stochastic loading method, a new stochastic user equilibrium (SUE) model can be developed. The SUE problem can be expressed as a mathematical programming problem, and its solution found by an iterative search procedure similar to that of the Frank–Wolfe algorithm commonly used to solve the UE problem. The method is made practicable because quantities calculated during the stochastic loading process make the SUE objective function easy to compute. As a consequence, at each iteration, the optimal step length along the search direction can be estimated using a simple interpolation method. The algorithm is demonstrated by applying it successfully to a number of test problems, in which the algorithm shows good behaviour. It is shown that, as the values of parameters describing the variability and degree of capacity restraint are varied, the SUE solution moves smoothly between the UE and pure stochastic solutions.