Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 37 / 2 / a / Solution

For a steady flow network, let each origin-destination class $k$ have fixed demand $d_k$. Route flows $h_r\ge 0$ satisfy ${\sum }_{r\in R_k}{h}_r=d_k$. The link flow is $y_j={\sum }_r{A}_{jr}{h}_r$, and a link has continuous travel delay ${\ell }_j(y_j)$. Its route delay is $c_r(h)={\sum }_j{A}_{jr}{\ell }_j(y_j)$.

A Wardrop equilibrium describes nonatomic traffic: each traveler is too small to alter aggregate delays by changing route. For each class there is a minimum delay ${\lambda }_k$ such thatThus all used routes of one class have equal minimum cost, and an unused route cannot offer a shorter trip. The condition concerns private travel time, rather than total network delay.

Equivalently, for every feasible route vector $\tilde{h}$,Indeed, each used route has class minimum cost, so reallocating demand cannot reduce the cost evaluated at the original flows; conversely a positive flow on a non-minimum route gives an improving transfer. This variational inequality form remains meaningful even when route costs are not separable link functions.