The BPR Latency Function

The Bureau of Public Roads (BPR) latency function is the standard cost model for congested links in routing problems. For an air-route link carrying flow $x$ (aircraft per hour), with capacity $c$ and free-flow travel time $t_0,$ the travel time on the link is

$$t(x) = t_0\left(1 + \alpha\left(\dfrac{x}{c}\right)^{\beta}\right),$$

where $\alpha > 0$ and $\beta > 0$ are shape parameters. The ratio $x/c$ is called the volume-to-capacity ratio.

Notice that $t(0) = t_0,$ so an empty link runs at free-flow time. As $x$ grows, the term $\alpha(x/c)^{\beta}$ inflates the travel time, and $t(x)$ is strictly increasing in $x.$

For example, with $t_0 = 20$ min, $c = 50$ ac/hr, $\alpha = 0.2,$ $\beta = 3,$ and $x = 25$ ac/hr:

$$t(25) = 20\left(1 + 0.2\left(\dfrac{25}{50}\right)^3\right) = 20(1 + 0.2 \cdot 0.125) = 20 \cdot 1.025 = 20.5\text{ min.}$$