The railway between the towns of A and B spans 800 km through mountainous terrain. The total one-way travel time, t r , is 20 hrs and currently the...

The railway between the towns of A and B spans 800 km through mountainous terrain. The total one-way travel time, tr, is 20 hrs and currently the fare, cr, is £600/ton. As the service is used at low capacity tr is a constant, independent of the traffic volume Vr.

There is a lorry service competing with the railway in an approximately parallel route; its average speed is 50 km/hr and it charges a fare of £950/ton. There is a project to build a highway in order to replace the present road; it is expected that most of its traffic will continue to be heavy trucks.

The level-of-service function of the new highway has been estimated as:

tt = 7 + 0.08Vt (hours)

where Vt is the total flow of trucks per hour.

On the other hand the railway has estimated its demand function as follows:

(Vr/Vt) = 0.83(tr/tt)−0.8(cr/ct)−1.6

and it is expected that the total volume transported between the two towns, Vr + Vt, will remain constant and equal to 200 truckloads/hr in the medium term.

(a) Estimate the current modal split (i.e. volumes transported by rail and lorry).