Investment planning of a road link

Abstract

The purpose of this paper is to determine the optimal investments of an existing road link over time. The problem is formulated in terms of optimal control and solved by Pontryagin's maximum principle. Three state variables of the road are considered: the smoothness of the road pavement surface, the volume of traffic, and the capacity. The control variables are: investment in smoothness and in capacity. Optimality is considered to be that investment programme for smoothness and capacity which maximizes the integral of net benefits over a finite or infinite time horizon.

The time path of the investment in smoothness is uniquely determined by a saddle point solution. There are three possible solutions for the investment in capacity. Either the road will be widened at the initial time of the system, or at a later point in time, or never. This depends on the time path of the shadow price of capacity relative to the constant marginal cost to invest in capacity. Finally, a budget constraint to the Ministry of Transport is imposed. As a result, the pattern of the time paths does not change in general.