Inland waterway efficiency through skipper collaboration and joint speed optimization

We address the problem of minimizing the aggregated fuel consumption by the vessels in an inland waterway, e.g., a river, with a single lock. The fuel consumption of a vessel depends on its velocity and the slower it moves, the less fuel it consumes. Given entry times of the vessels into the waterway and the deadlines before which they need to leave the waterway, we start from the optimal velocities of the vessels that minimize their private fuel consumption, where we assume selfish behavior of the skippers. Presence of the lock and possible congestion on the waterway make the problem computationally challenging. First, we prove that in general, a Nash equilibrium might not exist, i.e., if there is no supervision on the vessels’ velocities, there might not exist a strategy profile from which no vessel can unilaterally deviate to decrease its private fuel consumption. Next, we introduce simple supervision methods to guarantee the existence of a Nash equilibrium. Unfortunately, though a Nash equilibrium can be computed, the aggregated fuel consumption of such a stable solution can be high compared to the social optimum, where the total fuel consumption is minimized. Therefore, we propose a mechanism involving payments between vessels, guaranteeing a Nash equilibrium while minimizing the fuel consumption. This mechanism is studied for both the offline setting, where all information is known beforehand, and online setting, where we only know the entry time and deadline of a vessel when it enters the waterway.