Resumen
In the current literature about queuing pricing, the optimal time-varying toll and adjustable step toll schemes have been developed for the Suez Canal to eliminate or alleviate the inefficiency of a large number of ships waiting in line to enter the canal. The purpose of the above two tolling schemes is to solve the serious queuing problem at the canal anchorage, which is completely different from the current navigation toll levied by the Suez Canal to recover the operation and maintenance costs incurred by ships passing through the canal. Because the two developed tolling schemes are complicated in model structure and inconvenient in practical operation, to solve these problems, this paper uses a new approach of the mid-point theorem to establish the optimal step toll scheme as alternative pricing to ease queuing for the Suez Canal. All variables for the optimal step toll scheme are able to be solved by the mid-point theorem, and these solutions show obvious regularities as the number of tolling steps increases progressively. By analyzing these regularities, we obtain three critical formulas for these solutions, such as the toll amount and the corresponding start and end times for each step of the optimal step toll scheme. The contents of the three formulas are extremely concise. These not only reinforce the theoretical basis of the existing queuing pricing model but can also promote implementing electronic toll collection with the optimal step toll scheme. Finally, we use the three formulas to provide a numerical analysis of the optimal triple-step toll scheme for the Suez Canal. This simulation example could be a practical reference for the canal authorities.