New Models for Truck Appointment Problem and Extensions

The problematic issues surrounding gate congestion at marine container terminals have been well documented. Random truck arrivals at maritime container terminals are one of the primary reasons for gate congestion. Gate congestion negatively affects the terminal’s and drayage firms’ productivity and the surrounding communities in terms of air pollution and noise. To alleviate gate congestion, more and more terminals in the U.S. are utilizing a truck appointment system (TAS). The first study proposes a novel approach for designing a Truck Appointment System (TAS) intended to serve both the marine container terminal operator and drayage operators. The aim of the proposed TAS is to minimize the impact to both terminal and drayage operations. In regard to terminal operations, the TAS seeks to distribute the truck arrivals evenly throughout the day to avoid gate and yard congestion. In regard to drayage operations, the TAS explicitly considers the drayage truck tours and seeks to provide appointment times such that trucks do not have to deviate greatly from their original schedule. The proposed TAS is formulated as a mixed integer nonlinear program (MINLP) and the model is solved using the Lingo commercial software. Experimental results indicate that the proposed TAS reduces the drayage operation cost by 11.5% compared to a TAS where its aim is only to minimize gate queuing time by making truck arrivals uniform throughout the day. The second study proposes a novel approach to modeling the TAS to better capture the multi-player game (i.e., interplay) between the terminal and drayage firms regarding appointments. A multi-player bi-level programming model is proposed with the terminal functions as the leader at the upper-level and the drayage firms function as followers at the lower-level. The objective of the leader (the terminal) is to minimize the gate waiting cost of trucks by spreading out the truck arrivals, and the objective of the followers (drayage firms) is to minimize their own drayage cost. To make the model tractable, the bi-level model is transformed to a single-level problem by replacing the lower-level problem with its equivalent Karush–Kuhn–Tucker (KKT) conditions. For comparison purposes, a single-player version of the TAS model is also developed. Experimental results indicate that the proposed multi-player model yields a lower gate waiting cost compared to the single-player model and that it yields higher cost savings for the drayage firms as the number of appointments per truck increases. Moreover, the solution of the of multi-player model is less sensitive to objective function coefficients across problem sizes compared to the single-player model. Lastly, the third study develops a truck appointment system (TAS) considering variability in turn time at the container terminals. The consideration of this operational characteristic is crucial for optimal drayage scheduling. The TAS is formulated as a stochastic model and solved using the Sample Average Approximation (SAA) algorithm. Using turn time distributions obtained from actual data from a U.S. port, a series of experiments is designed to evaluate the effectiveness of the proposed stochastic TAS model compared to the deterministic version where an average turn time is used instead of a distribution. Numerical experiment results demonstrate the benefit of the stochastic TAS model given its lower drayage cost error by 3.9% compared to the deterministic TAS model. This result implies that the schedules produced by the stochastic TAS model are more robust and are able to accommodate a wider range of turn time scenarios. Another key takeaway from the experiment results is that the stochastic TAS model is more beneficial to utilize when the ratio of quotas to requested appointments is lower. Thus, in practice, when this ratio is more likely to be on the lower end, drayage companies would benefit more if the appointment schedule adopts the stochastic approach described in this paper