New solution approaches for the train load planning problem

Abstract

The present paper faces the train load planning problem in container terminals. The problem consists of assigning containers to rail wagons while maximizing the total priority of the containers loaded and minimizing the number of rehandles executed in the terminal yard. Two different heuristic approaches, based on an innovative way to compute weight limitations and on two 0/1 integer programming models, are proposed and compared on the basis of specific key performance indicators. The heuristic approaches are compared using random generated instances based on real-world data. An extensive computational analysis has been performed.

Appendix 1. The lever principles

Stability conditions are derived from lever principles, which state that two unequal forces, when acting in opposite directions, arrive at an equilibrium when the product of the magnitude of a generic force $\overrightarrow{F_1}$ and its lever arm $e_1$ (the distance of its point of application from the fulcrum), is equal to the product of the magnitude of a second force $\overrightarrow{F_2}$ with its corresponding lever arm $e_2$ ($\overrightarrow{F_1} \cdot e_1=\overrightarrow{F_2} \cdot e_2$). Note that a bogie—also called railroad truck or wheel truck—represents a structure underneath a train to which axles (and, hence, wheels) are attached through bearings.

To better clarify lever principles, refer to Fig. 12, levers of containers $c_1$ and $c_2$ ($e_1$ and $e_2$, respectively) are determined as the distance between their center of gravity—which should be in the middle of the container—and the attachment of one of the two bogies (note that containers’ levers are calculated in reference to the same bogie). Moreover, the distance (d) between bogies is known; finally, it is assumed that the tare mass of the wagon is equally distributed on the two wagon bogies.

Fig. 12

Lever principle for weight distribution on wagon bogies