Truck scheduling in cross-docking terminals with fixed outbound departures

Abstract

At a cross-docking terminal, inbound shipments are directly transshipped across the terminal to designated outbound trucks, so that delays and inventories are kept as low as possible. We consider an operational truck scheduling problem, where a dock door and a start time have to be assigned to each inbound truck. A set of outbound trucks is scheduled beforehand and, therefore, departure times are fixed. If a shipment is not unloaded, transshipped to the outbound gate and loaded onto the designated outbound truck before its departure, we consider the shipments’s value as lost profit. The objective is to minimize total lost profit. This paper at hand formalizes the resulting truck scheduling problem. We settle its computational complexity and develop heuristics (namely, decomposition procedures and simulated annealing) in order to tackle the problem. We show the efficiency of these heuristics by means of a computational study. Last but not least, a case study is presented.

Appendix: A simulated annealing procedure for TSFD

Our straightforward simulated annealing (SA) procedure for TSFD operates on a solution representation $\Omega$ containing $|G|$ ordered lists $\Omega _g$ of queuing trucks. Clearly, $\bigcup _{g \in G} \Omega _g = I$ and $\Omega _g \cap \Omega _{g^{\prime }} = \emptyset \, \forall g,g^{\prime } \in G$ with $g\ne g^{\prime }$ must hold. In order to determine the objective value $Z(\Omega )$ of a solution $\Omega$ the trucks only need to be successively scheduled in their given order at the respective gates $g \in G$. The initial solution is randomly drawn while equally partitioning the trucks among gates best as possible. A neighboring solution is determined by three types of moves, which are chosen with equal probability:

• a shift move assigns a randomly chosen truck to a new sequence position at its current door,

• a reassignment move randomly determines a new sequence position at another gate for a randomly chosen truck, and

• a swap move exchanges the positions of two randomly chosen trucks at different doors.

Whether or not a neighboring solution $\Omega ^{\prime }$ obtained by a shift move is accepted is decided according to traditional probability schemes (Aarts et al. 1997):

$$\begin{aligned} \mathrm{Prob}(\Omega ^{\prime } \text{ replacing} \Omega )= {\left\{ \begin{array}{ll} 1,&\text{ if} Z(\Omega ^{\prime }) \le Z(\Omega ) \\ \mathrm{exp}\left( \frac{Z(\Omega )-Z(\Omega ^{\prime })}{C}\right),&\text{ otherwise}\\ \end{array}\right.} \end{aligned}$$

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If accepted, the current solution $\Omega$ is replaced by $\Omega ^{\prime }$ as the starting point for further local search moves.

Our SA is steered by a simple static cooling schedule (see Kirkpatrick et al. 1983). The initial value for control parameter $C$ is calculated as $C=\frac{Z(\Omega ^\mathrm{start})}{3}$, where $\Omega ^\mathrm{start}$ denotes the first randomly determined solution. Subsequently, this value $C$ is continuously decreased in the course of the procedure by multiplying it with factor 0.99995 in each iteration. If after 5,000 successive iterations no new best solution value was determined the SA is restarted with a new random solution and a re-initialized parameter $C$. SA is stopped after a predetermined number of iterations and the solution with the minimum objective function value $Z(\Omega )$ is returned. In our computational study, we have invariably used control parameter values as described above. Note that preliminary studies have indicated that this parameter constellation outperforms other settings.