Preemptive depot returns for dynamic same-day delivery

Abstract

In this paper, we explore same-day delivery routing and particularly how same-day delivery vehicles can better integrate dynamic requests into delivery routes by taking advantage of preemptive depot returns. A preemptive depot return occurs when a delivery vehicle returns to the depot before delivering all of the packages currently on-board the vehicle. In this paper, we assume that a vehicle serves requests in a particular delivery area. Beginning the day with some known deliveries, the vehicle seeks to serve the known requests as well as additional new requests that are received throughout the day. To serve the new requests, the vehicle must return to the depot to pick up the packages for delivery. In contrast to previous work on same-day delivery routing, in this paper, we allow the vehicle to return to the depot before serving all loaded packages. To solve the problem, we couple an approximation of the value of choosing any particular subset of requests for delivery with a routing heuristic. Our approximation procedure is based on approximate dynamic programming and allows us to capture both the current value of a subset selection decision and its impact on future rewards. Using extensive computational tests, we demonstrate the value of preemptive depot returns and the value of the proposed approximation scheme in supporting preemptive returns. We also identify characteristics of instances for which preemptive depot returns are most likely to offer improvement.

Appendix

In the Appendix, we present instance generation details, the PDR-algorithm as well as the results and parameters for every instance setting.

1.1 Instance generation details

In the following, we describe how the realizations for the computational evaluation are generated. The number of customers and the order times for a realization are generated by a Poisson process $\mathfrak {P}$. With $c_0$ the expected number of IOs, the number of IOs is generated by $\mathfrak {P}(c_0)$. The spatial and temporal probability distribution for order times and locations is divided into two independent probability distributions. The times of SO occurrences are (discretely) uniformly distributed $t \sim U_\mathbb {Z}[1,t_{\text {max}}-1]$. Customer locations $f(C) \in \mathcal {A}$ are realizations $f \sim \mathcal {F}$ of the spatial probability distribution $\mathcal {F}: \mathcal {A}\rightarrow [0,1]$. A realization of the order time is again conducted by a Poisson process $\mathfrak {P}$ for every minute $0<t<t_{\text {max}}$. Given two points of time $0<t^j<t^h<t_{\text {max}}$, this results in an expected number of customers of $c^{t^h}_{t^j}=\mathbb {E}_{\omega \in \Omega }\left| \{C^{\omega }_i \in \mathcal {C}^{\omega }_+:t^j<t_i\le t^h\}\right|$ ordering in times $t^j<t_i\le t^h$ as described in Eq. (9).

$$\begin{aligned} c^{t^h}_{t^j} = \textit{dod}\cdot c \cdot \frac{t^h-t^j}{T-2} \end{aligned}$$

(9)

1.2 Preemptive depot returns: algorithm

This section presents a detailed algorithm for the PDR routing heuristic that was described in Sect. 4.3. Let $\mathcal {D}$ denote the depot, $\mathcal {P}_k$ the vehicle’s position, $C_l$ the loaded IOs. Further, we let $C_n$ represent the assigned unloaded SOs. The current planned tour can then be described as

$$\begin{aligned} \theta _k=(\mathcal {P}_k, C_l,\dots ,C_l,\mathcal {D}, C_{n}, C_{l}, C_{n},\dots , C_{l},\mathcal {D}). \end{aligned}$$

Let $\theta ^j_k$ refer to the jth component of $\theta _k$, e.g., $\theta ^1_k=\mathcal {P}_k$. Further, let $\mathcal {C}_r=\{C^1_r,\dots ,C^h_r\}$ be the subset of new SOs to assign. PDR first removes the depot from $\theta _k$ leading to an infeasible tour $\bar{\theta }$. In this infeasible tour, the customers $\mathcal {C}_r$ are subsequently inserted via CI at the cheapest position. Procedure $\textit{Insert}(\bar{\theta },\theta ^*,C^*)$ inserts the new order $C^*$ after $\theta ^*$ in tour $\bar{\theta }$. When all new customers are inserted, the depot is inserted between the current position and the first not loaded customer ($C_n$ or $C_r$) via CI resulting in a tour $\theta ^x_k$. If $\theta ^x_k$ does not violate the time limit, the tour is feasible. We assume an initial tour $\theta _0=(\mathcal {D},\mathcal {D})$ without customers, starting and ending at the depot.

1.3 Detailed results

In this section, we present the detailed results for the computational experiments discussed in Sect. 6. The first table presents the average number of assignments for APDR and each benchmark over all realizations for each instance setting. The second table presents the average number of depot returns, the average time required for a depot return, and the initial free time budget over all realizations for each instance setting for APDR.

Table 3 Average number of assignments

Table 4 Parameters for APDR