Optimization of Next-Day Delivery Coverage using Constraint Programming and Random Key Optimizers

TL;DR

This work tackles maximizing next-day delivery coverage in a retailer's middle-mile network by optimizing truck departure times along network legs. It develops two complementary solver families: exact constraint programming (CP) and heuristic random-key optimization (RKO), each augmented with problem-specific constraints such as rolling capacity, labor efficiency, and dispatch spacing, and evaluated against a black-box demand evaluator. Key contributions include incorporating intermediate sort centers, multi-wave legs, and hybrid solver configurations that couple CP or RKO with local search. The experiments on a large EU-scale network show that hybrid solvers can outperform a bespoke greedy baseline in 1D delivery coverage, with RKO hybrids offering the largest gains at the cost of longer runtimes, while CP hybrids achieve competitive improvements with moderate runtimes. The results provide actionable insights for fast-and-slow optimization strategies in complex logistics networks and point to promising directions for integrating proxy and black-box objectives, warm-starting, and decomposition techniques in future work.

Abstract

We consider the logistics network of an e-commerce retailer, specifically the so-called "middle mile" network, that routes inventory from supply warehouses to distribution stations to be ingested into the terminal ("last mile") delivery network. The speed of packages through this middle mile network is a key determinant for the ultimate delivery speed to the end user. An important target for a retailer is to maximize the fraction of user orders that can be serviced within one day, i.e., next-day delivery. As such, we formulate the maximization of expected next-day delivery coverage within the middle-mile network as an optimization problem, involving a set of temporal and capacity-based constraints on the network and requiring the use of a black-box model to evaluate the objective function. We design both exact constraint programming (CP) and heuristic random-key optimizer (RKO) approaches, the former of which uses a proxy objective function. We perform experiments on large-scale, real-world problem instances and show that both approaches have merit, in that they can match or outperform the baseline solution, a bespoke greedy solver with integrated local search, in expected next-day delivery coverage. Our experiments focus on two high-level problem definitions, starting with a base problem and then adding more complexity, and also explore the generalization of the solvers across a range of problem instance sizes. We find that a hybrid model using RKO and a bespoke local search protocol performs best on the full problem definition with respect to expected next-day delivery (increase of +50 basis points [bps] over baseline) but can take days to run, whereas the hybrid model using CP and local search is slightly less competitive (+20 bps) but takes only hours to run.

Figures (3)

• Figure 1: Example middle mile network with $|\mathcal{V}|=7$ and $|\mathcal{A}|=11$. The supply warehouses (SW) serve as path origins, connecting to downstream sort centers (SC) or distribution stations (DS). The travel time between nodes $i$ and $j$ is denoted as $\delta_{(i,j)}$. The network has both direct paths (e.g., $(SW2,DS3)$) and paths with multiple legs (e.g., $((SW1, SC1), (SC1, DS1))$.
• Figure 2: Example (direct) SW-DS connection. The truck departure time (TDT) from the SW node is $x_{(SW,DS),1}$ on Day 0 with transition time $\delta_{(SW,DS)}$ and processing time at the DS of $p_{DS}$. If the truck arrival time ($x_{(SW,DS),1} + \delta_{(SW,DS)} + p_{DS}$) on the next day is prior to the same day cutoff time, the volume carried by the truck is delivered with next-day (i.e., 1-day) promise. If it is later than the same day cutoff time, it is delivered with 2-day promise.
• Figure 3: Schematic illustration of the RKO decoder design, with native handling of IB/OB constraints. The input is given in terms of a random key $\chi$ with element $\chi_{n} \in [0,1)$ encoding the TDT $x_{(i,j),w}$ for leg $n=(i,j)$. The decoder maps $\chi_{n}$ to $x_{(i,j),w}$ while ensuring compatibility with the opening hours of the relevant inbound (IB) and outbound (OB) nodes. To this end, the key $\chi_{n}$ is binned such that bins can only map to a TDT assignment labeled as $\bot$ (where no TDT is placed) or times compatible with OB opening hours and IB opening hours (adjusted accordingly by the corresponding travel time $\delta_{(i,j)}$), as indicated by the green and blue bars, respectively. The decoder ensures mapping to IB/OB-feasible time domains only via leg-specific bins, as illustrated here for two different legs.

Theorems & Definitions (2)

• Example 2.1: Direct path
• Example 2.2: Sort center path