Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Last Truck Scheduling for Middle-mile Next-day Delivery Coverage

Konstantinos Benidis, Georgios Paschos, Martin Gross, George Iosifidis

Abstract

We consider an e-commerce retailer operating a supply chain that consists of middle- and last-mile transportation, and study its ability to deliver products stored in warehouses within a day from customer's order time. Successful next-day delivery requires inventory availability and timely truck schedules in the middle-mile and in this paper we assume a fixed inventory position and focus on optimizing the middle-mile last truck schedule. We formulate a novel optimization problem which decides the departure of the last truck at each (potential) network connection in order to maximize the number of customer orders that are served with next-day promise. We show that the respective next-day delivery optimization is a combinatorial problem that is NP-hard to approximate within $(1-1/e)opt\approx 0.632opt$, hence every retailer that offers one-day deliveries has to deal with this complexity barrier. We study three variants of the problem motivated by operational constraints that different retailers encounter, and propose solutions schemes tailored to each problem's properties. To that end, we rely on greedy submodular maximization, pipage rounding techniques, and Lagrangian heuristics. The algorithms are scalable, offer worst-case optimality gap guarantees, and evaluated in realistic datasets and network scenarios were found to achieve even near-optimal results.

Last Truck Scheduling for Middle-mile Next-day Delivery Coverage

Abstract

We consider an e-commerce retailer operating a supply chain that consists of middle- and last-mile transportation, and study its ability to deliver products stored in warehouses within a day from customer's order time. Successful next-day delivery requires inventory availability and timely truck schedules in the middle-mile and in this paper we assume a fixed inventory position and focus on optimizing the middle-mile last truck schedule. We formulate a novel optimization problem which decides the departure of the last truck at each (potential) network connection in order to maximize the number of customer orders that are served with next-day promise. We show that the respective next-day delivery optimization is a combinatorial problem that is NP-hard to approximate within , hence every retailer that offers one-day deliveries has to deal with this complexity barrier. We study three variants of the problem motivated by operational constraints that different retailers encounter, and propose solutions schemes tailored to each problem's properties. To that end, we rely on greedy submodular maximization, pipage rounding techniques, and Lagrangian heuristics. The algorithms are scalable, offer worst-case optimality gap guarantees, and evaluated in realistic datasets and network scenarios were found to achieve even near-optimal results.
Paper Structure (54 sections, 7 theorems, 29 equations, 11 figures, 5 algorithms)

This paper contains 54 sections, 7 theorems, 29 equations, 11 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Literature Review
  5. Problem Definition
  6. System Model
  7. Network:
  8. Products:
  9. Demand:
  10. Arrival and departure deadlines:
  11. Decision Variables and Covered Demand
  12. Decision variables:
  13. Covered demand:
  14. Optimization Model
  15. Problem Variations
  16. ...and 39 more sections

Key Result

Theorem 1

The worst-case complexity of NDD problems is characterized as follows:

Figures (11)

  • Figure 1: Example of a single FC-DS connection. The cutoff time of the DS (for NDD) is $t^\text{(ad)}=30$ (or next day 6am), the transit time $\delta = 13$ and the deadline departure $t^\text{(dd)}=17$ (5pm).
  • Figure 2: Coverage of orders with next-day delivery from individual and combined warehouses. Horizontal axis depicts the departure of the last-truck for each (of the two) FC, and vertical axis measures the number of orders that each FC can cover.
  • Figure 3: Relative excess coverage $E_{cov}(\mathbf{X})$ of all algorithms for $\mathcal{P}^{\text{(ob)}}$. Optimality gap is measured with respect to the solution returned by a commercial MIP solver on a cloud server, with a time limit of 5 hours.
  • Figure 4: Running time of all algorithms for $\mathcal{P}^{\text{(ob)}}$.
  • Figure 5: Relative excess coverage of all algorithms for $\mathcal{P}^{\text{(ib)}}$.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Corollary 2
  • Lemma 3
  • Definition 1: Pipage step
  • Lemma 4