A framework for window design in delivery schedules

TL;DR

The paper tackles time-window design in delivery with a fixed route under stochastic travel and service times. It develops a two-tier framework: a fast static design (WOS and UWOS) using a per-customer decomposition and probabilistic thresholds, and a dynamic DWOS scheme that updates windows during execution based on remaining lead time and limited communications. Key contributions include explicit first-order conditions yielding unique solutions, closed-form results under normal travel times, practical convolution-based and normal-approximation methods for general distributions, and substantial performance gains from dynamic updates demonstrated in synthetic experiments and real-world Last-Mile data. The work offers a scalable, communication-efficient approach for improving delivery reliability and customer experience in modern parcel networks.

Abstract

This paper develops a structured framework for the design and dynamic updating of service time windows in delivery and appointment-based systems. We consider a single-server setting with stochastic service and travel times, where customers are promised a time window in which the provider will arrive. The first part of the paper introduces a static window construction method based on a probabilistic threshold criterion, using an analytical approximation of residual travel and service time distributions. Building on this, we develop a dynamic update mechanism that monitors residual system uncertainty, where time windows are revised during execution only when the remaining time until the window's start falls below a predefined threshold. This threshold-based approach enables communication-efficient scheduling while substantially improving delivery accuracy. Numerical experiments demonstrate significant performance gains of the dynamic approach in both stylized and real-world settings.

Figures (14)

• Figure 1: Centered WOS time windows for normally distributed travel times with parameters $\mu_i = 10$ and $\sigma_i = 2.5$. The three rows correspond to $\omega=0.25, 0.5, 0.75$, respectively, and the three columns to $\alpha = 0.1, 0.25, 0.5$, respectively.
• Figure 2: Centered WOS time windows using numerical convolutions; we use the same mean ($\mu_i = 10$) and standard deviation ($\sigma_i = 2.5$) for all distribution functions.
• Figure 3: Difference of time windows for Weibull and Lognormal distribution, relative to time windows for normal distribution, for $\omega = 0.25$. (i) Red: the $t_i$ based on the normal distribution minus those based on the Weibull distribution, (ii) red dotted: same, but then for the $t_i+\Delta_i$, (iii) blue: the $t_i$ based on the normal distribution minus those based on the Lognormal distribution, (iv) blue dotted: same, but then for the $t_i+\Delta_i$.
• Figure 4: Boxplot assessing accuracy of the normal approximation over a broad range of heterogeneous scenarios. The hinges of the box correspond to the 25- and 75-percentiles, the whiskers to the 5- and 95-percentiles. The predefined parameter sets used in experiments are: $\omega\in \{0.25, 0.5, 0.75\}$, $\alpha\in\{0.01, 0.05, 0.1, 0.15\}$, $\beta\in\{1.1, 1.2, 1.3, 1.4, 1.5\}$, $\mu\in\{10, 12, 15\}$.
• Figure 5: Time windows UWOS vs. WOS. Per panel there are 12 lines: (a) the bullets correspond to WOS, the crosses to UWOS; (b) the solid lines correspond to the start of the windows, the dashed lines to the end of the windows; (c) the three colors correspond to the three values of $\beta$.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Definition 1: Window-based optimal schedule
• Definition 2: Window-based optimal uniform schedule
• Remark 3
• Theorem 1
• Lemma 1
• Remark 4