Vehicle Routing with Time-Dependent Travel Times: Theory, Practice, and Benchmarks

TL;DR

This work addresses vehicle routing with time-dependent travel times modeled by piecewise linear arrival time functions $a$, respecting the FIFO property and enabling timing-aware scheduling. It develops a cohesive framework combining fast ATF operations, an $O(m\log n)$ method for the ATF minimum, a monotonicity-preserving Imai–Iri simplification, linear-time tour scheduling, and efficient data structures for insertions/deletions, all integrated into a practical local-search algorithm (BonnTour) augmented by road-network contraction hierarchies. The authors introduce new real-world benchmarks derived from OpenStreetMap and Uber speed data to demonstrate the importance of time-dependent travel times, and show competitive results on standard VRP benchmarks while achieving substantial improvements on their time-dependent instances. The approach yields more reliable tours, particularly under tight time windows, and provides a scalable, production-ready toolkit for practice, including publicly available benchmarks for future research. Overall, the paper advances theory, algorithms, and benchmarks for time-dependent VRP, delivering practical impact through a robust, scalable local-search framework and validated real-world data.

Abstract

We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results. First, we study basic operations on piecewise linear arrival time functions. In particular, we devise a faster algorithm to compute the pointwise minimum of a set of piecewise linear functions and a monotonicity-preserving variant of the Imai-Iri algorithm to approximate an arrival time function with fewer breakpoints. Next, we show how to evaluate insertion and deletion operations in tours efficiently and update the underlying data structure faster than previously known when a tour changes. Evaluating a tour also requires a scheduling step which is non-trivial in the presence of time windows and time-dependent travel times. We show how to perform this in linear time. Based on these results, we develop a local search heuristic to solve real-world vehicle routing problems with various constraints efficiently and report experimental results on classical benchmarks. Since most of these do not have time-dependent travel times, we generate and publish new benchmark instances that are based on real-world data. This data also demonstrates the importance of considering time-dependent travel times in instances with tight time windows.

Figures (6)

• Figure 1: Taking the minimum of two ATFs may increase the number of inner breakpoints by a factor of 2.
• Figure 2: The intervals $I_{i,j}$ (for $k=3$ and $n=7$) in the proof of Theorem \ref{['thm:atf_minimum_computation']}, and a possible assignment of the seven piecewise linear functions shown at the bottom to these intervals.
• Figure 3: Illustrating Definition \ref{['def:visibility']} and Algorithm \ref{['alg:imai_iri_atfs']}. The algorithm progresses through the corridor between $f$ and $f+\varepsilon$ by constructing visibility polygons (shaded blue, green, orange) and their windows on the right boundary. The blue area is weakly visible from infinitely far to the left; $P_1$ is the remaining polygon. The green area is weakly visible from the blue area, and in fact from the window $W_1$. There are three invisibility polygons from $W_1$; the relevant one, $P_2$, is to the right of the window $W_2$ (the window from $W_1$ to the line segment $\overline{p_bp_b^+}$ in $P_1$). The orange area is the part of $P_2$ that is weakly visible from $W_2$. A possible output $g$ is the ATF shown red and dashed, with breakpoints $q_1,q_2,q_3$.
• Figure 4: A balanced binary search tree. To compute $a_{i,j}$ we find the lowest common ancestor $h$ of $i$ and $j$ and compose $a_{i,h}$ and $a_{h,j}$. These ATFs are precomputed because $h$ is an ancestor of $i$ and $j$.
• Figure 5: The map on the left shows three paths in the 500-customer New York instance that are fastest at some point between 3 pm and 10 pm from some customer in Manhattan (A) to a customer in Brooklyn (B). On the right-hand side the corresponding travel time function is depicted. The segments are colored in the color of the path that is the fastest at this departure time.

Theorems & Definitions (27)

• Definition 1
• Proposition 2
• proof
• Corollary 3
• proof
• Proposition 4
• proof
• Lemma 5
• proof
• Lemma 6