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Split Algorithm in Linear Time for the Vehicle Routing Problem with Simultaneous Pickup and Delivery and Time Windows

Ethan Gibbons, Mario Ventresca, Beatrice M. Ombuki-Berman

TL;DR

This work extends the linear Split algorithm, originally developed for the CVRP, to a broad class of VRP variants, culminating in a linear-time, optimal Split for the VRP with simultaneous pickups and deliveries and time windows (VRPSPDTW). It introduces a unifying graph framework \mathcal{G}_{h,a} and constant-time feasibility checks that enable amortized linear runtime under the triangle inequality, while also accommodating soft capacity penalties and time-warp penalties. The authors present specialized linear Splits for VRPSPD and VRPTW, including a soft-capacity extension and a time-warp–aware variant, all backed by detailed proofs and runtime analyses. Computational experiments demonstrate substantial speedups over the traditional Bellman-based Split across hard and soft variants, highlighting practical gains for state-of-the-art VRP solvers that rely on long-tour representations and order-first splits.

Abstract

For many kinds of vehicle routing problems (VRPs), a popular heuristic approach involves constructing a Traveling Salesman Problem (TSP) solution, referred to as a long tour, then partitioning segments of the solution into routes for different vehicles with respect to problem constraints. Previously, a Split algorithm with a worst-case runtime of $Θ(n)$ was proposed for the capacitated VRP (CVRP) that finds the most cost-efficient partition of customers, given a long tour. This was an improvement over the previously fastest-known Split algorithm with a worst-case runtime of $Θ(n^2)$ that was based on Bellman's shortest path algorithm. While this linear Split has been an integral part of modern state-of-the-art CVRP approaches, little progress has been made in extending this algorithm to handle additional VRP variants, limiting the general applicability of the algorithm. In this work, we propose an extension of the linear Split that handles two cardinal VRP variants simultaneously: (i) simultaneous pickups and deliveries (VRPSPD) and (ii) time windows (VRPTW). The resulting $Θ(n)$ algorithm is guaranteed to be optimal, assuming travel times between nodes satisfy the triangle inequality. Additionally, we extend the linear Split to handle a capacity penalty for the VRPSPD. For the VRPTW, we extend the linear Split to handle the CVRP capacity penalty in conjunction with the popular time warp penalty function. Computational experiments are performed to empirically validate the speed gains of these linear Splits against their $Θ$($n^2$) counterparts.

Split Algorithm in Linear Time for the Vehicle Routing Problem with Simultaneous Pickup and Delivery and Time Windows

TL;DR

This work extends the linear Split algorithm, originally developed for the CVRP, to a broad class of VRP variants, culminating in a linear-time, optimal Split for the VRP with simultaneous pickups and deliveries and time windows (VRPSPDTW). It introduces a unifying graph framework \mathcal{G}_{h,a} and constant-time feasibility checks that enable amortized linear runtime under the triangle inequality, while also accommodating soft capacity penalties and time-warp penalties. The authors present specialized linear Splits for VRPSPD and VRPTW, including a soft-capacity extension and a time-warp–aware variant, all backed by detailed proofs and runtime analyses. Computational experiments demonstrate substantial speedups over the traditional Bellman-based Split across hard and soft variants, highlighting practical gains for state-of-the-art VRP solvers that rely on long-tour representations and order-first splits.

Abstract

For many kinds of vehicle routing problems (VRPs), a popular heuristic approach involves constructing a Traveling Salesman Problem (TSP) solution, referred to as a long tour, then partitioning segments of the solution into routes for different vehicles with respect to problem constraints. Previously, a Split algorithm with a worst-case runtime of was proposed for the capacitated VRP (CVRP) that finds the most cost-efficient partition of customers, given a long tour. This was an improvement over the previously fastest-known Split algorithm with a worst-case runtime of that was based on Bellman's shortest path algorithm. While this linear Split has been an integral part of modern state-of-the-art CVRP approaches, little progress has been made in extending this algorithm to handle additional VRP variants, limiting the general applicability of the algorithm. In this work, we propose an extension of the linear Split that handles two cardinal VRP variants simultaneously: (i) simultaneous pickups and deliveries (VRPSPD) and (ii) time windows (VRPTW). The resulting algorithm is guaranteed to be optimal, assuming travel times between nodes satisfy the triangle inequality. Additionally, we extend the linear Split to handle a capacity penalty for the VRPSPD. For the VRPTW, we extend the linear Split to handle the CVRP capacity penalty in conjunction with the popular time warp penalty function. Computational experiments are performed to empirically validate the speed gains of these linear Splits against their () counterparts.
Paper Structure (48 sections, 21 theorems, 65 equations, 5 figures, 2 tables, 7 algorithms)

This paper contains 48 sections, 21 theorems, 65 equations, 5 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Vehicle Routing Problem with Simultaneous Pickup and Delivery with Time Windows
  4. CVRP Formulation
  5. VRPSPD Formulation
  6. VRPSPDTW Formulation
  7. Bellman and Linear Split for the CVRP
  8. Bellman CVRP Split
  9. Linear CVRP Split
  10. Generalized Queue-Based Split
  11. Split for the VRPSPDTW
  12. $\mathcal{G}_{h,a}$ Definition and Properties
  13. Constant-Time VRPSPD Evaluation
  14. Constant-Time VRPTW Evaluation
  15. VRPSPDTW Split Algorithm
  16. ...and 33 more sections

Key Result

Lemma 1

On graph $\mathcal{G}_{h,a}$, for any two fixed indices $i,j$, we have $load(x,i,y) - load(x,j,y) = L_{i,j}$, where $L_{i,j}\in \mathbb{R}$ is some constant, for any $x,y$ as long as $0 \leq x \leq i < j \leq y \leq n$. Additionally, the constant $L_{i,j}$ does not depend on $d_i$ or $p_i$.

Figures (5)

  • Figure 1: Speedup factors of the VRPSPDTW linear Split over the Bellman-based algorithm for all 105 instances. Each graph corresponds to different $Q$ and $b_i$ multiplier pairs. The X-axis indicates the instance size. A logarithmic scale is used for both axes.
  • Figure 2: Speedup factors for the VRPSPD with a capacity penalty factor.
  • Figure 3: Speedup factors for the VRPTW with a capacity penalty factor and time warp factor.
  • Figure 4: A visualization of the directed acyclic graph $G_\Pi$ based on a permutation of the customers of a CVRP instance. In this example, the instance has $n=10$ customers. The Bellman Split finds the shortest path from $0_0$ to $0_{10}$, subject to CVRP capacity constraints.
  • Figure 5: The directed acyclic graph $\mathcal{G}_{d}$ created using $G_\Pi$ shown in Figure \ref{['fig:dag']} with instance information from Table \ref{['tab:instance-info']}. For this graph, arc $(0_i,0_j)$ is generated only if the route $(0,i+1,i+2,\dots,j,0)$ is a feasible CVRP route. All arcs originating from the same node are colour-coded the same. The Bellman Split implicitly considers these arcs and their associated weights as needed to perform a typical Bellman shortest path algorithm from $0_0$ to $0_n$. Arc weights are calculated with the function $c(i,j)$ for arc $(0_i,0_j)$.

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Corollary 1
  • Lemma 7
  • Lemma 8
  • Corollary 2
  • ...and 11 more