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09 / routing

Vehicle Routing

VRP with Time Windows + Capacity
Binary ILP / QUBO logistics VRP real-world
Instances 55
Optimally solved 54 / 55
Variable range 441–441
Objective minimize

Overview

The Vehicle Routing Problem (VRP) is a classic logistics optimization problem combining aspects of the Traveling Salesman Problem, time window scheduling, and knapsack constraints. It has direct applications in delivery services, supply chain management, and transportation planning.

Problem Description

Given:

  • A fleet of $k$ vehicles, each with capacity $X$
  • A central depot
  • A set of customers $C = \{1,\ldots,n\}$ with demands $d_i$ for $i \in C$
  • Time windows for each customer
  • Distance/cost matrix between all locations

Objective: Determine routes for all vehicles to serve all customers while:

  • Respecting vehicle capacity constraints
  • Satisfying time window requirements
  • Minimizing total distance or cost

Instance Parameters:

  • $k = 4$ vehicles
  • $n = 20$ customers

Performance

Runtime to reach best-known objective

Sorted instances vs total runtime. A point (x, y) means x instances were solved within y seconds. Solid line + filled circle = proven exact; dashed line + open diamond = heuristic. Lower-right is better.

Classical (38 · heuristic)
Cactus plot: cumulative number of instances solved (horizontal) versus total runtime in seconds on a log scale (vertical), one line per method group; lower-right is better.1251020071421283538instances solved →runtime (s, log)Classical · heuristic · XSH-n20-k4-16 · 1.57 sClassical · heuristic · XSH-n20-k4-25 · 3.58 sClassical · heuristic · XSH-n20-k4-04 · 3.96 sClassical · heuristic · XSH-n20-k4-01 · 4.23 sClassical · heuristic · XSH-n20-k4-54 · 4.47 sClassical · heuristic · XSH-n20-k4-02 · 4.7 sClassical · heuristic · XSH-n20-k4-50 · 4.71 sClassical · heuristic · XSH-n20-k4-23 · 5.19 sClassical · heuristic · XSH-n20-k4-32 · 6.75 sClassical · heuristic · XSH-n20-k4-24 · 6.93 sClassical · heuristic · XSH-n20-k4-13 · 7.37 sClassical · heuristic · XSH-n20-k4-49 · 7.73 sClassical · heuristic · XSH-n20-k4-33 · 7.77 sClassical · heuristic · XSH-n20-k4-39 · 8 sClassical · heuristic · XSH-n20-k4-29 · 8.24 sClassical · heuristic · XSH-n20-k4-55 · 8.37 sClassical · heuristic · XSH-n20-k4-40 · 8.67 sClassical · heuristic · XSH-n20-k4-28 · 8.79 sClassical · heuristic · XSH-n20-k4-37 · 9.02 sClassical · heuristic · XSH-n20-k4-22 · 9.2 sClassical · heuristic · XSH-n20-k4-07 · 9.3 sClassical · heuristic · XSH-n20-k4-15 · 9.8 sClassical · heuristic · XSH-n20-k4-44 · 10.4 sClassical · heuristic · XSH-n20-k4-42 · 10.9 sClassical · heuristic · XSH-n20-k4-45 · 11.1 sClassical · heuristic · XSH-n20-k4-30 · 11.2 sClassical · heuristic · XSH-n20-k4-35 · 12 sClassical · heuristic · XSH-n20-k4-12 · 12.2 sClassical · heuristic · XSH-n20-k4-17 · 12.9 sClassical · heuristic · XSH-n20-k4-20 · 13.1 sClassical · heuristic · XSH-n20-k4-51 · 13.3 sClassical · heuristic · XSH-n20-k4-18 · 13.8 sClassical · heuristic · XSH-n20-k4-48 · 14.9 sClassical · heuristic · XSH-n20-k4-19 · 15.2 sClassical · heuristic · XSH-n20-k4-26 · 15.5 sClassical · heuristic · XSH-n20-k4-53 · 17.1 sClassical · heuristic · XSH-n20-k4-03 · 19.4 sClassical · heuristic · XSH-n20-k4-41 · 19.9 s

Time-to-solution (TTS) to reach best-known objective

Same as the runtime cactus but uses the reported Time-to-Solution rather than total runtime. Solid = exact, dashed = heuristic.

Classical (38 · heuristic)
Cactus plot: cumulative number of instances solved (horizontal) versus time-to-solution in seconds on a log scale (vertical), one line per method group; lower-right is better.110071421283538instances solved →time-to-solution (s, log)Classical · heuristic · XSH-n20-k4-25 · 0.36 sClassical · heuristic · XSH-n20-k4-16 · 0.62 sClassical · heuristic · XSH-n20-k4-50 · 0.7 sClassical · heuristic · XSH-n20-k4-28 · 1.12 sClassical · heuristic · XSH-n20-k4-23 · 1.94 sClassical · heuristic · XSH-n20-k4-04 · 2.48 sClassical · heuristic · XSH-n20-k4-01 · 2.76 sClassical · heuristic · XSH-n20-k4-02 · 2.84 sClassical · heuristic · XSH-n20-k4-07 · 3.14 sClassical · heuristic · XSH-n20-k4-54 · 3.25 sClassical · heuristic · XSH-n20-k4-39 · 3.28 sClassical · heuristic · XSH-n20-k4-55 · 3.31 sClassical · heuristic · XSH-n20-k4-22 · 3.5 sClassical · heuristic · XSH-n20-k4-48 · 3.59 sClassical · heuristic · XSH-n20-k4-13 · 4.62 sClassical · heuristic · XSH-n20-k4-49 · 4.75 sClassical · heuristic · XSH-n20-k4-32 · 5.29 sClassical · heuristic · XSH-n20-k4-30 · 5.37 sClassical · heuristic · XSH-n20-k4-29 · 5.55 sClassical · heuristic · XSH-n20-k4-24 · 5.61 sClassical · heuristic · XSH-n20-k4-03 · 5.66 sClassical · heuristic · XSH-n20-k4-37 · 5.79 sClassical · heuristic · XSH-n20-k4-33 · 5.99 sClassical · heuristic · XSH-n20-k4-15 · 6.05 sClassical · heuristic · XSH-n20-k4-20 · 6.13 sClassical · heuristic · XSH-n20-k4-35 · 6.37 sClassical · heuristic · XSH-n20-k4-26 · 7.12 sClassical · heuristic · XSH-n20-k4-44 · 7.26 sClassical · heuristic · XSH-n20-k4-18 · 7.29 sClassical · heuristic · XSH-n20-k4-40 · 7.36 sClassical · heuristic · XSH-n20-k4-12 · 7.56 sClassical · heuristic · XSH-n20-k4-45 · 7.84 sClassical · heuristic · XSH-n20-k4-53 · 7.84 sClassical · heuristic · XSH-n20-k4-42 · 8.77 sClassical · heuristic · XSH-n20-k4-51 · 11 sClassical · heuristic · XSH-n20-k4-17 · 11.1 sClassical · heuristic · XSH-n20-k4-19 · 13.6 sClassical · heuristic · XSH-n20-k4-41 · 14.3 s

Solution quality (performance profile)

Share of instances each group brings within a given optimality gap of the best-known objective. Higher is better; the value at “best” is the share solved exactly.

Classical (55)
Performance profile: share of instances (vertical) reached within a given optimality gap of the best-known objective (horizontal), one line per method group; higher is better.0%25%50%75%100%best+0.26%+0.58%+0.99%+1.5%optimality gap from best-known →instances solved (%)Classical · within best · 69% · XSH-n20-k4-01, XSH-n20-k4-02, XSH-n20-k4-03, XSH-n20-k4-04, XSH-n20-k4-07, XSH-n20-k4-12, XSH-n20-k4-13, XSH-n20-k4-15, XSH-n20-k4-16, XSH-n20-k4-17, XSH-n20-k4-18, XSH-n20-k4-19, XSH-n20-k4-20, XSH-n20-k4-22, XSH-n20-k4-23, XSH-n20-k4-24, XSH-n20-k4-25, XSH-n20-k4-26, XSH-n20-k4-28, XSH-n20-k4-29, XSH-n20-k4-30, XSH-n20-k4-32, XSH-n20-k4-33, XSH-n20-k4-35, XSH-n20-k4-37, XSH-n20-k4-39, XSH-n20-k4-40, XSH-n20-k4-41, XSH-n20-k4-42, XSH-n20-k4-44, XSH-n20-k4-45, XSH-n20-k4-48, XSH-n20-k4-49, XSH-n20-k4-50, XSH-n20-k4-51, XSH-n20-k4-53, XSH-n20-k4-54, XSH-n20-k4-55Classical · within +0.12% · 71% · XSH-n20-k4-21Classical · within +0.14% · 73% · XSH-n20-k4-10Classical · within +0.14% · 75% · XSH-n20-k4-14Classical · within +0.2% · 76% · XSH-n20-k4-38Classical · within +0.22% · 78% · XSH-n20-k4-34Classical · within +0.28% · 80% · XSH-n20-k4-08Classical · within +0.34% · 82% · XSH-n20-k4-47Classical · within +0.37% · 84% · XSH-n20-k4-27Classical · within +0.48% · 85% · XSH-n20-k4-52Classical · within +0.57% · 87% · XSH-n20-k4-09Classical · within +0.57% · 89% · XSH-n20-k4-05Classical · within +0.86% · 91% · XSH-n20-k4-46Classical · within +1% · 93% · XSH-n20-k4-06Classical · within +1.1% · 95% · XSH-n20-k4-11Classical · within +1.1% · 96% · XSH-n20-k4-36Classical · within +1.3% · 98% · XSH-n20-k4-31Classical · within +1.5% · 100% · XSH-n20-k4-43

Runtime scaling with instance size

Fastest feasible runtime (log scale) per instance versus Customers — shows how each group scales.

Classical (55)
Scaling plot: fastest feasible runtime in seconds on a log scale (vertical) versus Customers (horizontal), one series per method group.11010020Customers →runtime (s, log)Classical · XSH-n20-k4-01 · Customers 20 · 4.23 sClassical · XSH-n20-k4-02 · Customers 20 · 4.7 sClassical · XSH-n20-k4-03 · Customers 20 · 19.4 sClassical · XSH-n20-k4-04 · Customers 20 · 3.96 sClassical · XSH-n20-k4-05 · Customers 20 · 8.51 sClassical · XSH-n20-k4-06 · Customers 20 · 10.7 sClassical · XSH-n20-k4-07 · Customers 20 · 9.3 sClassical · XSH-n20-k4-08 · Customers 20 · 5.31 sClassical · XSH-n20-k4-09 · Customers 20 · 4.07 sClassical · XSH-n20-k4-10 · Customers 20 · 4.47 sClassical · XSH-n20-k4-11 · Customers 20 · 6.55 sClassical · XSH-n20-k4-12 · Customers 20 · 12.2 sClassical · XSH-n20-k4-13 · Customers 20 · 7.37 sClassical · XSH-n20-k4-14 · Customers 20 · 24.4 sClassical · XSH-n20-k4-15 · Customers 20 · 9.8 sClassical · XSH-n20-k4-16 · Customers 20 · 1.57 sClassical · XSH-n20-k4-17 · Customers 20 · 12.9 sClassical · XSH-n20-k4-18 · Customers 20 · 13.8 sClassical · XSH-n20-k4-19 · Customers 20 · 15.2 sClassical · XSH-n20-k4-20 · Customers 20 · 13.1 sClassical · XSH-n20-k4-21 · Customers 20 · 16.4 sClassical · XSH-n20-k4-22 · Customers 20 · 9.2 sClassical · XSH-n20-k4-23 · Customers 20 · 5.19 sClassical · XSH-n20-k4-24 · Customers 20 · 6.93 sClassical · XSH-n20-k4-25 · Customers 20 · 3.58 sClassical · XSH-n20-k4-26 · Customers 20 · 15.5 sClassical · XSH-n20-k4-27 · Customers 20 · 29.9 sClassical · XSH-n20-k4-28 · Customers 20 · 8.79 sClassical · XSH-n20-k4-29 · Customers 20 · 8.24 sClassical · XSH-n20-k4-30 · Customers 20 · 11.2 sClassical · XSH-n20-k4-31 · Customers 20 · 16.5 sClassical · XSH-n20-k4-32 · Customers 20 · 6.75 sClassical · XSH-n20-k4-33 · Customers 20 · 7.77 sClassical · XSH-n20-k4-34 · Customers 20 · 70.3 sClassical · XSH-n20-k4-35 · Customers 20 · 12 sClassical · XSH-n20-k4-36 · Customers 20 · 6.8 sClassical · XSH-n20-k4-37 · Customers 20 · 9.02 sClassical · XSH-n20-k4-38 · Customers 20 · 7.11 sClassical · XSH-n20-k4-39 · Customers 20 · 8 sClassical · XSH-n20-k4-40 · Customers 20 · 8.67 sClassical · XSH-n20-k4-41 · Customers 20 · 19.9 sClassical · XSH-n20-k4-42 · Customers 20 · 10.9 sClassical · XSH-n20-k4-43 · Customers 20 · 13.5 sClassical · XSH-n20-k4-44 · Customers 20 · 10.4 sClassical · XSH-n20-k4-45 · Customers 20 · 11.1 sClassical · XSH-n20-k4-46 · Customers 20 · 6.25 sClassical · XSH-n20-k4-47 · Customers 20 · 21.7 sClassical · XSH-n20-k4-48 · Customers 20 · 14.9 sClassical · XSH-n20-k4-49 · Customers 20 · 7.73 sClassical · XSH-n20-k4-50 · Customers 20 · 4.71 sClassical · XSH-n20-k4-51 · Customers 20 · 13.3 sClassical · XSH-n20-k4-52 · Customers 20 · 10.5 sClassical · XSH-n20-k4-53 · Customers 20 · 17.1 sClassical · XSH-n20-k4-54 · Customers 20 · 4.47 sClassical · XSH-n20-k4-55 · Customers 20 · 8.37 s

Submissions (1)

Method Submitter Type Date Instances
MemeticGA_LoopUntilFeasible Othmane El Yaakoubi Classical 2026-07-21 110

Instances (55)

55 of 55
Name Customers Vehicles Best objective Source Status Download
XSH-n20-k4-01 20 4 646 Reference solution Optimal ↓ raw
XSH-n20-k4-02 20 4 650 Reference solution Optimal ↓ raw
XSH-n20-k4-03 20 4 508 Reference solution Optimal ↓ raw
XSH-n20-k4-04 20 4 776 Reference solution Optimal ↓ raw
XSH-n20-k4-05 20 4 702 Reference solution Optimal ↓ raw
XSH-n20-k4-06 20 4 690 Reference solution Optimal ↓ raw
XSH-n20-k4-07 20 4 730 Reference solution Optimal ↓ raw
XSH-n20-k4-08 20 4 718 Reference solution Optimal ↓ raw
XSH-n20-k4-09 20 4 707 Reference solution Optimal ↓ raw
XSH-n20-k4-10 20 4 737 Reference solution Optimal ↓ raw
XSH-n20-k4-11 20 4 914 Reference solution Optimal ↓ raw
XSH-n20-k4-12 20 4 709 Reference solution Optimal ↓ raw
XSH-n20-k4-13 20 4 628 Reference solution Optimal ↓ raw
XSH-n20-k4-14 20 4 696 Reference solution Optimal ↓ raw
XSH-n20-k4-15 20 4 780 Reference solution Optimal ↓ raw
XSH-n20-k4-16 20 4 830 Reference solution Optimal ↓ raw
XSH-n20-k4-17 20 4 605 Reference solution Optimal ↓ raw
XSH-n20-k4-18 20 4 997 Reference solution Optimal ↓ raw
XSH-n20-k4-19 20 4 976 Reference solution Optimal ↓ raw
XSH-n20-k4-20 20 4 648 Reference solution Optimal ↓ raw
XSH-n20-k4-21 20 4 842 Reference solution Best known ↓ raw
XSH-n20-k4-22 20 4 760 Reference solution Optimal ↓ raw
XSH-n20-k4-23 20 4 851 Reference solution Optimal ↓ raw
XSH-n20-k4-24 20 4 725 Reference solution Optimal ↓ raw
XSH-n20-k4-25 20 4 462 Reference solution Optimal ↓ raw
XSH-n20-k4-26 20 4 688 Reference solution Optimal ↓ raw
XSH-n20-k4-27 20 4 540 Reference solution Optimal ↓ raw
XSH-n20-k4-28 20 4 669 Reference solution Optimal ↓ raw
XSH-n20-k4-29 20 4 658 Reference solution Optimal ↓ raw
XSH-n20-k4-30 20 4 685 Reference solution Optimal ↓ raw
XSH-n20-k4-31 20 4 445 Reference solution Optimal ↓ raw
XSH-n20-k4-32 20 4 882 Reference solution Optimal ↓ raw
XSH-n20-k4-33 20 4 568 Reference solution Optimal ↓ raw
XSH-n20-k4-34 20 4 891 Reference solution Optimal ↓ raw
XSH-n20-k4-35 20 4 707 Reference solution Optimal ↓ raw
XSH-n20-k4-36 20 4 610 Reference solution Optimal ↓ raw
XSH-n20-k4-37 20 4 650 Reference solution Optimal ↓ raw
XSH-n20-k4-38 20 4 991 Reference solution Optimal ↓ raw
XSH-n20-k4-39 20 4 717 Reference solution Optimal ↓ raw
XSH-n20-k4-40 20 4 700 Reference solution Optimal ↓ raw
XSH-n20-k4-41 20 4 736 Reference solution Optimal ↓ raw
XSH-n20-k4-42 20 4 998 Reference solution Optimal ↓ raw
XSH-n20-k4-43 20 4 465 Reference solution Optimal ↓ raw
XSH-n20-k4-44 20 4 973 Reference solution Optimal ↓ raw
XSH-n20-k4-45 20 4 1,071 Reference solution Optimal ↓ raw
XSH-n20-k4-46 20 4 812 Reference solution Optimal ↓ raw
XSH-n20-k4-47 20 4 583 Reference solution Optimal ↓ raw
XSH-n20-k4-48 20 4 544 Reference solution Optimal ↓ raw
XSH-n20-k4-49 20 4 1,045 Reference solution Optimal ↓ raw
XSH-n20-k4-50 20 4 627 Reference solution Optimal ↓ raw
XSH-n20-k4-51 20 4 491 Reference solution Optimal ↓ raw
XSH-n20-k4-52 20 4 622 Reference solution Optimal ↓ raw
XSH-n20-k4-53 20 4 967 Reference solution Optimal ↓ raw
XSH-n20-k4-54 20 4 791 Reference solution Optimal ↓ raw
XSH-n20-k4-55 20 4 617 Reference solution Optimal ↓ raw