Classic VRPs¶

This notebook shows how to use PyVRP to solve two classic variants of the VRP: the capacitated vehicle routing problem (CVRP), and the vehicle routing problem with time windows (VRPTW). It builds on the tutorial by solving much larger instances, and going into more detail about the various plotting tools and diagnostics available in PyVRP.

A CVRP instance is defined on a complete graph \(G=(V,A)\), where \(V\) is the vertex set and \(A\) is the arc set. The vertex set \(V\) is partitioned into \(V=\{0\} \cup V_c\), where \(0\) represents the depot and \(V_c=\{1, \dots, n\}\) denotes the set of \(n\) customers. Each arc \((i, j) \in A\) has a weight \(d_{ij} \ge 0\) that represents the travel distance from \(i \in V\) to \(j \in V\). Each customer \(i \in V_c\) has a demand \(q_{i} \ge 0\). The objective is to find a feasible solution that minimises the total distance.

A VRPTW instance additionally incorporates time aspects into the problem. For the sake of exposition we assume the travel duration \(t_{ij} \ge 0\) is equal to the travel distance \(d_{ij}\) in this notebook. Each customer \(i \in V_c\) has a service time \(s_{i} \ge 0\) and a (hard) time window \(\left[e_i, l_i\right]\) that denotes the earliest and latest time that service can start. A vehicle is allowed to arrive at a customer location before the beginning of the time window, but it must wait for the window to open to start the delivery. Each vehicle must return to the depot before the end of the depot time window \(H\). The objective is to find a feasible solution that minimises the total distance.

Let’s first import what we will use in this notebook.

[1]:

import matplotlib.pyplot as plt
from tabulate import tabulate
from vrplib import read_solution

from pyvrp import Model, read
from pyvrp.plotting import (
    plot_coordinates,
    plot_instance,
    plot_result,
    plot_route_schedule,
)
from pyvrp.stop import MaxIterations, MaxRuntime

The capacitated VRP¶

Reading the instance¶

We will solve the X-n439-k37 instance, which is part of the X instance set that is widely used to benchmark CVRP algorithms. The function pyvrp.read reads the instance file and converts it to a ProblemData instance. We pass the argument round_func="round" to compute the Euclidean distances rounded to the nearest integral, which is the convention for the X benchmark set. We also load the best known solution to evaluate our solver later on.

[2]:

INSTANCE = read("data/X-n439-k37.vrp", round_func="round")
BKS = read_solution("data/X-n439-k37.sol")

Let’s plot the instance and see what we have.

[3]:

_, ax = plt.subplots(figsize=(8, 8))
plot_coordinates(INSTANCE, ax=ax)
plt.tight_layout()

Solving the instance¶

We will again use the Model interface to solve the instance. The Model interface supports a convenient from_data method that can be used to instantiate a model from a known ProblemData object.

[4]:

model = Model.from_data(INSTANCE)
result = model.solve(stop=MaxIterations(2000), seed=42, display=False)
print(result)

Solution results
================
    # routes: 37
   # clients: 438
   objective: 36693
    distance: 36693
    duration: 36693
# iterations: 2000
    run-time: 66.41 seconds

Routes
------
Route #1: 105 2 169 8 311 434 362 187 136 59 236 217
Route #2: 3 370 133 425 223 349 267 410 411 348 260 26
Route #3: 72 97 400 386 299 392 57 139 200 145 206 281
Route #4: 326 155 92 275 41 406 270 308 202 149 172
Route #5: 404 381 250 211 347 43 296 375 218 239 42 335
Route #6: 71 228 346 162 435 166 345 385 438 312 280 249
Route #7: 227 377 433 337 242 342 221 393 237 325 121 195
Route #8: 83 412 17 403 384 366 122 407 416 418 421 245
Route #9: 360 241 303 110 409 388 225 352 264 372 391 423
Route #10: 437 233 324 229 268 380 115 353 422 44
Route #11: 252 101 402 428 89 293 339 66 126 297 86 315
Route #12: 253 289 266 319 329 351 432 396 420 413 321 243
Route #13: 383 257 271 338 309 285 193 215 153 15 159 65
Route #14: 283 323 246 138 376 47 286 350 341 98 137 251
Route #15: 25 79 189 61 344 197 7 154 431 144 334 146
Route #16: 91 118 173 88 210 22 56 75 196 116 113 274
Route #17: 333 31 5 426 367 1 6 80 130 131 176 204
Route #18: 134 140 90 109 62 371 390 184 395 287 397 387
Route #19: 207 58 161 361 220 331 330 19 248 430 28 401
Route #20: 175 24 343 389 327 152 408 302 255 340 240 108
Route #21: 73 157 135 177 34 230 67 16 112 378 21 84
Route #22: 244 13 103 117 63 74 354 209 11 181 4 132
Route #23: 124 77 265 313 174 291 160 292 382 192 235 165
Route #24: 417 368 290 356 190 125 222 310 178 30 259 364
Route #25: 171 394 379 301 37 10 64 123 357 363 300 424
Route #26: 99 216 106 399 316 114 120 332 269 284 150 183
Route #27: 320 234 405 168 272 214 198 52 95 328 213 279
Route #28: 205 219 282 224 273 188 369 419 304 336 232 142
Route #29: 179 96 256 78 182 170 163 151 199 128 20 53
Route #30: 186 164 51 100 54 27 141 208 374 322 35 39
Route #31: 107 306 314 148 85 12 48 180 81
Route #32: 167 247 40 46 212 94 49 45 277 69 68 111
Route #33: 70 129 191 158 29 87 102 127 119 76 201 203
Route #34: 36 373 436 18 307 14 93 258 261 262 50 194
Route #35: 355 263 55 288 298 33 278 226 276 317 398 415
Route #36: 414 231 358 295 359 365 429 9 254 427 156 38
Route #37: 238 305 32 82 147 143 60 104 185 294 318 23

[5]:

gap = 100 * (result.cost() - BKS["cost"]) / BKS["cost"]
print(f"Found a solution with cost: {result.cost()}.")
print(f"This is {gap:.1f}% worse than the best known", end=" ")
print(f"solution, which is {BKS['cost']}.")

Found a solution with cost: 36693.
This is 0.8% worse than the best known solution, which is 36391.

We’ve managed to find a very good solution quickly!

The Result object also contains useful statistics about the optimisation. We can now plot these statistics as well as the final solution use plot_result.

[6]:

fig = plt.figure(figsize=(15, 9))
plot_result(result, INSTANCE, fig)
fig.tight_layout()

PyVRP internally uses a genetic algorithm consisting of a population of feasible and infeasible solutions. These solutions are iteratively combined into new offspring solutions, that should result in increasingly better solutions. Of course, the solutions should not all be too similar: then there is little to gain from combining different solutions. The top-left Diversity plot tracks the average diversity of solutions in each of the feasible and infeasible solution populations. The Objectives plot gives an overview of the best and average solution quality in the current population. The bottom-left figure shows iteration runtimes in seconds. Finally, the Solution plot shows the best observed solution.

The VRP with time windows¶

Reading the instance¶

We start with a basic example that loads an instance and solves it using the standard configuration used by the Model interface. For the basic example we use one of the well-known Solomon instances.

We again use the function pyvrp.read. We pass the argument round_func="dimacs" following the DIMACS VRP challenge convention, this computes distances and durations truncated to one decimal place.

[7]:

INSTANCE = read("data/RC208.vrp", round_func="dimacs")
BKS = read_solution("data/RC208.sol")

Let’s plot the instance and see what we have. The function plot_instance will plot time windows, delivery demands and coordinates, which should give us a good impression of what the instance looks like. These plots can also be produced separately by calling the appropriate plot_* function: see the API documentation for details.

[8]:

fig = plt.figure(figsize=(12, 6))
plot_instance(INSTANCE, fig)

Solving the instance¶

We will again use the Model interface to solve the instance.

[9]:

model = Model.from_data(INSTANCE)
result = model.solve(stop=MaxIterations(1000), seed=42, display=False)
print(result)

Solution results
================
    # routes: 4
   # clients: 100
   objective: 7761
    distance: 7761
    duration: 17761
# iterations: 1000
    run-time: 3.70 seconds

Routes
------
Route #1: 90 65 82 99 52 83 64 49 19 18 48 21 23 25 77 58 75 97 59 87 74 86 57 24 22 20 66
Route #2: 94 92 95 67 62 50 34 31 29 27 26 28 30 32 33 76 89 63 85 51 84 56 91 80
Route #3: 61 42 44 39 38 36 35 37 40 43 41 72 71 93 96 54 81
Route #4: 69 98 88 2 6 7 79 73 78 12 14 47 17 16 15 13 9 11 10 53 60 8 46 4 45 5 3 1 70 100 55 68

[10]:

cost = result.cost() / 10
gap = 100 * (cost - BKS["cost"]) / BKS["cost"]
print(f"Found a solution with cost: {cost}.")
print(f"This is {gap:.1f}% worse than the optimal solution,", end=" ")
print(f"which is {BKS['cost']}.")

Found a solution with cost: 776.1.
This is 0.0% worse than the optimal solution, which is 776.1.

We’ve managed to find a (near) optimal solution in a few seconds!

[11]:

fig = plt.figure(figsize=(15, 9))
plot_result(result, INSTANCE, fig)
fig.tight_layout()

We can also inspect some statistics of the different routes, such as route distance, various durations, the number of stops and total delivery amount.

[12]:

solution = result.best
routes = solution.routes()

data = [
    {
        "num_stops": len(route),
        "distance": route.distance(),
        "service_duration": route.service_duration(),
        "wait_duration": route.wait_duration(),
        "time_warp": route.time_warp(),
        "delivery": route.delivery(),
    }
    for route in routes
]

tabulate(data, headers="keys", tablefmt="html")

[12]:

We can inspect the routes in more detail using the plot_route_schedule function. This will plot distance on the x-axis, and time on the y-axis, separating actual travel/driving time from waiting and service time. The clients visited are plotted as grey vertical bars indicating their time windows. In some cases, there is slack in the route indicated by a semi-transparent region on top of the earliest time line. The grey background indicates the remaining load of the truck during the route, where the (right) y-axis ends at the vehicle capacity.

[13]:

fig, axarr = plt.subplots(2, 2, figsize=(15, 9))
for idx, (ax, route) in enumerate(zip(axarr.flatten(), routes)):
    plot_route_schedule(
        INSTANCE,
        route,
        title=f"Route {idx}",
        ax=ax,
        legend=idx == 0,
    )

fig.tight_layout()

Each route begins at a given start_time, that can be obtained as follows. Note that this start time is typically not zero, that is, routes do not have to start immediately at the beginning of the time horizon.

[14]:

solution = result.best
shortest_route = min(solution.routes(), key=len)

shortest_route.start_time()

[14]:

2991

Some of the statistics presented in the plots above can also be obtained from the route schedule, as follows:

[15]:

data = [
    {
        "start_service": visit.start_service,
        "end_service": visit.end_service,
        "service_duration": visit.service_duration,
        "wait_duration": visit.wait_duration,  # if vehicle arrives early
    }
    for visit in shortest_route.schedule()
]

tabulate(data, headers="keys", tablefmt="html")

[15]:

Solving a larger VRPTW instance¶

To show that PyVRP can also handle much larger instances, we will solve one of the largest Gehring and Homberger VRPTW benchmark instances. The selected instance - RC2_10_5 - has 1000 clients.

[16]:

INSTANCE = read("data/RC2_10_5.vrp", round_func="dimacs")
BKS = read_solution("data/RC2_10_5.sol")

[17]:

fig = plt.figure(figsize=(15, 9))
plot_instance(INSTANCE, fig)

Here, we will use a runtime-based stopping criterion: we give the solver 30 seconds to compute.

[18]:

model = Model.from_data(INSTANCE)
result = model.solve(stop=MaxRuntime(30), seed=42, display=False)

[19]:

cost = result.cost() / 10
gap = 100 * (cost - BKS["cost"]) / BKS["cost"]
print(f"Found a solution with cost: {cost}.")
print(f"This is {gap:.1f}% worse than the best-known solution,", end=" ")
print(f"which is {BKS['cost']}.")

Found a solution with cost: 27368.6.
This is 6.1% worse than the best-known solution, which is 25797.5.

[20]:

plot_result(result, INSTANCE)
plt.tight_layout()

Conclusion¶

In this notebook, we used PyVRP’s Model interface to solve a CVRP instance with 438 clients to near-optimality, as well as several VRPTW instances, including a large 1000 client instance. Moreover, we demonstrated how to use the plotting tools to visualise the instance and statistics collected during the search procedure.