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
     # trips: 37
   # clients: 438
   objective: 36627
    distance: 36627
    duration: 36627
# iterations: 2000
    run-time: 37.90 seconds

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

[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: 36627.
This is 0.6% 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
     # trips: 4
   # clients: 100
   objective: 7761
    distance: 7761
    duration: 17761
# iterations: 1000
    run-time: 2.83 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 = [
    {
        "location": visit.location,  # Client or depot location of visit
        "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: 27254.5.
This is 5.6% 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.