# 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: 36634
distance: 36634
duration: 36634
# iterations: 2000
run-time: 51.57 seconds
Routes
------
Route #1: 414 217 236 434 8 311 133 370 3 169 2 105
Route #2: 348 411 410 349 425 223 299 386 267 400 97 72
Route #3: 326 26 260 92 275 41 406 270 308 202 149 172
Route #4: 237 312 250 211 347 43 296 375 218 239 42 335
Route #5: 155 71 228 346 162 166 345 385 438 381 404 195
Route #6: 435 281 206 57 392 139 200 145 122 366 418 421
Route #7: 393 280 303 225 388 409 110 245 241 360 342 221
Route #8: 44 324 115 229 268 380 227 249 325 121 353 422
Route #9: 377 433 337 242 372 391 423 396 420 413 321 243
Route #10: 83 412 416 407 384 403 17 89 293 428 193 285
Route #11: 264 352 315 86 297 126 66 339 402 101 252 309
Route #12: 383 257 253 289 271 338 319 329 266 351 432 233
Route #13: 65 154 7 197 344 118 91 215 153 15 159 283
Route #14: 251 137 98 341 350 286 47 376 138 246 323 437
Route #15: 1 140 274 113 116 196 75 56 22 210 88 173
Route #16: 204 31 5 176 79 144 334 25 146
Route #17: 431 189 61 130 80 6 131 367 426 134 207 333
Route #18: 58 161 287 395 184 390 371 109 90 62 397 387
Route #19: 430 248 19 330 108 340 240 220 331 361 28 401
Route #20: 73 157 135 177 152 408 302 255 327 389 343 24
Route #21: 175 132 4 34 230 67 16 112 378 21 181 84
Route #22: 382 244 11 209 354 74 63 117 103 13 171 216
Route #23: 265 313 174 160 291 178 222 310 292 192 235 165
Route #24: 356 363 357 123 64 10 37 301 379 394 125 190
Route #25: 399 364 259 30 417 368 290 300 424 114 316 106
Route #26: 124 99 77 182 78 256 96 179 81
Route #27: 188 234 405 168 272 214 198 52 95 328 213 279
Route #28: 170 183 205 150 282 224 273 320 284 269 120 332
Route #29: 219 369 336 419 304 322 374 208 35 232 39 142
Route #30: 53 128 186 54 27 141 100 51 164 199 151 163
Route #31: 107 306 314 148 20 111 68 69 85 12 48 180
Route #32: 38 167 203 201 40 247 49 46 212 94 45 277
Route #33: 76 119 127 102 87 158 29 129 191 70 258 93
Route #34: 194 50 18 307 262 261 14 436 373 36 427 156
Route #35: 355 263 55 288 298 33 278 226 276 317 398 415
Route #36: 238 305 32 82 147 143 60 104 185 294 318 23
Route #37: 231 59 136 187 362 358 295 359 365 429 9 254
```

```
[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: 36634.
This is 0.7% 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: 7765
distance: 7765
duration: 17765
# iterations: 1000
run-time: 3.10 seconds
Routes
------
Route #1: 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
Route #2: 90 82 99 52 57 49 19 18 48 21 23 25 77 58 75 97 59 87 86 74 24 22 20 66
Route #3: 65 83 64 51 85 63 76 89 33 30 32 28 26 27 29 31 34 50 62 84 56 91
Route #4: 92 95 67 71 72 41 39 38 37 35 36 40 43 44 42 61 81 54 96 93 94 80
```

```
[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: 776.5.
This is 0.1% 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
]
header = list(data[0].keys())
rows = [datum.values() for datum in data]
tabulate(rows, header, tablefmt="html")
```

```
[12]:
```

num_stops | distance | service_duration | wait_duration | time_warp | delivery |
---|---|---|---|---|---|

32 | 2266 | 3200 | 0 | 0 | 5920 |

24 | 1970 | 2400 | 0 | 0 | 4180 |

22 | 1886 | 2200 | 0 | 0 | 3630 |

22 | 1643 | 2200 | 0 | 0 | 3510 |

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. We can see a jump to the start of the time window in the main (earliest) time line when a vehicle arrives early at a customer and has to wait. 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.reshape(-1), routes)):
plot_route_schedule(
INSTANCE,
route,
title=f"Route {idx}",
ax=ax,
legend=idx == 0,
)
fig.tight_layout()
```

## 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.

```
[14]:
```

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

```
[15]:
```

```
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.

```
[16]:
```

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

```
[17]:
```

```
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: 27199.0.
This is 5.4% worse than the best-known solution, which is 25797.5.
```

```
[18]:
```

```
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.