VRPBench: A Vehicle Routing Benchmark Tool
October 19, 2016
Guilherme A. Zeni1 , Mauro Menzori1, P. S. Martins1, Luis A. A. Meira1
Abstract
The number of optimization techniques in the combinatorial do- main is large and diversified. Nevertheless, there is still a lack of real benchmarks to validate optimization algorithms. In this work we in- troduce VRPBench, a tool to create instances and visualize solutions to the Vehicle Routing Problem (VRP) in a planar graph embedded in the Euclidean 2D space. We use VRPBench to model a real-world mail delivery case of the city of Artur Nogueira. Such scenarios were char- acterized as a multi-objective optimization of the VRP. We extracted a weighted graph from a digital map of the city to create a challeng- ing benchmark for the VRP. Each instance models one generic day of mail delivery with hundreds to thousands of delivery points, thus al- lowing both the comparison and validation of optimization algorithms for routing problems. arXiv:1610.05402v1 [cs.AI] 18 Oct 2016
Keywords: Benchmark, VRP, Graphs and Combinatorial Optimisa- tion, Free Software Tool.
1School of Technology, University of Campinas, Limeira – SP – Brazil
1 1 Introduction
Benchmarks are found in various fields of science, such as geology [13], economy [25], climatology [36], among other areas. Specifically in computer science, benchmarks play a central role, e.g. in image processing [15, 29, 22], hardware performance [7] and optimization [32, 28,5]. In the context of optimization, David S. Johnson [24] divided algorithm analysis in three approaches: the worst-case, the average-case, and the ex- perimental analysis. Relative to experimental papers, he identifies four situ- ations: (i) to solve a real problem; (ii) to provide evidence that an algorithm is superior than the others; (iii) to better understand a problem; and (iv) to study the average-case. He suggests the use of well-established benchmarks to provide evidence of the superiority of an algorithm (item ii). Such papers are called horse race papers. Johnson highlights that reproducibility and comparability are essential aspects present in any experimental paper. He also advocates the use of instances that lead to general conclusions. The author mentions the diffi- culty in justifying experiments on problems with no direct application. Such problems have no real instances and the researcher is forced to generate the data in a vacuum. Johnson cautions against a pitfall: the researcher starts by using randomly-generated instances to evaluate the algorithms and ends up using the algorithm to explore the properties of the randomly-generated instances. According to him, another pitfall is spending time processing useless experiments that attempt to answer the wrong questions. Our work deals with a variant of the Vehicle Routing Problem (VRP) based on a real mail delivery case of the city of Artur Nogueira (see the contributions at the end of this section). One of the first references to the VRP dates back to 1959 [14] under the name Truck Dispaching Problem, a generalization of the Traveler Salesman Problem (TSP). The term VRP was first seen in the paper by Christophides [9], in 1976.
2 Christophides defines VRP as a generic name, given to a class of problems that involves the visit of “customers” using vehicles. Real world aspects may impose variants of the problem. For example, the Capacitaded-VRP (CVRP) [18] considers a limit to the vehicle capacity, the VRP with Time Windows (VRPTW) [26] accounts for delivery time windows, and the Multi-Depot VRP (MDVRP) [34] extends the number of depots. Other variants may be easily found in the literature. In 1991, Reinelt [32] created a benchmark for the TSP, known as TSPLib. In his work, he consolidated non-solved instances from twenty distinct pa- pers. His repository (TSPLIB95) [33] has instances of both the symmetric and the asymmetric traveling salesman problem (TSP/aTSP) as well as three related problems: (i) CVRP; (ii) Sequential Ordered Problem (SOP); and (iii) Hamiltonian Cycle Problem (HCP). The number of instances is 113, 19, 16, 41, 9 for TSP, aTSP, CVRP, SOP, and HCP, respectively. The number of vertices varies from 14 to 85900 for the TSP, 17 to 443 for the aTSP, 7 to 262 for the CVRP, 7 to 378 for the SOP, and from 1000 to 5000 for the HCP. The optimum of all TSPLib instances was finally achieved in 2007, after sixteen years of notable progress in the development of algorithms. The optimum of the d15112 instance was found in 2001 [2]. This instance con- tains 15112 German cities and required 22.6 years of processing split across 110 500MHz processors [12]. The instance pla33810 was solved [2] in March 2004. The pla33810 instance represents a printed circuit board with 33810 nodes and it was solved in 15.7 years of processing [16]. The last instance of the TSPLib, called pla85900, was solved in 2006 [2]. This instance contains 85900 nodes representing a VLSI application. Solomon [35] created a benchmark for the VRPTW in 1987. It is com- posed by 56 instances partitioned in 6 sets. The number of customers is 100 in all instances. The vehicle has a fixed capacity and the customers
3 have demand or weight. The number of vehicles is not fixed, but it derives the fact that there is a limit of capacity. In this view, the problem can be considered multi-objective. It aims to minimize the route and the number of vehicles. The first optimum solution was published in 1999 [27]. In 2005, Chabrier [6] managed to solve 17 of the instances that still had remained unsolved in the benchmark. In 2010, Amini et. al [1] obtained solutions very close to optimum, considering the first 25 customers only. In july 2015, after 28 years of the benchmark launching, Jawarneh and Abdullah [23] published in PlosONE a Bee Colony Optimization metaheuristic. Such al- gorithm reached 11 new best results in Solomon’s 56 VRPTW 100 customer instances. It is really surprising that such small instances have an internal structure so complex to be optimized. In Figure 1, there is a simple instance of Solomon’s composed by 100 customers and its solution considering 3 ve-
Table 1 hicles.
RC207 NO. XCOORD. YCOORD. DEMAND READY TIME DUE DATE SERVICE TIME 90
VEHICLE 0 40 50 0 0 240 0
NUMBER CAPACITY 85 20 49 191 10
25 1000 75 30 30 168 10
3 22 85 10 95 152 10 67.5 CUSTOMER 4 20 80 40 69 193 10 CUST NO. XCOORD. YCOORD. DEMAND READY TIME DUE DATE SERVICE TIME
0 40 50 0 0 960 0
65 35 40 16 11 570 10
83 37 31 14 19 326 10 45 64 45 30 17 20 497 10
95 56 37 6 20 299 10
67 64 42 9 112 543 10
31 88 35 20 102 347 10
29 90 35 10 131 376 10 22.5 28 92 30 10 68 349 10
30 88 30 10 114 497 10
33 85 25 10 51 673 10
63 65 20 6 39 353 10
76 60 12 31 162 493 10 0 51 55 20 19 281 490 10 0 25 50 75 100 19 42 10 40 205 422 10
21 40 5 10 158 391 10 18 44 5 20 338 451 10 Figure 1: The Solomon’s RC207 instance composed by 100 customers and 23 38 5 30 45 522 10
75 5 5 16 183 400 10 59 10 20 19 42 466 10 1 depot. Bent and Hentenryck proposed the solution containing 3 routes, 87 12 24 13 365 472 10
74 20 20 8 427 913 10 86 21 24 28 332 485 10 in 2001 [4]. The picture does not show the capacity and the time windows 57 30 25 23 347 566 10
22 40 15 40 263 606 10
20 42 15 10 511 682 10 constraints.
49 42 12 10 423 576 10
25 35 5 20 348 904 10
77 23 3 7 360 900 10 58 15 10 20 380 902 10 Despite of their complexity, the TSPLib and the Solomon benchmarks 97 4 18 35 546 667 10
13 5 35 10 359 911 10
10 10 40 30 470 687 10
17 0 45 20 359 909 10
60 15 60 17 613 894 10 55 30 60 16 421 896 10 4 100 31 67 3 262 930 10
70 35 69 23 211 930 10
68 40 60 21 441 902 10
0 40 50 0 0 960 0
82 27 43 9 85 338 10
99 26 35 15 170 565 10
52 25 30 3 109 360 10
9 10 35 20 236 625 10
11 8 40 40 33 510 10
15 2 40 20 39 538 10
16 0 40 20 214 415 10
47 2 45 10 38 366 10
14 5 45 10 35 457 10
12 8 45 20 166 399 10
53 20 50 5 284 585 10
78 8 56 27 195 654 10
73 2 60 5 255 436 10
79 6 68 30 258 565 10
7 15 75 20 281 438 10
6 18 75 20 60 159 10
8 15 80 10 272 581 10
46 18 80 10 435 662 10
4 20 80 40 418 913 10
45 20 82 10 37 677 10
3 22 85 10 473 588 10
1 25 85 20 591 874 10
43 55 85 20 549 718 10
36 65 85 40 43 712 10
35 67 85 20 396 905 10
37 65 82 10 530 703 10
54 55 60 16 568 753 10
0 40 50 0 0 960 0
69 31 52 27 9 532 10
98 26 52 9 109 354 10
88 24 58 19 100 539 10
2 22 75 30 73 350 10
5 20 85 20 40 390 10
42 55 80 10 33 624 10
44 55 82 10 62 491 10
40 60 85 30 233 544 10
38 62 80 30 131 544 10
39 60 80 10 36 800 10
41 58 75 20 207 664 10
72 63 65 8 27 382 10
71 65 55 14 201 400 10
93 61 52 3 298 928 10
81 49 58 10 369 460 10
61 45 65 9 230 405 10
90 37 47 6 324 513 10
56 50 35 19 404 829 10
84 57 29 18 304 605 10
85 63 23 2 318 489 10
94 57 48 23 385 518 10
96 55 54 26 538 705 10
92 53 43 14 14 669 10
62 65 35 3 29 587 10
50 72 35 30 471 662 10
34 85 35 30 462 631 10
27 95 35 20 57 664 10
26 95 30 30 509 668 10
32 87 30 10 564 797 10
89 67 5 25 565 897 10
48 42 5 10 636 904 10
24 38 15 10 598 883 10
66 41 37 16 423 828 10
91 49 42 13 651 906 10
80 47 47 13 426 942 10
0 40 50 0 0 960 0