Solving CVRP with ACO
Minimizing Travel Cost for Complex Delivery Problems
This scenario involves the Capacitated Vehicle Routing Problem,
solved using the meta-heuristics algorithm Ant Colony Optimization. Basically, VRP is a network consisting of a number of nodes
(sometimes called cities) and arcs connecting one to all others along with the corresponding costs.
Mostly, the aim is to minimize the cost in visiting each customer once and only once. The term
"capacitated" is added due to some capacity constraints on the vehicles (vcap).
Enter the problem. Some company wants to deliver loads to a number of customers. In this case, we
have 24 nodes based on the location of Germany's train stations (don't ask why). The delivery
always starts from and ends at the depot, visiting a list of customers in other cities. And then
a number of questions arise:
- How do we minimize the travel cost in terms of distance?
- How many trucks are required?
- Which cities are visited by the truck #1, #2. etc.?
- depot: [0..23], def = 0
- vcap: [200..400], def = 400
There is a way to set all the demands, but I don't think you are ready for that. 😉
VCAP: 400 vol.
ACTIVE: 20 customers
- Düsseldorf Hbf (30 vol.)
- Hannover Hbf (20 vol.)
- Aachen Hbf (90 vol.)
- Stuttgart Hbf (50 vol.)
- Dresden Hbf (50 vol.)
- Hamburg Hbf (45 vol.)
- München Hbf (100 vol.)
- Leipzig Hbf (90 vol.)
- Dortmund Hbf (60 vol.)
- Nürnberg Hbf (70 vol.)
- Karlsruhe Hbf (25 vol.)
- Ulm Hbf (45 vol.)
- Köln Hbf (70 vol.)
- Mannheim Hbf (90 vol.)
- Kiel Hbf (55 vol.)
- Mainz Hbf (50 vol.)
- Würzburg Hbf (50 vol.)
- Saarbrücken Hbf (85 vol.)
- Osnabrück Hbf (25 vol.)
- Freiburg Hbf (20 vol.)
Tour 1
COST: 1367.719 km
LOAD: 395 vol.
- Hannover Hbf | 20 vol.
- Hamburg Hbf | 45 vol.
- Kiel Hbf | 55 vol.
- Osnabrück Hbf | 25 vol.
- Dortmund Hbf | 60 vol.
- Düsseldorf Hbf | 30 vol.
- Köln Hbf | 70 vol.
- Aachen Hbf | 90 vol.
Tour 2
COST: 1347.71 km
LOAD: 370 vol.
- Mannheim Hbf | 90 vol.
- Karlsruhe Hbf | 25 vol.
- Stuttgart Hbf | 50 vol.
- Freiburg Hbf | 20 vol.
- Saarbrücken Hbf | 85 vol.
- Mainz Hbf | 50 vol.
- Würzburg Hbf | 50 vol.
Tour 3
COST: 1418.796 km
LOAD: 355 vol.
- Leipzig Hbf | 90 vol.
- Dresden Hbf | 50 vol.
- Nürnberg Hbf | 70 vol.
- München Hbf | 100 vol.
- Ulm Hbf | 45 vol.
LOAD: 395 vol.
- Hannover Hbf | 20 vol.
- Hamburg Hbf | 45 vol.
- Kiel Hbf | 55 vol.
- Osnabrück Hbf | 25 vol.
- Dortmund Hbf | 60 vol.
- Düsseldorf Hbf | 30 vol.
- Köln Hbf | 70 vol.
- Aachen Hbf | 90 vol.
LOAD: 370 vol.
- Mannheim Hbf | 90 vol.
- Karlsruhe Hbf | 25 vol.
- Stuttgart Hbf | 50 vol.
- Freiburg Hbf | 20 vol.
- Saarbrücken Hbf | 85 vol.
- Mainz Hbf | 50 vol.
- Würzburg Hbf | 50 vol.
LOAD: 355 vol.
- Leipzig Hbf | 90 vol.
- Dresden Hbf | 50 vol.
- Nürnberg Hbf | 70 vol.
- München Hbf | 100 vol.
- Ulm Hbf | 45 vol.
#generations: 10 for global, 5 for local
#ants: 5 times #active_customers
ACO
Rel. importance of pheromones α = 1.0
Rel. importance of visibility β = 10.0
Trail persistance ρ = 0.5
Pheromone intensity Q = 10
See this wikipedia page to learn more.
NETWORK Depo: [0] Kassel-Wilhelmshöhe | Number of cities: 24 | Total loads: 1120 vol. | Vehicle capacity: 400 vol. Loads: [0, 0, 30, 0, 20, 90, 50, 50, 45, 100, 0, 90, 60, 70, 25, 45, 70, 90, 55, 50, 50, 85, 25, 20] ITERATION Generation: #1 Best cost: 4296.691 | Path: [0, 2, 16, 5, 12, 22, 4, 8, 18, 0, 20, 13, 9, 15, 6, 14, 19, 0, 17, 21, 23, 7, 11, 0] Best cost: 4241.979 | Path: [0, 4, 8, 18, 22, 12, 2, 16, 5, 0, 20, 13, 9, 15, 6, 14, 23, 0, 19, 17, 21, 7, 11, 0] Best cost: 4236.668 | Path: [0, 21, 23, 14, 6, 15, 9, 13, 0, 12, 2, 16, 5, 19, 17, 0, 22, 4, 8, 18, 11, 7, 20, 0] Generation: #8 Best cost: 4218.164 | Path: [0, 4, 8, 18, 22, 12, 2, 16, 5, 0, 19, 17, 21, 14, 6, 15, 23, 0, 20, 13, 9, 7, 11, 0] Best cost: 4196.914 | Path: [0, 4, 8, 18, 22, 12, 2, 16, 5, 0, 20, 6, 14, 17, 19, 21, 23, 0, 11, 7, 13, 9, 15, 0] OPTIMIZING each tour... Current: [[0, 4, 8, 18, 22, 12, 2, 16, 5, 0], [0, 20, 6, 14, 17, 19, 21, 23, 0], [0, 11, 7, 13, 9, 15, 0]] [2] Cost: 1410.399 to 1347.710 | Optimized: [0, 17, 14, 6, 23, 21, 19, 20, 0] ACO RESULTS [1/395 vol./1367.719 km] Kassel-Wilhelmshöhe -> Hannover Hbf -> Hamburg Hbf -> Kiel Hbf -> Osnabrück Hbf -> Dortmund Hbf -> Düsseldorf Hbf -> Köln Hbf -> Aachen Hbf --> Kassel-Wilhelmshöhe [2/370 vol./1347.710 km] Kassel-Wilhelmshöhe -> Mannheim Hbf -> Karlsruhe Hbf -> Stuttgart Hbf -> Freiburg Hbf -> Saarbrücken Hbf -> Mainz Hbf -> Würzburg Hbf --> Kassel-Wilhelmshöhe [3/355 vol./1418.796 km] Kassel-Wilhelmshöhe -> Leipzig Hbf -> Dresden Hbf -> Nürnberg Hbf -> München Hbf -> Ulm Hbf --> Kassel-Wilhelmshöhe OPTIMIZATION RESULT: 3 tours | 4134.225 km.