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: 19 customers
- Berlin Hbf (90 vol.)
- Frankfurt Hbf (40 vol.)
- Hannover Hbf (40 vol.)
- Aachen Hbf (80 vol.)
- Stuttgart Hbf (95 vol.)
- Hamburg Hbf (30 vol.)
- München Hbf (70 vol.)
- Bremen Hbf (25 vol.)
- Leipzig Hbf (70 vol.)
- Dortmund Hbf (40 vol.)
- Nürnberg Hbf (30 vol.)
- Ulm Hbf (65 vol.)
- Köln Hbf (30 vol.)
- Mannheim Hbf (35 vol.)
- Kiel Hbf (65 vol.)
- Mainz Hbf (25 vol.)
- Würzburg Hbf (20 vol.)
- Saarbrücken Hbf (20 vol.)
- Freiburg Hbf (30 vol.)
Tour 1
COST: 1660.096 km
LOAD: 395 vol.
- Hannover Hbf | 40 vol.
- Hamburg Hbf | 30 vol.
- Kiel Hbf | 65 vol.
- Bremen Hbf | 25 vol.
- Dortmund Hbf | 40 vol.
- Köln Hbf | 30 vol.
- Aachen Hbf | 80 vol.
- Mainz Hbf | 25 vol.
- Frankfurt Hbf | 40 vol.
- Würzburg Hbf | 20 vol.
Tour 2
COST: 1540.816 km
LOAD: 345 vol.
- Nürnberg Hbf | 30 vol.
- München Hbf | 70 vol.
- Ulm Hbf | 65 vol.
- Stuttgart Hbf | 95 vol.
- Freiburg Hbf | 30 vol.
- Saarbrücken Hbf | 20 vol.
- Mannheim Hbf | 35 vol.
Tour 3
COST: 849.361 km
LOAD: 160 vol.
- Berlin Hbf | 90 vol.
- Leipzig Hbf | 70 vol.
LOAD: 395 vol.
- Hannover Hbf | 40 vol.
- Hamburg Hbf | 30 vol.
- Kiel Hbf | 65 vol.
- Bremen Hbf | 25 vol.
- Dortmund Hbf | 40 vol.
- Köln Hbf | 30 vol.
- Aachen Hbf | 80 vol.
- Mainz Hbf | 25 vol.
- Frankfurt Hbf | 40 vol.
- Würzburg Hbf | 20 vol.
LOAD: 345 vol.
- Nürnberg Hbf | 30 vol.
- München Hbf | 70 vol.
- Ulm Hbf | 65 vol.
- Stuttgart Hbf | 95 vol.
- Freiburg Hbf | 30 vol.
- Saarbrücken Hbf | 20 vol.
- Mannheim Hbf | 35 vol.
LOAD: 160 vol.
- Berlin Hbf | 90 vol.
- Leipzig Hbf | 70 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: 900 vol. | Vehicle capacity: 400 vol. Loads: [0, 90, 0, 40, 40, 80, 95, 0, 30, 70, 25, 70, 40, 30, 0, 65, 30, 35, 65, 25, 20, 20, 0, 30] ITERATION Generation: #1 Best cost: 4880.951 | Path: [0, 1, 11, 4, 10, 8, 18, 3, 19, 0, 12, 16, 5, 21, 17, 6, 15, 13, 0, 20, 9, 23, 0] Best cost: 4255.822 | Path: [0, 3, 19, 17, 21, 23, 6, 15, 9, 20, 0, 12, 16, 5, 4, 10, 8, 18, 1, 0, 11, 13, 0] Best cost: 4224.861 | Path: [0, 18, 8, 4, 10, 12, 16, 5, 21, 19, 3, 0, 20, 13, 9, 15, 6, 17, 23, 0, 11, 1, 0] Best cost: 4169.019 | Path: [0, 10, 8, 18, 4, 12, 16, 5, 19, 3, 20, 0, 17, 21, 23, 6, 15, 9, 13, 0, 11, 1, 0] Best cost: 4164.910 | Path: [0, 10, 8, 18, 4, 12, 16, 5, 21, 19, 3, 0, 20, 13, 9, 15, 6, 17, 23, 0, 11, 1, 0] Generation: #2 Best cost: 4114.228 | Path: [0, 18, 8, 10, 4, 12, 16, 5, 21, 17, 19, 0, 20, 13, 9, 15, 6, 23, 3, 0, 11, 1, 0] Generation: #3 Best cost: 4065.567 | Path: [0, 4, 8, 18, 10, 12, 16, 5, 19, 3, 20, 0, 17, 21, 23, 6, 15, 9, 13, 0, 11, 1, 0] OPTIMIZING each tour... Current: [[0, 4, 8, 18, 10, 12, 16, 5, 19, 3, 20, 0], [0, 17, 21, 23, 6, 15, 9, 13, 0], [0, 11, 1, 0]] [2] Cost: 1556.083 to 1540.816 | Optimized: [0, 13, 9, 15, 6, 23, 21, 17, 0] [3] Cost: 849.388 to 849.361 | Optimized: [0, 1, 11, 0] ACO RESULTS [1/395 vol./1660.096 km] Kassel-Wilhelmshöhe -> Hannover Hbf -> Hamburg Hbf -> Kiel Hbf -> Bremen Hbf -> Dortmund Hbf -> Köln Hbf -> Aachen Hbf -> Mainz Hbf -> Frankfurt Hbf -> Würzburg Hbf --> Kassel-Wilhelmshöhe [2/345 vol./1540.816 km] Kassel-Wilhelmshöhe -> Nürnberg Hbf -> München Hbf -> Ulm Hbf -> Stuttgart Hbf -> Freiburg Hbf -> Saarbrücken Hbf -> Mannheim Hbf --> Kassel-Wilhelmshöhe [3/160 vol./ 849.361 km] Kassel-Wilhelmshöhe -> Berlin Hbf -> Leipzig Hbf --> Kassel-Wilhelmshöhe OPTIMIZATION RESULT: 3 tours | 4050.273 km.