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: 15 customers
- Düsseldorf Hbf (25 vol.)
- Hannover Hbf (75 vol.)
- Aachen Hbf (50 vol.)
- Dresden Hbf (20 vol.)
- Hamburg Hbf (25 vol.)
- München Hbf (50 vol.)
- Bremen Hbf (25 vol.)
- Leipzig Hbf (100 vol.)
- Nürnberg Hbf (60 vol.)
- Ulm Hbf (25 vol.)
- Köln Hbf (20 vol.)
- Kiel Hbf (30 vol.)
- Würzburg Hbf (65 vol.)
- Saarbrücken Hbf (50 vol.)
- Osnabrück Hbf (30 vol.)
Tour 1
COST: 1919.221 km
LOAD: 400 vol.
- Köln Hbf | 20 vol.
- Aachen Hbf | 50 vol.
- Düsseldorf Hbf | 25 vol.
- Osnabrück Hbf | 30 vol.
- Bremen Hbf | 25 vol.
- Hamburg Hbf | 25 vol.
- Kiel Hbf | 30 vol.
- Hannover Hbf | 75 vol.
- Leipzig Hbf | 100 vol.
- Dresden Hbf | 20 vol.
Tour 2
COST: 1310.988 km
LOAD: 250 vol.
- Würzburg Hbf | 65 vol.
- Nürnberg Hbf | 60 vol.
- München Hbf | 50 vol.
- Ulm Hbf | 25 vol.
- Saarbrücken Hbf | 50 vol.
LOAD: 400 vol.
- Köln Hbf | 20 vol.
- Aachen Hbf | 50 vol.
- Düsseldorf Hbf | 25 vol.
- Osnabrück Hbf | 30 vol.
- Bremen Hbf | 25 vol.
- Hamburg Hbf | 25 vol.
- Kiel Hbf | 30 vol.
- Hannover Hbf | 75 vol.
- Leipzig Hbf | 100 vol.
- Dresden Hbf | 20 vol.
LOAD: 250 vol.
- Würzburg Hbf | 65 vol.
- Nürnberg Hbf | 60 vol.
- München Hbf | 50 vol.
- Ulm Hbf | 25 vol.
- Saarbrücken Hbf | 50 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: 650 vol. | Vehicle capacity: 400 vol. Loads: [0, 0, 25, 0, 75, 50, 0, 20, 25, 50, 25, 100, 0, 60, 0, 25, 20, 0, 30, 0, 65, 50, 30, 0] ITERATION Generation: #1 Best cost: 3325.823 | Path: [0, 2, 16, 5, 22, 4, 10, 8, 18, 7, 11, 0, 20, 13, 9, 15, 21, 0] Best cost: 3249.464 | Path: [0, 5, 16, 2, 22, 10, 8, 18, 4, 11, 7, 0, 20, 13, 9, 15, 21, 0] Best cost: 3246.088 | Path: [0, 7, 11, 4, 8, 18, 10, 22, 2, 16, 5, 0, 20, 13, 9, 15, 21, 0] OPTIMIZING each tour... Current: [[0, 7, 11, 4, 8, 18, 10, 22, 2, 16, 5, 0], [0, 20, 13, 9, 15, 21, 0]] [1] Cost: 1935.100 to 1919.221 | Optimized: [0, 16, 5, 2, 22, 10, 8, 18, 4, 11, 7, 0] ACO RESULTS [1/400 vol./1919.221 km] Kassel-Wilhelmshöhe -> Köln Hbf -> Aachen Hbf -> Düsseldorf Hbf -> Osnabrück Hbf -> Bremen Hbf -> Hamburg Hbf -> Kiel Hbf -> Hannover Hbf -> Leipzig Hbf -> Dresden Hbf --> Kassel-Wilhelmshöhe [2/250 vol./1310.988 km] Kassel-Wilhelmshöhe -> Würzburg Hbf -> Nürnberg Hbf -> München Hbf -> Ulm Hbf -> Saarbrücken Hbf --> Kassel-Wilhelmshöhe OPTIMIZATION RESULT: 2 tours | 3230.209 km.