
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: 300 vol.
ACTIVE: 15 customers
- Kassel-Wilhelmshöhe (50 vol.)
- Düsseldorf Hbf (60 vol.)
- Aachen Hbf (90 vol.)
- Stuttgart Hbf (60 vol.)
- Dresden Hbf (40 vol.)
- Hamburg Hbf (85 vol.)
- Bremen Hbf (85 vol.)
- Leipzig Hbf (40 vol.)
- Dortmund Hbf (75 vol.)
- Nürnberg Hbf (100 vol.)
- Karlsruhe Hbf (40 vol.)
- Ulm Hbf (45 vol.)
- Köln Hbf (45 vol.)
- Würzburg Hbf (20 vol.)
- Osnabrück Hbf (50 vol.)
Tour 1
COST: 1520.946 km
LOAD: 290 vol.
- Dortmund Hbf | 75 vol.
- Düsseldorf Hbf | 60 vol.
- Köln Hbf | 45 vol.
- Aachen Hbf | 90 vol.
- Würzburg Hbf | 20 vol.
Tour 2
COST: 1233.851 km
LOAD: 300 vol.
- Dresden Hbf | 40 vol.
- Leipzig Hbf | 40 vol.
- Osnabrück Hbf | 50 vol.
- Bremen Hbf | 85 vol.
- Hamburg Hbf | 85 vol.
Tour 3
COST: 1526.532 km
LOAD: 295 vol.
- Kassel-Wilhelmshöhe | 50 vol.
- Karlsruhe Hbf | 40 vol.
- Stuttgart Hbf | 60 vol.
- Ulm Hbf | 45 vol.
- Nürnberg Hbf | 100 vol.

LOAD: 290 vol.
- Dortmund Hbf | 75 vol.
- Düsseldorf Hbf | 60 vol.
- Köln Hbf | 45 vol.
- Aachen Hbf | 90 vol.
- Würzburg Hbf | 20 vol.

LOAD: 300 vol.
- Dresden Hbf | 40 vol.
- Leipzig Hbf | 40 vol.
- Osnabrück Hbf | 50 vol.
- Bremen Hbf | 85 vol.
- Hamburg Hbf | 85 vol.

LOAD: 295 vol.
- Kassel-Wilhelmshöhe | 50 vol.
- Karlsruhe Hbf | 40 vol.
- Stuttgart Hbf | 60 vol.
- Ulm Hbf | 45 vol.
- Nürnberg Hbf | 100 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: [1] Berlin Hbf | Number of cities: 24 | Total loads: 885 vol. | Vehicle capacity: 300 vol. Loads: [50, 0, 60, 0, 0, 90, 60, 40, 85, 0, 85, 40, 75, 100, 40, 45, 45, 0, 0, 0, 20, 0, 50, 0] ITERATION Generation: #1 Best cost: 5063.599 | Path: [1, 0, 12, 2, 16, 22, 20, 1, 11, 7, 13, 15, 6, 1, 8, 10, 5, 14, 1] Best cost: 4566.452 | Path: [1, 5, 2, 16, 12, 20, 1, 7, 11, 8, 10, 22, 1, 0, 14, 6, 15, 13, 1] Generation: #2 Best cost: 4496.628 | Path: [1, 12, 2, 16, 5, 20, 1, 7, 11, 10, 8, 22, 1, 0, 14, 6, 15, 13, 1] OPTIMIZING each tour... Current: [[1, 12, 2, 16, 5, 20, 1], [1, 7, 11, 10, 8, 22, 1], [1, 0, 14, 6, 15, 13, 1]] [2] Cost: 1449.150 to 1233.851 | Optimized: [1, 7, 11, 22, 10, 8, 1] ACO RESULTS [1/290 vol./1520.946 km] Berlin Hbf -> Dortmund Hbf -> Düsseldorf Hbf -> Köln Hbf -> Aachen Hbf -> Würzburg Hbf --> Berlin Hbf [2/300 vol./1233.851 km] Berlin Hbf -> Dresden Hbf -> Leipzig Hbf -> Osnabrück Hbf -> Bremen Hbf -> Hamburg Hbf --> Berlin Hbf [3/295 vol./1526.532 km] Berlin Hbf -> Kassel-Wilhelmshöhe -> Karlsruhe Hbf -> Stuttgart Hbf -> Ulm Hbf -> Nürnberg Hbf --> Berlin Hbf OPTIMIZATION RESULT: 3 tours | 4281.329 km.