
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: 17 customers
- Kassel-Wilhelmshöhe (25 vol.)
- Frankfurt Hbf (30 vol.)
- Dresden Hbf (90 vol.)
- Hamburg Hbf (40 vol.)
- Bremen Hbf (30 vol.)
- Dortmund Hbf (40 vol.)
- Nürnberg Hbf (75 vol.)
- Karlsruhe Hbf (55 vol.)
- Ulm Hbf (60 vol.)
- Köln Hbf (95 vol.)
- Mannheim Hbf (65 vol.)
- Kiel Hbf (25 vol.)
- Mainz Hbf (75 vol.)
- Würzburg Hbf (65 vol.)
- Saarbrücken Hbf (45 vol.)
- Osnabrück Hbf (50 vol.)
- Freiburg Hbf (85 vol.)
Tour 1
COST: 1758.553 km
LOAD: 280 vol.
- Frankfurt Hbf | 30 vol.
- Mannheim Hbf | 65 vol.
- Karlsruhe Hbf | 55 vol.
- Freiburg Hbf | 85 vol.
- Saarbrücken Hbf | 45 vol.
Tour 2
COST: 1388.501 km
LOAD: 290 vol.
- Würzburg Hbf | 65 vol.
- Ulm Hbf | 60 vol.
- Nürnberg Hbf | 75 vol.
- Dresden Hbf | 90 vol.
Tour 3
COST: 1435.948 km
LOAD: 285 vol.
- Kassel-Wilhelmshöhe | 25 vol.
- Mainz Hbf | 75 vol.
- Köln Hbf | 95 vol.
- Dortmund Hbf | 40 vol.
- Osnabrück Hbf | 50 vol.
Tour 4
COST: 959.498 km
LOAD: 95 vol.
- Hamburg Hbf | 40 vol.
- Bremen Hbf | 30 vol.
- Kiel Hbf | 25 vol.

LOAD: 280 vol.
- Frankfurt Hbf | 30 vol.
- Mannheim Hbf | 65 vol.
- Karlsruhe Hbf | 55 vol.
- Freiburg Hbf | 85 vol.
- Saarbrücken Hbf | 45 vol.

LOAD: 290 vol.
- Würzburg Hbf | 65 vol.
- Ulm Hbf | 60 vol.
- Nürnberg Hbf | 75 vol.
- Dresden Hbf | 90 vol.

LOAD: 285 vol.
- Kassel-Wilhelmshöhe | 25 vol.
- Mainz Hbf | 75 vol.
- Köln Hbf | 95 vol.
- Dortmund Hbf | 40 vol.
- Osnabrück Hbf | 50 vol.

LOAD: 95 vol.
- Hamburg Hbf | 40 vol.
- Bremen Hbf | 30 vol.
- Kiel Hbf | 25 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: 950 vol. | Vehicle capacity: 300 vol. Loads: [25, 0, 0, 30, 0, 0, 0, 90, 40, 0, 30, 0, 40, 75, 55, 60, 95, 65, 25, 75, 65, 45, 50, 85] ITERATION Generation: #1 Best cost: 7019.464 | Path: [1, 0, 22, 10, 8, 18, 20, 3, 1, 7, 13, 17, 14, 1, 12, 16, 19, 21, 1, 23, 15, 1] Best cost: 6714.955 | Path: [1, 3, 19, 17, 14, 15, 1, 7, 13, 20, 0, 12, 1, 8, 10, 22, 16, 21, 18, 1, 23, 1] Best cost: 6543.258 | Path: [1, 7, 13, 20, 3, 0, 1, 8, 18, 22, 10, 12, 16, 1, 17, 14, 15, 21, 19, 1, 23, 1] Best cost: 6480.146 | Path: [1, 10, 8, 18, 22, 12, 16, 1, 7, 13, 20, 3, 0, 1, 15, 14, 17, 19, 21, 1, 23, 1] Best cost: 6344.037 | Path: [1, 12, 16, 19, 3, 14, 1, 7, 0, 22, 10, 8, 18, 1, 13, 20, 17, 21, 1, 23, 15, 1] Best cost: 6208.092 | Path: [1, 14, 17, 19, 3, 20, 1, 7, 0, 12, 16, 22, 1, 8, 18, 10, 23, 21, 15, 1, 13, 1] Best cost: 6161.130 | Path: [1, 23, 14, 17, 19, 1, 7, 13, 20, 3, 0, 1, 8, 18, 10, 22, 12, 16, 1, 15, 21, 1] Best cost: 6153.876 | Path: [1, 7, 13, 20, 3, 0, 1, 8, 18, 10, 22, 12, 16, 1, 19, 17, 14, 23, 1, 15, 21, 1] Best cost: 6133.083 | Path: [1, 23, 14, 17, 3, 20, 1, 7, 13, 15, 21, 0, 1, 8, 18, 10, 22, 12, 16, 1, 19, 1] Best cost: 6125.671 | Path: [1, 16, 12, 22, 10, 8, 18, 1, 7, 13, 20, 3, 0, 1, 19, 17, 14, 23, 1, 15, 21, 1] Best cost: 6021.907 | Path: [1, 21, 17, 14, 23, 3, 1, 7, 13, 20, 15, 1, 8, 18, 10, 22, 12, 16, 1, 0, 19, 1] Best cost: 5999.231 | Path: [1, 7, 15, 14, 17, 3, 1, 10, 8, 18, 22, 12, 16, 1, 20, 19, 21, 23, 0, 1, 13, 1] Best cost: 5961.873 | Path: [1, 17, 14, 23, 21, 3, 1, 7, 20, 13, 15, 1, 8, 18, 10, 22, 12, 16, 1, 0, 19, 1] Best cost: 5727.030 | Path: [1, 17, 14, 23, 21, 3, 1, 7, 13, 20, 15, 1, 0, 22, 12, 16, 19, 1, 8, 18, 10, 1] OPTIMIZING each tour... Current: [[1, 17, 14, 23, 21, 3, 1], [1, 7, 13, 20, 15, 1], [1, 0, 22, 12, 16, 19, 1], [1, 8, 18, 10, 1]] [1] Cost: 1763.369 to 1758.553 | Optimized: [1, 3, 17, 14, 23, 21, 1] [2] Cost: 1445.364 to 1388.501 | Optimized: [1, 20, 15, 13, 7, 1] [3] Cost: 1543.232 to 1435.948 | Optimized: [1, 0, 19, 16, 12, 22, 1] [4] Cost: 975.065 to 959.498 | Optimized: [1, 8, 10, 18, 1] ACO RESULTS [1/280 vol./1758.553 km] Berlin Hbf -> Frankfurt Hbf -> Mannheim Hbf -> Karlsruhe Hbf -> Freiburg Hbf -> Saarbrücken Hbf --> Berlin Hbf [2/290 vol./1388.501 km] Berlin Hbf -> Würzburg Hbf -> Ulm Hbf -> Nürnberg Hbf -> Dresden Hbf --> Berlin Hbf [3/285 vol./1435.948 km] Berlin Hbf -> Kassel-Wilhelmshöhe -> Mainz Hbf -> Köln Hbf -> Dortmund Hbf -> Osnabrück Hbf --> Berlin Hbf [4/ 95 vol./ 959.498 km] Berlin Hbf -> Hamburg Hbf -> Bremen Hbf -> Kiel Hbf --> Berlin Hbf OPTIMIZATION RESULT: 4 tours | 5542.500 km.