
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: 20 customers
- Kassel-Wilhelmshöhe (20 vol.)
- Düsseldorf Hbf (20 vol.)
- Frankfurt Hbf (70 vol.)
- Hannover Hbf (60 vol.)
- Aachen Hbf (55 vol.)
- Dresden Hbf (85 vol.)
- Hamburg Hbf (90 vol.)
- Bremen Hbf (90 vol.)
- Leipzig Hbf (50 vol.)
- Nürnberg Hbf (90 vol.)
- Karlsruhe Hbf (80 vol.)
- Ulm Hbf (80 vol.)
- Köln Hbf (95 vol.)
- Mannheim Hbf (85 vol.)
- Kiel Hbf (20 vol.)
- Mainz Hbf (90 vol.)
- Würzburg Hbf (90 vol.)
- Saarbrücken Hbf (45 vol.)
- Osnabrück Hbf (100 vol.)
- Freiburg Hbf (60 vol.)
Tour 1
COST: 1370.993 km
LOAD: 295 vol.
- Frankfurt Hbf | 70 vol.
- Nürnberg Hbf | 90 vol.
- Leipzig Hbf | 50 vol.
- Dresden Hbf | 85 vol.
Tour 2
COST: 1414.495 km
LOAD: 290 vol.
- Osnabrück Hbf | 100 vol.
- Düsseldorf Hbf | 20 vol.
- Köln Hbf | 95 vol.
- Aachen Hbf | 55 vol.
- Kassel-Wilhelmshöhe | 20 vol.
Tour 3
COST: 972.057 km
LOAD: 260 vol.
- Hannover Hbf | 60 vol.
- Bremen Hbf | 90 vol.
- Hamburg Hbf | 90 vol.
- Kiel Hbf | 20 vol.
Tour 4
COST: 1313.53 km
LOAD: 265 vol.
- Mainz Hbf | 90 vol.
- Mannheim Hbf | 85 vol.
- Würzburg Hbf | 90 vol.
Tour 5
COST: 1841.269 km
LOAD: 265 vol.
- Ulm Hbf | 80 vol.
- Karlsruhe Hbf | 80 vol.
- Freiburg Hbf | 60 vol.
- Saarbrücken Hbf | 45 vol.

LOAD: 295 vol.
- Frankfurt Hbf | 70 vol.
- Nürnberg Hbf | 90 vol.
- Leipzig Hbf | 50 vol.
- Dresden Hbf | 85 vol.

LOAD: 290 vol.
- Osnabrück Hbf | 100 vol.
- Düsseldorf Hbf | 20 vol.
- Köln Hbf | 95 vol.
- Aachen Hbf | 55 vol.
- Kassel-Wilhelmshöhe | 20 vol.

LOAD: 260 vol.
- Hannover Hbf | 60 vol.
- Bremen Hbf | 90 vol.
- Hamburg Hbf | 90 vol.
- Kiel Hbf | 20 vol.

LOAD: 265 vol.
- Mainz Hbf | 90 vol.
- Mannheim Hbf | 85 vol.
- Würzburg Hbf | 90 vol.

LOAD: 265 vol.
- Ulm Hbf | 80 vol.
- Karlsruhe Hbf | 80 vol.
- Freiburg Hbf | 60 vol.
- Saarbrücken 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: [1] Berlin Hbf | Number of cities: 24 | Total loads: 1375 vol. | Vehicle capacity: 300 vol. Loads: [20, 0, 20, 70, 60, 55, 0, 85, 90, 0, 90, 50, 0, 90, 80, 80, 95, 85, 20, 90, 90, 45, 100, 60] ITERATION Generation: #1 Best cost: 7744.240 | Path: [1, 0, 4, 10, 22, 2, 1, 11, 7, 13, 3, 1, 8, 18, 16, 5, 1, 20, 19, 17, 1, 15, 14, 23, 21, 1] Best cost: 7135.673 | Path: [1, 5, 16, 2, 0, 22, 1, 11, 7, 3, 19, 1, 4, 10, 8, 18, 1, 13, 20, 15, 1, 17, 14, 21, 23, 1] Best cost: 7077.070 | Path: [1, 22, 2, 16, 5, 0, 1, 7, 11, 13, 3, 1, 8, 18, 10, 4, 1, 17, 14, 23, 21, 1, 20, 19, 15, 1] Best cost: 7054.852 | Path: [1, 0, 5, 16, 2, 22, 1, 11, 7, 20, 3, 1, 8, 18, 10, 4, 1, 19, 17, 14, 21, 1, 13, 15, 23, 1] Best cost: 7047.993 | Path: [1, 5, 16, 2, 22, 0, 1, 11, 7, 13, 3, 1, 4, 10, 8, 18, 1, 20, 19, 17, 1, 15, 14, 23, 21, 1] Generation: #3 Best cost: 7000.135 | Path: [1, 22, 2, 16, 5, 0, 1, 11, 7, 20, 3, 1, 8, 18, 10, 4, 1, 17, 14, 23, 21, 1, 13, 15, 19, 1] Generation: #10 Best cost: 6994.857 | Path: [1, 11, 7, 13, 3, 1, 22, 16, 2, 5, 0, 1, 4, 10, 8, 18, 1, 20, 19, 17, 1, 15, 14, 23, 21, 1] OPTIMIZING each tour... Current: [[1, 11, 7, 13, 3, 1], [1, 22, 16, 2, 5, 0, 1], [1, 4, 10, 8, 18, 1], [1, 20, 19, 17, 1], [1, 15, 14, 23, 21, 1]] [1] Cost: 1403.656 to 1370.993 | Optimized: [1, 3, 13, 11, 7, 1] [2] Cost: 1442.725 to 1414.495 | Optimized: [1, 22, 2, 16, 5, 0, 1] [4] Cost: 1335.150 to 1313.530 | Optimized: [1, 19, 17, 20, 1] ACO RESULTS [1/295 vol./1370.993 km] Berlin Hbf -> Frankfurt Hbf -> Nürnberg Hbf -> Leipzig Hbf -> Dresden Hbf --> Berlin Hbf [2/290 vol./1414.495 km] Berlin Hbf -> Osnabrück Hbf -> Düsseldorf Hbf -> Köln Hbf -> Aachen Hbf -> Kassel-Wilhelmshöhe --> Berlin Hbf [3/260 vol./ 972.057 km] Berlin Hbf -> Hannover Hbf -> Bremen Hbf -> Hamburg Hbf -> Kiel Hbf --> Berlin Hbf [4/265 vol./1313.530 km] Berlin Hbf -> Mainz Hbf -> Mannheim Hbf -> Würzburg Hbf --> Berlin Hbf [5/265 vol./1841.269 km] Berlin Hbf -> Ulm Hbf -> Karlsruhe Hbf -> Freiburg Hbf -> Saarbrücken Hbf --> Berlin Hbf OPTIMIZATION RESULT: 5 tours | 6912.344 km.