
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: 22 customers
- Berlin Hbf (50 vol.)
- Düsseldorf Hbf (35 vol.)
- Frankfurt Hbf (45 vol.)
- Hannover Hbf (35 vol.)
- Aachen Hbf (25 vol.)
- Stuttgart Hbf (25 vol.)
- Dresden Hbf (85 vol.)
- Hamburg Hbf (35 vol.)
- München Hbf (40 vol.)
- Bremen Hbf (65 vol.)
- Leipzig Hbf (35 vol.)
- Dortmund Hbf (100 vol.)
- Karlsruhe Hbf (30 vol.)
- Ulm Hbf (25 vol.)
- Köln Hbf (100 vol.)
- Mannheim Hbf (25 vol.)
- Kiel Hbf (95 vol.)
- Mainz Hbf (35 vol.)
- Würzburg Hbf (100 vol.)
- Saarbrücken Hbf (70 vol.)
- Osnabrück Hbf (100 vol.)
- Freiburg Hbf (65 vol.)
Tour 1
COST: 1453.615 km
LOAD: 400 vol.
- Hannover Hbf | 35 vol.
- Bremen Hbf | 65 vol.
- Hamburg Hbf | 35 vol.
- Kiel Hbf | 95 vol.
- Berlin Hbf | 50 vol.
- Dresden Hbf | 85 vol.
- Leipzig Hbf | 35 vol.
Tour 2
COST: 1078.084 km
LOAD: 395 vol.
- Dortmund Hbf | 100 vol.
- Düsseldorf Hbf | 35 vol.
- Köln Hbf | 100 vol.
- Aachen Hbf | 25 vol.
- Mainz Hbf | 35 vol.
- Mannheim Hbf | 25 vol.
- Karlsruhe Hbf | 30 vol.
- Frankfurt Hbf | 45 vol.
Tour 3
COST: 1530.559 km
LOAD: 325 vol.
- Saarbrücken Hbf | 70 vol.
- Freiburg Hbf | 65 vol.
- Stuttgart Hbf | 25 vol.
- Ulm Hbf | 25 vol.
- München Hbf | 40 vol.
- Würzburg Hbf | 100 vol.
Tour 4
COST: 348.872 km
LOAD: 100 vol.
- Osnabrück Hbf | 100 vol.

LOAD: 400 vol.
- Hannover Hbf | 35 vol.
- Bremen Hbf | 65 vol.
- Hamburg Hbf | 35 vol.
- Kiel Hbf | 95 vol.
- Berlin Hbf | 50 vol.
- Dresden Hbf | 85 vol.
- Leipzig Hbf | 35 vol.

LOAD: 395 vol.
- Dortmund Hbf | 100 vol.
- Düsseldorf Hbf | 35 vol.
- Köln Hbf | 100 vol.
- Aachen Hbf | 25 vol.
- Mainz Hbf | 35 vol.
- Mannheim Hbf | 25 vol.
- Karlsruhe Hbf | 30 vol.
- Frankfurt Hbf | 45 vol.

LOAD: 325 vol.
- Saarbrücken Hbf | 70 vol.
- Freiburg Hbf | 65 vol.
- Stuttgart Hbf | 25 vol.
- Ulm Hbf | 25 vol.
- München Hbf | 40 vol.
- Würzburg Hbf | 100 vol.

LOAD: 100 vol.
- Osnabrück 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: [0] Kassel-Wilhelmshöhe | Number of cities: 24 | Total loads: 1220 vol. | Vehicle capacity: 400 vol. Loads: [0, 50, 35, 45, 35, 25, 25, 85, 35, 40, 65, 35, 100, 0, 30, 25, 100, 25, 95, 35, 100, 70, 100, 65] ITERATION Generation: #1 Best cost: 6223.752 | Path: [0, 1, 11, 7, 4, 22, 10, 5, 0, 12, 2, 16, 3, 19, 17, 14, 6, 0, 20, 15, 9, 21, 23, 8, 0, 18, 0] Best cost: 5064.856 | Path: [0, 2, 16, 5, 12, 22, 4, 0, 3, 19, 17, 14, 6, 15, 9, 20, 21, 0, 10, 8, 18, 1, 11, 7, 0, 23, 0] Best cost: 4638.142 | Path: [0, 11, 7, 1, 8, 18, 10, 4, 0, 12, 16, 2, 5, 21, 17, 14, 0, 19, 3, 20, 6, 15, 9, 23, 0, 22, 0] Best cost: 4629.680 | Path: [0, 8, 18, 10, 22, 12, 0, 3, 19, 17, 14, 6, 15, 9, 11, 7, 1, 0, 16, 2, 5, 21, 23, 20, 0, 4, 0] Generation: #2 Best cost: 4550.593 | Path: [0, 1, 11, 7, 9, 15, 6, 14, 17, 3, 19, 0, 12, 2, 16, 5, 21, 23, 0, 22, 10, 8, 18, 4, 0, 20, 0] Generation: #3 Best cost: 4518.948 | Path: [0, 11, 7, 1, 8, 18, 10, 4, 0, 22, 12, 2, 16, 5, 19, 0, 3, 17, 14, 6, 15, 9, 23, 21, 0, 20, 0] Generation: #4 Best cost: 4504.780 | Path: [0, 4, 10, 8, 18, 1, 7, 11, 0, 12, 2, 16, 5, 19, 3, 17, 14, 0, 20, 6, 15, 9, 23, 21, 0, 22, 0] OPTIMIZING each tour... Current: [[0, 4, 10, 8, 18, 1, 7, 11, 0], [0, 12, 2, 16, 5, 19, 3, 17, 14, 0], [0, 20, 6, 15, 9, 23, 21, 0], [0, 22, 0]] [2] Cost: 1105.005 to 1078.084 | Optimized: [0, 12, 2, 16, 5, 19, 17, 14, 3, 0] [3] Cost: 1597.288 to 1530.559 | Optimized: [0, 21, 23, 6, 15, 9, 20, 0] ACO RESULTS [1/400 vol./1453.615 km] Kassel-Wilhelmshöhe -> Hannover Hbf -> Bremen Hbf -> Hamburg Hbf -> Kiel Hbf -> Berlin Hbf -> Dresden Hbf -> Leipzig Hbf --> Kassel-Wilhelmshöhe [2/395 vol./1078.084 km] Kassel-Wilhelmshöhe -> Dortmund Hbf -> Düsseldorf Hbf -> Köln Hbf -> Aachen Hbf -> Mainz Hbf -> Mannheim Hbf -> Karlsruhe Hbf -> Frankfurt Hbf --> Kassel-Wilhelmshöhe [3/325 vol./1530.559 km] Kassel-Wilhelmshöhe -> Saarbrücken Hbf -> Freiburg Hbf -> Stuttgart Hbf -> Ulm Hbf -> München Hbf -> Würzburg Hbf --> Kassel-Wilhelmshöhe [4/100 vol./ 348.872 km] Kassel-Wilhelmshöhe -> Osnabrück Hbf --> Kassel-Wilhelmshöhe OPTIMIZATION RESULT: 4 tours | 4411.130 km.