
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: 14 customers
- Kassel-Wilhelmshöhe (50 vol.)
- Hannover Hbf (40 vol.)
- Stuttgart Hbf (20 vol.)
- Dresden Hbf (85 vol.)
- München Hbf (35 vol.)
- Bremen Hbf (20 vol.)
- Karlsruhe Hbf (55 vol.)
- Ulm Hbf (50 vol.)
- Mannheim Hbf (100 vol.)
- Kiel Hbf (65 vol.)
- Mainz Hbf (65 vol.)
- Würzburg Hbf (70 vol.)
- Saarbrücken Hbf (80 vol.)
- Freiburg Hbf (25 vol.)
Tour 1
COST: 1393.019 km
LOAD: 290 vol.
- Mainz Hbf | 65 vol.
- Mannheim Hbf | 100 vol.
- Karlsruhe Hbf | 55 vol.
- Würzburg Hbf | 70 vol.
Tour 2
COST: 2034.774 km
LOAD: 295 vol.
- Dresden Hbf | 85 vol.
- München Hbf | 35 vol.
- Ulm Hbf | 50 vol.
- Stuttgart Hbf | 20 vol.
- Freiburg Hbf | 25 vol.
- Saarbrücken Hbf | 80 vol.
Tour 3
COST: 1251.004 km
LOAD: 175 vol.
- Kassel-Wilhelmshöhe | 50 vol.
- Hannover Hbf | 40 vol.
- Bremen Hbf | 20 vol.
- Kiel Hbf | 65 vol.

LOAD: 290 vol.
- Mainz Hbf | 65 vol.
- Mannheim Hbf | 100 vol.
- Karlsruhe Hbf | 55 vol.
- Würzburg Hbf | 70 vol.

LOAD: 295 vol.
- Dresden Hbf | 85 vol.
- München Hbf | 35 vol.
- Ulm Hbf | 50 vol.
- Stuttgart Hbf | 20 vol.
- Freiburg Hbf | 25 vol.
- Saarbrücken Hbf | 80 vol.

LOAD: 175 vol.
- Kassel-Wilhelmshöhe | 50 vol.
- Hannover Hbf | 40 vol.
- Bremen Hbf | 20 vol.
- Kiel Hbf | 65 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: 760 vol. | Vehicle capacity: 300 vol. Loads: [50, 0, 0, 0, 40, 0, 20, 85, 0, 35, 20, 0, 0, 0, 55, 50, 0, 100, 65, 65, 70, 80, 0, 25] ITERATION Generation: #1 Best cost: 6151.926 | Path: [1, 0, 4, 10, 18, 19, 14, 1, 7, 20, 17, 6, 23, 1, 15, 9, 21, 1] Best cost: 5719.888 | Path: [1, 4, 10, 18, 0, 19, 14, 1, 7, 20, 17, 6, 23, 1, 9, 15, 21, 1] Best cost: 5653.535 | Path: [1, 6, 14, 17, 19, 0, 1, 7, 10, 4, 18, 20, 1, 9, 15, 21, 23, 1] Best cost: 5630.979 | Path: [1, 7, 0, 19, 17, 1, 4, 10, 18, 20, 6, 14, 23, 1, 9, 15, 21, 1] Best cost: 5178.309 | Path: [1, 9, 15, 6, 14, 17, 23, 1, 7, 4, 10, 18, 0, 1, 20, 19, 21, 1] Best cost: 4983.932 | Path: [1, 9, 15, 6, 14, 17, 23, 1, 7, 20, 19, 21, 1, 4, 10, 18, 0, 1] Best cost: 4896.932 | Path: [1, 9, 15, 6, 14, 17, 23, 1, 7, 0, 4, 10, 18, 1, 20, 19, 21, 1] Best cost: 4886.206 | Path: [1, 9, 15, 6, 14, 17, 23, 1, 7, 0, 4, 10, 18, 1, 19, 21, 20, 1] Best cost: 4808.470 | Path: [1, 9, 15, 6, 14, 17, 23, 1, 7, 20, 19, 21, 1, 0, 4, 10, 18, 1] Generation: #4 Best cost: 4739.902 | Path: [1, 20, 19, 17, 14, 1, 7, 9, 15, 6, 23, 21, 1, 18, 10, 4, 0, 1] OPTIMIZING each tour... Current: [[1, 20, 19, 17, 14, 1], [1, 7, 9, 15, 6, 23, 21, 1], [1, 18, 10, 4, 0, 1]] [1] Cost: 1453.342 to 1393.019 | Optimized: [1, 19, 17, 14, 20, 1] [3] Cost: 1251.786 to 1251.004 | Optimized: [1, 0, 4, 10, 18, 1] ACO RESULTS [1/290 vol./1393.019 km] Berlin Hbf -> Mainz Hbf -> Mannheim Hbf -> Karlsruhe Hbf -> Würzburg Hbf --> Berlin Hbf [2/295 vol./2034.774 km] Berlin Hbf -> Dresden Hbf -> München Hbf -> Ulm Hbf -> Stuttgart Hbf -> Freiburg Hbf -> Saarbrücken Hbf --> Berlin Hbf [3/175 vol./1251.004 km] Berlin Hbf -> Kassel-Wilhelmshöhe -> Hannover Hbf -> Bremen Hbf -> Kiel Hbf --> Berlin Hbf OPTIMIZATION RESULT: 3 tours | 4678.797 km.