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: 17 customers
- Berlin Hbf (30 vol.)
- Düsseldorf Hbf (95 vol.)
- Frankfurt Hbf (25 vol.)
- Hannover Hbf (35 vol.)
- Aachen Hbf (100 vol.)
- Stuttgart Hbf (40 vol.)
- Dresden Hbf (50 vol.)
- Hamburg Hbf (25 vol.)
- München Hbf (35 vol.)
- Bremen Hbf (30 vol.)
- Leipzig Hbf (35 vol.)
- Karlsruhe Hbf (35 vol.)
- Köln Hbf (95 vol.)
- Mannheim Hbf (35 vol.)
- Mainz Hbf (35 vol.)
- Saarbrücken Hbf (50 vol.)
- Freiburg Hbf (65 vol.)
Tour 1
COST: 1013.074 km
LOAD: 400 vol.
- Frankfurt Hbf | 25 vol.
- Mannheim Hbf | 35 vol.
- Saarbrücken Hbf | 50 vol.
- Aachen Hbf | 100 vol.
- Köln Hbf | 95 vol.
- Düsseldorf Hbf | 95 vol.
Tour 2
COST: 2341.961 km
LOAD: 380 vol.
- Hannover Hbf | 35 vol.
- Bremen Hbf | 30 vol.
- Hamburg Hbf | 25 vol.
- Berlin Hbf | 30 vol.
- Dresden Hbf | 50 vol.
- Leipzig Hbf | 35 vol.
- München Hbf | 35 vol.
- Stuttgart Hbf | 40 vol.
- Karlsruhe Hbf | 35 vol.
- Freiburg Hbf | 65 vol.
Tour 3
COST: 450.422 km
LOAD: 35 vol.
- Mainz Hbf | 35 vol.
LOAD: 400 vol.
- Frankfurt Hbf | 25 vol.
- Mannheim Hbf | 35 vol.
- Saarbrücken Hbf | 50 vol.
- Aachen Hbf | 100 vol.
- Köln Hbf | 95 vol.
- Düsseldorf Hbf | 95 vol.
LOAD: 380 vol.
- Hannover Hbf | 35 vol.
- Bremen Hbf | 30 vol.
- Hamburg Hbf | 25 vol.
- Berlin Hbf | 30 vol.
- Dresden Hbf | 50 vol.
- Leipzig Hbf | 35 vol.
- München Hbf | 35 vol.
- Stuttgart Hbf | 40 vol.
- Karlsruhe Hbf | 35 vol.
- Freiburg Hbf | 65 vol.
LOAD: 35 vol.
- Mainz Hbf | 35 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: 815 vol. | Vehicle capacity: 400 vol. Loads: [0, 30, 95, 25, 35, 100, 40, 50, 25, 35, 30, 35, 0, 0, 35, 0, 95, 35, 0, 35, 0, 50, 0, 65] ITERATION Generation: #1 Best cost: 5077.098 | Path: [0, 1, 7, 11, 17, 14, 6, 9, 3, 19, 21, 10, 0, 2, 16, 5, 4, 8, 0, 23, 0] Best cost: 4368.904 | Path: [0, 2, 16, 5, 17, 14, 6, 0, 4, 10, 8, 1, 7, 11, 3, 19, 21, 23, 0, 9, 0] Best cost: 4340.812 | Path: [0, 4, 8, 10, 16, 2, 5, 0, 3, 19, 17, 14, 6, 9, 21, 23, 7, 1, 0, 11, 0] Best cost: 4161.679 | Path: [0, 5, 16, 2, 3, 19, 17, 0, 4, 10, 8, 1, 11, 7, 9, 6, 14, 23, 0, 21, 0] Best cost: 4076.750 | Path: [0, 11, 7, 1, 8, 10, 4, 2, 16, 0, 3, 19, 17, 14, 6, 23, 21, 5, 0, 9, 0] Best cost: 4076.521 | Path: [0, 11, 7, 1, 8, 10, 4, 16, 2, 0, 3, 19, 17, 14, 6, 21, 23, 9, 0, 5, 0] Best cost: 4052.616 | Path: [0, 7, 11, 1, 8, 10, 4, 2, 16, 0, 3, 19, 17, 14, 6, 9, 23, 21, 0, 5, 0] Best cost: 3860.155 | Path: [0, 16, 2, 5, 21, 17, 3, 0, 4, 10, 8, 1, 11, 7, 9, 6, 14, 23, 0, 19, 0] Generation: #2 Best cost: 3814.407 | Path: [0, 2, 16, 5, 21, 17, 3, 0, 4, 10, 8, 1, 7, 11, 9, 6, 14, 23, 0, 19, 0] OPTIMIZING each tour... Current: [[0, 2, 16, 5, 21, 17, 3, 0], [0, 4, 10, 8, 1, 7, 11, 9, 6, 14, 23, 0], [0, 19, 0]] [1] Cost: 1022.024 to 1013.074 | Optimized: [0, 3, 17, 21, 5, 16, 2, 0] ACO RESULTS [1/400 vol./1013.074 km] Kassel-Wilhelmshöhe -> Frankfurt Hbf -> Mannheim Hbf -> Saarbrücken Hbf -> Aachen Hbf -> Köln Hbf -> Düsseldorf Hbf --> Kassel-Wilhelmshöhe [2/380 vol./2341.961 km] Kassel-Wilhelmshöhe -> Hannover Hbf -> Bremen Hbf -> Hamburg Hbf -> Berlin Hbf -> Dresden Hbf -> Leipzig Hbf -> München Hbf -> Stuttgart Hbf -> Karlsruhe Hbf -> Freiburg Hbf --> Kassel-Wilhelmshöhe [3/ 35 vol./ 450.422 km] Kassel-Wilhelmshöhe -> Mainz Hbf --> Kassel-Wilhelmshöhe OPTIMIZATION RESULT: 3 tours | 3805.457 km.