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: 15 customers
- Düsseldorf Hbf (50 vol.)
- Frankfurt Hbf (25 vol.)
- Aachen Hbf (65 vol.)
- Stuttgart Hbf (50 vol.)
- Dresden Hbf (20 vol.)
- München Hbf (85 vol.)
- Bremen Hbf (40 vol.)
- Leipzig Hbf (100 vol.)
- Dortmund Hbf (90 vol.)
- Karlsruhe Hbf (65 vol.)
- Mannheim Hbf (75 vol.)
- Kiel Hbf (30 vol.)
- Würzburg Hbf (45 vol.)
- Saarbrücken Hbf (85 vol.)
- Freiburg Hbf (85 vol.)
Tour 1
COST: 1496.639 km
LOAD: 300 vol.
- Bremen Hbf | 40 vol.
- Dortmund Hbf | 90 vol.
- Düsseldorf Hbf | 50 vol.
- Leipzig Hbf | 100 vol.
- Dresden Hbf | 20 vol.
Tour 2
COST: 1611.912 km
LOAD: 295 vol.
- Frankfurt Hbf | 25 vol.
- Mannheim Hbf | 75 vol.
- Karlsruhe Hbf | 65 vol.
- Saarbrücken Hbf | 85 vol.
- Würzburg Hbf | 45 vol.
Tour 3
COST: 2123.459 km
LOAD: 285 vol.
- München Hbf | 85 vol.
- Stuttgart Hbf | 50 vol.
- Freiburg Hbf | 85 vol.
- Aachen Hbf | 65 vol.
Tour 4
COST: 701.943 km
LOAD: 30 vol.
- Kiel Hbf | 30 vol.
LOAD: 300 vol.
- Bremen Hbf | 40 vol.
- Dortmund Hbf | 90 vol.
- Düsseldorf Hbf | 50 vol.
- Leipzig Hbf | 100 vol.
- Dresden Hbf | 20 vol.
LOAD: 295 vol.
- Frankfurt Hbf | 25 vol.
- Mannheim Hbf | 75 vol.
- Karlsruhe Hbf | 65 vol.
- Saarbrücken Hbf | 85 vol.
- Würzburg Hbf | 45 vol.
LOAD: 285 vol.
- München Hbf | 85 vol.
- Stuttgart Hbf | 50 vol.
- Freiburg Hbf | 85 vol.
- Aachen Hbf | 65 vol.
LOAD: 30 vol.
- Kiel Hbf | 30 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: 910 vol. | Vehicle capacity: 300 vol. Loads: [0, 0, 50, 25, 0, 65, 50, 20, 0, 85, 40, 100, 90, 0, 65, 0, 0, 75, 30, 0, 45, 85, 0, 85] ITERATION Generation: #1 Best cost: 7004.752 | Path: [1, 2, 12, 5, 21, 1, 11, 7, 9, 6, 20, 1, 10, 18, 3, 17, 14, 1, 23, 1] Best cost: 6752.814 | Path: [1, 3, 17, 14, 6, 20, 10, 1, 7, 11, 9, 23, 1, 18, 12, 2, 5, 1, 21, 1] Best cost: 6424.190 | Path: [1, 6, 14, 17, 3, 20, 7, 1, 11, 10, 18, 5, 2, 1, 9, 23, 21, 1, 12, 1] Best cost: 6226.940 | Path: [1, 18, 10, 12, 2, 5, 3, 1, 7, 11, 20, 6, 14, 1, 17, 21, 23, 1, 9, 1] Best cost: 6221.677 | Path: [1, 3, 17, 14, 6, 20, 7, 1, 11, 2, 12, 10, 1, 18, 5, 21, 23, 1, 9, 1] Best cost: 6152.125 | Path: [1, 9, 6, 14, 17, 3, 1, 11, 7, 20, 23, 2, 1, 12, 5, 21, 10, 1, 18, 1] Best cost: 6039.823 | Path: [1, 7, 11, 3, 17, 14, 1, 18, 10, 12, 2, 5, 1, 20, 6, 23, 21, 1, 9, 1] Generation: #10 Best cost: 5982.271 | Path: [1, 7, 11, 10, 12, 2, 1, 21, 14, 17, 3, 20, 1, 9, 6, 23, 5, 1, 18, 1] OPTIMIZING each tour... Current: [[1, 7, 11, 10, 12, 2, 1], [1, 21, 14, 17, 3, 20, 1], [1, 9, 6, 23, 5, 1], [1, 18, 1]] [1] Cost: 1536.744 to 1496.639 | Optimized: [1, 10, 12, 2, 11, 7, 1] [2] Cost: 1620.125 to 1611.912 | Optimized: [1, 3, 17, 14, 21, 20, 1] ACO RESULTS [1/300 vol./1496.639 km] Berlin Hbf -> Bremen Hbf -> Dortmund Hbf -> Düsseldorf Hbf -> Leipzig Hbf -> Dresden Hbf --> Berlin Hbf [2/295 vol./1611.912 km] Berlin Hbf -> Frankfurt Hbf -> Mannheim Hbf -> Karlsruhe Hbf -> Saarbrücken Hbf -> Würzburg Hbf --> Berlin Hbf [3/285 vol./2123.459 km] Berlin Hbf -> München Hbf -> Stuttgart Hbf -> Freiburg Hbf -> Aachen Hbf --> Berlin Hbf [4/ 30 vol./ 701.943 km] Berlin Hbf -> Kiel Hbf --> Berlin Hbf OPTIMIZATION RESULT: 4 tours | 5933.953 km.