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: 16 customers
- Kassel-Wilhelmshöhe (100 vol.)
- Düsseldorf Hbf (75 vol.)
- Frankfurt Hbf (80 vol.)
- Aachen Hbf (100 vol.)
- Stuttgart Hbf (90 vol.)
- Hamburg Hbf (55 vol.)
- Bremen Hbf (25 vol.)
- Leipzig Hbf (65 vol.)
- Nürnberg Hbf (60 vol.)
- Karlsruhe Hbf (35 vol.)
- Ulm Hbf (20 vol.)
- Köln Hbf (50 vol.)
- Kiel Hbf (85 vol.)
- Mainz Hbf (80 vol.)
- Würzburg Hbf (55 vol.)
- Osnabrück Hbf (80 vol.)
Tour 1
COST: 1531.582 km
LOAD: 280 vol.
- Würzburg Hbf | 55 vol.
- Frankfurt Hbf | 80 vol.
- Karlsruhe Hbf | 35 vol.
- Stuttgart Hbf | 90 vol.
- Ulm Hbf | 20 vol.
Tour 2
COST: 1290.063 km
LOAD: 290 vol.
- Leipzig Hbf | 65 vol.
- Kassel-Wilhelmshöhe | 100 vol.
- Düsseldorf Hbf | 75 vol.
- Köln Hbf | 50 vol.
Tour 3
COST: 1107.833 km
LOAD: 245 vol.
- Osnabrück Hbf | 80 vol.
- Bremen Hbf | 25 vol.
- Hamburg Hbf | 55 vol.
- Kiel Hbf | 85 vol.
Tour 4
COST: 1564.894 km
LOAD: 240 vol.
- Aachen Hbf | 100 vol.
- Mainz Hbf | 80 vol.
- Nürnberg Hbf | 60 vol.
LOAD: 280 vol.
- Würzburg Hbf | 55 vol.
- Frankfurt Hbf | 80 vol.
- Karlsruhe Hbf | 35 vol.
- Stuttgart Hbf | 90 vol.
- Ulm Hbf | 20 vol.
LOAD: 290 vol.
- Leipzig Hbf | 65 vol.
- Kassel-Wilhelmshöhe | 100 vol.
- Düsseldorf Hbf | 75 vol.
- Köln Hbf | 50 vol.
LOAD: 245 vol.
- Osnabrück Hbf | 80 vol.
- Bremen Hbf | 25 vol.
- Hamburg Hbf | 55 vol.
- Kiel Hbf | 85 vol.
LOAD: 240 vol.
- Aachen Hbf | 100 vol.
- Mainz Hbf | 80 vol.
- Nürnberg Hbf | 60 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: 1055 vol. | Vehicle capacity: 300 vol. Loads: [100, 0, 75, 80, 0, 100, 90, 0, 55, 0, 25, 65, 0, 60, 35, 20, 50, 0, 85, 80, 55, 0, 80, 0] ITERATION Generation: #1 Best cost: 7165.223 | Path: [1, 0, 2, 16, 10, 14, 1, 11, 13, 20, 3, 15, 1, 8, 18, 22, 19, 1, 5, 6, 1] Best cost: 6360.164 | Path: [1, 2, 16, 5, 14, 15, 1, 11, 13, 20, 3, 10, 1, 8, 18, 22, 19, 1, 0, 6, 1] Best cost: 5828.307 | Path: [1, 8, 18, 10, 22, 16, 1, 11, 0, 20, 13, 15, 1, 6, 14, 3, 19, 1, 2, 5, 1] Best cost: 5763.153 | Path: [1, 13, 15, 6, 14, 3, 1, 11, 20, 19, 0, 1, 10, 8, 18, 22, 16, 1, 2, 5, 1] Best cost: 5530.329 | Path: [1, 18, 8, 10, 22, 16, 1, 11, 0, 3, 20, 1, 13, 15, 6, 14, 19, 1, 2, 5, 1] Best cost: 5526.854 | Path: [1, 18, 8, 10, 22, 16, 1, 11, 0, 3, 20, 1, 19, 14, 6, 15, 13, 1, 2, 5, 1] Best cost: 5524.044 | Path: [1, 15, 6, 14, 3, 20, 1, 11, 0, 2, 16, 1, 8, 18, 10, 22, 1, 13, 19, 5, 1] OPTIMIZING each tour... Current: [[1, 15, 6, 14, 3, 20, 1], [1, 11, 0, 2, 16, 1], [1, 8, 18, 10, 22, 1], [1, 13, 19, 5, 1]] [1] Cost: 1534.102 to 1531.582 | Optimized: [1, 20, 3, 14, 6, 15, 1] [3] Cost: 1132.488 to 1107.833 | Optimized: [1, 22, 10, 8, 18, 1] [4] Cost: 1567.391 to 1564.894 | Optimized: [1, 5, 19, 13, 1] ACO RESULTS [1/280 vol./1531.582 km] Berlin Hbf -> Würzburg Hbf -> Frankfurt Hbf -> Karlsruhe Hbf -> Stuttgart Hbf -> Ulm Hbf --> Berlin Hbf [2/290 vol./1290.063 km] Berlin Hbf -> Leipzig Hbf -> Kassel-Wilhelmshöhe -> Düsseldorf Hbf -> Köln Hbf --> Berlin Hbf [3/245 vol./1107.833 km] Berlin Hbf -> Osnabrück Hbf -> Bremen Hbf -> Hamburg Hbf -> Kiel Hbf --> Berlin Hbf [4/240 vol./1564.894 km] Berlin Hbf -> Aachen Hbf -> Mainz Hbf -> Nürnberg Hbf --> Berlin Hbf OPTIMIZATION RESULT: 4 tours | 5494.372 km.