Solving CVRP with ACO

Minimizing Travel Cost for Complex Delivery Problems

Start With The Story

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.?
Such questions are addressed by employing the ants.

How to use this page?
Two parameters can be adjusted:
  • depot: [0..23], def = 0
  • vcap: [200..400], def = 400
Calling this page without parameter will get the defaults. Otherwise, just try something like this

There is a way to set all the demands, but I don't think you are ready for that. 😉
Map
DEPOT: Berlin Hbf
VCAP: 300 vol.
ACTIVE: 22 customers
  1. Kassel-Wilhelmshöhe (70 vol.)
  2. Düsseldorf Hbf (30 vol.)
  3. Frankfurt Hbf (95 vol.)
  4. Hannover Hbf (70 vol.)
  5. Aachen Hbf (95 vol.)
  6. Stuttgart Hbf (20 vol.)
  7. Dresden Hbf (50 vol.)
  8. Hamburg Hbf (45 vol.)
  9. Bremen Hbf (90 vol.)
  10. Leipzig Hbf (100 vol.)
  11. Dortmund Hbf (85 vol.)
  12. Nürnberg Hbf (25 vol.)
  13. Karlsruhe Hbf (95 vol.)
  14. Ulm Hbf (50 vol.)
  15. Köln Hbf (65 vol.)
  16. Mannheim Hbf (85 vol.)
  17. Kiel Hbf (85 vol.)
  18. Mainz Hbf (60 vol.)
  19. Würzburg Hbf (25 vol.)
  20. Saarbrücken Hbf (55 vol.)
  21. Osnabrück Hbf (30 vol.)
  22. Freiburg Hbf (80 vol.)
Result
OVERALL | #TOURS: 5 | COST: 6745.345 km | LOAD: 1405 vol. | VCAP: 300 vol.
Tour 1
COST: 1534.551 km
LOAD: 300 vol.

  1. Nürnberg Hbf | 25 vol.
  2. Ulm Hbf | 50 vol.
  3. Stuttgart Hbf | 20 vol.
  4. Karlsruhe Hbf | 95 vol.
  5. Mannheim Hbf | 85 vol.
  6. Würzburg Hbf | 25 vol.

Tour 2
COST: 1047.518 km
LOAD: 290 vol.

  1. Dresden Hbf | 50 vol.
  2. Leipzig Hbf | 100 vol.
  3. Kassel-Wilhelmshöhe | 70 vol.
  4. Hannover Hbf | 70 vol.

Tour 3
COST: 1308.428 km
LOAD: 275 vol.

  1. Dortmund Hbf | 85 vol.
  2. Düsseldorf Hbf | 30 vol.
  3. Köln Hbf | 65 vol.
  4. Aachen Hbf | 95 vol.

Tour 4
COST: 1107.833 km
LOAD: 250 vol.

  1. Osnabrück Hbf | 30 vol.
  2. Bremen Hbf | 90 vol.
  3. Hamburg Hbf | 45 vol.
  4. Kiel Hbf | 85 vol.

Tour 5
COST: 1747.015 km
LOAD: 290 vol.

  1. Frankfurt Hbf | 95 vol.
  2. Mainz Hbf | 60 vol.
  3. Saarbrücken Hbf | 55 vol.
  4. Freiburg Hbf | 80 vol.

ANTS
#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.

What kind of cost?
Directed driving distance, obtained through Google API. The visualization does not display that since the idea is VRP. Adding such feature is very easy, but not a priority for this case.
Can we use any address?
Yes, absolutely. What we need is the geo-coordinates of the addresses, and the distance matrix, which is not a problem. See my oldie master thesis here, implemented using PHP/MySQL for a delivery case in Darmstadt city, Germany.
Travel time as the cost?
Just replace the distance matrix with a duration matrix, then it is done. Please keep in mind, this feature is not intended for realtime use. But regarding the idea, not an issue.
Up to how many nodes?
There is no definitive answer for that. However, if a large number of nodes involved, a good strategy is required. Actually, for this one, a suitable technique is already implemented instead of a "plain" ACO. 

NETWORK
Depo: [1] Berlin Hbf | Number of cities: 24 | Total loads: 1405 vol. | Vehicle capacity: 300 vol.
Loads: [70, 0, 30, 95, 70, 95, 20, 50, 45, 0, 90, 100, 85, 25, 95, 50, 65, 85, 85, 60, 25, 55, 30, 80]

ITERATION
Generation: #1
Best cost: 8126.542 | Path: [1, 0, 12, 2, 16, 22, 6, 1, 11, 7, 13, 20, 3, 1, 8, 18, 10, 4, 1, 14, 17, 19, 21, 1, 15, 23, 5, 1]
Best cost: 7530.816 | Path: [1, 2, 16, 5, 12, 20, 1, 11, 7, 13, 15, 6, 21, 1, 8, 18, 10, 22, 1, 4, 0, 3, 19, 1, 17, 14, 23, 1]
Best cost: 7447.956 | Path: [1, 6, 14, 17, 19, 20, 1, 7, 11, 13, 3, 2, 1, 8, 18, 10, 4, 1, 16, 5, 12, 22, 1, 0, 15, 23, 21, 1]
Best cost: 7320.856 | Path: [1, 11, 7, 0, 16, 1, 18, 8, 10, 4, 1, 22, 12, 2, 5, 19, 1, 13, 20, 3, 17, 21, 1, 23, 14, 6, 15, 1]
Best cost: 7304.626 | Path: [1, 20, 6, 14, 17, 19, 1, 7, 11, 0, 22, 2, 1, 18, 8, 10, 4, 1, 13, 15, 23, 21, 12, 1, 3, 16, 5, 1]
Best cost: 7152.209 | Path: [1, 18, 8, 10, 4, 1, 7, 11, 6, 14, 20, 1, 0, 12, 2, 16, 22, 1, 13, 17, 19, 3, 1, 15, 23, 21, 5, 1]
Best cost: 7132.811 | Path: [1, 4, 10, 8, 18, 1, 7, 11, 13, 20, 3, 1, 22, 2, 16, 5, 0, 1, 19, 17, 14, 6, 1, 12, 21, 23, 15, 1]
Best cost: 7057.246 | Path: [1, 23, 14, 6, 15, 20, 13, 1, 7, 11, 0, 2, 22, 1, 4, 10, 8, 18, 1, 3, 19, 17, 21, 1, 12, 16, 5, 1]
Generation: #4
Best cost: 6982.860 | Path: [1, 20, 13, 15, 6, 14, 17, 1, 11, 7, 0, 4, 1, 12, 2, 16, 5, 1, 8, 18, 10, 22, 1, 3, 19, 21, 23, 1]

OPTIMIZING each tour...
Current: [[1, 20, 13, 15, 6, 14, 17, 1], [1, 11, 7, 0, 4, 1], [1, 12, 2, 16, 5, 1], [1, 8, 18, 10, 22, 1], [1, 3, 19, 21, 23, 1]]
[1] Cost: 1652.399 to 1534.551 | Optimized: [1, 13, 15, 6, 14, 17, 20, 1]
[2] Cost: 1142.530 to 1047.518 | Optimized: [1, 7, 11, 0, 4, 1]
[4] Cost: 1132.488 to 1107.833 | Optimized: [1, 22, 10, 8, 18, 1]

ACO RESULTS
[1/300 vol./1534.551 km] Berlin Hbf -> Nürnberg Hbf -> Ulm Hbf -> Stuttgart Hbf -> Karlsruhe Hbf -> Mannheim Hbf -> Würzburg Hbf --> Berlin Hbf
[2/290 vol./1047.518 km] Berlin Hbf -> Dresden Hbf -> Leipzig Hbf -> Kassel-Wilhelmshöhe -> Hannover Hbf --> Berlin Hbf
[3/275 vol./1308.428 km] Berlin Hbf -> Dortmund Hbf -> Düsseldorf Hbf -> Köln Hbf -> Aachen Hbf --> Berlin Hbf
[4/250 vol./1107.833 km] Berlin Hbf -> Osnabrück Hbf -> Bremen Hbf -> Hamburg Hbf -> Kiel Hbf --> Berlin Hbf
[5/290 vol./1747.015 km] Berlin Hbf -> Frankfurt Hbf -> Mainz Hbf -> Saarbrücken Hbf -> Freiburg Hbf --> Berlin Hbf
OPTIMIZATION RESULT: 5 tours | 6745.345 km.