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: Kassel-Wilhelmshöhe
VCAP: 400 vol.
ACTIVE: 21 customers
  1. Berlin Hbf (70 vol.)
  2. Düsseldorf Hbf (70 vol.)
  3. Frankfurt Hbf (40 vol.)
  4. Hannover Hbf (60 vol.)
  5. Aachen Hbf (30 vol.)
  6. Stuttgart Hbf (100 vol.)
  7. Dresden Hbf (80 vol.)
  8. München Hbf (70 vol.)
  9. Bremen Hbf (85 vol.)
  10. Leipzig Hbf (55 vol.)
  11. Nürnberg Hbf (50 vol.)
  12. Karlsruhe Hbf (100 vol.)
  13. Ulm Hbf (90 vol.)
  14. Köln Hbf (50 vol.)
  15. Mannheim Hbf (80 vol.)
  16. Kiel Hbf (70 vol.)
  17. Mainz Hbf (80 vol.)
  18. Würzburg Hbf (90 vol.)
  19. Saarbrücken Hbf (35 vol.)
  20. Osnabrück Hbf (80 vol.)
  21. Freiburg Hbf (70 vol.)
Result
OVERALL | #TOURS: 4 | COST: 4913.935 km | LOAD: 1455 vol. | VCAP: 400 vol.
Tour 1
COST: 1818.8 km
LOAD: 390 vol.

  1. Aachen Hbf | 30 vol.
  2. Bremen Hbf | 85 vol.
  3. Kiel Hbf | 70 vol.
  4. Berlin Hbf | 70 vol.
  5. Dresden Hbf | 80 vol.
  6. Leipzig Hbf | 55 vol.

Tour 2
COST: 958.445 km
LOAD: 380 vol.

  1. Frankfurt Hbf | 40 vol.
  2. Mainz Hbf | 80 vol.
  3. Köln Hbf | 50 vol.
  4. Düsseldorf Hbf | 70 vol.
  5. Osnabrück Hbf | 80 vol.
  6. Hannover Hbf | 60 vol.

Tour 3
COST: 1078.186 km
LOAD: 400 vol.

  1. Nürnberg Hbf | 50 vol.
  2. München Hbf | 70 vol.
  3. Ulm Hbf | 90 vol.
  4. Stuttgart Hbf | 100 vol.
  5. Würzburg Hbf | 90 vol.

Tour 4
COST: 1058.504 km
LOAD: 285 vol.

  1. Mannheim Hbf | 80 vol.
  2. Karlsruhe Hbf | 100 vol.
  3. Freiburg Hbf | 70 vol.
  4. Saarbrücken Hbf | 35 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: [0] Kassel-Wilhelmshöhe | Number of cities: 24 | Total loads: 1455 vol. | Vehicle capacity: 400 vol.
Loads: [0, 70, 70, 40, 60, 30, 100, 80, 0, 70, 85, 55, 0, 50, 100, 90, 50, 80, 70, 80, 90, 35, 80, 70]

ITERATION
Generation: #1
Best cost: 5719.338 | Path: [0, 1, 11, 7, 13, 20, 3, 0, 4, 10, 22, 2, 16, 5, 0, 19, 17, 14, 6, 21, 0, 9, 15, 23, 18, 0]
Best cost: 5237.581 | Path: [0, 2, 16, 5, 19, 3, 17, 21, 0, 22, 10, 4, 18, 1, 0, 20, 13, 9, 15, 6, 0, 14, 23, 7, 11, 0]
Best cost: 5221.207 | Path: [0, 21, 14, 17, 19, 3, 16, 0, 20, 13, 9, 15, 6, 0, 4, 10, 22, 2, 5, 23, 0, 11, 7, 1, 18, 0]
Best cost: 5184.810 | Path: [0, 20, 13, 9, 15, 6, 0, 4, 10, 22, 2, 16, 5, 0, 3, 19, 17, 14, 23, 0, 11, 7, 1, 18, 21, 0]
Best cost: 5108.514 | Path: [0, 7, 11, 1, 18, 10, 5, 0, 22, 4, 2, 16, 19, 3, 0, 20, 13, 9, 15, 6, 0, 17, 14, 23, 21, 0]
Best cost: 5065.728 | Path: [0, 23, 14, 17, 3, 19, 5, 0, 4, 10, 22, 2, 16, 21, 0, 20, 13, 9, 15, 6, 0, 11, 7, 1, 18, 0]
Generation: #2
Best cost: 5024.510 | Path: [0, 7, 11, 1, 18, 10, 5, 0, 4, 22, 2, 16, 19, 3, 0, 20, 13, 9, 15, 6, 0, 17, 14, 23, 21, 0]
Best cost: 4984.502 | Path: [0, 22, 10, 4, 2, 16, 5, 0, 20, 13, 9, 15, 6, 0, 17, 14, 23, 21, 19, 0, 11, 7, 1, 18, 3, 0]
Best cost: 4933.883 | Path: [0, 11, 7, 1, 18, 10, 5, 0, 4, 22, 2, 16, 19, 3, 0, 20, 13, 9, 15, 6, 0, 17, 14, 23, 21, 0]

OPTIMIZING each tour...
Current: [[0, 11, 7, 1, 18, 10, 5, 0], [0, 4, 22, 2, 16, 19, 3, 0], [0, 20, 13, 9, 15, 6, 0], [0, 17, 14, 23, 21, 0]]
[1] Cost: 1823.609 to 1818.800 | Optimized: [0, 5, 10, 18, 1, 7, 11, 0]
[2] Cost:  966.443 to  958.445 | Optimized: [0, 3, 19, 16, 2, 22, 4, 0]
[3] Cost: 1085.327 to 1078.186 | Optimized: [0, 13, 9, 15, 6, 20, 0]

ACO RESULTS
[1/390 vol./1818.800 km] Kassel-Wilhelmshöhe -> Aachen Hbf -> Bremen Hbf -> Kiel Hbf -> Berlin Hbf -> Dresden Hbf -> Leipzig Hbf --> Kassel-Wilhelmshöhe
[2/380 vol./ 958.445 km] Kassel-Wilhelmshöhe -> Frankfurt Hbf -> Mainz Hbf -> Köln Hbf -> Düsseldorf Hbf -> Osnabrück Hbf -> Hannover Hbf --> Kassel-Wilhelmshöhe
[3/400 vol./1078.186 km] Kassel-Wilhelmshöhe -> Nürnberg Hbf -> München Hbf -> Ulm Hbf -> Stuttgart Hbf -> Würzburg Hbf --> Kassel-Wilhelmshöhe
[4/285 vol./1058.504 km] Kassel-Wilhelmshöhe -> Mannheim Hbf -> Karlsruhe Hbf -> Freiburg Hbf -> Saarbrücken Hbf --> Kassel-Wilhelmshöhe
OPTIMIZATION RESULT: 4 tours | 4913.935 km.