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: 20 customers
  1. Kassel-Wilhelmshöhe (45 vol.)
  2. Düsseldorf Hbf (35 vol.)
  3. Frankfurt Hbf (25 vol.)
  4. Aachen Hbf (25 vol.)
  5. Stuttgart Hbf (80 vol.)
  6. Dresden Hbf (80 vol.)
  7. Hamburg Hbf (60 vol.)
  8. München Hbf (75 vol.)
  9. Leipzig Hbf (90 vol.)
  10. Dortmund Hbf (45 vol.)
  11. Nürnberg Hbf (40 vol.)
  12. Karlsruhe Hbf (80 vol.)
  13. Ulm Hbf (90 vol.)
  14. Köln Hbf (50 vol.)
  15. Mannheim Hbf (100 vol.)
  16. Kiel Hbf (90 vol.)
  17. Mainz Hbf (75 vol.)
  18. Saarbrücken Hbf (90 vol.)
  19. Osnabrück Hbf (35 vol.)
  20. Freiburg Hbf (100 vol.)
Result
OVERALL | #TOURS: 5 | COST: 7375.913 km | LOAD: 1310 vol. | VCAP: 300 vol.
Tour 1
COST: 1664.455 km
LOAD: 270 vol.

  1. Frankfurt Hbf | 25 vol.
  2. Saarbrücken Hbf | 90 vol.
  3. Aachen Hbf | 25 vol.
  4. Köln Hbf | 50 vol.
  5. Düsseldorf Hbf | 35 vol.
  6. Dortmund Hbf | 45 vol.

Tour 2
COST: 1351.104 km
LOAD: 285 vol.

  1. Dresden Hbf | 80 vol.
  2. Leipzig Hbf | 90 vol.
  3. Nürnberg Hbf | 40 vol.
  4. München Hbf | 75 vol.

Tour 3
COST: 1244.584 km
LOAD: 230 vol.

  1. Kassel-Wilhelmshöhe | 45 vol.
  2. Osnabrück Hbf | 35 vol.
  3. Hamburg Hbf | 60 vol.
  4. Kiel Hbf | 90 vol.

Tour 4
COST: 1394.307 km
LOAD: 255 vol.

  1. Mainz Hbf | 75 vol.
  2. Mannheim Hbf | 100 vol.
  3. Karlsruhe Hbf | 80 vol.

Tour 5
COST: 1721.463 km
LOAD: 270 vol.

  1. Ulm Hbf | 90 vol.
  2. Stuttgart Hbf | 80 vol.
  3. Freiburg Hbf | 100 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: 1310 vol. | Vehicle capacity: 300 vol.
Loads: [45, 0, 35, 25, 0, 25, 80, 80, 60, 75, 0, 90, 45, 40, 80, 90, 50, 100, 90, 75, 0, 90, 35, 100]

ITERATION
Generation: #1
Best cost: 8051.110 | Path: [1, 0, 12, 2, 16, 5, 19, 3, 1, 11, 7, 13, 6, 1, 8, 18, 22, 17, 1, 9, 15, 14, 1, 21, 23, 1]
Best cost: 8008.353 | Path: [1, 2, 16, 5, 12, 0, 22, 8, 1, 7, 11, 13, 9, 1, 18, 3, 19, 17, 1, 6, 14, 23, 1, 15, 21, 1]
Best cost: 7734.970 | Path: [1, 3, 19, 17, 14, 1, 11, 7, 13, 6, 1, 8, 18, 22, 12, 2, 5, 1, 0, 16, 21, 23, 1, 9, 15, 1]
Best cost: 7700.726 | Path: [1, 6, 14, 17, 3, 1, 11, 7, 13, 9, 1, 8, 18, 22, 12, 2, 5, 1, 0, 19, 21, 16, 1, 15, 23, 1]
Best cost: 7689.897 | Path: [1, 6, 14, 17, 3, 1, 11, 7, 13, 9, 1, 18, 8, 22, 12, 2, 5, 1, 0, 19, 21, 16, 1, 15, 23, 1]
Best cost: 7644.176 | Path: [1, 19, 3, 17, 14, 1, 7, 11, 13, 9, 1, 8, 18, 22, 12, 2, 5, 1, 0, 16, 21, 23, 1, 15, 6, 1]
Best cost: 7572.255 | Path: [1, 23, 14, 17, 1, 11, 7, 13, 9, 1, 8, 18, 22, 12, 2, 5, 1, 0, 3, 19, 21, 16, 1, 15, 6, 1]
Best cost: 7547.585 | Path: [1, 23, 14, 17, 1, 7, 11, 13, 9, 1, 8, 18, 22, 12, 2, 5, 1, 0, 3, 19, 21, 16, 1, 6, 15, 1]
Best cost: 7434.199 | Path: [1, 12, 2, 16, 5, 21, 3, 1, 11, 7, 13, 9, 1, 8, 18, 22, 0, 1, 19, 17, 14, 1, 15, 6, 23, 1]
Generation: #3
Best cost: 7407.977 | Path: [1, 12, 2, 16, 5, 21, 3, 1, 7, 11, 13, 9, 1, 8, 18, 22, 0, 1, 19, 17, 14, 1, 15, 6, 23, 1]

OPTIMIZING each tour...
Current: [[1, 12, 2, 16, 5, 21, 3, 1], [1, 7, 11, 13, 9, 1], [1, 8, 18, 22, 0, 1], [1, 19, 17, 14, 1], [1, 15, 6, 23, 1]]
[1] Cost: 1670.644 to 1664.455 | Optimized: [1, 3, 21, 5, 16, 2, 12, 1]
[3] Cost: 1270.459 to 1244.584 | Optimized: [1, 0, 22, 8, 18, 1]

ACO RESULTS
[1/270 vol./1664.455 km] Berlin Hbf -> Frankfurt Hbf -> Saarbrücken Hbf -> Aachen Hbf -> Köln Hbf -> Düsseldorf Hbf -> Dortmund Hbf --> Berlin Hbf
[2/285 vol./1351.104 km] Berlin Hbf -> Dresden Hbf -> Leipzig Hbf -> Nürnberg Hbf -> München Hbf --> Berlin Hbf
[3/230 vol./1244.584 km] Berlin Hbf -> Kassel-Wilhelmshöhe -> Osnabrück Hbf -> Hamburg Hbf -> Kiel Hbf --> Berlin Hbf
[4/255 vol./1394.307 km] Berlin Hbf -> Mainz Hbf -> Mannheim Hbf -> Karlsruhe Hbf --> Berlin Hbf
[5/270 vol./1721.463 km] Berlin Hbf -> Ulm Hbf -> Stuttgart Hbf -> Freiburg Hbf --> Berlin Hbf
OPTIMIZATION RESULT: 5 tours | 7375.913 km.