Multiple Depot Vehicle Routing Problem

1. Multiple Depot Vehicle Routing Problem Bo Sun Jonathan Mee April 7th, 2009

2. Contents • Where the Problem Comes From • Introduction • VRP Description • MDVPR Description • Motivation • Abstraction (Problem Formulation) • NP Proof • NP-C Proof • Strong or Weak?

3. Where the Problem Comes From • Each day at Sears Home Appliance Repair, a fleet of technicians must have routes made for them for the next day in order to service customers

4. Introduction • Vehicle Routing Problem (VRP) • Originally formulated as “The Truck Dispatching Problem” by Dantzig and R.H. Ramser, 1959 • Routes must be made for multiple vehicles to drop off goods or services at multiple destinations, constrained on total distance (but could be some other “cost”).

5. VRP Depot Legend Service Destination

6. VRP Route 1 Depot Route 3 Route 2 Legend Service Destination Route (Path)

7. MDVRP • Multiple Depot Vehicle Routing Problem (MDVRP) • Variant of VRP • Same as VRP but with more than one depot Depot Depot Depot Legend Service Destination Depot

8. MDVRP • A Solution might look something like this • Notice that solutions allows revisiting depots Route 1 Route 2 Depot Depot Depot Legend Service Destination Depot Route 3 Route 4

9. Motivation • “Real-world” applicable: transportation, distribution, and logistics [1] • Appliance Repair • Parcel Delivery • Good routes save money • More competitive businesses • Savings passed down to the buyer • Morally, we should save resources

10. Abstraction • MDVRP problem can be modeled in terms of a Graph with weighted edges • Vertices are service destinations and depots • Edges connect any two vertices and has some weight • There is one vehicle per depot

11. MDVRP Problem Formulation • Given • Directed Graph G=(V,E) • S = { all service destinations } • D = { all depots } • V = S ∪ D • E = { weighted positive cost between any two distinct v ∈ V } • W(e), is the weight for edge e ∈ E • Question • Does there exist a set of closed walks C, such that, ∀ s ∈ S implies s ∈ c, for some c ∈ C, AND sum{ W(c) }, ∀ c ∈ C, is less than or equal to some k? Assume each Depot has one vehicle

12. NP Proof • MDVTP can be answered by “yes” OR “no” making it a decision problem • A witness can be provided (the set containing closed walks C) which we can verify in polynomial time with respect to k to have the following properties: • ∀ s ∈ S implies s ∈ c, for some c ∈ C, • sum{ cost(c) }, ∀ c ∈ C, is less than or equal to some k • Simple iteration through S and C will suffice

13. NP-Complete Proof • Show that MSVRP is NP (last slide) • Show that a polynomial transformation from some known NP-C problem to MSVRP exists • Traveling Salesman Problem (TSP) will be used

14. TSP • Given • A undirected graph G’=(V’,E’) • V’ = { all cities } • E’ = { weighted postive cost between any two distinct v’ ∈ V’ } • W(e’), is the weight for edge ‘e ∈ E’ • Question • Is there a Hamiltonian Cycle C’ with sum { W(c’) }, c’ ∈ C’, less than or equal to some k’? V corresponds with the cities, E corresponds with distances between cities

15. Construction • Construct an instance of MDVRP for each instance of TSP such that • MDVRP answers “yes” iff TSP answers “yes” • MDVRP answers “no” iff TSP answers “no”

16. Transformation • For an instance of TSP: G’=(V’,E’) and k’ •  v’  V’ create vin  Vinand vout  Vout • Vin Vout = V •  e’ with endpoints v’i and v’j create a directed edge from the corresponding vi out to vj in and a directed edge from vj out to vi in such that |E| = 2|E’| • Now create |V’| edges with weight k’ going from each vin to its corresponding vout so the new |E| = 2|E’| + |V’| • k = k’(|V’| + 1) • Randomly select one element of V to be D so that |D| = 1 and all other elements of V are in the set S so S  D = V

17. Polynomial Sized Reduction • The G(V,E) and k are created from G(V’,E’) and k’ • |V| = 2|V’| so vertex creation is polynomial with respect to V’ • |E| = 2|E’| + |V’| and since the maximum number of edges in a TSP is limited by |V’|2, |E| = 2|V’|2 + |V’| so edge creation is polynomial with respect to V’ • k is created in linear time so the reduction is polynomial with respect to V’

18. Euclidian k = 76 Euclidian i = 19 7 19 7 19 6 7 6 19 5 6 5 non-Euclidian i = 24 non-Euclidian k = 96 12 24 12 24 6 12 6 24 5 6 5 Yes Instances • If a TSP returns yes a Hamiltonian Circuit was found with weight less than k’ • The MDVRP is always capable of following the same graph as the TSP because the edges are identical whether the graph is Euclidian or not.

19. k = 84 i = 21 5 5 5 5 21 21 21 5 5 No Instances • The TSP yields a no if the instance requires a vertex to be visited more than once or if it cannot complete with a weight less than or equal to k’ • In case a non-Hamiltonian cycle is required the MDVRP reduction will also fail because a vin to vout edge will be traversed more than once causing k to be exceeded.

20. End • Thanks for listening • Question? • Bo Sun (obisunk@gmail.com) • Jonathan Mee (howdyfromtn@hotmail.com)

21. References • [1] G. B. Dantzig and R.H. Ramser. "The Truck Dispatching Problem". Management Science 6, 80–91. 1959