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## Multiple Depot Vehicle Routing Problem

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**Multiple Depot Vehicle Routing Problem**Bo Sun Jonathan Mee April 7th, 2009**Contents**• Where the Problem Comes From • Introduction • VRP Description • MDVPR Description • Motivation • Abstraction (Problem Formulation) • NP Proof • NP-C Proof • Strong or Weak?**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**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”).**VRP**Depot Legend Service Destination**VRP**Route 1 Depot Route 3 Route 2 Legend Service Destination Route (Path)**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**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**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**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**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**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**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**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**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”**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**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’**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.**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.**End**• Thanks for listening • Question? • Bo Sun (obisunk@gmail.com) • Jonathan Mee (howdyfromtn@hotmail.com)**References**• [1] G. B. Dantzig and R.H. Ramser. "The Truck Dispatching Problem". Management Science 6, 80–91. 1959