Vehicle Routing Problems

Traveling Salesperson Problem (TSP) vs. Vehicle Routing Problem (VRP) In TSP we sought a cycle through each node of a graph that had the minimum total edge weights TSP appropriate for case of determining shortest route of ONE delivery person to ALL possible locations If we allow MULTIPLE delivery people, all starting from the same location, we have VRP

Assumptions for Basic VRP Vehicles have the same capacity Vehicles based at a single depot station Vehicles serve many different customers Each customer’s demand is delivered by exactly one vehicle Goal is to find minimum cost collection of vehicle routes, all starting and ending at same depot, that contain all customers and do not violate vehicle capacities

Mathematical Description of VRP Defined on undirected graph G = (V,E) V={0,1,…,n} is set of nodes, vertex 0 is the depot, and remaining nodes are customers E is set of edges on the graph Fleet of m identical vehicles, each with capacity D, is based at the depot Each customer i has demand di Cost cij for traveling route from i to j

VRP Constraints and Objective VRP seeks a set of m vehicle routes such that Each route begins and ends at the depot 0 Each customer is included on exactly one route Total demand of each route does not exceed D Total cost associated with each route is minimized

Example 4.1 A local pizza shop received 10 late orders for delivery last night; unfortunately, only three delivery persons are working. The shop uses a coordinate system to mark where houses are located (using the nearest intersection as locations). The 10 deliveries are to go to the following places: 1 2 3 4 5 6 7 8 9 10 E/W 20 40 180 130 160 50 30 100 90 75 N/S 70 80 60 120 15 All streets in this town go either north-south or east-west, so distance must be measured rectilinearly. Assuming that the pizza shop is located at position (0,0) and that each driver can deliver at most five orders, how should the delivery routes be determined in order to minimize the total travel distance?

Pizza Shop and Customer Locations

Pizza Delivery Routes Objective Value = 1100

Pizza Delivery Routes Objective Value = 1120

Pizza Delivery Routes Objective Value = 1140

Pizza Delivery Routes Objective Value = 1140

Optimal Pizza Delivery Routes Optimal Objective Value = 1140

VRP Formulation

VRP Formulation Assumptions Every vehicle visits at least two customers Direction a route is traversed does not change the cost (symmetric case) Model can be modified to include case where a vehicle may visit only one customer and to asymmetric case

Some VRP Variations Each customer must be visited only within a specified time window (VRP with time windows) Vehicles start from one of multiple depots (multidepot VRP) Customers can possibly be served by more than one vehicle (split delivery VRP)