1. Vehicle Routing Problems

2. 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

3. 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

4. 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

5. 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

6. 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?

7. Pizza Shop and Customer Locations

8. Pizza Delivery Routes Objective Value = 1100

9. Pizza Delivery Routes Objective Value = 1120

10. Pizza Delivery Routes Objective Value = 1140

11. Pizza Delivery Routes Objective Value = 1140

12. Optimal Pizza Delivery Routes Optimal Objective Value = 1140

13. VRP Formulation

14. 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

15. 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)