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
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If we allow MULTIPLE delivery people, all starting from the same location, we have VRP

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Vehicles have the same capacity
•
Vehicles based at a single depot station
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Vehicles serve many different customers
• Each customer’s demand is delivered by exactly one vehicle
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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

•
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
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Fleet of m identical vehicles, each with capacity D , is based at the depot
•
Each customer i has demand d i
•
Cost c ij for traveling route from i to j

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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

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 90 70 20 100 10 80 50 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
140
120 9
100
80
1
4
2
6
60
40
7
8
20
0
0
0
20 40 60
10
80 100 120 140 160
5
180
3
200

Pizza Delivery Routes
Objective Value = 1100
140
120 9
100
80
1
4
2
6
60
40
7
8
20
0
0
0
20 40 60
10
80 100 120 140 160
5
180
3
200

Pizza Delivery Routes
Objective Value = 1120
140
120 9
100
80
1
4
2
6
60
40
7
8
20
0
0
0
20 40 60
10
80 100 120 140 160
5
180
3
200

Pizza Delivery Routes
Objective Value = 1140
140
120 9
100
80
1
4
2
6
60
40
7
8
20
0
0
0
20 40 60
10
80 100 120 140 160
5
180
3
200

Pizza Delivery Routes
Objective Value = 1140
140
120 9
100
80
1
4
2
6
60
40
7
8
20
0
0
0
20 40 60
10
80 100 120 140 160
5
180
3
200

Optimal Pizza Delivery Routes
Optimal Objective Value = 1140
140
120 9
100
80
1
4
2
6
60
40
7
8
20
0
0
0
20 40 60
10
80 100 120 140 160
5
180
3
200

min j

)

E c x ij ij j

 x ij
 


{ 0} x
0 j

2 m x ji


S x i j

S


2, i {0}
( ), S
 
{0}, S

3 x i j
 j )

E

•
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

•
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)