MGTSC 352
Lecture 18:
Distribution Planning
Rent-A-Dent example: a transportation
problem
Oil example: a minimum cost flow problem
Henderson Food Company: shortest path
problem
Distribution Planning
• Rent-a-Dent example:
Transportation problem
• Oil example:
Minimum cost network flow problem
• Henderson Food Company example:
Shortest path problem
Transportation Problem
“arcs” / “links”
Supply nodes 
3 10
$54
15
1
$17
4 10
$18
$24
18
$23
2
5
$19
10
$30
$31
6 3
 Demand nodes
Minimum Cost Flow Problem (pg. 118)
Problem Data
-17
+22
$16
1
$14
$12
4
$4
3
0
$12
$14
$8
2
5
$13
+8
-13
Capacity of each arc: 10 units
To Excel
Task:
Send flow from supply (-) to
demand (+) nodes
Goal:
Minimize transport cost
Changing cells:
flow on each link
Constraints:
flows  arc capacities
flow in + supply  flow out + demand
flows  0
Min Cost Flow: Testing Your Intuition
• What if total supply < total demand, and we use
flow in + supply  flow out + demand?
• What if total supply > total demand, and we use
flow in + supply = flow out + demand
(= instead of )?
Rent-a-Dent vs Oil
• Rent-a-Dent (transportation problem)
– Every node is a supply or demand node
– “Arcs” correspond to routes
– Finds From-To shipments
• Oil (minimum cost flow problem)
– Full network
– Arcs correspond to route segments
– Finds flows on each segment
Relations
• Transportation problem is a special
case of the min cost flow problem
• Shortest path problem is (also) a special
case of min cost flow problem.
– What?
Find the “cheapest” path from
Node 1 to Node 5
• Can you manipulate the problem data so
as to “trick” Excel?
-17
-1
+22
$16
1
$14
$12
4
$4
3
0
$12
$14
$8
2
5
$13
-13
+1
+8
to Excel
Pg. 125
Quickest route from
Lethbridge to Fort McMurray that...
• ..does not go through Calgary?
• ..goes through Rocky Mountain House?
• ..includes the Drumheller-Red Deer arc?
• Suppose you are interested in the shortest
route from Lethbridge to Fort McMurray,
not the fastest route. Now what?
Shortest Path from Vancouver to
Miami that goes through Toronto
Vancouver
Toronto
What’s wrong
with this picture?
Miami