1. Many-to-Many Models Multicommodity Flows
John H. Vande Vate
Spring, 2001

2. Outline
Single vs Multi commodity problems
Edge vs Path Formulations

3. Single Commodity Flows
Single Commodity: A demon could secretly swap items in transit from one vehicle to another and no one would care.
Few things are truly single-commodity
Distinguished by
Obvious features
Origin
Destination
...

4. At Strategic Level Sometimes combine to single commodity
Example: Ford Service Parts
We used an “average” product
Did not consider individual parts
Danger for 1-to-Many
Different “dimensions” of product
Size
Weight
Cost/Value
Service requirements

5. Single Commodity Models
Built on Network Flow models
Variables are volume moving from point to point
These are “easy”, but complicated by...
Binary Fixed charge/Shut down variables
Did Denver ship to the warehouse?
Did we open the terminal?

6. Economies of Scale f3 v3 v2 f2 v1 f1 b1 b2 b3 Total Cost
Total = fixed + variable*volume f3 v3
Variable v2 f2
Total Cost v1 f1 b1 b2 b3
Volume Shipped

7. Capturing Economies Binary Objective: Minimize
hivol
hiuse
medvol
meduse
lowvol
lowuse
Objective: Minimize
f1*lowuse+f2*meduse+f3*hiuse +
v1*lowuse +v2*meduse+v3*hiuse
Binary

8. Constraints lowvol  b1*lowuse lowvol  b2*lowuse medvol  b2*meduse
hivol
hiuse
medvol
meduse
lowvol
lowuse
lowvol  b1*lowuse
lowvol  b2*lowuse
medvol  b2*meduse
medvol  b3*meduse
hivol  b3*hiuse
hivol  M*hiuse
lowuse+meduse+hiuse =1

9. Multi Commodity Models
Material balance for each commodity
Otherwise
ship consoles from Denver to the warehouse
send them to customers as CPUs!
Several network flow models combined
What ties them together?

10. Shared Capacity On lanes Through facilities Network flow 1
flows flows  Capacity

11. Shared Economy
Combining flows of separate commodities reduces unit transportation cost for all.

12. Path Formulation Variables are Typically huge numbers of variables!
Volume of a commodity
moving from an ultimate origin
to an ultimate destination
along a specific path
E.g., Volume of CPUs from Green Bay to DC 51 via warehouse in Indianapolis.
Typically huge numbers of variables!

13. Column Generation Solved by Column Generation
Solve LP with some paths
Use shadow prices to identify attractive paths
Generate variables for these new paths
Repeat…
Typically reduces size of problems.
This is only important for really large problems

14. Combining Flows Different commodities sharing a vehicle
Easy to figure vehicle capacity for single commodity
How to figure vehicle capacity for mixed loads?

15. Typical Approach
10 items
20 items
40%
60%
4 items
12 items

16. Formulation Blue Vol/10 + Red Vol/20  1 sum{c in commodities}
Volume[c]/Load[c]  1;
More often this is used to calculate the number of vehicles required to carry the given volumes: Assumes full loads!
Vehicles = Blue Vol/10 + Red Vol/20

17. Several Capacities Weight Limit Space or Cube
Ensure loads meet each limit
Vehicles  Blue Vol/10 + Red Vol/20
Vehicles  Blue Vol/8 + Red Vol/25
Number that reaches the weight limit
Number that fills the cube