Many-to-Many Models Multicommodity Flows

1
Many-to-Many ModelsMulticommodity 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
Total fixed variablevolume
f3
v3
Variable
v2
f2
Total Cost
v1
f1
b1
b2
b3
Volume Shipped
7
Capturing Economies
hivol
hiuse
medvol
meduse
lowvol
lowuse

• Objective Minimize
• f1lowusef2medusef3hiuse
• v1lowuse v2medusev3hiuse

Binary
8
Constraints
hivol
hiuse
medvol
meduse
lowvol
lowuse

• lowvol ? b1lowuse
• lowvol ? b2lowuse
• medvol ? b2meduse
• medvol ? b3meduse
• hivol ? b3hiuse
• hivol ? Mhiuse

lowusemedusehiuse 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
Network flow 2
flows 1 flows 2 ?
Capacity
11
Shared Economy

• Combining flows of separate commodities reduces
  unit transportation cost for all.

12
Path Formulation

• Variables are
• 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
• sumc in commodities
• Volumec/Loadc ? 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