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

Outline Single vs Multi commodity problems Edge vs Path Formulations

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 ...

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

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?

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

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

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

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?

Shared Capacity On lanes Through facilities Network flow 1 flows 1 + flows 2  Capacity

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

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!

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

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

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

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

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