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Many-to-Many Models Multicommodity Flows

Many-to-Many Models Multicommodity Flows. John H. Vande Vate Spring, 2001. Single vs Multi commodity problems Edge vs Path Formulations. Outline. Single Commodity: A demon could secretly swap items in transit from one vehicle to another and no one would care.

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Many-to-Many Models Multicommodity Flows

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

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

  3. 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 ... Single Commodity Flows

  4. 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 At Strategic Level

  5. 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? Single Commodity Models

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

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

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

  9. 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? Multi Commodity Models

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

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

  12. 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! Path Formulation

  13. 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 Column Generation

  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. 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 Formulation

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

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