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Modeling Flows and Information

This guide covers modeling flows and information for single and multicommodity flow problems, including using Excel Solver, transportation cost optimization, and building Solver models. Learn to implement models using algebraic languages for better results.

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Modeling Flows and Information

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  1. Modeling Flows and Information Single commodity flow problems Multicommodity Flow Conservation Weight and Cube Pass through vs Unload/Reload Tracking Information Column Generation 15.057 Spring 02 Vande Vate 1

  2. Modeling Tools • We will rely on Solver in Excel • Instructions on installing and using Solver accessible from the class homepage • Convenient: • Everyone is familiar with Excel • Minimal fixed costs of installing, accessing and learning syntax for a new application • Inconvenient: • Solver is unreliable. • It says models are not linear when they are. • It says it has found a solution when it hasn't, etc. • It combines the problem data with the model • Limited problem sizes • Solver is not an industrial strength tool. • So, I encourage you to learn an algebraic modeling language and implement the models we discuss in class in that language. Claudio is an excellent resource for help in developing these models. 15.057 Spring 02 Vande Vate 2

  3. Netherlands Amsterdam 500 * 800 The Hague * Germany 500 Tilburg * 700 * Antwerp Leipzig * Belgium 400 * Liege 200 Nancy * 900 Miles 0 50 100 Motor Distribution 500 800 500 400 700 200 900 15.057 Spring 02 Vande Vate 3

  4. Transportation Costs Unit transportation costs from harbors to plants Minimize the transportation costs involved in moving the motors from the harbors to the plants 15.057 Spring 02 Vande Vate 4

  5. A Transportation Model Spreadsheet embedded in presentation 15.057 Spring 02 Vande Vate 5

  6. Challenge • Find a best answer 15.057 Spring 02 Vande Vate 6

  7. Building a Solver Model • Tools | Solver… • Set Target Cell: The cell holding the value you want to minimize (cost) or maximize (revenue) • Equal to: Choose Max to maximize or Min to minimize this • By Changing Cells: The cells or variables the model is allowed to adjust In the Transportation spreadsheet that’s G19 - the total transportation cost In the Transportation spreadsheet we choose Min to minimize transport cost In the Transportation spreadsheet that is C9:F11 - the Shipment volumes 15.057 Spring 02 Vande Vate 7

  8. Solver Model Cont’d • Subject to the Constraints: The constraints that limit the choices of the values of the adjustables. • Click on Add • Cell Reference is a cell that holds a value calculated from the adjustables • Constraint is a cell that holds a value that constraints the Cell Reference. • <=, =, => is the sense of the constraint. Choose one. In the Transportation spreadsheet for example, G9 is the total volume shipped out of Amsterdam In the Transportation spreadsheet for example, H9 is the total volume we can ship out of Amsterdam 15.057 Spring 02 Vande Vate 8 <= in this case. Don’t ship more than we have in AMS

  9. What are the Constraints? • Supply Constraints • Amsterdam: G9 <= H9 • Antwerp: G10 <= H10 • The Hague: G11 <= H11 • Demand Constraints • Leipzig: C12 => C13 • Nancy: D12 => D13 • Liege: E12 => E13 • Tilburg: F12 => F13 G9 is the total volume shipped from Amsterdam Short cut: G9:G11 <= H9:H11 C12 is the total volume shipped to Leipzig Short cut: C12:F12 => C13:F13 15.057 Spring 02 Vande Vate 9

  10. The Model $G$9 <= $H$9 15.057 Spring 02 Vande Vate 10

  11. What’s Missing? 15.057 Spring 02 Vande Vate 11

  12. Options 15.057 Spring 02 Vande Vate 12

  13. Netherlands Amsterdam 500 * 800 The Hague * Germany 500 Tilburg * 700 * Antwerp Leipzig * Belgium 400 * Liege 200 Nancy * 900 Miles 0 50 100 Multiple Products • Same costs • Different products • Different supply • Different Demand Product 1 Product 2 800 500 500 800 400 500 300 400 700 200 500 200 900 500 15.057 Spring 02 Vande Vate 13

  14. Question • Can we just combine the products: • Total Supply at each source • Total Demand at each destination • The Hague • Total Supply: 1,100 = 800 + 300 • Tilburg • Total Demand: 1,000 = 500 + 500 15.057 Spring 02 Vande Vate 14

  15. Question • Can we just solve two separate problems: • One Problem for Product 1 • One Problem for Product 2? 15.057 Spring 02 Vande Vate 15

  16. Question • Can we just solve two separate problems: • One Problem for Product 1 • One Problem for Product 2? • When won’t this work? • Different Costs? • Capacities on Transportation • ... 15.057 Spring 02 Vande Vate 16

  17. Shared Capacities 15.057 Spring 02 Vande Vate 17

  18. Transshipments • 2 PRODUCTS • 3 plants • 2 distribution centers • 2 customers • Minimize shipping costs 15.057 Spring 02 Vande Vate 18

  19. Challenge • Formulate a model 15.057 Spring 02 Vande Vate 19

  20. Flow Conservation • Conserve Flow at the DC • Product by Product • Otherwise… • Alchemy: Turn lead into gold 15.057 Spring 02 Vande Vate 20

  21. AMPL Model set PRODUCTS; set PLANTS; set DCS; set CUSTS; set EDGES within (PLANTS cross DCS) union (DCS cross CUSTS); param Supply{PRODUCTS, PLANTS}; param Demand{PRODUCTS, CUSTS}; param Cost{EDGES}; param Capacity{EDGES}; 15.057 Spring 02 Vande Vate 21

  22. var Flow{PRODUCTS, EDGES} >= 0; minimize TotalCost: sum {p in PRODUCTS, (f, t) in EDGES} Cost[f,t]*Flow[p, f, t]; s.t. ObserveSupply {prd in PRODUCTS, plant in PLANTS}: sum {(plant, toloc) in EDGES} Flow[prd, plant, toloc] <= Supply[prd, plant]; s.t. MeetDemand{prd in PRODUCTS, cust in CUSTS}: sum{(fromloc, cust) in EDGES} Flow[prd, fromloc, cust] >= Demand[prd, cust]; s.t. ConserveFlow{prd in PRODUCTS, dc in DCS}: sum{(fromloc, dc) in EDGES} Flow[prd, fromloc, dc] = sum{(dc, toloc) in EDGES} Flow[prd, dc, toloc]; s.t. JointCapacity{ (fromloc, toloc) in EDGES}: sum{prd in PRODUCTS} Flow[prd, fromloc, toloc] <= Capacity[fromloc, toloc]; 15.057 Spring 02 Vande Vate 22

  23. Weight & Cube • More realistic version of “shared capacity” • Product 1: • 10 lbs/unit • 1 cubic ft/unit • Pay by Truckload • Weight Limit: 4,000 lbs • Cubic Capacity: 1,000 Cu. Ft. • Joint demands on transportation capacity • Product 2: • 1 lbs/unit • 10 cubic ft/unit 15.057 Spring 02 Vande Vate 23

  24. Weight & Cube • How many vehicles? 15.057 Spring 02 Vande Vate 24

  25. Modeling Weight & Cube • The number of vehicles required is… The larger of the number required • for the weight and • for the cube • Conveyances = Max(TotalWeight/WeightLimit, TotalCube/CubeCap) 15.057 Spring 02 Vande Vate 25

  26. Challenge • Formulate a model • Do NOT use MAX 15.057 Spring 02 Vande Vate 26

  27. More Detail • Container at a port • Container at a distribution center • Container at a port • Conserve flow of containers • Goods in the containers don’t change • Container at a distribution center • Conserve flow of goods • Loads change from in-bound to outbound 15.057 Spring 02 Vande Vate 27

  28. Challenge • Formulate a model with Transloading at DC 15.057 Spring 02 Vande Vate 28

  29. Without Transloading • Same number of conveyances in and out • Same products in those conveyances • Lose potential to improve capacity utilization • How to model? 15.057 Spring 02 Vande Vate 29

  30. Flows on Paths • Product 1 from Plant 1 to Customer 1 via DC 1 • Product 2 from Plant 2 to Customer 1 via DC 1 15.057 Spring 02 Vande Vate 30

  31. Challenge • Formulate a model WITHOUT Transloading at the DC 15.057 Spring 02 Vande Vate 31

  32. AMPL Model set PLANTS; set DCS; set CUSTS; set PRODS; set EDGES := (PLANTS cross DCS) union (DCS cross CUSTS); param Cost{EDGES}; param Weight{PRODS}; param Cube{PRODS}; param Supply {PRODS, PLANTS} default 0; param Demand {PRODS, CUSTS} default 0; param WeightLimit; param CubicCapacity; 15.057 Spring 02 Vande Vate 32

  33. var PathFlow {PRODS, PLANTS, DCS, CUSTS} >= 0; var Conveys{PLANTS, DCS, CUSTS} >= 0; minimize FreightCost: sum{plant in PLANTS, dc in DCS, cust in CUSTS} (Cost[plant, dc] + Cost[dc, cust])*Conveys[plant, dc, cust]; s.t. ObserveSupply {prd in PRODS, plant in PLANTS}: sum{dc in DCS, cust in CUSTS} PathFlow[prd, plant, dc, cust] <= Supply[prd, plant]; s.t. MeetDemand {prd in PRODS, cust in CUSTS}: sum{dc in DCS, plant in PLANTS} PathFlow[prd, plant, dc, cust] >= Demand[prd, cust]; 15.057 Spring 02 Vande Vate 33

  34. AMPL Model s.t. ConveysByWeight {plant in PLANTS, dc in DCS, cust in CUSTS}: sum{prd in PRODS} Weight[prd]*PathFlow[prd, plant, dc, cust] <= Conveys[plant, dc, cust]*WeightLimit; s.t. ConveysByCube {plant in PLANTS, dc in DCS, cust in CUSTS}: sum{prd in PRODS} Cube[prd]*PathFlow[prd, plant, dc, cust] <= Conveys[plant, dc, cust]*CubicCapacity; 15.057 Spring 02 Vande Vate 34

  35. Other Uses • Additional Information about the products history • Where it was made for • Duties • Pipeline Inventory • How long it has been traveling • … • Extreme cases: very long involved paths 15.057 Spring 02 Vande Vate 35

  36. Column Generation • Airlines Crew Scheduling • Complex pay and duty rules • Unions • FAA • Tour of duty for a crew • Several legs returning to base • Millions of possible tours of duty • Each is a variable: • 1 = use it in schedule • 0 = do not. 15.057 Spring 02 Vande Vate 36

  37. Solution Strategy • Initial Tours of Duty • Used in previous solutions • Fillers to ensure feasibility • Repeat until Done • Solve small problem • Use Shadow Prices to generate new tours that will improve solution • When no better tours are found -- done 15.057 Spring 02 Vande Vate 37

  38. Column Generation Example • Multicommodity Flow 15.057 Spring 02 Vande Vate 38

  39. Traditional Model • Minimize SS cij xkij • s.t. Sj xkij – Sj xkji = {0, supply, -demand • s.t. Sk xkij capacityij • xkij 0 • xkij is the volume of commodity k we send directly from node i to node j 15.057 Spring 02 Vande Vate 39

  40. Flow on Paths Model This is the sum over all the paths for commodity k • Pk is the set of paths from the source for commodity k to the destination. • xp is the flow on path p • cp is the cost of one unit of flow on path p • For simplicity assume each commodity has a unique source and a unique destination • Minimize S cp xp • s.t. S xp = supply for commodity k • s.t. S xp capacity of edge (i,j) • xp 0 This is the sum over all the paths that cross edge (i,j) 15.057 Spring 02 Vande Vate 40

  41. Flows On Paths Formulation • Variable for • Product 1: Each path from 1 to 5 • Product 2: Each path from 2 to 4 • Constraints: Capacity on each edge: • Sum of flows on paths that use the edge do not exceed capacity of the edge • Cost: Each unit of flow on a path pays the cost of all the edges on the path • Why no Flow Conservation Constraints? 15.057 Spring 02 Vande Vate 41

  42. Column Generation Approach • Start with 2 paths • Product 1: The edge from 1 to 5 • Product 2: The edge from 2 to 4 • Solve and get Shadow Prices on the capacity constraints • Calculate new (shortest) paths with the best reduced costs • Repeat. 15.057 Spring 02 Vande Vate 42

  43. Quick LP review Is there a path whose revised cost is less than the price for the commodity • A solution is optimal if no variable is a candidate to enter the basis. That is if no variable prices out so that increasing the variable decreases the solution cost. • To price out a variable, compare its cost with the resources it consumes • cp - S shadow prices*coefficents < 0 • cp - S shadow prices on edges of P – price for the commodity’s demand constraint 15.057 Spring 02 Vande Vate 43

  44. Column Generation Example • Multicommodity Flow 15.057 Spring 02 Vande Vate 44

  45. AMPL Commands • In AMPL • Modeling languages do more than model. • They also drive the solver. ColumnGeneration.mod 15.057 Spring 02 Vande Vate 45

  46. Summary • Conserve Flow at lowest level of detail • Distinction between transloading and container handling • Involves additional information in “Edges”, e.g., what port it passed through • Extreme cases: Additional information is the entire path • Very Technical: Column Generation can handle these cases (sometimes). 15.057 Spring 02 Vande Vate 46

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