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OPS 301 Module C Transportation Models and Network Models

OPS 301 Module C Transportation Models and Network Models. Dr. Steven Harrod. Topics. Definition of Transportation Models Transportation Assignment Transshipment Formulation Tips Word Problem Practice CAUTION: Significant material not in your text!. Network Models.

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OPS 301 Module C Transportation Models and Network Models

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  1. OPS 301Module CTransportation Modelsand Network Models Dr. Steven Harrod

  2. Topics • Definition of Transportation Models • Transportation • Assignment • Transshipment • Formulation Tips • Word Problem Practice CAUTION: Significant material not in your text!

  3. Network Models • Model the flow or transportation of items as arrows between nodes or locations. • Can be formulated as a linear program • Related to graph theory in mathematics

  4. Three Models • Transportation Model • Minimize cost of transportation • Simple origin to destination transportation • Transshipment Model • Add intermediate junctions to transportation • Flow conservation constraints at junctions • Assignment Model • Minimize cost, time, etc. of resource assignment • Binary variables (0,1; integer)

  5. Transportation Model • A special class of linear programming • Procedure that finds the least costly means of moving products from a series of sources to a series of destinations • Can be used to help resolve distribution and location decisions

  6. Need to Know The origin points and the capacity or supply at each The destination points and the demand per period at each The cost of shipping one unit from each origin to each destination

  7. Boston (200 units demand) Cleveland (200 units demand) Des Moines (100 units supply) Albuquerque (300 units demand) Evansville (300 units supply) Fort Lauderdale (300 units supply) Transportation Problem Figure C.1

  8. Example Cost Data Table C.1

  9. Formulate • What is the goal? • Serve all customers at minimum cost • What do we control? • Quantities shipped on each route • What are the limits? • Quantities available at sources • Quantities demanded at destinations

  10. Variables • Amount shipped from origin to destination • Designate with serial index

  11. Objective Function Minimize 5 X1 + 4 X2 + 3 X3 + 8 X4 + 4 X5 + 3 X6 + 9 X7 + 7 X8 + 5 X9

  12. Boston (200 units demand) Cleveland (200 units demand) Des Moines (100 units supply) Albuquerque (300 units demand) Evansville (300 units supply) Fort Lauderdale (300 units supply) Limits: Supply and Demand X2 X5 X3 X6 X1 X8 X9 X4 X7 Figure C.1

  13. Subject to X1 + X2 + X3 <= 100 (supply) X4 + X5 + X6 <= 300 X7 + X8 + X9<= 300 X1 + X4 + X7 = 300 (demand) X2 + X5 + X8 = 200 X3 + X6 + X9= 200

  14. Solve • 9 variables • 6 constraints

  15. Answer • Objective 3900

  16. Transshipment • Multi-step shipment route • Requires “flow conservation” constraints

  17. Change Prior Example • Suppose cheap transportation available Ft. Lauderdale to Des Moines • Suppose Ft. Lauderdale preferred manufacturing location • Convert Des Moines into warehouse • Increase production at Ft. Lauderdale

  18. Real World • Tropicana juice unit train • Solid train of orange refrigerated boxcars • Frozen concentrate juice • Much cheaper than truck

  19. Add Transfer Station Boston (200 units demand) Cleveland (200 units demand) Des Moines (transfer) X2 X5 X3 X6 X1 Albuquerque (300 units demand) Evansville (300 units supply) X8 X9 X4 X7 Fort Lauderdale (400 units supply) Figure C.1 X10

  20. New Objective Function Minimize 5 X1 + 4 X2 + 3 X3 + 8 X4 + 4 X5 + 3 X6 + 9 X7 + 7 X8 + 5 X9 + 2 X10

  21. Subject to X1 + X2 + X3 = X10 (Des Moines flow conservation) X4 + X5 + X6 <= 300 X7 + X8 + X9 + X10<= 400 (new Ft. Lauderdale) X1 + X4 + X7 = 300 (demand) X2 + X5 + X8 = 200 X3 + X6 + X9= 200

  22. Modify for POM Rewrite X1 + X2 + X3 = X10 as X1 + X2 + X3 - X10 = 0 X4 + X5 + X6 <= 300 X7 + X8 + X9 + X10<= 400 (new Ft. Lauderdale) X1 + X4 + X7 = 300 (demand) X2 + X5 + X8 = 200 X3 + X6 + X9= 200

  23. Subject to (revised) X1 + X2 + X3 - X10 = 0 (Des Moines flow conservation) X4 + X5 + X6 <= 300 X7 + X8 + X9 + X10<= 400 (new Ft. Lauderdale) X1 + X4 + X7 = 300 (demand) X2 + X5 + X8 = 200 X3 + X6 + X9= 200

  24. Solve • 10 variables • 6 constraints • New trick: Format:Insert/Delete

  25. Answer • Objective 3700

  26. Assignment Model • Time available?

  27. Problem • Each Leader can manage one project • What assignments? • Estimated time to complete in days

  28. Objective Function Minimize 10 X1 + 15 X2 + 9 X3 + 9 X4 + 18 X5 + 5 X6 + 6 X7 + 14 X8 + 3 X9

  29. Subject to X1 + X2 + X3 <= 1 X4 + X5 + X6 <= 1 X7 + X8 + X9<= 1 X1 + X4 + X7 = 1 X2 + X5 + X8 = 1 X3 + X6 + X9= 1 All X’s binary, 0 or 1

  30. Answer • Would you have guessed this answer?

  31. Conclusion • Define 3 Network Models • Transportation • Transshipment • Assignment • Formulate Problems • Solve in POM

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