1. © 2006 Prentice Hall, Inc.C – 1 OPS 301 Module C Transportation Models and Network Models Dr. Steven Harrod

2. © 2006 Prentice Hall, Inc.C – 2 Topics Definition of Transportation ModelsDefinition of Transportation Models –Transportation –Assignment –Transshipment Formulation TipsFormulation Tips Word Problem PracticeWord Problem Practice CAUTION: Significant material not in your text!

3. © 2006 Prentice Hall, Inc.C – 3 Network Models Model the flow or transportation of items as arrows between nodes or locations.Model the flow or transportation of items as arrows between nodes or locations. Can be formulated as a linear programCan be formulated as a linear program Related to graph theory in mathematicsRelated to graph theory in mathematics

4. © 2006 Prentice Hall, Inc.C – 4 Three Models Transportation ModelTransportation Model –Minimize cost of transportation –Simple origin to destination transportation Transshipment ModelTransshipment Model –Add intermediate junctions to transportation –Flow conservation constraints at junctions Assignment ModelAssignment Model –Minimize cost, time, etc. of resource assignment –Binary variables (0,1; integer)

5. © 2006 Prentice Hall, Inc.C – 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. © 2006 Prentice Hall, Inc.C – 6 Need to Know 1.The origin points and the capacity or supply at each 2.The destination points and the demand per period at each 3.The cost of shipping one unit from each origin to each destination

7. © 2006 Prentice Hall, Inc.C – 7 Transportation Problem Albuquerque (300 units demand) Des Moines (100 units supply) Evansville (300 units supply) Fort Lauderdale (300 units supply) Cleveland (200 units demand) Boston (200 units demand) Figure C.1

8. © 2006 Prentice Hall, Inc.C – 8 Example Cost Data To FromAlbuquerqueBostonCleveland Des Moines $5$4$3 Evansville$8$4$3 Fort Lauderdale $9$7$5 Table C.1

9. © 2006 Prentice Hall, Inc.C – 9 Formulate What is the goal?What is the goal? –Serve all customers at minimum cost What do we control?What do we control? –Quantities shipped on each route What are the limits?What are the limits? –Quantities available at sources –Quantities demanded at destinations

10. © 2006 Prentice Hall, Inc.C – 10 Variables Amount shipped from origin to destinationAmount shipped from origin to destination Designate with serial indexDesignate with serial index To FromAlbuquerqueBostonCleveland Des Moines X1X1X1X1 X2X2X2X2 X3X3X3X3 Evansville X4X4X4X4 X5X5X5X5 X6X6X6X6 Fort Lauderdale X7X7X7X7 X8X8X8X8 X9X9X9X9

11. © 2006 Prentice Hall, Inc.C – 11 Objective Function Minimize 5 X 1 + 4 X 2 + 3 X 3 + 8 X 4 + 4 X 5 + 3 X 6 + 9 X 7 + 7 X 8 + 5 X 9

12. © 2006 Prentice Hall, Inc.C – 12 Limits: Supply and Demand Albuquerque (300 units demand) Des Moines (100 units supply) Evansville (300 units supply) Fort Lauderdale (300 units supply) Cleveland (200 units demand) Boston (200 units demand) Figure C.1 X9X9X9X9 X6X6X6X6 X3X3X3X3 X2X2X2X2 X1X1X1X1 X5X5X5X5 X4X4X4X4 X8X8X8X8 X7X7X7X7

13. © 2006 Prentice Hall, Inc.C – 13 Subject to X 1 + X 2 + X 3 <= 100 (supply) X 4 + X 5 + X 6 <= 300 X 7 + X 8 + X 9 <= 300 X 1 + X 4 + X 7 = 300 (demand) X 2 + X 5 + X 8 = 200 X 3 + X 6 + X 9 = 200

14. © 2006 Prentice Hall, Inc.C – 14 Solve 9 variables9 variables 6 constraints6 constraints

15. © 2006 Prentice Hall, Inc.C – 15 Answer Objective 3900Objective 3900 To FromAlbuquerqueBostonCleveland Des Moines 100 Evansville200100 Fort Lauderdale 200100

16. © 2006 Prentice Hall, Inc.C – 16 Transshipment Multi-step shipment routeMulti-step shipment route Requires “flow conservation” constraintsRequires “flow conservation” constraints

17. © 2006 Prentice Hall, Inc.C – 17 Change Prior Example Suppose cheap transportation available Ft. Lauderdale to Des MoinesSuppose cheap transportation available Ft. Lauderdale to Des Moines Suppose Ft. Lauderdale preferred manufacturing locationSuppose Ft. Lauderdale preferred manufacturing location Convert Des Moines into warehouseConvert Des Moines into warehouse Increase production at Ft. LauderdaleIncrease production at Ft. Lauderdale

18. © 2006 Prentice Hall, Inc.C – 18 Real World Tropicana juice unit trainTropicana juice unit train Solid train of orange refrigerated boxcarsSolid train of orange refrigerated boxcars Frozen concentrate juiceFrozen concentrate juice Much cheaper than truckMuch cheaper than truck

19. © 2006 Prentice Hall, Inc.C – 19 Add Transfer Station Albuquerque (300 units demand) Des Moines (transfer) Evansville (300 units supply) Fort Lauderdale (400 units supply) Cleveland (200 units demand) Boston (200 units demand) Figure C.1 X9X9X9X9 X6X6X6X6 X3X3X3X3 X2X2X2X2 X1X1X1X1 X5X5X5X5 X4X4X4X4 X8X8X8X8 X7X7X7X7 X 10

20. © 2006 Prentice Hall, Inc.C – 20 New Objective Function Minimize 5 X 1 + 4 X 2 + 3 X 3 + 8 X 4 + 4 X 5 + 3 X 6 + 9 X 7 + 7 X 8 + 5 X 9 + 2 X 10

21. © 2006 Prentice Hall, Inc.C – 21 Subject to X 1 + X 2 + X 3 = X 10 (Des Moines flow conservation) X 4 + X 5 + X 6 <= 300 X 7 + X 8 + X 9 + X 10 <= 400 (new Ft. Lauderdale) X 1 + X 4 + X 7 = 300 (demand) X 2 + X 5 + X 8 = 200 X 3 + X 6 + X 9 = 200

22. © 2006 Prentice Hall, Inc.C – 22 Modify for POM Rewrite X 1 + X 2 + X 3 = X 10 as X 1 + X 2 + X 3 - X 10 = 0 X 4 + X 5 + X 6 <= 300 X 7 + X 8 + X 9 + X 10 <= 400 (new Ft. Lauderdale) X 1 + X 4 + X 7 = 300 (demand) X 2 + X 5 + X 8 = 200 X 3 + X 6 + X 9 = 200

23. © 2006 Prentice Hall, Inc.C – 23 Subject to (revised) X 1 + X 2 + X 3 - X 10 = 0 (Des Moines flow conservation) X 4 + X 5 + X 6 <= 300 X 7 + X 8 + X 9 + X 10 <= 400 (new Ft. Lauderdale) X 1 + X 4 + X 7 = 300 (demand) X 2 + X 5 + X 8 = 200 X 3 + X 6 + X 9 = 200

24. © 2006 Prentice Hall, Inc.C – 24 Solve 10 variables10 variables 6 constraints6 constraints New trick: Format:Insert/DeleteNew trick: Format:Insert/Delete

25. © 2006 Prentice Hall, Inc.C – 25 Answer Objective 3700Objective 3700 To FromAlbuquerqueBostonCleveland Des Moines 300100 Evansville100200 Fort Lauderdale Ft. L. to Des M. 400

26. © 2006 Prentice Hall, Inc.C – 26 Assignment Model Time available?Time available?

27. © 2006 Prentice Hall, Inc.C – 27 Problem Each Leader can manage one projectEach Leader can manage one project What assignments?What assignments? Estimated time to complete in daysEstimated time to complete in days Project LeaderABC Terry10159 Carle9185 McClymonds6143

28. © 2006 Prentice Hall, Inc.C – 28 Objective Function Minimize 10 X 1 + 15 X 2 + 9 X 3 + 9 X 4 + 18 X 5 + 5 X 6 + 6 X 7 + 14 X 8 + 3 X 9

29. © 2006 Prentice Hall, Inc.C – 29 Subject to X 1 + X 2 + X 3 <= 1 X 4 + X 5 + X 6 <= 1 X 7 + X 8 + X 9 <= 1 X 1 + X 4 + X 7 = 1 X 2 + X 5 + X 8 = 1 X 3 + X 6 + X 9 = 1 All X’s binary, 0 or 1

30. © 2006 Prentice Hall, Inc.C – 30 Answer Would you have guessed this answer?Would you have guessed this answer? Project LeaderABC Terryx Carlex McClymondsx

31. © 2006 Prentice Hall, Inc.C – 31 Conclusion Define 3 Network ModelsDefine 3 Network Models –Transportation –Transshipment –Assignment Formulate ProblemsFormulate Problems Solve in POMSolve in POM