1. Reid & Sanders, Operations Management © Wiley 2002 Solving Transportation Problems C SUPPLEMENT

2. Reid & Sanders, Operations Management © Wiley 2002 Page 2 Learning Objectives Define the problem & prepare the transportation tableau Obtain an initial feasible solution Identify the optimal solution Understand special situations

3. Reid & Sanders, Operations Management © Wiley 2002 Page 3 Transportation Problems Transportation problems determine how much of the demand at each one of several destinations is supplied by each one of several sources The goal is to minimize costs Terminology: –Points of demand: The destination that products are shipped to –Points of supply: Where products are shipped from

4. Reid & Sanders, Operations Management © Wiley 2002 Page 4 LP Notation

5. Reid & Sanders, Operations Management © Wiley 2002 Page 5 Preparing the Problem To use the stepping stone or modified distribution (MODI) methods, supply must equal demand. –If not, create a dummy source or destination to make up the difference. –In the solution, shipments from the dummy source represent unmet demand & deliveries to a dummy destination represent excess supply capacity.

6. Reid & Sanders, Operations Management © Wiley 2002 Page 6 Required Information Demand values for each destination (blue) Capacity level at each source (green) Cost of delivering 1 unit to each destination from each source (yellow) PlantDestinationsSources A2413300 B8265 C6142200 Demand200 300100

7. Reid & Sanders, Operations Management © Wiley 2002 Page 7 Initial Solutions Common heuristics (rules of thumb): –Select a cell & allocate as large a shipment as possible without violating capacity or demand constraints (this eliminates a row or column constraint) –Continue selecting new cells until all row & column constraints are satisfied Examples: –Northwest Corner Method (NWC) –Vogel’s Approximation Method (VAM)

8. Reid & Sanders, Operations Management © Wiley 2002 Page 8 Northwest Corner Method Begin in the upper left-hand corner of the tableau (the NW corner) Assign the largest shipment possible –If the column constraint is satisfied, move to the column on the right –If the row constraint is satisfied, move to the row below Continue until all row & column constraints are satisfied

9. Reid & Sanders, Operations Management © Wiley 2002 Page 9 NWC Example: Step 1 PlantDestinationsSources A2413300 B8265 C6142200 Demand200 300100 200 Column Satisfied

10. Reid & Sanders, Operations Management © Wiley 2002 Page 10 NWC Example: Step 2 PlantDestinationsSources A2413 300 – 200 = 100 B8265300 C6142200 Demand200 300100 200100 Row Satisfied

11. Reid & Sanders, Operations Management © Wiley 2002 Page 11 NWC Example: Step 3 PlantDestinationsSources A2413300 B8265 C6142200 Demand200 200 – 100 = 100 300100 200100 Column Satisfied

12. Reid & Sanders, Operations Management © Wiley 2002 Page 12 NWC Example: Step 4 PlantDestinationsSources A2413300 B8265 300 – 100 = 200 C6142 Demand200 300100 200100 200 Row Satisfied

13. Reid & Sanders, Operations Management © Wiley 2002 Page 13 NWC Example: Step 5 PlantDestinationsSources A2413300 B8265 C6142200 Demand200 300 – 200 = 100 100 200100 200 100 Column Satisfied

14. Reid & Sanders, Operations Management © Wiley 2002 Page 14 NWC Example: Step 6 PlantDestinationsSources A2413300 B8265 C6142 200 –100 = 100 Demand200 300100 200100 200 100

15. Reid & Sanders, Operations Management © Wiley 2002 Page 15 NWC Initial Solution PlantDestinationsSources A2413300 B8265 C6142200 Demand200 300100 200100 200 100

16. Reid & Sanders, Operations Management © Wiley 2002 Page 16 Limitations NW Corner Method ignores the objective function coefficients (costs) Solution often isn’t very good: Total cost: 200 units ($2) + 100 units ($4) + 100 units ($2) + 200 units ($6) + 100 units ($4) + 100 units ($2) = $2800 to transport the 800 units

17. Reid & Sanders, Operations Management © Wiley 2002 Page 17 Vogel’s Approximation Method Compute penalties for each row & column: –Compute penalties by subtracting the smallest c ij from the next smallest c ij Select the row or column with the largest penalty Select the cell with the lowest c ij Allocate as many units as possible to that cell Continue until all constraints are satisfied

18. Reid & Sanders, Operations Management © Wiley 2002 Page 18 VAM Example: Step 1 PlantDestinationsSourcesPenalties A24133001 B8265 3 C61422001 Demand200 300100 Penalties4131 200 Column Satisfied

19. Reid & Sanders, Operations Management © Wiley 2002 Page 19 VAM Example: Step 2 PlantDestinationsSourcesPenalties A24131002 B82653003 C61422001 Demand200 300100 Penalties131 200 Tied

20. Reid & Sanders, Operations Management © Wiley 2002 Page 20 Arbitrarily Chose 3 rd Destination PlantDestinationsSourcesPenalties A24131002 B82653003 C61422001 Demand200 300100 Penalties131 200100 Row Satisfied

21. Reid & Sanders, Operations Management © Wiley 2002 Page 21 VAM Example: Step 3 PlantDestinationsSourcesPenalties A2413300 B8265 3 C61422001 Demand200 100 Penalties123 200 Tied 100

22. Reid & Sanders, Operations Management © Wiley 2002 Page 22 Arbitrarily Chose 4th Destination PlantDestinationsSourcesPenalties A24133002 B8265 3 C61422001 Demand200 100 Penalties123 200100 Column Satisfied

23. Reid & Sanders, Operations Management © Wiley 2002 Page 23 VAM Example: Step 4 PlantDestinationsSourcesPenalties A24133002 B8265 4 C61421003 Demand200 100 Penalties12 200100 200 Column Satisfied

24. Reid & Sanders, Operations Management © Wiley 2002 Page 24 VAM Example: Step 5 (only 1 column left & only one feasible solution) PlantDestinationsSourcesPenalties A2413300 B82651003 C6142 1 Demand200 300100 Penalties2 200100 200 100

25. Reid & Sanders, Operations Management © Wiley 2002 Page 25 VAM Initial Solution PlantDestinationsSources A2413300 B8265 C6142200 Demand200 300100 200100 200 100

26. Reid & Sanders, Operations Management © Wiley 2002 Page 26 Better Initial Solution Total Costs: 200 units ($2) + 100 units ($1) + 200 units ($2) + 100 units ($6) +100 units ($4) + 100 units ($2) = $2100 to transport the 800 units Compared to $2800 using the Northwest Corner Method

27. Reid & Sanders, Operations Management © Wiley 2002 Page 27 Finding the Optimal Solution Initial solutions are feasible, but may not be optimal Use the Stepping Stone or Modified Distribution Method to identify improvements & confirm optimality

28. Reid & Sanders, Operations Management © Wiley 2002 Page 28 The End Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.