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### Chap 10

1. 1. Quantitative Analysis for Management Chapter 10 Transportation and Assignment Models To accompany 10-1 © 2000 by Prentice
2. 2. Chapter Outline 10.1 Introduction 10.2 Setting Up a Transportation Problem 10.2 Developing an Initial Solution:Northwest Corner Rule 10.4 Stepping-Stone Method: Finding a Least-Cost Solution 10.5 MODI Method To accompany 10-2 © 2000 by Prentice
3. 3. Chapter Outline - continued 10.6 Vogel’s Approximation Method: Another Way to Find an Initial Solution 10.7 Unbalanced Transportation Problems 10.8 Degeneracy in Transportation Problems 10.9 More Than One Optimal Solution 10.10 Facility Location Analysis To accompany 10-3 © 2000 by Prentice
4. 4. Learning Objectives Students will be able to ♣ Structure special linear programming problems using the transportation and assignment models. ♣ Use the northwest corner method and Vogel’s approximation method to find initial solutions to transportation problems. ♣ Apply the stepping-stone and MODI methods to find optimal solutions to transportation problems. To accompany 10-4 © 2000 by Prentice
5. 5. Learning Objectives - continued ♣ Solve the facility location problem and other application problems with the transportation model. ♣ Solve assignment problems with the Hungarian (matrix reduction) method. To accompany 10-5 © 2000 by Prentice
6. 6. Specialized Problems ♦ Transportation Problem ♣ Distribution of items from several sources to several destinations. Supply capacities and destination requirements known. ♦ Assignment Problem ♣ One to one assignment of people to jobs, etc. Specialized algorithms save time! To accompany 10-6 © 2000 by Prentice
7. 7. Transportation Problem Cleveland (200 units) required Des Moines (100 units) capacity Albuquerque (300 units) required Evansville (300 units) capacity Boston (200 units) required Ft. Lauderdale (300 units) capacity To accompany 10-7 © 2000 by Prentice
8. 8. Transportation Costs From (Sources) To (Destinations) Albuquerque Boston Cleveland Des Moines \$5 \$4 \$3 Evansville \$8 \$4 \$3 Fort Lauderdale \$9 \$7 \$5 To accompany 10-8 © 2000 by Prentice
9. 9. Unit Shipping Cost:1Unit, Factory to Warehouse Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 7 Factory Capacity 5 Warehouse Req. To accompany 10-9 © 2000 by Prentice
10. 10. Total Demand and Total Supply Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity Des Moines (D) 100 Evansville (E) 300 Ft Lauderdale (F) 300 Warehouse Req. To accompany 300 200 10-10 200 700 © 2000 by Prentice
11. 11. Transportation Table For Executive Furniture Corp. Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 7 5 Warehouse Req. To accompany 300 200 10-11 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
12. 12. Initial Solution Using the Northwest Corner Rule ♦ Start in the upper left-hand cell and allocate units to shipping routes as follows: ♣ Exhaust the supply (factory capacity) of each row before moving down to the next row. ♣ Exhaust the demand (warehouse) requirements of each column before moving to the next column to the right. ♣ Check that all supply and demand requirements are met. To accompany 10-12 © 2000 by Prentice
13. 13. Initial Solution North West Corner Rule Albuquerque (A) Des Moines (D) Evansville (E) 100 200 Warehouse Req. To accompany Cleveland (C) 5 4 3 8 4 3 7 5 9 Ft Lauderdale (F) Boston (B) 100 100 300 200 10-13 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
14. 14. The Stepping-Stone Method ♦ 1. Select any unused square to evaluate. ♦ 2. Begin at this square. Trace a closed path back to the original square via squares that are currently being used (only horizontal or vertical moves allowed). ♦ 3. Place + in unused square; alternate - and + on each corner square of the closed path. ♦ 4. Calculate improvement index: add together the unit cost figures found in each square containing a +; subtract the unit cost figure in each square containing a -. ♦ 5. Repeat steps 1 - 4 for each unused square. To accompany 10-14 © 2000 by Prentice
15. 15. Stepping-Stone Method - The Des Moines-to-Cleveland Route Albuquerque (A) Des Moines (D) Evansville (E) Ft Lauderdale (F) + Warehouse Req. To accompany 100 200 Boston (B) 5 Cleveland (C) 4 Start 3 + 8 - 9 + 300 100 100 200 10-15 4 3 7 5 - 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
16. 16. Stepping-Stone Method An Improved Solution Albuquerque (A) Des Moines (D) Evansville (E) 100 100 Ft Lauderdale (F) 100 Warehouse Req. Boston (B) 5 4 3 8 4 3 7 5 200 9 300 To accompany Cleveland (C) 200 10-16 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
17. 17. Third and Final Solution Albuquerque (A) Boston (B) Cleveland (C) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 200 9 Warehouse Req. 300 Des Moines (D) To accompany 100 200 7 200 10-17 100 100 200 5 Factory Capacity 100 300 300 700 © 2000 by Prentice
18. 18. MODI Method: 5 Steps 1. Compute the values for each row and column: set Ri + Kj = Cij for those squares currently used or occupied. 2. After writing all equations, set R1 = 0. 3. Solve the system of equations for Ri and Kj values. 4. Compute the improvement index for each unused square by the formula improvement index: Cij - Ri - Kj 5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method. To accompany 10-18 © 2000 by Prentice
19. 19. Vogel’s Approximation 1. For each row/column of table, find difference between two lowest costs. (Opportunity cost) 1. Find greatest opportunity cost. 1. Assign as many units as possible to lowest cost square in row/column with greatest opportunity cost. 1. Eliminate row or column which has been completely satisfied. 1. Begin again, omitting eliminated rows/columns. To accompany 10-19 © 2000 by Prentice
20. 20. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 7 Warehouse Req. 200 1 To accompany 300 3 10-20 200 5 200 Factory Capacity 100 1 300 4 300 2 700 2 © 2000 by Prentice
21. 21. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) Des Moines (D) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 Warehouse Req. 100 7 300 200 1 To accompany 200 5 100 1 300 4 300 2 3 10-21 200 Factory Capacity 700 © 2000 by Prentice
22. 22. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 9 Des Moines (D) Warehouse Req. 100 100 7 300 200 4 To accompany 200 5 100 1 300 300 2 3 10-22 200 Factory Capacity 700 © 2000 by Prentice
23. 23. Vogel’s Approximation Albuquerque (A) Boston (B) Cleveland (C) 5 4 3 Evansville (E) 8 4 3 Ft Lauderdale (F) 200 9 Warehouse Req. 300 Des Moines (D) To accompany 100 100 100 200 10-23 200 7 5 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
24. 24. Special Problems in Transportation Method ♦ Unbalanced Problem ♣ Demand Less than Supply ♣ Demand Greater than Supply ♦ Degeneracy ♦ More Than One Optimal Solution To accompany 10-24 © 2000 by Prentice
25. 25. Unbalanced Problem Demand Less than Supply Customer 1 Customer 2 Dummy Factory Capacity Factory 1 8 5 16 Factory 2 15 10 7 Factory 3 3 9 10 Customer Requirements To accompany 150 80 10-25 150 170 130 80 380 © 2000 by Prentice
26. 26. Unbalanced Problem Supply Less than Demand Customer 1 Customer 2 Customer 3 Factory 1 8 5 16 Factory 2 15 10 7 Dummy 3 9 10 Customer Requirements To accompany 150 80 10-26 150 Factory Capacity 170 130 80 380 © 2000 by Prentice
27. 27. Degeneracy Customer 1 Customer 2 Customer 3 5 4 3 Factory 2 8 4 3 Factory 3 9 Factory 1 Customer Requirements To accompany 100 100 100 7 100 10-27 20 80 100 5 Factory Capacity 100 120 80 300 © 2000 by Prentice
28. 28. Degeneracy - Coming Up! Customer 1 Customer 2 Factory 1 Factory 2 Factory 3 Customer Requirements To accompany 70 50 30 Customer 3 8 5 16 15 10 7 9 10 80 3 150 80 10-28 50 50 Factory Capacity 70 130 80 280 © 2000 by Prentice
29. 29. Stepping-Stone Method - The Des Moines-to-Cleveland Route Albuquerque (A) Des Moines (D) Evansville (E) Ft Lauderdale (F) + Warehouse Req. To accompany 100 200 Boston (B) 5 Cleveland (C) 4 Start 3 + 8 - 9 + 300 100 100 200 10-29 4 3 7 5 - 200 200 Factory Capacity 100 300 300 700 © 2000 by Prentice
30. 30. The Assignment Method 1. subtract the smallest number in each row from every number in that row ♣ subtract the smallest number in each column from every number in that column 2. draw the minimum number of vertical and horizontal straight lines necessary to cover zeros in the table ♣ if the number of lines equals the number of rows or columns, then one can make an optimal assignment (step 4) To accompany 10-30 © 2000 by Prentice
31. 31. The Assignment Method continued 3. if the number of lines does not equal the number of rows or columns ♣ subtract the smallest number not covered by a line from every other uncovered number ♣ add the same number to any number lying at the intersection of any two lines ♣ return to step 2 4. make optimal assignments at locations of zeros within the table To accompany 10-31 PG 10.13b © 2000 by Prentice
32. 32. Hungarian Method Person Adams Initial Table Project 1 2 11 14 3 6 Brown 8 10 11 Cooper 9 12 7 To accompany 10-32 © 2000 by Prentice
33. 33. Hungarian Method Row Reduction Person Project 1 2 Adams 5 8 3 0 Brown 0 2 3 Cooper 2 5 0 To accompany 10-33 © 2000 by Prentice
34. 34. Hungarian Method Column Reduction Person Project 1 2 Adams 5 6 3 0 Brown 0 0 3 Cooper 2 3 0 To accompany 10-34 © 2000 by Prentice
35. 35. Hungarian Method Person Adams 1 5 Testing Project 2 6 Covering Line 2 3 0 Brown 0 0 3 Cooper 2 3 Covering Line 1 0 To accompany 10-35 © 2000 by Prentice
36. 36. Hungarian Method Revised Opportunity Cost Table Person Project 1 2 3 Adams 3 4 0 Brown 0 0 5 Cooper 0 1 0 To accompany 10-36 © 2000 by Prentice
37. 37. Hungarian Method Person Adams Testing Covering Line 1 Project 1 2 3 4 Covering Line 3 3 0 Brown 0 0 5 Cooper 0 1 0 To accompany 10-37 Covering Line 2 © 2000 by Prentice
38. 38. Hungarian Method Assignments Person 1 Project 2 Adams Brown Cooper To accompany 3 6 10 9 10-38 © 2000 by Prentice