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Chapter 10 Transportation and Assignment Models

Chapter 10 Transportation and Assignment Models. Learning Objectives. Students will be able to Structure special LP problems using the transportation and assignment models. Use the N.W. corner, VAM, MODI, and stepping-stone method.

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Chapter 10 Transportation and Assignment Models

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  1. Chapter 10 Transportation and Assignment Models 10-1

  2. Learning Objectives Students will be able to • Structure special LP problems using the transportation and assignment models. • Use the N.W. corner, VAM, MODI, and stepping-stone method. • Solve facility location and other application problems with transportation methods. • Solve assignment problems with the Hungarian (matrix reduction) method 10-2

  3. 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 10.6 Vogel’s Approximation Method 10.7 Unbalanced Transportation Problems 10-3

  4. Chapter Outline - continued 10.8 Degeneracy in Transportation Problems 10.9 More Than One Optimal Solution 10.10 Maximization Transportation Problems 10.11 Unacceptable or Prohibited Routes 10.12 Facility Location Analysis 10.13 Approach of the Assignment Model 10.14 Unbalanced Assignment Models 10.15 Maximization Assignment Problems 10-4

  5. 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! 10-5

  6. Fewer, less complicated, computations than with simplex Less computer memory required Produce integer solutions Importance of Special Purpose Algorithms 10-6

  7. Transportation Problem Des Moines (100 units) capacity Cleveland (200 units) required Boston (200 units) required Albuquerque (300 units) required Evansville (300 units) capacity Ft. Lauderdale (300 units) capacity 10-7

  8. Transportation Costs To (Destinations) From (Sources) Albuquerque Cleveland Boston Des Moines Evansville Fort Lauderdale $5 $8 $9 $4 $4 $7 $3 $3 $5 10-8

  9. Unit Shipping Cost:1Unit, Factory to Warehouse Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity 5 4 3 Des Moines (D) 3 8 4 Evansville (E) 9 7 5 Fort Lauderdale (F) Warehouse Req. 10-9

  10. Total Demand and Total Supply Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity Des Moines (D) 100 Evansville (E) 300 Fort Lauderdale (F) 300 Warehouse Req. 300 200 200 700 10-10

  11. Transportation Table For Executive Furniture Corp. Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity 5 4 3 Des Moines (D) 100 8 4 3 Evansville (E) 300 Fort Lauderdale (F) 9 7 5 300 Warehouse Req. 300 200 200 700 10-11

  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. 10-12

  13. Initial SolutionNorth West Corner Rule Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity 5 4 3 Des Moines (D) 100 100 8 4 3 Evansville (E) 200 300 100 Fort Lauderdale (F) 9 7 5 300 200 100 Warehouse Req. 300 200 200 700 10-13

  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. 10-14

  15. Stepping-Stone Method - The Des Moines-to-Cleveland Route Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity 5 4 3 Des Moines (D) Start 100 100 - + 8 4 3 Evansville (E) 300 200 100 - + Fort Lauderdale (F) 9 7 5 100 200 300 + - Warehouse Req. 300 200 200 700 10-15

  16. Stepping-Stone MethodAn Improved Solution Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity 5 4 3 Des Moines (D) 100 100 8 4 3 Evansville (E) 300 100 200 Fort Lauderdale (F) 9 7 5 200 300 100 Warehouse Req. 300 200 200 700 10-16

  17. Third and Final Solution Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 5 4 3 Des Moines (D) 100 100 8 4 3 Evansville (E) 300 100 200 9 7 5 Ft Lauderdale (F) 100 300 200 Warehouse Req. 300 200 200 700 10-17

  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 Kjvalues. 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. 10-18

  19. Vogel’s Approximation 1. For each row/column of table, find difference between two lowest costs. (Opportunity cost) 2. Find greatest opportunity cost. 3. Assign as many units as possible to lowest cost square in row/column with greatest opportunity cost. 4. Eliminate row or column which has been completely satisfied. 4. Begin again, omitting eliminated rows/columns. 10-19

  20. Special Problems in Transportation Method • Unbalanced Problem • Demand Less than Supply • Demand Greater than Supply • Degeneracy • More Than One Optimal Solution 10-20

  21. Unbalanced ProblemDemand Less than Supply Customer 1 Dummy Customer 2 Factory Capacity 8 5 0 Factory 1 170 15 10 0 Factory 2 130 3 9 0 Factory 3 80 Customer Requirements 150 80 150 380 10-21

  22. Unbalanced ProblemSupply Less than Demand Customer 2 Customer 1 Customer 3 Factory Capacity 8 5 16 Factory 1 170 15 10 7 Factory 2 130 0 0 0 Dummy 80 Customer Requirements 150 80 150 380 10-22

  23. Degeneracy Customer 2 Customer 1 Customer 3 Factory Capacity 5 4 3 Factory 1 100 100 8 4 3 Factory 2 120 20 100 9 7 5 Factory 3 80 80 Customer Requirements 100 100 100 300 10-23

  24. Degeneracy - Coming Up! Customer 2 Customer 1 Customer 3 Factory Capacity 8 5 16 Factory 1 70 70 15 10 7 Factory 2 130 50 80 3 9 10 Factory 3 50 80 30 Customer Requirements 150 80 50 280 10-24

  25. Stepping-Stone Method - The Des Moines-to-Cleveland Route Albuquerque (A) Cleveland (C) Boston (B) Factory Capacity 5 4 Start 3 Des Moines (D) 100 100 - + 8 4 3 Evansville (E) 300 200 100 - + 9 7 5 Fort Lauderdale (F) 100 200 300 + - Warehouse Req. 300 200 200 700 10-25

  26. The Assignment Problem 10-26

  27. 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) 10-27

  28. 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 10-28 PG 10.13b

  29. Hungarian Method Initial Table Person Project 1 2 3 Adams 11 14 6 Brown 8 10 11 Cooper 9 12 7 10-29

  30. Hungarian Method Person Project 1 2 3 Adams 5 8 0 Brown 0 2 3 2 5 0 Cooper Row Reduction 10-30

  31. Hungarian Method Column Reduction Person Project 1 2 3 Adams 5 6 0 Brown 0 0 3 Cooper 2 3 0 10-31

  32. Hungarian Method Covering Line 2 CoveringLine 1 Testing Person Project 1 2 3 Adams 5 6 0 Brown 0 0 3 Cooper 2 3 0 10-32

  33. Hungarian Method Revised Opportunity Cost Table Person Project 1 2 3 Adams 3 4 0 Brown 0 0 5 Cooper 0 1 0 10-33

  34. Hungarian Method Testing Covering Line 1 Covering Line 3 Person Project 1 2 3 Adams 3 4 0 Brown Covering Line 2 0 0 5 Cooper 0 1 0 10-34

  35. Hungarian Method Person Project 1 2 3 6 Adams Brown 10 Cooper 9 Assignments 10-35

  36. Maximization Assignment Problem 10-36

  37. Maximization Assignment Problem 10-37

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