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# Pa 906.transportation problem and algorithms final

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### Pa 906.transportation problem and algorithms final

1. 1. Transportation & Assignment Problems Russel Ann L. Rica, MPA Student PA 906: Quantitative Methods BA 906: Quantitative Analysis Assignment for the Transpo Topic can be found in Slide #52
2. 2. PART I. Transportation Problem Russel Ann L. Rica, AdDU MPA Student 2
3. 3. DEFINITION • deals with the distribution of goods from several points of supply (origins or sources) to a number of points of demand (destinations). Usually we are given a capacity (supply) of goods at each source, a requirement (demand) for goods at each destination and the shipping cost per unit from each source of destination Russel Ann L. Rica, AdDU MPA Student 3
4. 4. HISTORY • In 1941, the use of transportation models to minimize the cost of shipping from a number of sources to a number of destinations was first proposed The study was called “ The Distribution of a Product from Several Sources of to Numerous Localities” was written by FL Hitchcock Russel Ann L. Rica, AdDU MPA Student 4
5. 5. • Six years later, the 2nd major contribution was independently produced by TC Koopmans, a report titled “Optimum Utilization of the Transportation System” • A. Charnes and WW Cooper developed the stepping-stone method in the year 1953 • In 1955, the quicker computational approach came about, known as “Modified Distribution (MODI)” Russel Ann L. Rica, AdDU MPA Student 5 HISTORY
6. 6. The Problem (page 342, Quantitative Analysis for Mgt., Render, et al) The Executive Furniture Corporation is faced with the transportation problem shown in Figure 9.1. The company would like to minimize the transportation costs while meeting the demand at each destination and not exceeding the supply at each source. In formulating this as a linear program, there are three supply constraints (one for each source) and three demand constraints (one for each destination). The decisions to be made are the number of units to ship on each route, so there is one decision variable for each arc (arrow) in the network. Russel Ann L. Rica, AdDU MPA Student 6
7. 7. Fort Lauderdale (Source 3) Albuquerque (Destination 1) Boston (Destination 2) Cleveland (Destination 3) \$9 \$7 \$5 SUPPLY SOURCE DESTINATION DEMAND 300 300 200 200 Evansville (Source 2) \$8 \$4 \$3 Des Moines (Source 1) \$5 \$4 \$3 100 300 Russel Ann L. Rica, AdDU MPA Student 7
8. 8. Linear Program for the Example Transportation Problem FACTS: • The name of the company is the Executive Furniture Corporation • EFC would like to minimize the transportation costs, • while meeting the demand at each destination and not exceeding the supply at each source (Figure 9.1, Chapter 9, Quantitative Analysis for Management) Russel Ann L. Rica, AdDU MPA Student 8
9. 9. Fort Lauderdale (Source 3) Albuquerque (Destination 1) Boston (Destination 2) Cleveland (Destination 3) \$9 \$7 \$5 SUPPLY SOURCE DESTINATION DEMAND 300 300 200 200 Evansville (Source 2) \$8 \$4 \$3 Des Moines (Source 1) \$5 \$4 \$3 100 300 Russel Ann L. Rica, AdDU MPA Student 9
10. 10. FACTS: • 3 Supply Points (Des Moines, Evansville, Fort Lauderdale) • 3 Demand Constraints (Albuquerque, Boston, Cleveland) • 9 Transportation Costs (for each source to destination) Russel Ann L. Rica, AdDU MPA Student 10 Linear Program for the Example Transportation Problem cont…
11. 11. Xij = number of units from source i to destination j where: i = 1, 2, 3, with 1 = DM, 2 = E, and 3 – FL j = 1, 2, 3, with 1 = A, 2 = B, and 3 – C Russel Ann L. Rica, AdDU MPA Student 11 Linear Program for the Example Transportation Problem cont…
12. 12. The LP Formulation is to: MINIMIZE the Total Cost Total Cost = 5X11 + 4X12 + 3X13 + 8X21 + 4X22 + 3X23 + 9X31 + 7X32 + 5X33 Russel Ann L. Rica, AdDU MPA Student 12 Linear Program for the Example Transportation Problem cont… Cost of Transportation Source 1 Destination 1
13. 13. Subject to: X11 + X12 + X13 ≤ 100 X21 + X22 + X23 ≤ 300 X31 + X32 + X33 ≤ 300 X11 + X12 + X13 = 300 X21 + X22 + X23 = 300 X31 + X32 + X33 = 200 Xij ≥ 0 for all i and j Russel Ann L. Rica, AdDU MPA Student 13 Linear Program for the Example Transportation Problem cont… Des Moines Supply Evansville Supply Fort Lauderdale Supply Albuquerque demand Boston demand Cleveland demand
14. 14. Transportation Table for EFC (Table 9.2) Russel Ann L. Rica, AdDU MPA Student 14
15. 15. General LP Model for Transportation Problem Variables 3 Sources X 3 Destinations = 9 Constraints 3 Sources (Supply) + 3 Destinations (Demand) = 6 Russel Ann L. Rica, AdDU MPA Student 15 Linear Program for the Example Transportation Problem cont…
16. 16. The Linear Programming model is Minimize cost = Subject to Russel Ann L. Rica, AdDU MPA Student 16 Linear Program for the Example Transportation Problem cont…
17. 17. Solving the Transportation Problem with Transportation Algorithms • Northwest Corner Rule • Stepping-Stone Method Russel Ann L. Rica, AdDU MPA Student 17
18. 18. NORTHWEST CORNER RULE Russel Ann L. Rica, AdDU MPA Student 18
19. 19. • Is a systematic procedure which requires us to start in the upper-left-hand cell (or northwest corner) of the table and allocate units to shipping routes Russel Ann L. Rica, AdDU MPA Student 19 NORTHWEST CORNER RULE
20. 20. 1. Exhaust the supply (factory capacity) at each row before moving down to the next row. 2. Exhaust the (warehouse) requirements of each column before moving to the right to the next column. 3. Check that all supply and demands are met. Russel Ann L. Rica, AdDU MPA Student 20 NORTHWEST CORNER RULE
21. 21. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 21 Start here!
22. 22. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 22 100 100 100200 200 EXHAUSTED
23. 23. Initial Solution: ROUTE UNITS SHIPPED X PER UNIT COST (\$) = TOTAL COST (\$)FROM TO Des Moines Albuquerque 100 5 \$ 500 Evansville Albuquerque 200 8 1,600 Evansville Boston 100 4 400 Fort Lauderdale Boston 100 7 700 Fort Lauderdale Cleveland 200 5 1,000 TOTAL \$ 4,200 Russel Ann L. Rica, AdDU MPA Student 23
24. 24. STEPPING-STONE METHOD Russel Ann L. Rica, AdDU MPA Student 24
25. 25. STEPPING STONE METHOD • Iterative technique for moving from an initial feasible solution to an OPTIMAL feasible solution –This process has two distinct parts: 1. The first involves testing the current solution to determine if improvement is possible, and, 2. The second part involves changes to the current solution in order to obtain an improved solution Russel Ann L. Rica, AdDU MPA Student 25
26. 26. STEPPING STONE METHOD • RULE on shipping routes when using the SS method in Transportation Problem: The number of occupied routes (or squares) must always be equal to one less than the sum of the number of rows plus the number of columns. Russel Ann L. Rica, AdDU MPA Student 26
27. 27. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 27 100 100 100200 200 Start here!
28. 28. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 28 100 100 100200 200 + + - - DM to Boston = (+) \$4 (–) \$5 (+) \$8 (–) \$4 = (+) \$3 cost of transportation increase
29. 29. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 29 100 100 100200 200 + + - - DM to Cleveland = (+) \$3 (–) \$5 (+) \$8 (–) \$4 (+) \$7 (-) \$5 = (+) \$4 cost of transportation increase + -
30. 30. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 30 100 100 100200 200 + - Evansville to Cleveland = (+) \$3 (–) \$4 (+) \$7 (–) \$5 = (+) \$1 cost of transportation increase + -
31. 31. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 31 100 100 100200 200 + - Fort to Albuquerque = (+) \$9 (–) \$7 (+) \$4 (–) \$8 = (-) \$2 cost of transportation decrease + -
32. 32. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 32 100 100 100200 200 + - + - 0 200100 100
33. 33. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 33 100 200100 200100 TSC = 100(\$5) + 100 (\$8) + 100 (\$9) + 200 (\$4) + 200 (\$5) = \$ 4,000
34. 34. Second Solution: ROUTE UNITS SHIPPED X PER UNIT COST (\$) = TOTAL COST (\$)FROM TO Des Moines Albuquerque 100 5 \$ 500 Evansville Albuquerque 100 8 800 Evansville Boston 200 4 800 Fort Lauderdale Albuquerque 100 9 900 Fort Lauderdale Cleveland 200 5 1,000 TOTAL \$ 4,000 Russel Ann L. Rica, AdDU MPA Student 34
35. 35. The Solution derived from the Stepping Stone Method may or may not be the OPTIMAL SOLUTION To check it again, we just have to go back from the first five steps. Russel Ann L. Rica, AdDU MPA Student 35
36. 36. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 36 100 200100 200100 +- + - Evansville to Cleveland = (+) \$3 (–) \$8 (+) \$9 (–) \$5 = (-) \$1
37. 37. Russel Ann L. Rica, AdDU MPA Student 37 Transportation Table for EFC
38. 38. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 38 100 200100 200+ - + - 100200 0 100 100
39. 39. Transportation Table for EFC Russel Ann L. Rica, AdDU MPA Student 39 100 200 100200 100 TSC = 100(\$5) + 200 (\$9) + 200 (\$4) + 100 (\$3) + 100 (\$5) = \$ 3,900
40. 40. Third Solution: ROUTE UNITS SHIPPED X PER UNIT COST (\$) = TOTAL COST (\$)FROM TO Des Moines Albuquerque 100 5 \$ 500 Evansville Boston 200 4 800 Evansville Cleveland 100 3 300 Fort Lauderdale Albuquerque 200 9 1,800 Fort Lauderdale Cleveland 100 5 500 TOTAL \$ 3,900 Russel Ann L. Rica, AdDU MPA Student 40
41. 41. The Solution derived from the 2nd run of the Stepping Stone Method may or may not be the OPTIMAL SOLUTION To check it again, we just have to go back from the first five steps. Russel Ann L. Rica, AdDU MPA Student 41
42. 42. SUMMARY (checking) Russel Ann L. Rica, AdDU MPA Student 42 Therefore, the Third Sol’n contains the optimal shipping assignments because each improvement index that can be computed is either greater than or EQUAL to zero.
43. 43. Special Situations with the Transportation Algorithm Russel Ann L. Rica, AdDU MPA Student 43
44. 44. Special Situations 1. Unbalanced Transportation Problems a. Demand less than Supply b. Supply less than Demand Russel Ann L. Rica, AdDU MPA Student 44 To address unbalanced problems, dummy sources or destinations are being used.
45. 45. EXAMPLE: Initial Sol’n to an UNBALANCED PROBLEM (Demand < Supply) Russel Ann L. Rica, AdDU MPA Student 45
46. 46. SPECIAL SITUATIONS with the Transportation Algorithm 1. Unbalanced Transportation Problems •Demand LESS than Supply •Demand GREATER than Supply Russel Ann L. Rica, AdDU MPA Student 46 To solve this, dummy sources/ destinations are created where transportation cost is equal to ZERO
47. 47. SPECIAL SITUATIONS with the Transportation Algorithm 2. Degeneracy in Transportation Problems This occurs when the number of OCCUPIED squares or routes in a transportation table solution is less than the number of rows plus the number of columns MINUS 1 Russel Ann L. Rica, AdDU MPA Student 47
48. 48. Russel Ann L. Rica, AdDU MPA Student 48 • To handle degenerate problems, we create an artificially occupied cell—that is, we place a zero (representing a fake shipment) in one of the unused squares and then treat that square as if it were occupied. SPECIAL SITUATIONS with the Transportation Algorithm
49. 49. 3. More Than One Optimal Solution This is indicated when one or more of the improvement indices that we calculate for each unused square is zero in the optimal solution. • This means that it is possible to design alternative shipping routes with the same total shipping cost. Russel Ann L. Rica, AdDU MPA Student 49 SPECIAL SITUATIONS with the Transportation Algorithm
50. 50. 4. Maximization Transportation Problem • If the objective of the transportation problem is to maximize profit, a minor change is required in the transportation algorithm. Since the improvement index for an empty cell indicates how the objective function value will change if one unit is placed in that empty cell, the optimal solution is reached when all the improvement indices are negative or zero. Russel Ann L. Rica, AdDU MPA Student 50 SPECIAL SITUATIONS with the Transportation Algorithm
51. 51. 5. Unacceptable or Prohibited Routes • At times there are transportation problems in which one of the sources is unable to ship to one or more of the destinations---when this occurs, the problem is called an unacceptable or prohibited route. • In a minimization problem, such a prohibited route is assigned a very high cost to prevent this route from ever being used in the optimal solution. Russel Ann L. Rica, AdDU MPA Student 51 SPECIAL SITUATIONS with the Transportation Algorithm
52. 52. ASSIGNMENT • Using the Northwest Corner Rule and Stepping-Stone Method, solve the Transportation below: Present your answer in a separate sheet of short bond paper, handwritten.
53. 53. -end-