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- 1. Operations Management Module C – Transportation Models PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e© 2008 Prentice Hall, Inc. C–1
- 2. Outline Transportation Modeling Developing an Initial Solution The Northwest-Corner Rule The Intuitive Lowest-Cost Method The Stepping-Stone Method Special Issues in Modeling Demand Not Equal to Supply Degeneracy© 2008 Prentice Hall, Inc. C–2
- 3. Learning Objectives When you complete this module you should be able to: 1. Develop an initial solution to a transportation models with the northwest-corner and intuitive lowest-cost methods 2. Solve a problem with the stepping- stone method 3. Balance a transportation problem 4. Solve a problem with degeneracy© 2008 Prentice Hall, Inc. C–3
- 4. Transportation Modeling An interactive 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© 2008 Prentice Hall, Inc. C–4
- 5. Transportation Modeling A special class of linear programming Need to know 1. The origin points and the capacity or supply per period 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© 2008 Prentice Hall, Inc. C–5
- 6. Transportation Problem To From Albuquerque Boston Cleveland Des Moines $5 $4 $3 Evansville $8 $4 $3 Fort Lauderdale $9 $7 $5 Table C.1© 2008 Prentice Hall, Inc. C–6
- 7. Transportation Problem Boston Cleveland (200 units (200 units required) Des Moines required) (100 units capacity) Albuquerque (300 units required) Evansville (300 units capacity) Fort Lauderdale (300 units Figure C.1 capacity)© 2008 Prentice Hall, Inc. C–7
- 8. Transportation Matrix Figure C.2 To Factory Des Moines Albuquerque Boston Cleveland capacity capacity From constraint $5 $4 $3 Des Moines 100 Cell representing $8 $4 $3 a possible Evansville 300 source-to- destination shipping $9 $7 $5 assignment Fort Lauderdale 300 (Evansville to Cleveland) Warehouse requirement 300 200 200 700 Cost of shipping 1 unit from Fort Cleveland Total demand Lauderdale factory to Boston warehouse warehouse demand and total supply© 2008 Prentice Hall, Inc. C–8
- 9. Northwest-Corner Rule Start in the upper left-hand cell (or northwest corner) of the table and allocate units to shipping routes as follows: 1. Exhaust the supply (factory capacity) of each row before moving down to the next row 2. Exhaust the (warehouse) requirements of each column before moving to the next column 3. Check to ensure that all supplies and demands are met© 2008 Prentice Hall, Inc. C–9
- 10. Northwest-Corner Rule 1. Assign 100 tubs from Des Moines to Albuquerque (exhausting Des Moines’s supply) 2. Assign 200 tubs from Evansville to Albuquerque (exhausting Albuquerque’s demand) 3. Assign 100 tubs from Evansville to Boston (exhausting Evansville’s supply) 4. Assign 100 tubs from Fort Lauderdale to Boston (exhausting Boston’s demand) 5. Assign 200 tubs from Fort Lauderdale to Cleveland (exhausting Cleveland’s demand and Fort Lauderdale’s supply)© 2008 Prentice Hall, Inc. C – 10
- 11. Northwest-Corner Rule To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 $9 $7 $5 (F) Fort Lauderdale 100 200 300 Warehouse requirement 300 200 200 700 Means that the firm is shipping 100 Figure C.3 bathtubs from Fort Lauderdale to Boston© 2008 Prentice Hall, Inc. C – 11
- 12. Northwest-Corner Rule Computed Shipping Cost Route From To Tubs Shipped Cost per Unit Total Cost D A 100 $5 $ 500 E A 200 8 1,600 E B 100 4 400 F B 100 7 700 F C 200 5 $1,000 Total: $4,200 This is a feasible solution Table C.2 but not necessarily the lowest cost alternative© 2008 Prentice Hall, Inc. C – 12
- 13. Intuitive Lowest-Cost Method 1. Identify the cell with the lowest cost 2. Allocate as many units as possible to that cell without exceeding supply or demand; then cross out the row or column (or both) that is exhausted by this assignment 3. Find the cell with the lowest cost from the remaining cells 4. Repeat steps 2 and 3 until all units have been allocated© 2008 Prentice Hall, Inc. C – 13
- 14. Intuitive Lowest-Cost Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 300 $9 $7 $5 (F) Fort Lauderdale 300 Warehouse requirement 300 200 200 700 First, $3 is the lowest cost cell so ship 100 units from Des Moines to Cleveland and cross off the first row as Des Moines is satisfied Figure C.4© 2008 Prentice Hall, Inc. C – 14
- 15. Intuitive Lowest-Cost Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 100 300 $9 $7 $5 (F) Fort Lauderdale 300 Warehouse requirement 300 200 200 700 Second, $3 is again the lowest cost cell so ship 100 units from Evansville to Cleveland and cross off column C as Cleveland is satisfied Figure C.4© 2008 Prentice Hall, Inc. C – 15
- 16. Intuitive Lowest-Cost Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 $9 $7 $5 (F) Fort Lauderdale 300 Warehouse requirement 300 200 200 700 Third, $4 is the lowest cost cell so ship 200 units from Evansville to Boston and cross off column B and row E as Evansville and Boston are satisfied Figure C.4© 2008 Prentice Hall, Inc. C – 16
- 17. Intuitive Lowest-Cost Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 $9 $7 $5 (F) Fort Lauderdale 300 300 Warehouse requirement 300 200 200 700 Finally, ship 300 units from Albuquerque to Fort Lauderdale as this is the only remaining cell to complete the allocations Figure C.4© 2008 Prentice Hall, Inc. C – 17
- 18. Intuitive Lowest-Cost Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 $9 $7 $5 (F) Fort Lauderdale 300 300 Warehouse requirement 300 200 200 700 Total Cost = $3(100) + $3(100) + $4(200) + $9(300) = $4,100 Figure C.4© 2008 Prentice Hall, Inc. C – 18
- 19. Intuitive Lowest-Cost Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity ThisDes Moines $5 $4 $3 (D) is a feasible solution, 100 100 and an improvement over the previous solution, but$8 $4 $3 not necessarily the lowest 200 (E) Evansville 100 300 cost alternative $9 $7 $5 (F) Fort Lauderdale 300 300 Warehouse requirement 300 200 200 700 Total Cost = $3(100) + $3(100) + $4(200) + $9(300) = $4,100 Figure C.4© 2008 Prentice Hall, Inc. C – 19
- 20. Stepping-Stone Method 1. Select any unused square to evaluate 2. Beginning at this square, trace a closed path back to the original square via squares that are currently being used 3. Beginning with a plus (+) sign at the unused corner, place alternate minus and plus signs at each corner of the path just traced© 2008 Prentice Hall, Inc. C – 20
- 21. Stepping-Stone Method 4. Calculate an improvement index by first adding the unit-cost figures found in each square containing a plus sign and subtracting the unit costs in each square containing a minus sign 5. Repeat steps 1 though 4 until you have calculated an improvement index for all unused squares. If all indices are ≥ 0, you have reached an optimal solution.© 2008 Prentice Hall, Inc. C – 21
- 22. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 - + 100 Des Moines- (E) Evansville 200 $8 100 $4 $3 300 Boston index + - (F) Fort Lauderdale $9 100 $7 200 $5 300 = $4 - $5 + $8 - $4 = +$3 Warehouse requirement 300 200 200 700 99 $5 1 $4 100 - + 201 + $8 99 - $4 Figure C.5 200 100© 2008 Prentice Hall, Inc. C – 22
- 23. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 Start $3 (D) Des Moines 100 100 - + $8 $4 $3 (E) Evansville 200 100 300 + - $9 $7 $5 (F) Fort Lauderdale 100 200 300 + - Warehouse requirement 300 200 200 700 Des Moines-Cleveland index Figure C.6 = $3 - $5 + $8 - $4 + $7 - $5 = +$4© 2008 Prentice Hall, Inc. C – 23
- 24. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville Evansville-Cleveland index 200 100 300 = $3 - $4 + $7 - $5 = +$1 $9 $7 $5 (F) Fort Lauderdale 100 200 300 (Closed path = EC - EB + FB - FC) Warehouse requirement Fort Lauderdale-Albuquerque index 300 200 200 700 = $9 - $7 + $4 - $8 = -$1 (Closed path = FA - FB + EB - EA)© 2008 Prentice Hall, Inc. C – 24
- 25. Stepping-Stone Method 1. If an improvement is possible, choose the route (unused square) with the largest negative improvement index 2. On the closed path for that route, select the smallest number found in the squares containing minus signs 3. Add this number to all squares on the closed path with plus signs and subtract it from all squares with a minus sign© 2008 Prentice Hall, Inc. C – 25
- 26. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 200 100 300 - + $9 $7 $5 (F) Fort Lauderdale 100 200 300 + - Warehouse requirement 300 200 200 700 1. Add 100 units on route FA 2. Subtract 100 from routes FB 3. Add 100 to route EB Figure C.7 4. Subtract 100 from route EA© 2008 Prentice Hall, Inc. C – 26
- 27. Stepping-Stone Method To (A) (B) (C) Factory From Albuquerque Boston Cleveland capacity $5 $4 $3 (D) Des Moines 100 100 $8 $4 $3 (E) Evansville 100 200 300 $9 $7 $5 (F) Fort Lauderdale 100 200 300 Warehouse requirement 300 200 200 700 Total Cost = $5(100) + $8(100) + $4(200) + $9(100) + $5(200) = $4,000 Figure C.8© 2008 Prentice Hall, Inc. C – 27
- 28. Special Issues in Modeling Demand not equal to supply Called an unbalanced problem Common situation in the real world Resolved by introducing dummy sources or dummy destinations as necessary with cost coefficients of zero© 2008 Prentice Hall, Inc. C – 28
- 29. Special Issues in 50($3) + 150($5) + 150(0) Total Cost = 250($5) + 50($8) + 200($4) + Modeling = $3,350 To (A) (B) (C) Factory Dummy capacity From Albuquerque Boston Cleveland $5 $4 $3 0 (D) Des Moines 250 250 $8 $4 $3 0 (E) Evansville 50 200 50 300 $9 $7 $5 0 (F) Fort Lauderdale 150 150 300 Warehouse requirement 300 200 200 150 850 New Figure C.9 Des Moines capacity© 2008 Prentice Hall, Inc. C – 29
- 30. Special Issues in Modeling Degeneracy To use the stepping-stone methodology, the number of occupied squares in any solution must be equal to the number of rows in the table plus the number of columns minus 1 If a solution does not satisfy this rule it is called degenerate© 2008 Prentice Hall, Inc. C – 30
- 31. Special Issues in Modeling To Customer Customer Customer Warehouse From 1 2 3 supply $8 $2 $6 Warehouse 1 100 100 $10 $9 $9 Warehouse 2 0 100 20 120 $7 $10 $7 Warehouse 3 80 80 Customer demand 100 100 100 300 Initial solution is degenerate Place a zero quantity in an unused square Figure C.10 and proceed computing improvement indices© 2008 Prentice Hall, Inc. C – 31

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