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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. C1

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.C2 3. Learning Objectives When you complete this module you should be able to: 1. Develop an initial solution to atransportation models with thenorthwest-corner and intuitivelowest-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. C3 4. Transportation Modeling An interactive procedure that findsthe least costly means of movingproducts from a series of sourcesto a series of destinations Can be used tohelp resolvedistributionand locationdecisions 2008 Prentice Hall, Inc.C4 5. Transportation Modeling A special class of linear programming Need to know 1. The origin points and the capacityor supply per period at each 2. The destination points and thedemand per period at each 3. The cost of shipping one unit fromeach origin to each destination 2008 Prentice Hall, Inc. C5 6. Transportation Problem ToFromAlbuquerque Boston ClevelandDes Moines$5$4$3Evansville$8$4$3Fort Lauderdale $9$7$5Table C.1 2008 Prentice Hall, Inc. C6 7. Transportation Problem Boston Cleveland (200 units (200 units required) Des Moines required)(100 units capacity) Albuquerque(300 units required) Evansville (300 unitscapacity)Fort Lauderdale (300 units Figure C.1 capacity) 2008 Prentice Hall, Inc.C7 8. Transportation MatrixFigure C.2 ToFactoryDes MoinesAlbuquerque BostonClevelandcapacity capacityFromconstraint $5$4 $3Des Moines 100Cellrepresenting $8$4 $3a possibleEvansville 300source-to-destinationshipping $9$7 $5assignmentFort Lauderdale300(Evansvilleto Cleveland)Warehouserequirement300 200200700 Cost of shipping 1 unit from FortClevelandTotal demand Lauderdale factory to Boston warehouse warehouse demand and total supply 2008 Prentice Hall, Inc. C8 9. Northwest-Corner Rule Start in the upper left-hand cell (ornorthwest corner) of the table and allocateunits to shipping routes as follows: 1. Exhaust the supply (factory capacity) of eachrow before moving down to the next row 2. Exhaust the (warehouse) requirements ofeach column before moving to the nextcolumn 3. Check to ensure that all supplies anddemands are met 2008 Prentice Hall, Inc.C9 10. Northwest-Corner Rule1. Assign 100 tubs from Des Moines to Albuquerque (exhausting Des Moiness supply)2. Assign 200 tubs from Evansville to Albuquerque (exhausting Albuquerques demand)3. Assign 100 tubs from Evansville to Boston (exhausting Evansvilles supply)4. Assign 100 tubs from Fort Lauderdale to Boston (exhausting Bostons demand)5. Assign 200 tubs from Fort Lauderdale to Cleveland (exhausting Clevelands demand and Fort Lauderdales supply) 2008 Prentice Hall, Inc.C 10 11. Northwest-Corner Rule To (A)(B)(C) FactoryFromAlbuquerque Boston Clevelandcapacity$5$4 $3(D) Des Moines100100$8$4 $3(E) Evansville200 100300$9$7 $5(F) Fort Lauderdale 100200 300Warehouserequirement300200200 700 Means that the firm is shipping 100Figure C.3 bathtubs from Fort Lauderdale to Boston 2008 Prentice Hall, Inc. C 11 12. Northwest-Corner Rule Computed Shipping CostRouteFrom To Tubs Shipped Cost per Unit Total CostD A 100 $5$ 500E A 2008 1,600E B 1004 400F B 1007 700F C 2005$1,000 Total: $4,200 This is a feasible solutionTable C.2but not necessarily thelowest 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 tothat cell without exceeding supply ordemand; then cross out the row orcolumn (or both) that is exhausted bythis assignment 3. Find the cell with the lowest cost fromthe remaining cells 4. Repeat steps 2 and 3 until all unitshave been allocated 2008 Prentice Hall, Inc.C 13 14. Intuitive Lowest-Cost Method To (A) (B)(C) FactoryFromAlbuquerque BostonClevelandcapacity $5$4 $3(D) Des Moines100 100 $8$4 $3(E) Evansville300 $9$7 $5(F) Fort Lauderdale 300Warehouserequirement300 200200 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) FactoryFromAlbuquerque BostonClevelandcapacity $5$4 $3(D) Des Moines100 100 $8$4 $3(E) Evansville100 300 $9$7 $5(F) Fort Lauderdale 300Warehouserequirement300 200200 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) FactoryFromAlbuquerque BostonClevelandcapacity $5$4 $3(D) Des Moines100 100 $8$4 $3(E) Evansville200 100 300 $9$7 $5(F) Fort Lauderdale 300Warehouserequirement300 200200 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) FactoryFromAlbuquerque BostonClevelandcapacity $5$4 $3(D) Des Moines100 100 $8$4 $3(E) Evansville200 100 300 $9$7 $5(F) Fort Lauderdale300300Warehouserequirement300 200200 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) FactoryFromAlbuquerque BostonClevelandcapacity $5$4 $3(D) Des Moines100 100 $8$4 $3(E) Evansville200 100 300 $9$7 $5(F) Fort Lauderdale300300Warehouserequirement300 200200 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 MethodTo(A) (B)(C) FactoryFromAlbuquerque BostonClevelandcapacity 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 Lauderdale300300Warehouserequirement300 200200 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 aclosed path back to the original squarevia squares that are currently beingused 3. Beginning with a plus (+) sign at theunused corner, place alternate minusand plus signs at each corner of thepath just traced 2008 Prentice Hall, Inc.C 20 21. Stepping-Stone Method 4. Calculate an improvement index byfirst adding the unit-cost figures foundin each square containing a plus signand subtracting the unit costs in eachsquare containing a minus sign 5. Repeat steps 1 though 4 until you havecalculated an improvement index for allunused squares. If all indices are 0,you have reached an optimal solution. 2008 Prentice Hall, Inc. C 21 22. Stepping-Stone MethodTo (A)(B) (C)FactoryFrom AlbuquerqueBoston Cleveland capacity $5 $4$3 (D) Des Moines 100 -+ 100 Des Moines-(E) Evansville 200 $8100$4$3 300Boston index +-(F) Fort Lauderdale $9100$7 200$5 300 = $4 - $5 + $8 - $4 = +$3 Warehouse requirement300 200 20070099$51$4 100-+ 201 +$899-$4Figure C.5 200 100 2008 Prentice Hall, Inc. C 22 23. Stepping-Stone Method To (A) (B) (C) FactoryFromAlbuquerqueBostonClevelandcapacity$5$4 Start $3(D) Des Moines100100-+$8$4 $3(E) Evansville200100 300+-$9$7 $5(F) Fort Lauderdale100 200 300 + -Warehouserequirement300200200 700 Des Moines-Cleveland indexFigure C.6 = $3 - $5 + $8 - $4 + $7 - $5 = +$4 2008 Prentice Hall, Inc. C 23 24. Stepping-Stone Method To (A)(B)(C)FactoryFromAlbuquerque Boston Cleveland capacity$5$4$3(D) Des Moines100 100$8$4$3(E) EvansvilleEvansville-Cleveland index200100300= $3 - $4 + $7 - $5 = +$1 $9 $7$5(F) Fort Lauderdale 100200300(Closed path = EC - EB + FB - FC)Warehouserequirement Fort Lauderdale-Albuquerque index 300200 200700= $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, choosethe route (unused square) with thelargest negative improvement index 2. On the closed path for that route,select the smallest number found in thesquares containing minus signs 3. Add this number to all squares on theclosed path with plus signs andsubtract it from all squares with aminus sign 2008 Prentice Hall, Inc.C 25 26. Stepping-Stone Method To (A) (B) (C) FactoryFromAlbuquerqueBostonClevelandcapacity$5$4 $3(D) Des Moines100100$8$4 $3(E) Evansville200100 300-+$9$7 $5(F) Fort Lauderdale100 200 300+-Warehouserequirement300200200 700 1.Add 100 units on route FA 2.Subtract 100 from routes FB 3.Add 100 to route EBFigure C.7 4.Subtract 100 from route EA 2008 Prentice Hall, Inc. C 26 27. Stepping-Stone Method To (A)(B) (C) FactoryFromAlbuquerque BostonClevelandcapacity $5$4 $3(D) Des Moines100100 $8$4 $3(E) Evansville100 200300 $9$7 $5(F) Fort Lauderdale 100 200300Warehouserequirement300 200200700 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 = $3,350To (A) (B) (C)Factory Dummycapacity FromAlbuquerqueBostonCleveland $5$4 $30 (D) Des Moines 250250 $8$4 $30 (E) Evansville 5020050300 $9$7 $50 (F) Fort Lauderdale150150 300 Warehouse requirement 300 200200 15