1. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-1 Modeling and Solving LP Problems in a Spreadsheet Chapter 3

2. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-2 Introduction u Solving LP problems graphically is only possible when there are two decision variables u Few real-world LP have only two decision variables

3. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-3 Spreadsheet Solvers u The company that makes the Solver in Excel, Lotus 1-2-3, and Quattro Pro is Frontline Systems, Inc. web site: http://www.frontsys.com u Other packages for solving MP problems: AMPLLINDO CPLEXMPSX

4. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-4 The Steps in Implementing an LP Model in a Spreadsheet 1.Organize the data for the model on the spreadsheet. 2.Reserve separate cells in the spreadsheet to represent each decision variable in the model. 3.Create a formula in a cell in the spreadsheet that corresponds to the objective function. 4.For each constraint, create a formula in a separate cell in the spreadsheet that corresponds to the left- hand side (LHS) of the constraint.

5. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-5 The Blue Ridge Hot Tubs Example... MAX: 350X 1 + 300X 2 } profit S.T.:1X 1 + 1X 2 <= 200} pumps 9X 1 + 6X 2 <= 1566} labor 12X 1 + 16X 2 <= 2880} tubing X 1, X 2 >= 0} nonnegativity

6. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-6 Implementing the Model See file Fig3-1.xls

7. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-7 How Solver Views the Model u Target cell - the cell in the spreadsheet that represents the objective function u Changing cells - the cells in the spreadsheet representing the decision variables u Constraint cells - the cells in the spreadsheet representing the LHS formulas on the constraints

8. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-8 Goals For Spreadsheet Design  Communication - A spreadsheet's primary business purpose is that of communicating information to managers.  Reliability - The output a spreadsheet generates should be correct and consistent.  Auditability - A manager should be able to retrace the steps followed to generate the different outputs from the model in order to understand the model and verify results.  Modifiability - A well-designed spreadsheet should be easy to change or enhance in order to meet dynamic user requirements.

9. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-9 Spreadsheet Design Guidelines u Organize the data, then build the model around the data. u Do not embed numeric constants in formulas. u Things which are logically related should be physically related. u Use formulas that can be copied. u Column/rows totals should be close to the columns/rows being totaled. u The English-reading eye scans left to right, top to bottom. u Use color, shading, borders and protection to distinguish changeable parameters from other model elements. u Use text boxes and cell notes to document various elements of the model.

10. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-10 Make vs. Buy Decisions: The Electro-Poly Corporation u Electro-Poly is a leading maker of slip-rings. u A $750,000 order has just been received. Model 1 Model 2Model 3 Number ordered3,0002,000900 Hours of wiring/unit21.53 Hours of harnessing/unit121 Cost to Make$50$83$130 Cost to Buy$61$97$145 u The company has 10,000 hours of wiring capacity and 5,000 hours of harnessing capacity.

11. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-11 Defining the Decision Variables M 1 = Number of model 1 slip rings to make in-house M 2 = Number of model 2 slip rings to make in-house M 3 = Number of model 3 slip rings to make in-house B 1 = Number of model 1 slip rings to buy from competitor B 2 = Number of model 2 slip rings to buy from competitor B 3 = Number of model 3 slip rings to buy from competitor

12. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-12 Defining the Objective Function Minimize the total cost of filling the order. MIN:50M 1 + 83M 2 + 130M 3 + 61B 1 + 97B 2 + 145B 3

13. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-13 Defining the Constraints u Demand Constraints M 1 + B 1 = 3,000} model 1 M 2 + B 2 = 2,000} model 2 M 3 + B 3 = 900} model 3 u Resource Constraints 2M 1 + 1.5M 2 + 3M 3 <= 10,000 } wiring 1M 1 + 2.0M 2 + 1M 3 <= 5,000 } harnessing u Nonnegativity Conditions M 1, M 2, M 3, B 1, B 2, B 3 >= 0

14. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-14 Implementing the Model See file Fig3-17.xls

15. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-15 An Investment Problem: Retirement Planning Services, Inc. u A client wishes to invest $750,000 in the following bonds. Years to CompanyReturn MaturityRating Acme Chemical8.65%111-Excellent DynaStar9.50%103-Good Eagle Vision10.00%64-Fair Micro Modeling8.75%101-Excellent OptiPro9.25%73-Good Sabre Systems9.00%132-Very Good

16. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-16 Investment Restrictions u No more than 25% can be invested in any single company. u At least 50% should be invested in long- term bonds (maturing in 10+ years). u No more than 35% can be invested in DynaStar, Eagle Vision, and OptiPro.

17. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-17 Defining the Decision Variables X 1 = amount of money to invest in Acme Chemical X 2 = amount of money to invest in DynaStar X 3 = amount of money to invest in Eagle Vision X 4 = amount of money to invest in MicroModeling X 5 = amount of money to invest in OptiPro X 6 = amount of money to invest in Sabre Systems

18. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-18 Defining the Objective Function Maximize the total annual investment return. MAX:.0865X 1 +.095X 2 +.10X 3 +.0875X 4 +.0925X 5 +.09X 6

19. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-19 Defining the Constraints u Total amount is invested X 1 + X 2 + X 3 + X 4 + X 5 + X 6 = 750,000  No more than 25% in any one investment X i <= 187,500, for all i u 50% long term investment restriction. X 1 + X 2 + X 4 + X 6 >= 375,000 u 35% Restriction on DynaStar, Eagle Vision, and OptiPro. X 2 + X 3 + X 5 <= 262,500 u Nonnegativity conditions X i >= 0 for all i

20. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-20 Implementing the Model See file Fig3-20.xls

21. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-21 A Transportation Problem: Tropicsun Mt. Dora 1 Eustis 2 Clermont 3 Ocala 4 Orlando 5 Leesburg 6 Distances (in miles) Capacity Supply 275,000 400,000 300,000 225,000 600,000 200,000 Groves Processing Plants 21 50 40 35 30 22 55 25 20

22. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-22 Defining the Decision Variables X ij = # of bushels shipped from node i to node j Specifically, the nine decision variables are: X 14 = # of bushels shipped from Mt. Dora (node 1) to Ocala (node 4) X 15 = # of bushels shipped from Mt. Dora (node 1) to Orlando (node 5) X 16 = # of bushels shipped from Mt. Dora (node 1) to Leesburg (node 6) X 24 = # of bushels shipped from Eustis (node 2) to Ocala (node 4) X 25 = # of bushels shipped from Eustis (node 2) to Orlando (node 5) X 26 = # of bushels shipped from Eustis (node 2) to Leesburg (node 6) X 34 = # of bushels shipped from Clermont (node 3) to Ocala (node 4) X 35 = # of bushels shipped from Clermont (node 3) to Orlando (node 5) X 36 = # of bushels shipped from Clermont (node 3) to Leesburg (node 6)

23. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-23 Defining the Objective Function Minimize the total number of bushel-miles. MIN:21X 14 + 50X 15 + 40X 16 + 35X 24 + 30X 25 + 22X 26 + 55X 34 + 20X 35 + 25X 36

24. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-24 Defining the Constraints u Capacity constraints X 14 + X 24 + X 34 <= 200,000} Ocala X 15 + X 25 + X 35 <= 600,000} Orlando X 16 + X 26 + X 36 <= 225,000} Leesburg  Supply constraints X 14 + X 15 + X 16 = 275,000} Mt. Dora X 24 + X 25 + X 26 = 400,000} Eustis X 34 + X 35 + X 36 = 300,000} Clermont u Nonnegativity conditions X ij >= 0 for all i and j

25. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-25 Implementing the Model See file Fig3-24.xls

26. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-26 A Blending Problem: The Agri-Pro Company u Agri-Pro has received an order for 8,000 pounds of chicken feed to be mixed from the following feeds. NutrientFeed 1Feed 2 Feed 3Feed 4 Corn30%5%20%10% Grain10%3%15%10% Minerals20%20%20%30% Cost per pound$0.25$0.30$0.32$0.15 Percent of Nutrient in u The order must contain at least 20% corn, 15% grain, and 15% minerals.

27. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-27 Defining the Decision Variables X 1 = pounds of feed 1 to use in the mix X 2 = pounds of feed 2 to use in the mix X 3 = pounds of feed 3 to use in the mix X 4 = pounds of feed 4 to use in the mix

28. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-28 Defining the Objective Function Minimize the total cost of filling the order. MIN: 0.25X 1 + 0.30X 2 + 0.32X 3 + 0.15X 4

29. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-29 Defining the Constraints u Produce 8,000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8,000 u Mix consists of at least 20% corn (0.3X 1 + 0.5X 2 + 0.2X 3 + 0.1X 4 )/8000 >= 0.2 u Mix consists of at least 15% grain (0.1X 1 + 0.3X 2 + 0.15X 3 + 0.1X 4 )/8000 >= 0.15 u Mix consists of at least 15% minerals (0.2X 1 + 0.2X 2 + 0.2X 3 + 0.3X 4 )/8000 >= 0.15 u Nonnegativity conditions X 1, X 2, X 3, X 4 >= 0

30. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-30 A Comment About Scaling  Notice that the coefficient for X 2 in the ‘corn’ constraint is 0.05/8000 = 0.00000625 u As Solver solves our problem, intermediate calculations must be done that make coefficients large or smaller. u Storage problems may force the computer to use approximations of the actual numbers. u Such ‘scaling’ problems sometimes prevents Solver from being able to solve the problem accurately. u Most problems can be formulated in a way to minimize scaling errors...

31. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-31 Re-Defining the Decision Variables X 1 = thousands of pounds of feed 1 to use in the mix X 2 = thousands of pounds of feed 2 to use in the mix X 3 = thousands of pounds of feed 3 to use in the mix X 4 = thousands of pounds of feed 4 to use in the mix

32. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-32 Re-Defining the Objective Function Minimize the total cost of filling the order. MIN: 250X 1 + 300X 2 + 320X 3 + 150X 4

33. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-33 Re-Defining the Constraints u Produce 8,000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8 u Mix consists of at least 20% corn (0.3X 1 + 0.5X 2 + 0.2X 3 + 0.1X 4 )/8 >= 0.2 u Mix consists of at least 15% grain (0.1X 1 + 0.3X 2 + 0.15X 3 + 0.1X 4 )/8 >= 0.15 u Mix consists of at least 15% minerals (0.2X 1 + 0.2X 2 + 0.2X 3 + 0.3X 4 )/8 >= 0.15 u Nonnegativity conditions X 1, X 2, X 3, X 4 >= 0

34. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-34 A Comment About Scaling  Earlier the largest coefficient in the constraints was 8,000 and the smallest is 0.05/8 = 0.00000625.  Now the largest coefficient in the constraints is 8 and the smallest is 0.05/8 = 0.00625. u The problem is now more evenly scaled.

35. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-35 The Assume Linear Model Option u The Solver Options dialog box has an option labeled “Assume Linear Model”. u When you select this option Solver performs some tests to verify that your model is in fact linear. u These test are not 100% accurate & often fail as a result of a poorly scaled model. u If Solver tells you a model isn’t linear when you know it is, try solving it again. If that doesn’t work, try re-scaling your model.

36. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-36 Implementing the Model See file Fig3-33.xls

37. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-37 A Production Planning Problem: The Upton Corporation u Upton is planning the production of their heavy-duty air compressors for the next 6 months. 12 3456 Unit Production Cost$240$250$265$285$280$260 Units Demanded1,0004,5006,0005,5003,5004,000 Maximum Production4,0003,5004,0004,5004,0003,500 Minimum Production2,0001,7502,0002,2502,0001,750 Month u Beginning inventory = 2,750 units u Safety stock = 1,500 units u Unit carrying cost = 1.5% of unit production cost u Maximum warehouse capacity = 6,000 units

38. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-38 Defining the Decision Variables P i = number of units to produce in month i, i=1 to 6 B i = beginning inventory month i, i=1 to 6

39. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-39 Defining the Objective Function Minimize the total cost production & inventory costs. MIN : 240P 1 + 250P 2 + 265P 3 + 285P 4 + 280P 5 + 260P 6 + 3.6(B 1 +B 2 )/2 + 3.75(B 2 +B 3 )/2 + 3.98(B 3 +B 4 )/2 + 4.28(B 4 +B 5 )/2 + 4.20(B 5 + B 6 )/2 + 3.9(B 6 +B 7 )/2 Note: The beginning inventory in any month is the same as the ending inventory in the previous month.

40. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-40 Defining the Constraints u Production levels 2,000 <= P 1 <= 4,000 } month 1 1,750 <= P 2 <= 3,500 } month 2 2,000 <= P 3 <= 4,000 } month 3 2,250 <= P 4 <= 4,500 } month 4 2,000 <= P 5 <= 4,000 } month 5 1,750 <= P 6 <= 3,500 } month 6 u Ending Inventory (EI = BI + P - D) 1,500 <= B 1 + P 1 - 1,000 <= 6,000} month 1 1,500 <= B 2 + P 2 - 4,500 <= 6,000} month 2 1,500 <= B 3 + P 3 - 6,000 <= 6,000} month 3 1,500 <= B 4 + P 4 - 5,500 <= 6,000} month 4 1,500 <= B 5 + P 5 - 3,500 <= 6,000} month 5 1,500 <= B 6 + P 6 - 4,000 <= 6,000} month 6

41. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-41 Defining the Constraints (cont’d) u Beginning Balances B 1 = 2750 B 2 = B 1 + P 1 - 1,000 B 3 = B 2 + P 2 - 4,500 B 4 = B 3 + P 3 - 6,000 B 5 = B 4 + P 4 - 5,500 B 6 = B 5 + P 5 - 3,500 B 7 = B 6 + P 6 - 4,000 Notice that the B i can be computed directly from the P i. Therefore, only the P i need to be identified as changing cells.

42. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-42 Implementing the Model See file Fig3-31.xls

43. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-43 A Multi-Period Cash Flow Problem: The Taco-Viva Sinking Fund - I u Taco-Viva needs to establish a sinking fund to pay $800,000 in building costs for a new restaurant in the next 6 months. u Payments of $250,000 are due at the end of months 2 and 4, and a final payment of $300,000 is due at the end of month 6. u The following investments may be used. InvestmentAvailable in MonthMonths to MaturityYield at Maturity A1, 2, 3, 4, 5, 611.8% B1, 3, 523.5% C1, 435.8% D1611.0%

44. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-44 Summary of Possible Cash Flows Investment1234567 A 1 -11.018 B 1 -1 1.035 C 1 -1 1.058 D 1 -1 1.11 A 2 -11.018 A 3 -11.018 B 3 -1 1.035 A 4 -11.018 C 4 -1 1.058 A 5 -11.018 B 5 -1 1.035 A 6 -11.018 Req’d Payments $0$0$250 $0$250$0$300 (in $1,000s) Cash Inflow/Outflow at the Beginning of Month

45. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-45 Defining the Decision Variables A i = amount (in $1,000s) placed in investment A at the beginning of month i=1, 2, 3, 4, 5, 6 B i = amount (in $1,000s) placed in investment B at the beginning of month i=1, 3, 5 C i = amount (in $1,000s) placed in investment C at the beginning of month i=1, 4 D i = amount (in $1,000s) placed in investment D at the beginning of month i=1

46. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-46 Defining the Objective Function Minimize the total cash invested in month 1. MIN :A 1 + B 1 + C 1 + D 1

47. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-47 Defining the Constraints u Cash Flow Constraints 1.018A 1 – 1A 2 = 0 } month 2 1.035B 1 + 1.018A 2 – 1A 3 – 1B 3 = 250 } month 3 1.058C 1 + 1.018A 3 – 1A 4 – 1C 4 = 0 } month 4 1.035B 3 + 1.018A 4 – 1A 5 – 1B 5 = 250 } month 5 1.018A 5 –1A 6 = 0 } month 6 1.11D 1 + 1.058C 4 + 1.035B 5 + 1.018A 6 = 300 } month 7  Nonnegativity Conditions A i, B i, C i, D i >= 0, for all i

48. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-48 Implementing the Model See file Fig3-35.xls

49. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-49 Risk Management: The Taco-Viva Sinking Fund - II u Assume the CFO has assigned the following risk ratings to each investment on a scale from 1 to 10 (10 = max risk) InvestmentRisk Rating A1 B3 C8 D6 u The CFO wants the weighted average risk to not exceed 5.

50. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-50 Defining the Constraints u Risk Constraints 1A 1 + 3B 1 + 8C 1 + 6D 1 <= 5 A 1 + B 1 + C 1 + D 1 } month 1 1A 2 + 3B 1 + 8C 1 + 6D 1 <= 5 A 2 + B 1 + C 1 + D 1 } month 2 1A 3 + 3B 3 + 8C 1 + 6D 1 <= 5 A 3 + B 3 + C 1 + D 1 } month 3 1A 4 + 3B 3 + 8C 4 + 6D 1 <= 5 A 4 + B 3 + C 4 + D 1 } month 4 1A 5 + 3B 5 + 8C 4 + 6D 1 <= 5 A 5 + B 5 + C 4 + D 1 } month 5 1A 6 + 3B 5 + 8C 4 + 6D 1 <= 5 A 6 + B 5 + C 4 + D 1 } month 6

51. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-51 An Alternate Version of the Risk Constraints u Equivalent Risk Constraints -4A 1 - 2B 1 + 3C 1 + 1D 1 <= 0} month 1 – 2B 1 + 3C 1 + 1D 1 – 4A 2 <= 0} month 2 3C 1 + 1D 1 – 4A 3 – 2B 3 <= 0} month 3 1D 1 – 2B 3 – 4A 4 + 3C 4 <= 0} month 4 1D 1 + 3C 4 – 4A 5 – 2B 5 <= 0} month 5 1D 1 + 3C 4 – 2B 5 – 4A 6 <= 0} month 6 Note that each coefficient is equal to the risk factor for the investment minus 5 (the maximum allowable weighted average risk).

52. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-52 Implementing the Model See file Fig3-38.xls

53. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-53 Data Envelopment Analysis (DEA): Steak & Burger u Steak & Burger needs to evaluate the performance (efficiency) of 12 units.  Outputs for each unit ( O ij ) include measures of: Profit, Customer Satisfaction, and Cleanliness  Inputs for each unit ( I ij ) include: Labor Hours, and Operating Costs  The “Efficiency” of unit i is defined as follows: Weighted sum of unit i’s outputs Weighted sum of unit i’s inputs =

54. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-54 Defining the Decision Variables w j = weight assigned to output j v j = weight assigned to input j A separate LP is solved for each unit, allowing each unit to select the best possible weights for itself.

55. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-55 Defining the Objective Function Maximize the weighted output for unit i : MAX :

56. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-56 Defining the Constraints u Efficiency cannot exceed 100% for any unit u Sum of weighted inputs for unit i must equal 1  Nonnegativity Conditions w j, v j >= 0, for all j

57. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-57 Important Point When using DEA, output variables should be expressed on a scale where “more is better” and input variables should be expressed on a scale where “less is better”.

58. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-58 Implementing the Model See file Fig3-41.xls

59. Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 2-59 End of Chapter 3