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Modeling and Solving LP Problems in a Spreadsheet Chapter 3 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An Investment Problem: Retirement Planning Services, Inc. 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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Investment Restrictions No more than 25% can be invested in any single company. At least 50% should be invested in long- term bonds (maturing in 10+ years). No more than 35% can be invested in DynaStar, Eagle Vision, and OptiPro. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Objective Function Maximize the total annual investment return: MAX:.0865X 1 +.095X 2 +.10X 3 +.0875X 4 +.0925X 5 +.09X 6 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints 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 50% long term investment restriction. X 1 + X 2 + X 4 + X 6 >= 375,000 35% Restriction on DynaStar, Eagle Vision, and OptiPro. X 2 + X 3 + X 5 <= 262,500 Nonnegativity conditions X i >= 0 for all i © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Implementing the Model Fig 3-22 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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A Blending Problem: The Agri-Pro Company 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 The order must contain at least 20% corn, 15% grain, and 15% minerals. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints Produce 8,000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8,000 Mix consists of at least 20% corn (0.3X 1 + 0.5X 2 + 0.2X 3 + 0.1X 4 )/8000 >= 0.2 Mix consists of at least 15% grain (0.1X 1 + 0.3X 2 + 0.15X 3 + 0.1X 4 )/8000 >= 0.15 Mix consists of at least 15% minerals (0.2X 1 + 0.2X 2 + 0.2X 3 + 0.3X 4 )/8000 >= 0.15 Nonnegativity conditions X 1, X 2, X 3, X 4 >= 0 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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A Comment About Scaling Notice the coefficient for X 2 in the ‘corn’ constraint is 0.05/8000 = 0.00000625 As Solver runs, intermediate calculations are made that make coefficients larger or smaller. Storage problems may force the computer to use approximations of the actual numbers. Such ‘scaling’ problems sometimes prevents Solver from being able to solve the problem accurately. Most problems can be formulated in a way to minimize scaling errors... © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Re-Defining the Objective Function Minimize the total cost of filling the order. MIN: 250X 1 + 300X 2 + 320X 3 + 150X 4 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Re-Defining the Constraints Produce 8,000 pounds of feed X 1 + X 2 + X 3 + X 4 = 8 Mix consists of at least 20% corn (0.3X 1 + 0.5X 2 + 0.2X 3 + 0.1X 4 )/8 >= 0.2 Mix consists of at least 15% grain (0.1X 1 + 0.3X 2 + 0.15X 3 + 0.1X 4 )/8 >= 0.15 Mix consists of at least 15% minerals (0.2X 1 + 0.2X 2 + 0.2X 3 + 0.3X 4 )/8 >= 0.15 Nonnegativity conditions X 1, X 2, X 3, X 4 >= 0 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Scaling: Before and After Before: –Largest constraint coefficient was 8,000 –Smallest constraint coefficient was 0.05/8 = 0.00000625. After: –Largest constraint coefficient is 8 –Smallest constraint coefficient is 0.05/8 = 0.00625. The problem is now more evenly scaled! © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Implementing the Model See file Fig3-30.xlsmFig3-30.xlsm © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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A Production Planning Problem: The Upton Corporation Upton is planning the production of their heavy-duty air compressors for the next 6 months. Beginning inventory = 2,750 units Safety stock = 1,500 units Unit carrying cost = 1.5% of unit production cost Maximum warehouse capacity = 6,000 units 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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints - I 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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints - II 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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints - III 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. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Implementing the Model See file Fig3-33.xlsmFig3-33.xlsm © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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A Multi-Period Cash Flow Problem: The Taco-Viva Sinking Fund - I Taco-Viva needs a sinking fund to pay $800,000 in building costs for a new restaurant in the next 6 months. 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. 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% © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Objective Function Minimize the total cash invested in month 1. MIN: A 1 + B 1 + C 1 + D 1 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints 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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Implementing the Model Fig3-37 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Risk Management: The Taco-Viva Sinking Fund - II 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 The CFO wants the weighted average risk to not exceed 5. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints 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 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An Alternate Version of the Risk Constraints 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 max. allowable weighted average risk). © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Implementing the Model See file Fig3-40.xlsmFig3-40.xlsm © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Data Envelopment Analysis (DEA): Steak & Burger 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 = © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Objective Function Maximize the weighted output for unit i : MAX: © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Defining the Constraints Efficiency cannot exceed 100% for any unit Sum of weighted inputs for unit i must equal 1 Nonnegativity Conditions w j, v j >= 0, for all j © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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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”. © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Implementing the Model See file Fig3-43.xlsmFig3-43.xlsm © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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The Analytic Solver Platform software featured in this book is provided by Frontline Systems. http://www.solver.com © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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