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.

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.

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.

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.

Defining the Objective Function Maximize the total annual investment return: MAX:.0865X X X X X X 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.

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.

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.

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.

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.

Defining the Objective Function Minimize the total cost of filling the order. MIN: 0.25X X X X 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.

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 X X X 4 )/8000 >= 0.2  Mix consists of at least 15% grain (0.1X X X X 4 )/8000 >= 0.15  Mix consists of at least 15% minerals (0.2X X X X 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.

A Comment About Scaling  Notice the coefficient for X 2 in the ‘corn’ constraint is 0.05/8000 =  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.

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.

Re-Defining the Objective Function Minimize the total cost of filling the order. MIN: 250X X X X 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.

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 X X X 4 )/8 >= 0.2  Mix consists of at least 15% grain (0.1X X X X 4 )/8 >= 0.15  Mix consists of at least 15% minerals (0.2X X X X 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.

Scaling: Before and After  Before: –Largest constraint coefficient was 8,000 –Smallest constraint coefficient was 0.05/8 =  After: –Largest constraint coefficient is 8 –Smallest constraint coefficient is 0.05/8 =  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.

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.

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 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.

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.

Defining the Objective Function Minimize the total cost production & inventory costs. MIN : 240P P P P P P (B 1 +B 2 )/ (B 2 +B 3 )/ (B 3 +B 4 )/ (B 4 +B 5 )/ (B 5 + B 6 )/ (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.

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.

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.

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.

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.

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.

Summary of Possible Cash Flows Investment A B C D A A B A C A B A 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.

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.

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.

Defining the Constraints  Cash Flow Constraints 1.018A 1 – 1A 2 = 0 } month B A 2 – 1A 3 – 1B 3 = 250 } month C A 3 – 1A 4 – 1C 4 = 0 } month B A 4 – 1A 5 – 1B 5 = 250 } month A 5 –1A 6 = 0 } month D C B A 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The Analytic Solver Platform software featured in this book is provided by Frontline 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.