Modeling and Solving LP Problems in a Spreadsheet Chapter 3 Modeling and Solving LP Problems in a Spreadsheet © 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.

Nutrient Feed 1 Feed 2 Feed 3 Feed 4 Corn 30% 5% 20% 10% Agri-Pro has received an order for 8,000 pounds of chicken feed to be mixed from the following feeds. Nutrient Feed 1 Feed 2 Feed 3 Feed 4 Corn 30% 5% 20% 10% Grain 10% 30% 15% 10% Minerals 20% 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.

X1 = pounds of feed 1 to use in the mix MIN: 0.25X1+ 0.30X2+ 0.32X3+ 0.15X4 © 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 8,000 pounds of feed X1 + X2 + X3 + X4 = 8,000 At least 20% corn (0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8000 >= 0.2 At least 15% grain (0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8000 >= 0.15 At least 15% minerals (0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8000 >= 0.15 Nonnegativity conditions X1, X2, X3, X4 >= 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 X2 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.

Re-Defining the Decision Variables X1 = thousands of pounds of feed 1 to use in the mix X2 = thousands of pounds of feed 2 to use in the mix X3 = thousands of pounds of feed 3 to use in the mix X4 = 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.

Fig3-30.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.

Invest $750,000 in the following bonds. No more than 25% in any single company. At least 50% in long-term bonds (10+ year). No more than 35% in Good and less than Good Years to Company Return Maturity Rating Acme Chemical 8.65% 11 1-Excellent DynaStar 9.50% 10 3-Good Eagle Vision 10.00% 6 4-Fair Micro Modeling 8.75% 10 1-Excellent OptiPro 9.25% 7 3-Good Sabre Systems 9.00% 13 2-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.

Defining the Decision Variables X1 = amount of money to invest in Acme Chemical X2 = amount of money to invest in DynaStar X3 = amount of money to invest in Eagle Vision X4 = amount of money to invest in MicroModeling X5 = amount of money to invest in OptiPro X6 = 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.

Formulation MAX: .0865X1+ .095X2+ .10X3+ .0875X4+ .0925X5+ .09X6 Total amount invested 750,000. X1 + X2 + X3 + X4 + X5 + X6 = 750,000 At most 25% in any one investment. Xi <= 0.25(750,000) =187,500, for all i At least 50% long term investment. X1 + X2 + X4 + X6 >= 0.5(750,000) = 375,000 At most 35% Good and less (3 and 4). X2 + X3 + X5 <= 0.35(750,000) = 262,500 Nonnegativity constraints Xi >= 0 for all i

Solver

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. Investment Available in Month Months to Maturity Yield at Maturity A 1, 2, 3, 4, 5, 6 1 1.8% B 1, 3, 5 2 3.5% C 1, 4 3 5.8% D 1 6 11.0% © 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 1 2 3 4 5 6 7 A1 -1 1.018 B1 -1 <_____> 1.035 C1 -1 <_____> <_____> 1.058 D1 -1 <_____> <_____> <_____> <_____> <_____> 1.11 A2 -1 1.018 A3 -1 1.018 B3 -1 <_____> 1.035 A4 -1 1.018 C4 -1 <_____> <_____> 1.058 A5 -1 1.018 B5 -1 <_____> 1.035 A6 -1 1.018 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.

Defining the Decision Variables Ai = amount (in $1,000s) placed in investment A at the beginning of month i=1, 2, 3, 4, 5, 6 Bi = amount (in $1,000s) placed in investment B at the beginning of month i=1, 3, 5 Ci = amount (in $1,000s) placed in investment C at the beginning of month i=1, 4 Di = 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.

Formulation Objective Function. MIN: A1 + B1 + C1 + D1 Ai, Bi, Ci, Di >= 0, for all I 1.018, 1.035, 1.058, 1.11 3:250, 5:250, 7:300 Beg. month 2 1.018A1 – 1A2 = 0 Beg. month 3 1.035B1 + 1.018A2 – 1A3 – 1B3 = 250 Beg. month 4 1.058C1 + 1.018A3 – 1A4 – 1C4 = 0 Beg. month 5 1.035B3 + 1.018A4 – 1A5 – 1B5 = 250 Beg. month 6 1.018A5 –1A6 = 0 Beg. month 7 1.11D1 + 1.058C4 + 1.035B5 + 1.018A6 = 300

Trial 1 Fig3-37 Just Type it in © 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.

Trial 2 Fig3-37 =IF($A8=B$5,-1,IF($A8=B$6,1+B$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.

Trial 3 =IF($A8=B$5,-1,IF($A8=B$6,1+B$4,IF(AND($A8>B$5,$A8<B$6),"||",""))) 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) Investment Risk Rating A 1 B 3 C 8 D 6 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 1A1 + 3B1 + 8C1 + 6D1 < 5 A1 + B1 + C1 + D1 } month 1 1A2 + 3B1 + 8C1 + 6D1 < 5 A2 + B1 + C1 + D1 } month 2 1A3 + 3B3 + 8C1 + 6D1 < 5 A3 + B3 + C1 + D1 } month 3 1A4 + 3B3 + 8C4 + 6D1 < 5 A4 + B3 + C4 + D1 } month 4 1A5 + 3B5 + 8C4 + 6D1 < 5 A5 + B5 + C4 + D1 } month 5 1A6 + 3B5 + 8C4 + 6D1 < 5 A6 + B5 + C4 + D1 } 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 -4A1 – 2B1 + 3C1 + 1D1 < 0 } month 1 -2B1 + 3C1 + 1D1 – 4A2 < 0 } month 2 3C1 + 1D1 – 4A3 – 2B3 < 0 } month 3 1D1 – 2B3 – 4A4 + 3C4 < 0 } month 4 1D1 + 3C4 – 4A5 – 2B5 < 0 } month 5 1D1 + 3C4 – 2B5 – 4A6 < 0 } 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 =IF(AND($A8>=B$5,$A8<B$6),B$7,"") -4A1 – 2B1 + 3C1 + 1D1 -2B1 + 3C1 + 1D1 – 4A2 3C1 + 1D1 – 4A3 – 2B3 1D1 – 2B3 – 4A4 + 3C4 1D1 + 3C4 – 4A5 – 2B5 1D1 + 3C4 – 2B5 – 4A6 =IF(B16="","",IF(B16<0,B16&B$15,"+"&B16&B$15)) © 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.

Fig3-40.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.

A Production Planning Problem: The Upton Corporation Upton is planning the production of their heavy-duty air compressors for the next 6 months. 1 2 3 4 5 6 Unit Production Cost $240 $250 $265 $285 $280 $260 Units Demanded 1,000 4,500 6,000 5,500 3,500 4,000 Maximum Production 4,000 3,500 4,000 4,500 4,000 3,500 Minimum Production 2,000 1,750 2,000 2,250 2,000 1,750 Month 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 © 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 Pi = number of units to produce in month i, i=1 to 6 Bi = 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: 240P1+250P2+265P3+285P4+280P5+260P6 + 3.6(B1+B2)/2 + 3.75(B2+B3)/2 + 3.98(B3+B4)/2 + 4.28(B4+B5)/2 + 4.20(B5+ B6)/2 + 3.9(B6+B7)/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 <= P1 <= 4,000 } month 1 1,750 <= P2 <= 3,500 } month 2 2,000 <= P3 <= 4,000 } month 3 2,250 <= P4 <= 4,500 } month 4 2,000 <= P5 <= 4,000 } month 5 1,750 <= P6 <= 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 < B1 + P1 - 1,000 < 6,000 } month 1 1,500 < B2 + P2 - 4,500 < 6,000 } month 2 1,500 < B3 + P3 - 6,000 < 6,000 } month 3 1,500 < B4 + P4 - 5,500 < 6,000 } month 4 1,500 < B5 + P5 - 3,500 < 6,000 } month 5 1,500 < B6 + P6 - 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 B1 = 2750 B2 = B1 + P1 - 1,000 B3 = B2 + P2 - 4,500 B4 = B3 + P3 - 6,000 B5 = B4 + P4 - 5,500 B6 = B5 + P5 - 3,500 B7 = B6 + P6 - 4,000 Notice that the Bi can be computed directly from the Pi. Therefore, only the Pi 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.

Fig3-33.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.

Data Envelopment Analysis (DEA): Steak & Burger Steak & Burger needs to evaluate the performance (efficiency) of 12 units. Outputs for each unit (Oij) include measures of: Profit, Customer Satisfaction, and Cleanliness Inputs for each unit (Iij) 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 wj = weight assigned to output j vj = 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 wj, vj >= 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.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. 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.