1. Modeling and Solving LP Problems in a Spreadsheet
Chapter 3
Modeling and Solving LP Problems in a Spreadsheet
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2. 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.
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3. X1 = pounds of feed 1 to use in the mix
MIN: X X X X4
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4. Defining the Constraints
8,000 pounds of feed
X1 + X2 + X3 + X4 = 8,000
At least 20% corn
(0.3X X X X4)/8000 >= 0.2
At least 15% grain
(0.1X X X X4)/8000 >= 0.15
At least 15% minerals
(0.2X X X X4)/8000 >= 0.15
Nonnegativity conditions
X1, X2, X3, X4 >= 0
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5. A Comment About Scaling
Notice the coefficient for X2 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...
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6. 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
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7. Fig3-30.xlsm
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8. 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
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9. 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
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10. Formulation
MAX: .0865X X2+ .10X X X5+ .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

11. Solver

12. 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, %
B 1, 3, %
C 1, %
D %
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13. Summary of Possible Cash Flows
Investment A
B1 -1 <_____> 1.035
C1 -1 <_____> <_____>
D1 -1 <_____> <_____> <_____> <_____> <_____> 1.11 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
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14. 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
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15. 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
1.035B A2 – 1A3 – 1B3 = 250
Beg. month 4
1.058C A3 – 1A4 – 1C4 = 0
Beg. month 5
1.035B A4 – 1A5 – 1B5 = 250
Beg. month 6
1.018A5 –1A6 = 0
Beg. month 7
1.11D C B A6 = 300

16. Trial 1 Fig3-37 Just Type it in
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17. Trial 2 Fig3-37 =IF($A8=B$5,-1,IF($A8=B$6,1+B$4,""))
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18. Trial 3
=IF($A8=B$5,-1,IF($A8=B$6,1+B$4,IF(AND($A8>B$5,$A8<B$6),"||","")))
Fig3-37
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19. Risk Management: The Taco-Viva Sinking Fund - II
Assume the CFO has assigned the following risk ratings to each investment on a scale from 1 to 10 (10 = max risk)
Investment Risk Rating
A 1
B 3
C 8
D 6
The CFO wants the weighted average risk to not exceed 5.
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20. Defining the Constraints
Risk Constraints
1A1 + 3B1 + 8C1 + 6D1
< 5
A1 + B1 + C1 + D1
} month 1
1A2 + 3B1 + 8C1 + 6D1
< 5
A2 + B1 + C1 + D1
} month 2
1A3 + 3B3 + 8C1 + 6D1
< 5
A3 + B3 + C1 + D1
} month 3
1A4 + 3B3 + 8C4 + 6D1
< 5
A4 + B3 + C4 + D1
} month 4
1A5 + 3B5 + 8C4 + 6D1
< 5
A5 + B5 + C4 + D1
} month 5
1A6 + 3B5 + 8C4 + 6D1
< 5
A6 + B5 + C4 + D1
} month 6
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21. An Alternate Version of the Risk Constraints
-4A1 – 2B1 + 3C1 + 1D1 < 0 } month 1
-2B1 + 3C1 + 1D1 – 4A2 < 0 } month 2
3C1 + 1D1 – 4A3 – 2B3 < 0 } month 3
1D1 – 2B3 – 4A4 + 3C4 < 0 } month 4
1D1 + 3C4 – 4A5 – 2B5 < 0 } month 5
1D1 + 3C4 – 2B5 – 4A6 < 0 } month 6
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22. An Alternate Version of the Risk Constraints
=IF(AND($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))
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23. Fig3-40.xlsm
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24. A Production Planning Problem: The Upton Corporation
Upton is planning the production of their heavy-duty air compressors for the next 6 months.
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
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25. 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
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26. Defining the Objective Function
Minimize the total cost production
& inventory costs.
MIN: 240P1+250P2+265P3+285P4+280P5+260P6
+ 3.6(B1+B2)/ (B2+B3)/ (B3+B4)/2
+ 4.28(B4+B5)/ (B5+ B6)/ (B6+B7)/2
Note: The beginning inventory in any month is the same as the ending inventory in the previous month.
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27. 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
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28. 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
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29. 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.
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30. Fig3-33.xlsm
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31. 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
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32. 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.
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33. Defining the Objective Function
Maximize the weighted output for unit i :
MAX:
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34. 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
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35. 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”.
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36. Implementing the Model
See file Fig3-43.xlsm
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37. The Analytic Solver Platform software featured in this book is provided by Frontline Systems.
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