A local company manufactures birdseed. One variety consists of wheat. The cpmpan
ID: 454204 • Letter: A
Question
A local company manufactures birdseed. One variety consists of wheat. The cpmpany is trying to determine the optimal mix of buckwheat (X1), sunflower (X2), and poppy (X3) (each in lbs.). Relevant information is provided in the table below. Also, the final mix is required to contain at least 500 lbs. of poppy. Lastly, the total weight of the buckwheat may not exceed the total weight of the sunflower in the final mix.
Nutritional Item
Proportional Content
Total Requirement
Buckwheat
Sunflower
Poppy
Fat
0.04
0.06
0.05
480
Protein
0.12
0.10
0.10
1200
Roughage
0.10
0.15
0.07
1500
Cost/lb.
$0.18
$0.10
$0.11
The output of the linear program is given on the following page.
LINEAR PROGRAMMING PROBLEM
MIN 0.18X1+0.1X2+0.11X3
S.T.
1) .04X1+.06X2+.05X3>480
2) .12X1+.1X2+.1X3>1200
3) .1X1+.15X2+.07X3<1500
4) 1X3>500
5) 1X1-1X2<0
OPTIMAL SOLUTION
Objective Function Value = 1237.500
Variable Value Reduced Costs
-------------- --------------- ------------------
X1 0.000 0.050
X2 8250.000 0.000
X3 3750.000 0.000
Constraint Slack/Surplus Dual Prices
-------------- --------------- ------------------
1 202.500 0.000
2 0.000 -1.188
3 0.000 0.125
4 3250.000 0.000
5 8250.000 0.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
X1 0.130 0.180 No Upper Limit
X2 No Lower Limit 0.100 0.110
X3 0.100 0.110 0.160
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
1 No Lower Limit 480.000 682.500
2 1026.667 1200.000 2142.857
3 840.000 1500.000 1760.000
4 No Lower Limit 500.000 3750.000
5 -8250.000 0.000 No Upper Limit
(15)
If this had been run as an integer program, we would obtain a different solution. (Check/shade if true.)
If we could reduce the fat requirement by 100 lbs., the optimal solution would not change. (Check/shade if true.)
A new customer wants a mix with at least 20% buckwheat. Would this change the optimal solution? If so, would it increase or decrease? Check/shade the following:
Nora in Accounting noted a glitch in her software, and stated that the cost estimates should be changed. She said the cost values should be $0.17 for buckwheat, $0.12 for sunflower, and $0.12 for poppy. Would this be a cause for concern? If so, which component(s) would be affected? Check/shade the following:
We should be concerned.
If you could relax the requirement on one nutritional item, which would be the best choice to achieve the lowest cost? Fill in the blank.
Nutritional Item
Proportional Content
Total Requirement
Buckwheat
Sunflower
Poppy
Fat
0.04
0.06
0.05
480
Protein
0.12
0.10
0.10
1200
Roughage
0.10
0.15
0.07
1500
Cost/lb.
$0.18
$0.10
$0.11
Explanation / Answer
If it is a integer program ,
MIN 0.18X1+0.1X2+0.11X3
S.T.
1) .04X1+.06X2+.05X3>480
2) .12X1+.1X2+.1X3>1200
3) .1X1+.15X2+.07X3<1500
4) 1X3>500
5) 1X1-1X2<0
If this is a integer program then the solution
04X1+.06X2+.05X3>380
The answer is changing the new solution
If the customer wants atleast 20% of combination of buckwheat then one more constraint is to be added
0.04x1+0.12x2+0.1x3>=0.2
The new solution will be
No change in asnswer
Objective Function 1237.5 Decision Variable x1 0 x2 8250 x3 3750 Constraints 5137.5 480 > 1200 1200 > 1500 1500 < 3750 500 > -8250 0 < If fat Requirement is reduced by 100 lbs then first constrainst will be04X1+.06X2+.05X3>380
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