Question

8.) Hungry Birds, Inc. manufactures birdseed.  One variety consists of wheat.  They are trying to determine the optimal mix of buckwheat (X1), sunflower (X2), and poppy (X3) (each in lbs.).  Relevant information is provided in the following table. In addition, the final mix is required to contain at least 500 lbs. of poppy.  Also, 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 below and 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

    

a.      If this had been run as an integer program, we would have obtained a different solution. (Check/shade if true.)

b.     If we could reduce the fat requirement by 100 lbs., the optimal solution would not change.  (Check/shade if true.)

c.     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:

Change?

Decrease?

d.      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.

Buckwheat?

Sunflower?

Poppy?

e.     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.

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