Question

1) The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year's program. Advertising alternatives include television, radio, and newspaper. Audience estimates, costs, and maximum media usage limitations are as shown. Constraint Television Radio Newspaper Audience per advertisement 100,000 18,000 40,000 Cost per advertisement $2000 $300 $600 Maximum media usage 10 20 10 To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized Using Excel (and attach file to you submission), show if the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached? By how much would audience contact increase if an extra $100 were allocated to the promotional budget?

Answer 1

Question 1:

a.

Objective Function Value =       1052000.000

      Variable             Value             Reduced Costs  

    --------------        ---------------      ------------------

          T                       4.000                  0.000

          R                    14.000                  0.000

          N                    10.000                  0.000

     Constraint        Slack/Surplus            Dual Prices   

      --------------         ---------------              ------------------

          1                        0.000                           51.304

          2                        6.000                             0.000

          3                        6.000                             0.000

          4                        0.000                    11826.087

          5                        0.000                       5217.391

          6                        1.200                             0.000

OBJECTIVE COEFFICIENT RANGES

   Variable       Lower Limit          Current Value    Upper Limit

------------       ---------------          ---------------              ---------------

       T             -18000.000               100000.000           120000.000

       R              15000.000                 18000.000    No Upper Limit

       N              28173.913                 40000.000    No Upper Limit

RIGHT HAND SIDE RANGES

  Constraint    Lower Limit       Current Value        Upper Limit

   ------------      -------------------    --------------------              ---------------

       1                  14750.000                18200.000                31999.996

       2                          4.000                      10.000       No Upper Limit

       3                        14.000                      20.000       No Upper Limit

       4                          0.000                      10.000                      12.339

       5                         -8.050                        0.000                        2.936

       6         No Lower Limit                        0.000                        1.200

b.         The dual price for the budget constraint is 51.30.  Thus, a $100 increase in budget should provide an increase in audience coverage of approximately 5,130. The right-hand-side range for the budget constraint will show this interpretation is correct.

Question 2:

a.  

Objective Function Value =        1484.96655

      Variable             Value             Reduced Costs  

    --------------       ---------------        ------------------

          F                1484.96655                0.00000

         G1                232.39356                0.00000

         G2                720.38782                0.00000

         S1                 329.40353                0.00000

         S2                 180.18611                0.00000

         S3                     0.00000                0.02077

         S4                     0.00000                0.01942

         S5                 442.30769                0.00000

         S6                     0.00000                0.78551

  

  Constraint        Slack/Surplus               Dual Prices   

    --------------            ---------------              ------------------

          1                        0.00000                       -1.00000

          2                        0.00000                       -0.96154

          3                        0.00000                       -0.92456

          4                        0.00000                       -0.86903

          5                        0.00000                       -0.81693

          6                        0.00000                       -0.78551

The current investment required is $1,484,967.  This calls for investing $232,394 in government security 1 and $720,388 in government security 2.  The amounts, placed in savings are $329,404, $180,186 and $442,308 for years 1,2 and 5 respectively.  No funds are placed in savings for years 3, 4 and 6.

b.         The dual price for constraint 6 indicates that each $1 reduction in the payment required at the beginning of year 6 will reduce the amount of money Hoxworth must pay the trustee by $0.78551.  The lower limit on the right-hand-side range is zero so a $60,000 reduction in the payment at the beginning of year 6 will save Hoxworth $60,000 (0.78551) = $47,131.

c.         The dual price for constraint 1 shows that every dollar of reduction in the initial payment is worth $1.00 to Hoxworth.  So Hoxworth should be willing to pay anything less than $40,000.

d.         To reformulate this problem, one additional variable needs to be added, the right-hand sides for the original constraints need to be shifted ahead by one, and the right-hand side of the first constraint needs to be set equal to zero.  The value of the optimal solution with this formulation is $1,417,739.  Hoxworth will save $67,228 by having the payments moved to the end of each year.

            The revised formulation is shown below:

MIN F

            S.T.

        1)             F - 1.055G1 - 1.000G2 - S1 = 0

        2)             .0675G1 + .05125G2 + 1.04S1 - S2 = 190

        3)             .0675G1 + .05125G2 + 1.04S2 - S3 = 215

        4)             1.0675G1 + .05125G2 + 1.04S3 - S4 = 240

        5)             1.05125G2 +1.04S4 - S5 = 285

        6)             1.04S5 - S6 = 315

        7)             1.04S6 - S7 = 460