Question

The XYZ Company plans to allocate some or all of its monthly advertising budget of $75,000 in the Mankato area. It can purchase local radio spots at $120 per spot, local TV spots at $500 per spot, and local newspaper advertising at $260 per insertion.

     The company's policy requirements specify that the company must spend at least $30,000 on TV and allow monthly newspaper expenditures up to $15,000. The company’s internal policy also requires that the company must buy at least 100 radio spots.

The payoff from each advertising medium is a function of the size of its audience. The general experience of the firm is that the values of insertions and spots in terms of "audience points" (arbitrary unit), are as given below:

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

         Radio                            150 audience points per spot

         TV                                180 audience points per spot

         Newspapers                   280 audience points per insertion

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

Let x1 = no. of Radio spots to be purchased,

        X2 = no. of TV spots to be purchased, and

        X3= no. of Newspaper insertions.

Max    150x1+ 180x2 + 280x3

s.t.

      (1)      120x1 + 500x2 + 260x3 <= 75,000     (Advertising Budget)

      (2)                      500x2                  ≥ 30000      (Expenditure on TV)

          (3)                                      260x3 <= 15000      (Expenditure on Newspaper)

          (4)                  x1                            ≥   100         (Number of radio spots)

              X1, x2, x3 >= 0

LINEAR PROGRAMMING PROBLEM

MAX 150X1+ 180X2 + 280X3

Subject to:

1. 120X1 + 500X2 + 260X3 < 75000
2. 500X2 > 30000
3. 260X3 < 15000
4. 1X1 > 100

      OPTIMAL SOLUTION

      Objective Function Value =       67050.000

                Variable                   Value                   Reduced Costs

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

                 

                    X1                        375.000                     0.000

                    X2                          60.000                     0.000

                    X3                            0.000                   45.000

               Constraint               Slack/Surplus            Dual Prices

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

                      1                             0.000                          1.250

                      2                             0.000                        - 0.89

                      3                     15000.000                          0.000

                      4                         275.000                          0.000

    

                            

  OBJECTIVE COEFFICIENT RANGES

      Variable               Lower Limit          Current Value            Upper Limit

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

            X1                            129.231                   150.000           No Upper Limit

            X2                 No Lower Limit                 180.000                        625.000

            X3                 No Lower Limit                 280.000                        325.000

RIGHT HAND SIDE RANGES

       Variable               Lower Limit          Current Value            Upper Limit

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

            1                        42000.000               75000.000           No Upper Limit

            2                                0.000               30000.000                    63000.000

            3                                0.000               15000.000           No Upper Limit

            4                 No Lower Limit                 100.000                        375.000

19. Note that the audience points of TV is currently 180 per spot. If you want to improve
     the audience points of TV, what is the maximum allowable increase without affecting
     the current optimal solution? Show all your work. (4 points)

20. Suppose that the company’s policy specifies that the company must spend at least   
      $40,000 on TV instead of $30,000. Answer the following questions by showing all
      your work.

    (a) What will happen to the dual price? Justify your answer by showing all your
      work.

   (b) Compute the total audience points (OV) if there is any change. Explain clearly by
      showing all your work.

Answer 1

19. As seen from the sensitivity report, the no. of audience spots are specified as 180 against variable X2 under Objective coefficient table. The upper limit is 625 against X2. Hence, If you want to improve the audience points of TV, 625 nos. is the maximum allowable increase without affecting the current optimal solution.

20. As given under the Constraint table, the constraint for the budget on TV is denoted by Variable 2. The upper limit is given as 63000. Hence, the Dual price will remain the same till the upper limit. Thus, if the budget is increase to 40,000 as specified by the new company policy, the dual price will remain the same.

Change in audience points = Dual Price * (Change in Budget constraint RHS) = -0.89 * (40000 - 30000) = (-8900)

Hence, the audience point will reduce by 8900 to (67050 - 8900) = 58,150. This will be the new objective value.

_______________________________________________________________________________________

In case of any doubt, please ask through the comment section before Upvote/downvote.