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

11. Suppose that the rating of the newspaper is increased from 280 to 330 audience points. Which one of the following is true? (4 points)

1. Current optimal solution remains unchanged because 330 is outside the range of optimality for x3.
2. Current optimal solution will be changed because 330 is outside the range of optimality for x3.
3. Current dual prices should be changed because 330 is outside the range of optimality for x3.
4. Current optimal solution remains unchanged because x3= 0.
5. The Optimal value of the objective function (i.e., 67,050 points) remains unchanged because x3 = 0.

12. Suppose that the company eliminates the requirement that the company must buy at

     least 100 radio spots (Constraint 4)? Which one of the following is true? (4 points)

1. Current optimal solution remains unchanged because Constraint 4 is non-binding.
2. Current optimal solution will be changed because Constraint 4 is non-binding.  
3. Current dual prices should be changed because Constraint 4 is non-binding. .
4. The sensitivity analysis cannot be conducted because Constraint 4 is non-binding.
5. The problem must be re-solved.

Answer 1

Question 11:

Answer: b. Current optimal solution will be changed because 330 is outside the range of optimality for x3.

The rating of the newspaper is covered under Objective coefficient ranges. Hence, shadow price is not applicable to this value. The optimum solution will change since allowable upper limit is 325 only and we are increasing beyond 325.

Question 12.

Answer: a. Current optimal solution remains unchanged because Constraint 4 is non-binding.

As seen from the OPTIMAL SOLUTION, there is a slack of 275 available in Constraint 4. Hence, this constraint is non-binding. Change in non-binding constraint does not have an impact if it is deleted.

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