Question

The XYZ Company plans to allocate some or all of its monthly advertising budget of $75,000 in the 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

1. Which constraint(s) is/are binding (active)?

2. Interpret the dual price of 1.25 for Constraint 1.

Answer 1

1. Binding constraints are those constraints where the slack or surplus is 0. We can see from below table that out of the 4 constraints, Constraint 1 and Constraint 2 have 0 slack/surplus. Hence constraint 1(Advertising budget) and constraint 2 (Expenditure on TV) are binding constraints.

2. Dual price tell us by how much the value of the objective function would change in case of a unit change in the availablity of a constraint with all other conditions kept unchanged. For the given case we can see that for constraint 1, the dual price is 1.25. This means that for every $1 increase in advertising budget the value of objective function would increase by 1.25. For example if the advertising budget available was $75001 instead of $75000, the value of objective function would be 67050+1.25 = 67051.25