The primal solution tells management the minimum-cost advertising mix. The dual problem results are equally valuable. Each dual shadow price indicates the change in cost that would accompany a one-unit change in the various audience exposure requirements. These prices show the marginal costs of increasing each audience exposure requirement by one unit. For example, VA is the marginal cost of reaching the last individual in the overall audience. If there were a one-person reduction in the total audience exposure requirement, a cost saving of VA = $0.40 would be realized. The marginal cost of increasing total audience exposure from 100,000 to 100,001 individuals would also be 40«!.
Shadow prices for the remaining constraint conditions are interpreted in a similar manner. The shadow price for reaching individuals with incomes of at least $50,000 is Vj = $0.20, or 20«. It would cost an extra 20« per person to reach more high-income individuals. A zero value for VS, the marital status shadow price, means that the proposed advertising campaign already reaches more than the 40,000 minimum required number of single persons. Thus, a small change in the marital status constraint has no effect on total costs.
By comparing these marginal costs with the benefits derived from additional exposure, management is able to judge the effectiveness of its media advertising campaign. If the expected profit per exposure exceeds 40«, it would prove profitable to design an advertising campaign for a larger audience. Likewise, if the expected return per exposure to high-income individuals is greater than 20«, promotion to this category of potential customers should be increased. Conversely, if marginal profitability is less than marginal cost, audience size and/or income requirements should be reduced.
Dual slack variables also have an interesting interpretation. They represent opportunity costs of using each advertising medium. LR measures the excess of cost over benefit associated with using radio, whereas L7V indicates the excess of cost over benefit for television. Since LR = LTV = 0, the marginal benefit derived just equals the marginal cost incurred for both media. Both radio and TV are included in the optimal media mix, as was indicated in the primal solution.
This example again demonstrates the symmetry of the primal and dual specifications of linear programming problems. Either specification can be used to describe and solve the same basic problem. Both primal and dual problem statements and solutions offer valuable insight for decision making.
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