Dual Problem

The dual to the advertising-mix problem is a constrained-maximization problem, because the primal is a minimization problem. The objective function of the dual is expressed in terms of shadow prices or implicit values for the primal constraint conditions. The dual objective function includes an implicit value, or shadow price, for the minimum audience exposure requirement, the audience income requirement, and the marital status requirement. Because constraint limits in the primal problem become the dual objective function coefficients, the dual objective function is

where VA, Vj, and VS are shadow prices for the minimum audience exposure, audience income, and marital status requirements.

Dual constraints are based on the two variables from the primal objective function. Thus, there are two constraint conditions in the dual, the first associated with radio advertisements and the second with television advertisements. Both constraints are of the less-than-or-equal-to type, because primal constraints are of the greater-than-or-equal-to type.

The radio advertising constraint limit is the $6,000 radio advertisements coefficient from the primal objective function. Coefficients for each shadow price in this constraint equation are given by the advertising effectiveness measures for a single radio advertisement. The coefficient for the audience exposure shadow price, VA, is 10,000, the number of individuals reached by a single radio advertisement. Similarly, the coefficient for Vj is 10,000 and that for VS is 8,000. Thus, the dual radio advertisements constraint is

The dual television advertising constraint is developed in the same fashion. Because each TV advertisement reaches a total audience of 20,000, this is the coefficient for the VA variable in the second dual constraint equation. Coefficients for Vj and VS are 10,000 and 4,000, respectively, because these are the numbers of high-income and single persons reached by one TV advertisement. The $10,000 cost of a television advertisement is the limit to the second dual constraint, which can be written

20,000VA + 10,000^ + 4,000VS < $10,000 Following the introduction of constraint slack variables, the dual programming problem is Maximize C* = 100,000VA + 80,000V + 40,000VS

subject to and

10,000VA + 10,000^ + 8,000VS + LR = $ 6,000 20,000Va + 10,000^ + 4,000VS + LTV = $10,000

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