Problem:
JL Foods is planning to increase its advertising campaign from $1.4 million to $2 million based, in part, in the introduction of a new product. JL Taco Sauce, to accompany its traditional products, JL Ketchup and JL Spaghetti Sauce. In the past, JL Foods promoted its two products individually, splitting its advertising budget equally between ketchup and spaghetti sauce.
From past experience, the marketing department estimates that each dollar spent advertising only ketchup increases ketchup sales by four bottles and each dollar spent advertising only spaghetti sauce increases its sales by 3.2 bottles. Since JL makes $0.30 per bottle of ketchup and $0.35 per bottle of spaghetti sauce sold (excluding the sunk cost of the given advertising budget), this amounts to a return of $1.20 (=4 * $0.30) per advertising dollar on ketchup and $1.12 (=3.2 * $0.35) per advertising dollar on spaghetti sauce. Because taco sauce is a new product, its initial return is projected to be only $0.10 per bottle, but each advertising dollar spent solely on taco sauce is estimated to increase sales by 11 bottles. The company also projects that sales of each product would increase by another 1.4 bottles for each dollar spent on joint advertising of the three products.
JL wishes to maximize its increase in profits this year from advertising while also "building for the future" by adhering to the following guidelines for this year's advertising spending:
- A maximum of $2 million total advertising
- At most $400,000 on joint advertising
- At least $100,000 on joint advertising
- At least $1 million promoting taco sauce, either individually or through joint advertising
- At least $250,000 promoting ketchup only
- At least $250,000 promoting spaghetti sauce only
- At least $750,000 promoting taco sauce only
- At least as much spent this year as last year promoting ketchup, either individually or by joint advertising
- At least 7.5 million total bottles of product sold
a) Construct, in equation form, the linear programming model that determines the optimal allocation of advertising dollars among the four advertising possibilities (each individual & joint).
b) Solve the problem using Solver
c) What is the total return per advertising dollar?
d) Suppose the constraint requiring that at least $750,000 be spent on promoting taco sauce were lowered to $700,000. How much would the profit increase?
Hint: to calculate the profit impact of $1 in advertising as one of the objective function coefficients, you have to use a weighted average of the profit from each type of product since they have increased sales of 1.4 bottles each.