The Royal Cola Company is considering developing a special new carbonated drink to add to its standard product line of drinks for a couple years or so (after which it probably would be replaced by another special drink). However, it is unclear whether the new drink would be profitable, so analysis is needed to determine whether to go ahead with the development of the drink. If so, once the development is completed, the new drink would be marketed in a small regional test market to assess how popular the drink would become. If the test market suggests that the drink should become profitable, it then would be marketed nationally.

Here are the relevant data. The cost of developing the drink and then arranging to test it in the test market is estimated to be $40 million. A total budget of $100 million has been allocated to advertising the drink in both the test market and nationally (if it goes national). A minimum of $5 million is needed for advertising in the test market and the maximum allowed for this purpose would be $10 million, which would leave between $90 million and $95 million for national advertising. To simplify the analysis, sales in either the test market or nationally is assumed to be proportional to the level of advertising there (while recognizing that the rate of additional sales would fall off after the amount of advertising reaches a saturation level). Excluding the fixed cost of $40 million, the net profit in the test market is expected to be half the level of advertising.

To further simplify the analysis, the outcome of testing the drink in the test market would fall into just three categories: (1) very favorable, (2) barely favorable, (3) unfavorable. The probabilities of these outcomes are estimated to be 0.25, 0.25, and 0.50, respectively. If the outcome were very favorable, the net profit after going national would be expected to be about twice the level of advertising. The corresponding net profit if the outcome were barely favorable would be about 0.2 times the level of advertising. If the outcome were unfavorable, the drink would be dropped and so would not be marketed nationally.

The decision variables are:

x1: test market advertising

x21: national campaign advertising if test market is very favorable

x22: national campaign advertising if test market is barely favorable

x23: national campaign advertising if test market is unfavorable

Fill in the blanks below:

Maximize Z = __ ( __x1 + __x21) + __( __x1 + __x22) + __( __x1) - 40

= __x1 + __x21 + __x22) - 40

subject to

x1 ___ 5

x1 ____

x1 + x21 ___ ____

x1 + x22 ____

x1 + x22 ___ ____

x1 + x23 ____

and x1 0, x21 0, x22 0, x23 0.