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 $0\mathrm{.}25,0\mathrm{.}25,$ and $0\mathrm{.}50,$ 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 $0\mathrm{.}2$ times the level of advertising. If the outcome were unfavorable, the drink would be dropped and so

would not be marketed nationally.

Use stochastic programming with recourse to formulate

a model for this problem. Assuming the company should go

ahead with developing the drink, solve the model to determine how much advertising should be done in the test market and then how much advertising should be done nationally $($if any$)$ under each of the three possible outcomes in

the test market. Finally, calculate the expected value $($in the

statistical sense$)$ of the total net profit from the drink, including the fixed cost if the company goes ahead with developing the drink, where the company should indeed go ahead only if the expected total net profit is positive. x$1$: test market advertising

x$21$: national campaign advertising if test market is very favorable

x$22$: national campaign advertising if test market is barely favorable

x$23$: national campaign advertising if test market is unfavorable

Maximize Z $=\_\_\_(\_\_\_$x$1+\_\_\_$ x$21)+\_\_\_(\_\_\_$x$1+\_\_\_$ x$22)+\_\_\_(\_\_\_$x$1)-40$

$=\_\_\_$x$1+\_\_\_$ x$21+\_\_\_$ x$22)-40$

subject to

x$1\_\_\_5$

x$1<=\_\_\_\_$

x$1+$ x$21\_\_\_\_\_\_\_$

x$1+$ x$22<=\_\_\_\_$

x$1+$ x$22\_\_\_\_\_\_\_$

x$1+$ x$23<=\_\_\_\_$

and

x$1>=0,$ x$21>=0,$ x$22>=0,$ x$23>=0.$ fill in the blanks