An advertising agency will conduct a campaign for a new soft drink. The agency follows the share of the market possessed by the soft drink, in increments of \(5 \%\), from month to month. Each month that advertising continues, there is a cost of \(c\) dollars to the drink manufacturer. But there is a reward of \(r\) dollars to the manufacturer for each \(5 \%\) of the market pos-

sessed by their product. If no advertising is used in any given month, the share of the market will not change. If there is advertising, the share of the market will either increase by \(5 \%\) (with probability \(1 / 2\) ) or stay the same (also with probability \(1 / 2\) ), until a maximum of \(30 \%\) is reached, where it will stay forever. Formulate the problem of finding the optimal advertising policy as a Markov decision problem, including a description of the state space, the admissible action sets, the transition matrices, the reward function, and the optimal value function.