Sam wants to start work in the Dairy industry in Arizona. He plans eventually to own a cattle farm and make yogurt with them. Sam knows that this is a big undertaking and that it will require more capital than he has at the present.

Therefore, Sam is planning to start small. This is necessitated by both his lack of capital and his inexperience in yogurt making on a large scale. To begin, Sam is going to use $12,000 in savings to finance the initial purchase of milk from which he will make his first batch of yogurt. He is also thinking about going to the Bank and asking for a loan. He has decided to make the preliminary planning horizon two years and would like to estimate the profit over that period. His most immediate task is to decide how much of the $12,000 should be allocated to purchasing milk for the first year and how much to purchasing milk for the second year. In addition, each year he must decide how much he should allocate to purchasing milk to make his favorite Traditional yogurt and how much to purchasing milk to make the more popular Greek yogurt that seems to have been capturing the attention of a wider market during the last few years in America.

In the first year, each cup of Traditional yogurt requires $0.5 worth of milk, and each cup of Greek yogurt uses $0.7 worth of milk. For the second year, the costs of the yogurt per cup are $0.45 and $0.8, respectively.

Sam anticipates that his Traditional yogurt will sell for $4.00 a cup in the first year and for $5.00 in the second year, while his Greek yogurt price remains the same in both years at $6.5 a cup.

Besides the decisions about the amounts of milk purchased in the two years, Sam must make estimates of the sales levels for the two yogurts during the two years. The local yogurt-making association has told him that marketing is the key to success in any yogurt business; generally, demand is directly proportional to the amount of effort spent on marketing. Thus, since Sam cannot afford to do any market research about sales levels due to his lack of capital, He is pondering how much money he should spend to promote each yogurt each year. The yogurt-making association has given him a rule of thumb that relates estimated demand to the amount of money spent on advertising. For instance, they estimate that for each dollar spent in the first year promoting the Traditional yogurt, a demand for seven cups will be created; and for each dollar spent in the second year, a demand for ten cups will result.

Similarly, for each dollar spent on advertising for the Greek yogurt in the first year, up to eight cups can be sold; and for each dollar spent in the second year, up to twelve cups can be sold.

The initial funds for the advertising will come from the $12,000 savings. Assume that the cash earned from yogurt sales in the first year is available in the second year.

A personal concern Sam has is that he maintains a proper balance of yogurt products so that he will be well positioned to expand his marketing capabilities. Thus, in his mind it is important to ensure that the number of cups of Traditional yogurt sold each year falls in the range between 40% and 70% of the overall number of cups sold.

Question: Sam needs help to decide how much milk to buy, how much money to spend on advertising, and how much profit he can expect to earn over the two-year period. Develop a mathematical model and then solve the model to help him.