Problem
Hansen Hosea owns a small business in Yarra Valley. He buys a whole range of fruits and processes them into gift fruit baskets or into fresh juice. Hansen always has an overall goal of maximising his profits. His production manager grades the fruits they buy on a scale from 1 (lowest quality) to 5 (highest quality). The company's latest purchase is oranges. The following table summarises company's current inventory of oranges of different grades.
Grade of Orange 1 2 3 4 5
Supply (kilogram) 60 35 80 110 70
The production manager is planning to use the oranges from the inventory to produce gift fruit baskets or fresh juice. Based on company's historical data, Hansen and his production manager note that each kilogram of oranges devoted to fruit baskets results in a marginal profit of $2.70.
On the other hand, each kilogram devoted to fresh juice results in a marginal profit of $2.30. These profit margins have already considered the cost of processing the oranges into each of these types of products. To maintain his company's reputation, Hansen wants the fruit in its gift baskets to have an average overall quality grade of at least 3.40 and its fresh juice to have an average overall quality grade of at least 3.10.
Formulate a linear programming model to determine how many kilograms each grade of oranges should Hansen use for processing the oranges into fruit basket or fresh juice if he wants to maximize his profits.