Question

optimization problems
scenario 4: fruit smoothie
to make their delicious fruit smoothies, happy planet imports organic oranges and mangoes from brazil. each batch of smoothie requires at least 2 barrels of mangoes for every barrel of oranges. due to a tough growing season in brazil, it only 8 barrels of oranges and 10 barrels of mangoes available for import. the cost for each barrel is $40 for oranges and for mangoes. write the objective function that relates barrels of oranges and mangoes to the production cost of the fruit smoothies. create a model that could be used to determine the combination of barrels of oranges and mangoes that will result in a minimum production cost.

Explanation:

Step1: Define variables

Let $x$ be the number of barrels of oranges and $y$ be the number of barrels of mangoes.

Step2: Determine the objective function

The cost for each barrel of oranges is $40$ and for each barrel of mangoes is also $40$. The objective is to minimize the production cost $C$. So the objective function is $C = 40x+40y$.

Step3: Set up constraints

There are constraints on the availability of fruits. We know that $x\leq8$ (only 8 barrels of oranges available), $y\leq10$ (10 barrels of mangoes available), and $y\geq2x$ (smoothie requires at least 2 barrels of mangoes for every barrel of oranges). Also, $x\geq0,y\geq0$ since the number of barrels cannot be negative.

Answer:

Objective function: $C = 40x + 40y$
Constraints: $x\leq8$, $y\leq10$, $y\geq2x$, $x\geq0,y\geq0$