Question

optimization problems
scenario a: fruit smoothie
to make their delicious fruit smoothies, happy planet imports organic oranges and mangoes from brazil. each batch of fruit smoothie requires at least 2 barrels of mangoes for every barrel of oranges. due to a tough growing season in brazil, there are only 8 barrels of oranges and 10 barrels of mangoes available for import. the cost for each barrel is $40 for oranges and $50 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 results 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 per barrel of oranges is $\$40$ and per barrel of mangoes is $\$50$. The objective is to minimize the production cost $C$. So the objective function is $C = 40x+50y$.

Step3: Set up constraints

• Availability constraint: There are only 8 barrels of oranges and 10 barrels of mangoes available for import. So $x\leq8$ and $y\leq 10$.
• Production - requirement constraint: To make smoothies, at least 2 barrels of mangoes are required for every barrel of oranges. So $y\geq2x$. Also, $x\geq0$ and $y\geq0$ since the number of barrels cannot be negative.

Answer:

The objective function is $C = 40x + 50y$ subject to the constraints $x\leq8$, $y\leq 10$, $y\geq2x$, $x\geq0$, $y\geq0$.