Question

A wine manufacturer can source its grapes from two vineyards: vineyard A and vineyard B. Grapes from vineyard A costs $5 per ton to process, and grapes from vineyard B costs $10 per ton to process. Costs must be kept to less than $80 per day. A previous contract with the vineyards require that the amount of grapes sourced from vineyard B cannot exceed twice the amount of grapes from vineyard A. In order to keep his plant running, the wine manufacturer must produce at least 6 tons of grapes each day. Grapes from vineyard A yield 7.5 bottles of wine per ton, and grapes from vineyard B yield 20 bottles of wine per ton. How many tons of grapes from both vineyards must be processed each day to maximize the amount of wine produced subject to the above constraints?

1. Formulate this problem as a linear program by following the procedure presented in the lecture. Explain carefully what the unknowns represent.

Answer 1

tons of A is a and tons of B is b

So a, b are decision variables:

The objective function:

max of Z=7.5a+20b

Such that

b <= 2a

a+b>=6

5a+10b<= 80

a,b>=0

The solution:

	A	B
Tons of grapes	3.2	6.4	9.6	>=	6	Min tons to process constraint
Cost to process	5	10	80	<=	80	Max amount constraint
			6.4	<=	6.4	B<= 2A
Wine processed	7.5	20	152	objective function
			Maximization