Question

O.J. Juice Company sells bags of oranges and cartons of orange juice. O.J. grades oranges on a scale of 1 (poor) to 10 (excellent). O.J. now has on hand 100,000 lb of grade 9 oranges and 120,000 lb of grade 6 oranges. The average quality of oranges sold in bags must be at least 7, and the average quality of the oranges used to produce orange juice must be at least 8. Each pound of oranges that is used for juice yields a revenue of $1.50 and incurs a variable cost (consisting of labor costs, variable overhead costs, inventory costs, and so on) of $1.05. Each pound of oranges sold in bags yields a revenue of 50¢ and incurs a variable cost of 20¢. Formulate an LP to help O.J. maximize profit.

Answer 1

Answer:

max z = $0.3x_{1} + 0.45x_{2} + 0.3y_{1} + 0.45y_{2}$

Constraints

$x_{1} + x_{2} \leq 100,000$ (Grade 9 oranges)

$y_{1} + y_{2} \leq 120,000$ (Grade 6 oranges)

$2x_{1} - y_{1} \geq 0$ (Avg oranges in bags)

$x_{2} - 2y_{2} \geq 0$ (Avg oranges in juice)

$x_{1}\geq 0, x_{2} \geq 0, y_{1}\geq 0, y_{2} \geq 0$

Explanation:

Let the variables used be x and y

x representing oranges of grade 9

y representing oranges of grade 6

Now, let $x_{1}$ be the oranges used in each bag in lbs, and $x_{2}$ be oranges used in juice of grade 9 each.

Similarly let $y_{1}$ will represent oranges of grade 6 in bags, and $y_{2}$ will represent oranges in juice of grade 6

Now total oranges sold in bags

= $x_{1}$ + $y_{1}$

And their revenue in $ = 0.5 revenue - 0.2 expense = 0.3

Total profit from bag shall be

0.3 ($x_{1} + y_{1}$) = 0.3 $x_{1}$ + 0.3$y_{1}$

Similarly total oranges in juice shall be

= $x_{2} + y_{2}$

Profit shall be

$1.50 - $1.05 = $0.45

Total profit from juice shall be

$0.45 ($x_{1} + y_{1}$) = $0.45 x_{2} + 0.45 y_{2}$

Profit shall maximise as

z = $0.3x_{1} + 0.45x_{2} + 0.3y_{1} + 0.45y_{2}$

Further constraint 1 shall be

Total amount of  grade 9  oranges used shall be max 100,000 lb

$x_{1} + x_{2} \leq 100,000$

Constraint 2

Total amount of  grade 6  oranges used shall be max 120,000 lb

$y_{1} + y_{2} \leq 120,000$

Constraint 3

Average quality of oranges sold in bag shall be 7

$\frac{9x_{1} + 6y_{1} }{x_{1} + y_{1} } \geq 7$

Accordingly,

$9x_{1} + 6y_{1} \geq 7x_{1} + 7y_{1}$

Simplifying:

$2x_{1} - y_{1} \geq 0$

Constraint 4

Average quality of oranges sold as juice shall be 8

$\frac{9x_{2} + 6y_{2} }{x_{2} + y_{2} } \geq 8$

Accordingly,

$9x_{2} + 6y_{2} \geq 8x_{2} + 8y_{2}$

Simplifying:

$x_{2} - 2y_{2} \geq 0$

Constraints shall be

$x_{1}\geq 0\\x_{2} \geq 0\\y_{1}\geq 0\\y_{2} \geq 0$