Formulate and then solve a linear programming model of this problem, to determine how many containers of each product to produce tomorrow to maximize profits. The company makes four juice products using orange, grapefruit, and pineapple juice.

Product Retail Price per Quart

Orange juice $1.00

Grapefruit juice $0.90

Pineapple juice $0.80

All-in-One $1.10

The All-in-One juice has equal parts of orange, grapefruit, and pineapple juice.Each product is produced in a one-quart size (there are four quarts in a gallon). On hand are 400 gallons of orange juice, 300 gallons of grapefruit juice, and 200 gallons of pineapple juice.The cost per gallon is $2.00 for orange juice, $1.60 for grapefruit juice, and $1.40 for pineapple juice.

In addition, the manager wants grapefruit juice to be used for no more than 30 percent of the number of containers produced. She wants the ratio of the number of containers of orange juice to the number of containers of pineapple juice to be at least 7 to 5.

Solution:

Draw a table for revenue, cost and profit per quart.

O Juice G Juice P Juice All-in-One

Revenue per quart $1.000 $0.900 $0.800 $1.100

Cost per quart $0.500 $0.400 $0.350 $0.417

Profit per quart $0.500 $0.500 $0.450 $0.683

Formulate the linear programming equation.

Maximise:

0.50_x1 + 0.50_x2 + 0.45_x3 + 0.683_x4

Subject to:

Juice O 1_x1 + 0.333_x4 1,600

Juice G1_x2 + 0.333_x4 1,200

Juice P 1_x3 + 0.333_x4 800

Juice G Container: 0.30_x1 + 0.70_x2 0.30_x3 0.30_x4 0

Ratio:5_x1 7_x3 0

_x1, _x2, _x3, _x4 0

Draw a table to calculate decision variables and profit.

Decision variables Profit

x1 x2 x3 x4 2,240.566

800 400 0 2,402

Coefficients:

Objective 0.5 0.5 0.45 0.683

Juice O 1 0.333

Juice G 1 0.333

Juice P 1 0.333

Container G 0.3 0.7 0.3 0.3

Ratio 5 7

Constraints:

Juice O 1,600 <= 1,600

Juice G 1,200 <= 1,200

Juice P 800 <= 800

Container G 680.7 <= 0

Ratio 4,000 >= 0

Linear programming model will be:

0.50_x1 + 0.50_x2 + 0.45_x3 + 0.683_x4

Quantity required to be produced for Juice O is 800, Juice G is 400, Juice P is 0 and Juice All-in-One is 2,402. Profit will be $2,240.566.

Question:

I don't know how to get the Ratio of 4,000 >= 0 and quantities to be produced of Juice O is 800, Juice G is 400, Juice P is 0 and Juice All-in-One is 2402. Please advise. Thank you!