Let
x1= number of containers (quarts) of orange juice produced.
x2= number of containers (quarts) of grapefruit juice produced.
x3= number of containers (quarts) of pineapple juice produced.
x4= number of containers (quarts) of All-in-One juice produced.
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).
Revenue= 1.00x1+0.90x2+0.80x3+1.10x4
Number of gallons of orange juice used: (x1+x4/3)/4=x1/4+x4/12
Number of gallons of orange juice available: 439 gallons = 1756 quarts
x1/4+x4/12<=439
Number of gallons of grapefruit juice used: (x2+x4/3)/4=x2/4+x4/12
Number of gallons of grapefruit juice available: 301 gallons=1204 quarts
x2/4+x4/12<=301
Number of gallons of pineapple juice used: (x3+x4/3)/4=x3/4+x4/12
Number of gallons of pineapple juice available: 249 gallons = 996 quarts
x3/4+x4/12<=249
Cost/gallon: $2.00 for orange juice, $1.60 for grapefruit juice, and $1.40
for pineapple juice.
Total cost=2.00*(x1/4+x4/12)+1.60*(x2/4+x4/12)+1.40*(x3/4+x4/12)=
Total cost=0.50x1+0.40x2+0.35x3+(5/12)x4
In addition, the manager wants grapefruit juice to be used for no more
than 30 percent of the number of containers produced.
Containers containing grapefruit juice: x2
Total number of containers produced: x1+x2+x3+x4
x2<=(3/10)(x1+x2+x3+x4) which can be rewritten as
10x2<=3x1+3x2+3x3+3x4  3x1-7x2+3x3+3x4>=0
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.
x1/x3>=7/5 5x1-7x3>=0
Profit= Revenue – Total cost = (from the two red equalities)
(1.00x1+0.90x2+0.80x3+1.10x4)-(.50x1+0.40x2+0.35x3+(5/12)x4
Profit=0.5x1+0.5x2+0.45x3+0.68x4 - objective function
Z= 0.50x1+0.50x2+0.45x3+0.68x4
Every quart of All-in-One makes a profit of 68 cents; more
than any other juice type. So, to maximize profit you want
to produce as many containers of All-in-One as possible.
You have only 996 quarts of pineapple juice so you cannot
produce more than 3*996= 2988 quarts of All-In One.
So x4=2988. In the process you used all the pineapple juice
so you cannot produce any containers of pineapple juice
anymore. Hence x3=0. You are left with 1756-996=760
quarts of orange juice and 1204 – 996= 208 quarts of
grapefruit juice.
That is, x1= 760 and x2=208.
The maximum profit is then
Z= 0.5*760+0.5*208+0.45*0+0.68*2988 = approx 2516
Notice that the inequalities 3x1-7x2+3x3+3x4>=0 and
5x1-7x3>=0 are satisfied by the values of x1,x2 x3,x4 above.
Therefore the optimal solution is:
z = 2516; x1 = 760, x2 = 208, x3 = 0, x4 = 2988