A company sells syrup, the kind that should be put on pancakes and waffles. They
ID: 468355 • Letter: A
Question
A company sells syrup, the kind that should be put on pancakes and waffles. They syrup is made of maple syrup, corn syrup solution, or both. The cost of water is negligible and an unlimited amount is available. The contract minimum and demand is for number of bottles per month. The contract minimum should always be met, and more can be made, but what is made should not exceed the demand. How many bottles of each product should be made to maximize profit?A company sells syrup, the kind that should be put on pancakes and waffles. They syrup is made of maple syrup, corn syrup solution, or both. The cost of water is negligible and an unlimited amount is available. The contract minimum and demand is for number of bottles per month. The contract minimum should always be met, and more can be made, but what is made should not exceed the demand. How many bottles of each product should be made to maximize profit?
Syrup
Cost per liter
Maximum available per month (liters)
Maple
$12
100
Corn
$3
190
Product
Revenue / bottle
Contract minimum
Demand
Volume of bottle (liters)
Ratio
Pure Maple Syrup
$17.00
60
100
0.9
100% Maple
Corn Syrup Solution
$5.00
160
260
0.6
80% Corn/
20% Water
Maple / Corn Syrup mix
$10.00
50
130
0.7
40% Maple/
60% Corn
(a)Write the linear program, in standard form, that is needed to solve the problem.
(b)Put the linear program from (a) into Excel's solver to find the optimal solution.
Syrup
Cost per liter
Maximum available per month (liters)
Maple
$12
100
Corn
$3
190
Explanation / Answer
(a)
Decision Variables: x, y , z be the number of bottles for each kinds of syrup respectively.
Constraints:
x, y, z >= 0
(non negativity)
x+0.8y+0.4z <= 100
(maple availability)
0.2 y+0.6 z <= 190
(corn availability)
x <= 100
(demand constraints)
y < = 260
z <= 130
x > = 60
(contract minimum)
y >= 160
z > = 50
Objective function: Maximize profit
Cost = 0.9 ( 12*x ) + 0.6 ( 12* 0.8 y + 3* 0.2 y) + 0.7 ( 12* 0.4 z + 3 * 0.6 z)
Revenue = 17 x+ 5 y + 10 z
Profit = Revenue - Cost
(b) Solver Solution:
Maple
Corn
no of bottles
Pure Maple
1
0
100
Corn Syrup
0.8
0.2
160
Maple/Corn Mix
0.4
0.6
130
Constraints:
LHS
Sign
RHS
100
>=
0
(non negativity)
160
>=
0
130
>=
0
100
<=
100
(demand constraints)
160
<=
260
130
<=
130
100
>=
60
(contract minimum)
160
>=
160
130
>=
50
Cost =
$ 2,659.80
Revenue =
$ 3,800.00
Profit =
$ 1,140.20
Optimal Solution:
Product
No. of Bottles
Pure Maple Syrup
100
Corn Syrup Solution
160
Maple / Corn Syrup mix
130
Decision Variables: x, y , z be the number of bottles for each kinds of syrup respectively.
Constraints:
x, y, z >= 0
(non negativity)
x+0.8y+0.4z <= 100
(maple availability)
0.2 y+0.6 z <= 190
(corn availability)
x <= 100
(demand constraints)
y < = 260
z <= 130
x > = 60
(contract minimum)
y >= 160
z > = 50
Objective function: Maximize profit
Cost = 0.9 ( 12*x ) + 0.6 ( 12* 0.8 y + 3* 0.2 y) + 0.7 ( 12* 0.4 z + 3 * 0.6 z)
Revenue = 17 x+ 5 y + 10 z
Profit = Revenue - Cost
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