Manufacturing A factory manufactures two products, each requiring the use of three machines. The first machine can be used at most 70 hours; the second machine at most 40 hours; and the third machine at most 90 hours. The first product requires 2 hours on machine 1, 1 hour on machine 2 and 1 hour on machine 3 the second product requires 1 hour each on machines 1 and 2 and 2 hours on machine 3 If the profit is $35 per unit for the first product and $60 per unit for the second product, how many units of each product should be manufactured to maximize profit?
If the 3 products are made in quantities x,y,z, then
maximize p=35x+60y subject to
2x+y <= 70
x+y <= 40
x+2y <= 90
maximum profit at (0,40) = 2400
good tool at
http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html
set the x- and y- ranges at 0-50 and it will show a decent graph of the region.
Why did the factory hire a clown? Because they wanted a funny business!
To determine the number of units of each product that should be manufactured to maximize profit, let's consider the constraints and the objective function:
Constraints:
- Machine 1: max 70 hours
- Machine 2: max 40 hours
- Machine 3: max 90 hours
Objective function:
- Profit per unit for the first product: $35
- Profit per unit for the second product: $60
Now let's find the optimal solution using linear programming techniques.
To solve this problem, we can use linear programming to find the optimal solution.
Let's define our decision variables:
- Let x represent the number of units of the first product.
- Let y represent the number of units of the second product.
Now let's formulate our objective function and constraints.
Objective function:
Maximize Profit = 35x + 60y
Constraints:
Machine 1: 2x + y ≤ 70 (The total number of hours used on machine 1 can't exceed 70)
Machine 2: x + y ≤ 40 (The total number of hours used on machine 2 can't exceed 40)
Machine 3: x + 2y ≤ 90 (The total number of hours used on machine 3 can't exceed 90)
Non-negativity constraint:
x ≥ 0 and y ≥ 0 (You can't manufacture a negative number of units)
Now we can solve these equations using the Simplex method or graphically to find the optimal solution.
To solve this problem, we can use linear programming to find the optimal solution that maximizes the profit. Here are the steps:
Step 1: Define the Decision Variables
Let's assign the following decision variables:
x = number of units of the first product
y = number of units of the second product
Step 2: Write the Objective Function
The objective is to maximize the profit. The profit for the first product is $35 per unit, and for the second product is $60 per unit. Therefore, the objective function can be written as:
Profit = 35x + 60y
Step 3: Write the Constraints
The given information provides the limitations for each machine.
For the first machine:
2x + y ≤ 70 (constraint 1)
For the second machine:
x + y ≤ 40 (constraint 2)
For the third machine:
x + 2y ≤ 90 (constraint 3)
Non-negativity constraint:
x, y ≥ 0 (since we can't manufacture negative units)
Step 4: Solve the Linear Programming Problem
Using these objective and constraint functions, we can now solve the linear programming problem to find the optimal solution.
Alternatively, you can use software such as Microsoft Excel or any other linear programming solver to find the optimal solution.