A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $3/unit of product A and a profit of $4/unit of product B. To manufacture 1 unit of product A requires 6 min on machine I and 5 min on machine II. To manufacture 1 unit of product B requires 9 min on machine I and 4 min on machine II. There are 10 hr of machine time available on machine I and 6 hr of machine time available on machine II in each work shift.
How many units of each product should be produced in each shift to maximize the company's profit?
product A _____ units
product B ______ units
What is the largest profit the company can realize?
$ _____________
Is there any time left unused on the machines?
machine I ________ min
machine II _______ min
To solve this problem, we can use a linear programming approach, specifically the method called the graphical method. We can represent the constraints and objective function graphically to find the optimal solution.
Step 1: Define the variables
Let x be the number of units of product A to be produced.
Let y be the number of units of product B to be produced.
Step 2: Write the constraints
The constraints in this problem are:
1. Machine I time constraint: 6x + 9y ≤ 10 hours.
2. Machine II time constraint: 5x + 4y ≤ 6 hours.
3. Non-negativity constraint: x, y ≥ 0.
Step 3: Create the objective function
The objective function we want to maximize is the profit:
Maximize Z = 3x + 4y.
Step 4: Graphical representation
Now, let's plot the feasible region graphically using these constraints.
To graph the constraint 6x + 9y ≤ 10, we can convert it into the equation:
6x + 9y = 10.
Solving for y in terms of x, we get: y = (10 - 6x) / 9.
Now, let's plot this equation on a graph.
To graph the constraint 5x + 4y ≤ 6, we convert it into the equation:
5x + 4y = 6.
Solving for y in terms of x, we get: y = (6 - 5x) / 4.
Now, let's plot this equation on the same graph.
Next, we need to identify the intersection points of the feasible region, which is the region of the graph where all the constraints are simultaneously satisfied.
Step 5: Search for the optimal solution
Evaluate the objective function at each intersection point. The combination of x and y that gives the maximum value of Z = 3x + 4y will be the optimal solution.
Step 6: Analyze the results
Once you have determined the values for x and y at the optimal solution, you can calculate the profit as well as check if any machine time is left unused on the machines.
x = __________ units (product A)
y = __________ units (product B)
Profit = $____________
Unused machine time:
Machine I = ________ minutes
Machine II = ________ minutes