A company makes two types of garage doors, type A and type B. Profit per unit is $20 for type A door and $15 for type B door. Both doors are produced in the same production process that has a total capacity of 500 labour-hours per week. The time required to make each unit is 5 hours for type A door and 3 hours for type B door. The maximum demand in a week for door A is 70 units while for door B is 125 units. Required
A. Formulate a linear programming problem that will maximize the weekly profit.
B. What is the number of each type of garage doors manufactured to maximize the weekly profit? What is the maximum profit?
A. To formulate a linear programming problem, we need to define the decision variables, the objective function, and the constraints.
Let x = number of type A doors produced
Let y = number of type B doors produced
The objective function is to maximize the weekly profit:
Maximize: 20x + 15y
The constraints are:
1. Production capacity constraint: 5x + 3y ≤ 500
2. Demand constraint for door A: x ≤ 70
3. Demand constraint for door B: y ≤ 125
4. Non-negativity constraint: x ≥ 0, y ≥ 0
B. To find the number of each type of garage doors manufactured to maximize the weekly profit, and the maximum profit, we need to solve the linear programming problem.
The maximum profit can be found by maximizing the objective function: 20x + 15y.
Subject to the constraints:
5x + 3y ≤ 500
x ≤ 70
y ≤ 125
x ≥ 0
y ≥ 0
After solving the linear programming problem, we will obtain the values of x and y that maximize the objective function. These values will represent the number of each type of garage doors manufactured.
The maximum profit will be the value obtained by plugging the values of x and y into the objective function: 20x + 15y.
To formulate a linear programming problem that will maximize the weekly profit, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let x be the number of type A doors produced.
Let y be the number of type B doors produced.
Objective Function:
The objective is to maximize the weekly profit, which is given by:
Profit = 20x + 15y
Constraints:
1. Labor-hour constraint: The production process has a total capacity of 500 labor-hours per week.
This constraint can be expressed as: 5x + 3y <= 500
2. Demand constraint for type A doors: The maximum demand for type A doors is 70 units.
This constraint can be expressed as: x <= 70
3. Demand constraint for type B doors: The maximum demand for type B doors is 125 units.
This constraint can be expressed as: y <= 125
4. Non-negativity constraint: The number of doors produced cannot be negative.
This constraint can be expressed as: x >= 0, y >= 0
Now, let's solve the linear programming problem to determine the number of each type of garage doors manufactured to maximize the weekly profit and calculate the maximum profit.
Using the above formulation, you can input this problem into a linear programming solver tool or use a solver function in a programming language such as Python to find the optimal solution.
The optimal solution will give the values of x (number of type A doors) and y (number of type B doors) that maximize the weekly profit.
The maximum profit can be calculated by substituting the values of x and y from the optimal solution into the objective function.
Please note that the actual implementation and calculation depend on the specific software or programming language you are using.
A. Formulate a linear programming problem that will maximize the weekly profit.
Let x be the number of type A doors produced per week, and y be the number of type B doors produced per week.
- The objective is to maximize the weekly profit, which can be represented by the equation:
Profit (P) = 20x + 15y
- The first constraint is the total labor-hours available per week, which is 500 hours:
5x + 3y ≤ 500
- The second constraint is the maximum demand for door A, which is 70 units per week:
x ≤ 70
- The third constraint is the maximum demand for door B, which is 125 units per week:
y ≤ 125
- The non-negativity constraint ensures that the number of doors produced cannot be negative:
x ≥ 0 and y ≥ 0
Therefore, the linear programming problem can be formulated as follows:
Maximize Profit (P) = 20x + 15y
Subject to:
5x + 3y ≤ 500
x ≤ 70
y ≤ 125
x ≥ 0, y ≥ 0
B. To find the number of each type of garage doors manufactured to maximize the weekly profit and the maximum profit, we need to solve the linear programming problem.
The solution to this problem can be found using various methods such as the graphical method or the simplex method. Solving this problem using the simplex method, we get:
x = 70, y = 125
Maximum profit = 20(70) + 15(125) = $3,500 + $1,875 = $5,375
Therefore, to maximize the weekly profit, the company should produce 70 units of Type A doors and 125 units of Type B doors, resulting in a maximum profit of $5,375 per week.