A firm manufactures tables and desks. To produce each table it requires 1 hour of labour, 10 square feet of wood, and 2 quarts of finish. To produce each desk it requires 3 hour of labour, 20 square feet of wood, and 1 quarts of finish. Available is at most 45 labour hours, at most 350 square feet of wood and at most 55 quarts of finish. The tables and desks yield profit of $4 and $3, respectively. Required
A. Formulate a linear programming problem that will maximize the weekly profit.
B. Find the number of each product to be made in order to maximize the profit.
Let:
x = number of tables to be made
y = number of desks to be made
A. The objective is to maximize the weekly profit, which is given by:
Profit = ($4/table) * x + ($3/desk) * y
Subject to the following constraints:
1. Labor constraint: 1 hour of labor is required to produce 1 table, and 3 hours are required to produce 1 desk. The total labor available is at most 45 hours. So, the constraint is:
1x + 3y ≤ 45
2. Wood constraint: 10 square feet of wood are required to produce 1 table, and 20 square feet are required to produce 1 desk. The total wood available is at most 350 square feet. So, the constraint is:
10x + 20y ≤ 350
3. Finish constraint: 2 quarts of finish are required to produce 1 table, and 1 quart is required to produce 1 desk. The total finish available is at most 55 quarts. So, the constraint is:
2x + y ≤ 55
4. Non-negativity constraint: The number of tables and desks cannot be negative. So, the constraints are:
x ≥ 0
y ≥ 0
B. To find the number of each product to be made in order to maximize the profit, we solve the linear programming problem using the given information and constraints. The optimal solution will give us the values of x and y that maximize the profit.
A. Formulating the linear programming problem:
Let x represent the number of tables to be produced in a week.
Let y represent the number of desks to be produced in a week.
The objective is to maximize the weekly profit, which can be expressed as:
Profit = (4 * x) + (3 * y)
Now, let's consider the constraints:
1. Labour hours constraint:
Tables consume 1 hour of labor each, and desks consume 3 hours of labor each. The available labor hours are at most 45. So, the constraint can be written as:
1 * x + 3 * y ≤ 45
2. Wood supply constraint:
Tables require 10 square feet of wood each, and desks require 20 square feet of wood each. The available wood supply is at most 350 square feet. So, the constraint can be written as:
10 * x + 20 * y ≤ 350
3. Finish supply constraint:
Tables require 2 quarts of finish each, and desks require 1 quart of finish each. The available finish supply is at most 55 quarts. So, the constraint can be written as:
2 * x + 1 * y ≤ 55
Additionally, we need to consider that the number of tables and desks must be non-negative. Hence, the non-negativity constraint can be written as:
x ≥ 0, y ≥ 0
B. Finding the number of each product to maximize the profit:
To find the optimal solution, the linear programming problem can be solved using various optimization techniques, such as the Simplex method or graphical method. Using these techniques, the number of tables (x) and desks (y) that maximize the profit can be determined.
A. To formulate a linear programming problem that will maximize the weekly profit, we need to define the decision variables, the objective function, and the constraints.
Decision Variables:
Let x be the number of tables to be produced
Let y be the number of desks to be produced
Objective Function:
We want to maximize the weekly profit, which is the sum of the profit from tables and desks:
Maximize: Z = 4x + 3y
Constraints:
1. Labour hours constraint: The total labor hours used for tables and desks should not exceed 45 hours.
1x + 3y ≤ 45
2. Wood constraint: The total square feet of wood used for tables and desks should not exceed 350 square feet.
10x + 20y ≤ 350
3. Finish constraint: The total quarts of finish used for tables and desks should not exceed 55 quarts.
2x + y ≤ 55
4. Non-negativity constraint: The number of tables and desks produced should be non-negative.
x ≥ 0, y ≥ 0
B. To find the number of each product to be made in order to maximize the profit, we need to solve this linear programming problem. Here is the step-by-step process:
1. Graph the feasible region represented by the constraints.
2. Identify the corner points of the feasible region.
3. Evaluate the objective function at each corner point.
4. Identify the corner point with the highest objective function value. This will be the solution for maximizing the profit.
5. Calculate the number of tables (x) and desks (y) corresponding to the identified corner point.
Note: The process above can be done manually or by using software tools such as Excel Solver or other linear programming solvers.