High School

A manufacturing company has three employees who make two different kinds of hand-crafted windows: wood-framed and aluminum-framed windows. The company earns $60 profit for each wood-framed window and $30 profit for each aluminum-framed window.

- The first employee makes the wood frames and can produce a maximum of six wood-framed windows per day.
- The second employee makes the aluminum frames and can produce a maximum of four aluminum-framed windows per day.
- The third employee forms and cuts the glass for both window types, with a capacity to make 48 square feet of glass per day.

Each wood-framed window uses 6 square feet of glass, and each aluminum-framed window uses 8 square feet of glass. Market research indicates that the demand for aluminum-framed windows cannot exceed 40% of the total demand.

The company wishes to determine the number of windows of each type to produce per day to maximize total profit. Formulate a Linear Programming (LP) model for the problem.

Answer :

The linear programming (LP) model for the problem can be formulated as follows:

Let x be the number of wood-framed windows produced per day.

Let y be the number of aluminum-framed windows produced per day.

Objective function:

Maximize Profit = 60x + 30y

Constraints:

The first employee can make a maximum of 6 wood-framed windows per day:

x ≤ 6

The second employee can make a maximum of 4 aluminum-framed windows per day:

y ≤ 4

The third employee can produce a maximum of 48 square feet of glass per day:

6x + 8y ≤ 48

The demand for aluminum-framed windows cannot exceed 40% of the total demand:

y ≤ 0.4(x + y)

In this LP model, the objective is to maximize the total profit by determining the optimal number of wood-framed and aluminum-framed windows to produce. The constraints ensure that the production quantities do not exceed the maximum capacities of the employees and the available glass. Additionally, the fourth constraint reflects the market research indicating that the demand for aluminum-framed windows should not exceed 40% of the total demand.

The LP model can be solved using various optimization techniques to determine the values of x and y that maximize the profit. The solution will provide the optimal production quantities for each window type, allowing the company to make informed decisions and maximize its profitability while considering the available resources and market demand.

Learn more about linear programming here:

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