The Whitt Window Company is a company with only three employees that makes two different kinds of hand-crafted windows: a wood-framed and an aluminum-framed window. They earn a $60 profit for each wood-framed window and a $30 profit for each aluminum-framed window. Doug makes the wood frames and can make at most 6 per day. Linda makes the aluminum frames and can make at most 4 per day. Bob forms and cuts the glass and can make at most 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. The company wishes to determine the number of windows of each type to produce per day to maximize total profit.

(a) Formulate a linear programming model for this problem.

(b) Use the graphical method to solve the problem.

Question

05-01-2024
Business

College

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The Whitt Window Company is a company with only three employees that makes two different kinds of hand-crafted windows: a wood-framed and an aluminum-framed window. They earn a $60 profit for each wood-framed window and a $30 profit for each aluminum-framed window. Doug makes the wood frames and can make at most 6 per day. Linda makes the aluminum frames and can make at most 4 per day. Bob forms and cuts the glass and can make at most 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. The company wishes to determine the number of windows of each type to produce per day to maximize total profit.

(a) Formulate a linear programming model for this problem.

(b) Use the graphical method to solve the problem.

Answer :

Final answer:

This question is a linear programming problem, where mathematical model is created to solve for maximum profit that can be achieved per day by considering constraints on manufacturing wood-framed and aluminium-framed windows. The constraints are determined by capacity of employees. This problem is best solved using a graphical method approach.

Explanation:

Let's start solving the problem step by step.

Step 1: Define the Decision Variables. Let 'X' be the number of wood-framed windows and 'Y' be the number of aluminum-framed windows.

Step 2: Define the Objective Function. The company wants to maximize profit, which is obtained by selling wood and aluminum windows. So, the objective function is 60*X + 30*Y.

Step 3: Define the Constraints. Doug can make at most 6 wood windows a day (X ≤ 6), Linda can make at most 4 aluminum windows per day (Y ≤ 4). Bob can handle at most 48 square feet of glass a day which equates to the constraints 6*X + 8*Y ≤ 48.

(b) To solve this problem graphically you need to plot these inequalities on a graph, the feasible region is the region that satisfies all these conditions.

Finally, find the corner points of this region, and substitute these values in objective function to find max profit.

Learn more about Linear Programming here:

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