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The Whitt Window Company is a company with only three employees which makes two different kinds of hand-crafted windows: a wood-framed and an aluminum-framed window. They earn $60 profit for each wood-framed window and $30 profit for each aluminum-framed window. Doug makes the wood frames, and can make 6 per day. Linda makes the aluminum frames, and 4 per day. Bob forms and cuts the glass, and can 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.

1. Determine the decision variables.

2. Determine the constraints.

3. Formulate a linear programming model.

4. Define:
a. Optimum solution
b. Optimum value
c. Slack variable
d. Range of Optimality
e. Range of Feasibility
f. Shadow Price
g. Binding Constraints
h. Non-binding constraints
i. Feasible Region

5. Transform the linear programming model in (3) to standard form.

6. Using graphical method, analyze the phrases defined in (4).

Answer :

The decision variables are the number of wood-framed windows and the number of aluminum-framed windows. The constraints include Doug's production limit, Linda's production limit, and the available glass. A linear programming model is formulated to maximize profit, subject to the constraints.

1) Determine the decision variables.

The decision variables in this problem are the number of wood-framed windows (x) and the number of aluminum-framed windows (y) that the company should produce.

2) Determine the constraints.

The constraints are as follows:

Constraint 1: Doug can make 6 wood-framed windows per day.

So, the number of wood-framed windows (x) should be less than or equal to 6.

Constraint 2: Linda can make 4 aluminum-framed windows per day.

So, the number of aluminum-framed windows (y) should be less than or equal to 4.

Constraint 3: Bob can form and cut 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.

So, the total square feet of glass used by the wood-framed windows (6x) and the aluminum-framed windows (8y) should be less than or equal to 48.

3) Formulate a linear programming model.

The linear programming model for this problem is:

Maximize profit: 60x + 30y

Subject to constraints: x ≤ 6, y ≤ 4, 6x + 8y ≤ 48

Non-negative constraints: x ≥ 0, y ≥ 0

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