Ash owns a window company with three employees (Annie, Jake, and Bobby Joe) that make two different kinds of hand-crafted windows: a wood-framed window and an aluminum-framed window. The company makes a profit of $180 for each wood-framed window and $90 for each aluminum-framed window.

- Annie makes the wood frames and can make 6 per day.
- Jake makes the aluminum frames and can make 4 per day.
- Bobby Joe 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. Ash wants to decide how many of each window to produce per day in order to maximize profit.

a) Formulate the dual linear program.

b) Identify the optimal basic feasible solution for the dual.

c) What is the interpretation of the optimal solution to the dual?

The dual linear program is formed by considering the resources used in the production process as the decision variables, and the profits from each type of window as the objective function coefficients. The optimal solution for the dual is to produce 6 wood-framed windows and 4 aluminum-framed windows per day. The interpretation of the optimal solution is that the company should maximize profit by producing this combination.

Explanation:

The dual linear program is formed by considering the resources used in the production process as the decision variables, and the profits from each type of window as the objective function coefficients. Let x1 be the number of wood-framed windows produced, x2 be the number of aluminum-framed windows produced, and z be the profit. The objective function of the dual linear program is max(180x1+90x2). The constraints of the dual linear program are:

x1 ≤ 6 (Annie's production limit)
x2 ≤ 4 (Jake's production limit)
6x1 + 8x2 ≤ 48 (Bobby Joe's production limit)
x1, x2 ≥ 0 (non-negativity constraint)

The optimal basic feasible solution for the dual is x1 = 6, x2 = 4, and z = 180*6 + 90*4 = $1,440. This means that the maximum profit can be achieved by producing 6 wood-framed windows and 4 aluminum-framed windows per day.

The interpretation of the optimal solution to the dual is that the company should produce 6 wood-framed windows and 4 aluminum-framed windows per day to maximize profit. The dual linear program provides information on the value of additional resources beyond what is currently being used.