A wood-products company produces four types of household furniture. Three basic operations are involved: cutting, sanding, and finishing. The plant’s capacity is such that there is a limit of 480 hours for finishing, 800 hours for sanding, and 900 hours for cutting. The firm’s objective function is to maximize profits, which it does by solving the following linear programming problem:

Max 90x ₁ + 160x ₂   + 40x ₃ + 100x ₄

subject to:

2x ₁ + 8x ₂ + 4x ₃ + 2x ₄ ≤ 480

5x ₁ + 4x ₂ + 8x ₃ + 5x ₄ ≤ 800

7x ₁ + 8x ₂ + 3x ₃ + 5x ₄ ≤ 900

x ₁ , x ₂ , x ₃ , x ₄ ≥ 0

Here, x _i is the number of units of product i that the firm produces. The final dictionary when this LP is solved by the simplex method is the following:

x ₂ = 25 − 1/3x ₃ − 5/32s ₁ + 1/16s ₂

x4 = 140 − x ₁ − 3/2x ₃ + 1/8s ₁ − 1/4s ₂  

s3 = 0 − 2x ₁ + 11/2x ₃ + 5/8s ₁ + 3/4s ₂  

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z = 18000 − 10x ₁ − 130x3 − 25/2s ₁ − 15s ₂

a) What are the shadow prices on each of the constraints.

(b) Write down an optimal solution to the dual problem.

(c) What profit for x3 would justify its production? (Find the range for c3 which is currently at 40.)

(d) What are the limits on sanding capacity that will allow the current basis to stay optimal?

(e) Suppose management has to decide whether or not to introduce a new product requiring 2 hours of finishing, 3 hours of sanding, and 20 hours of cutting, with an expected profit of $120. Should the product be produced?

(f) If another saw to perform cutting can be rented at $10 per hour, should it be procured? What about a finisher at the same price? If either is rented, what will be the gain from the first hour’s use?

(g) Suppose the firm cannot sell more than 160 units of all the products altogether. Will the optimal profit increase or decrease? Find the new optimal solution using the dual simplex method.