Mathematical linear programming?

Question

Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3. A desk contibutes $40 to profit, and a chair contibutes $25. Marketing restrictions require that the number of chairs produced be at least twice the number of desks produced. If 20 units of wood are available, formulate an LP to maximize Furnco's profit. Then graphically solve the L.P.

Answer 1

Let D be the number of desks produced.

Let C be the number of chairs produced.

Objective Function: Maximize Z=$40D+$25C subject to the following constraints.

4D + 3C <= 20  (Availability of constraint)

2D -  1C <= 0 (Marketing constraint)

C, D >=0 (Numbers produced must be positive)

C, D must be integer values

Graphing the feasible region yields a "triangular area" that has vertices at (0,0), (4,2) and (6,0) where the first coordinate is the number of chairs, C, and the second coordinate is the number of desks, D.

Linear programming theory states that the optimal solution must occur at one of these points.  Testing the objective function at each yields:

Z = $40(0) + $25(0) = $0

Z = $40(2) + $25(4) = $180

Z = $40(0) + $25(6) = $150

Therefore, produce 2 desks and 4 chairs at a profit of $180.