Math Question..for guru math folks~?
The bookholder co. manufactures two types of bookcases out of oak and walnut. Model 01 requires 5 board feet of oak and 2 board feet of walnut. Model 02 requires 4 board feet of oak and 3 board feet of walnut. A profit of $75 is made on each model 01 Book Case and a profit of $125 is made on each Model 02 bookcase. The company has a supply of 1000 board feet of oak and 600 board feet of walnut. the company has orders for 40 model 01 bookcases and 50 model 02 bookcases.
Write the set of constraints.
Write the objective function.
Graph the set of constraints.
Determine the number of bookcases of each type the company should manufacture in order to maximize profits.
Determine the maximum profit.
How did you come this analysis.
Let the required quantities are x1 (model 01) and x2 (model 02) /sorry, I have no subscripts in the font I’m writing!/ then the set of constraints is given by:
5*x1 + 4*x2 ≤ 1000 /left-side: oak used/;
2*x1 + 3*x2 ≤ 600 /left-side: walnut used/;
x1 ≥ 40; x2 ≥ 50; x1, x2 – integers.
The objective function to maximize is:
Profit = 75*x1+ 125*x2.
This is a standard integer linear programming problem with 2 unknowns and the set of constraints determines a convex quadrilateral with the following vertexes (40, 50) /right angle here!/, (40, 520/3), (600/7, 1000/7) and (160, 50) – you can easily imagine how it looks or graph it, I’ll not do that in text mode (take the above inequalities, change them into equations, draw the lines with these equations, find the intersection points etc.). The gradient vector for the profit has coordinates (75, 125) and the maximum is reached in the point (40, 173) – the nearest point, having integer coordinates to the upper vertex (40, 520/3), so the final solution is:
x1 = 40, x2 = 173, maximum profit = 75*40 + 125*173 = 24625.
P.S. (Edit) You can try it Yourself if You have MathCad, MathLab or some sort of linear programming software installed on Your computer.
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