Linear inequality analysis for Ozark Furniture production planning

Math Inequalities

Ozark furniture has 3000 board feet of lumber to manufacture their rockers. The classic rocker (C) takes 15 board feet of lumber. The modern rocker (M) takes 12 board feet of lumber. The linear inequality that results when one combines these two variables with the amount of lumber available to Ozark furnitures is: 15C + 12M

What the graph demonstrates is Ozark's manufacturing capabilities with its current amount of lumber. In order to understand these capabilities, one need only examine three different points on the line, all using the same x coordinate. The x coordinate is represented by the variable M. And demonstrates how many modern rockers will be created. The first point considered is on the line and is (125,100). This represents the maximum number of classic rockers that can be manufactured assuming that 100 modern rockers are manufactured. Plugging the numbers into the equation for the line, one finds that 15(100) + 12(125) = 3000, which is the maximum amount of available board-feet. However, the company does not have to produce to its maximum. If the company received an order for 125 modern rockers and 50 classic rockers, the point (125,30) would lie inside of the shaded area. The resulting equation would be 15(30) + 12(125) = 1950, which is less than or equal to the 3,000 board feet. However, it is critical to realize that the line serves to maximize the number of rockers that can be created in conjunction with another type of rocker. The point (125,200) falls outside of the shaded area created by the inequality. 15(200) + 12(125) = 4500. As 4,500 is not less than or equal to 3,000, it is clear that Ozark cannot manufacture this particular combination of rockers with its existing lumber supply.

When the chain furniture store faxes an order for 175 modern rocking chairs and 125 classic rocking chairs to Ozark, a quick glance at the graph demonstrates that Ozark cannot fulfill the order. The point (175,125) does not fall on the line or inside the shaded area created by the equation, but, instead, falls to the right of the line, indicating that the limits set by the amount of available board would be exceeded by the order. To verify this, one can test the equation for the total number of board feet used and see if the total number of board feet used would exceed the 3,000 board feet limit.