# Linear programming question: Sensitivity Analysis

I was given a problem where a home computer table sells for $$36$$ dollars and uses $$6$$ board ft of lumber, $$2$$ finishing hrs, and $$2$$ carpentry hrs. Should the company manufacture any home computer tables?

Given LP:

$$\max$$ $$z$$ = $$60x_1$$ + $$30x_2$$ + $$20x_3$$

$$st.$$

$$8x_1$$ + $$6x_2$$ + $$1x_3$$ $$\leq 48$$ (Lumber)

$$4x_1$$ + $$2x_2$$ + $$1.5x_3$$ $$\leq 20$$ (Finishing)

$$2x_1$$ + $$1.5x_2$$ + $$0.5x_3$$ $$\leq 8$$ (Carpentry)

$$x_1$$ , $$x_2$$, $$x_3$$ $$\geq 0$$

The only solution the textbook gave was that it is not possible to build home computer table. However, I didn't understand why it is. I did the tableau to show some of my findings of the final result

Problem given from textbook

$$\begin{array}{rrrrrrrr|r|r} row & z & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & BV \\ \hline 0 & 1 & -60 & -30 & -20 & 0 & 0 & 0 & z= 0 \\ \hline 1 & 0 & 8 & 6 & 1 & 1 & 0 & 0 & s_1 = 48 \\ 2 & 0 & 4 & 2 & 1.5 & 0 & 1 & 0 & s_2 = 20 \\ 3 & 0 & 2 & 1.5 & 0.5 & 0 & 0 & 1 & s_3 = 8 \end{array}$$

$$\begin{array}{rrrrrrrr|r|r} row & z & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & BV \\ \hline 0 & 1 & 0 & 15 & -5 & 0 & 0 & 30 & z= 240 \\ \hline 1 & 0 & 0 & 0 & -1 & 1 & 0 & -4 & s_1 = 16 \\ 2 & 0 & 0 & -1 & 0.5 & 0 & 1 & 2 & s_2 = 4 \\ 3 & 0 & 1 &-0.75 & 0.25 & 0 & 0 & 1/2 & s_3 = 4 \end{array}$$

$$\begin{array}{rrrrrrrr|r|r} row & z & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & BV \\ \hline 0 & 1 & 0 & 5 & 0 & 0 & 10 & 10 & z= 280 \\ \hline 1 & 0 & 0 & -2 & 0 & 1 & 2 & -8 & s_1 = 16 \\ 2 & 0 & 0 & -2 & 1 & 0 & 2 & -4 & s_2 = 4 \\ 3 & 0 & 1 & 1.25 & 0 & 0 & -0.5 & 1.5 & s_3 = 4 \end{array}$$

$$x_1$$ $$= 2$$ , $$x_3$$ $$= 8$$ $$and$$ $$Max$$ $$z$$ $$= 280$$