Pivot Row in Simplex Method

Question

If I am trying to solve a minimization problem without converting it to a maximization problem how do I decide which variable to pivot ?

I think it involves looking at the ratio of that variable with the RHS but unsure if I should choose the variable. Is the one with the smallest ratio or the largest.

Here is the problem that I have:

Minimize:

$x_1+x_2-4x_3$

Subject to:

$$ \begin{align} x_1+x_2+2x_3+x_4=9\\ x_1+x_2-x_3+x_5=2\\ -x_1+x_2+x_3+x_6=4\\ x_1,x_2,x_3,x_4,x_5,x_6 \geq 0\\ \end{align}$$

In this case I know the 4 needs to be pivoted to made negative but not sure if I need to use the $x_4,x_5,x_6$ as the first pivot.

Answer 1

Minimising $x_1+x_2-4x_3$ is equivalent to maximising it's additive inverse, so we can simply copy the coefficients in the original problem to the last row of the inital simplex tableau:

\begin{array}{r|rrrrrr|rr} & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & \text{RHS} & \text{ratio}\\ \hline x_4 & 1 & 1 & 2 & 1 & 0 & 0 & 9 & 9/2 \\ x_5 & 1 & 1 & -1 & 0 & 1 & 0 & 2 & - \\ x_6 & -1 & 1 & 1^* & 0 & 0 & 1 & 4 & 4\\ \hline & 1 & 1 & -4 & 0 & 0 & 0 & 0 \end{array}

Choose the most negative number at the $z$-row (-4 in this case.), just like what we do for a standard simplex maximisation problem.

Then pick the least nonnegative number at the "ratio" column. (You may consult my other post on choosing the leaving variable for further explanation.)

\begin{array}{r|rrrrrr|rr} & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & \text{RHS} & \text{ratio}\\ \hline x_4 & 3^* & -1 & 0 & 1 & 0 & -2 & 1 & 1/3\\ x_5 & 0 & 2 & 0 & 0 & 1 & 1 & 6 & - \\ x_3 & -1 & 1 & 1 & 0 & 0 & 1 & 4 & - \\ \hline & -3 & 5 & 0 & 0 & 0 & 4 & 16 \end{array}

\begin{array}{r|rrrrrr|r} & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & \text{RHS} \\ \hline x_1 & 1 & -1/3 & 0 & 1/3 & 0 & -2/3 & 1/3 \\ x_5 & 0 & 2 & 0 & 0 & 1 & 1 & 6 \\ x_3 & 0 & 2/3 & 1 & 1/3 & 0 & 1/3 & 13/3 \\ \hline & 0 & 4 & 0 & 1 & 0 & 2 & 17 \end{array}

Hence our optimal solution is $(x_1,x_2,x_3,x_4,x_5,x_6) = (1/3, 0, 13/3, 0 ,6, 0)$ with optimal value -17.