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I have a linear program with the following tableaux:

\begin{array}{crrrrrr|l} & x_1 & x_2 & s_1 & s_2 & s_3 & P & rhs \\ \hline & 67& 126& 52& 36& -7& 0& 988\\ & -24& -46& -19& -14& 3& 0& -358 \\ & -9& -17& -7& -5& 1& 0& -133\\ \hline & -40 & -30 & 0 & 0 & 0 & 1 & 0 \\ \hline \end{array}

where I want to maximize $P$. I want to apply the simplex method starting from the initial solution $(x_1,x_2,s_1,s_2,s_3)=(8,0,8,1,0)$, but I'm confused about how to pivot when nearly all elements of the matrix are nonzero.

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You have to make $x_1$, $s_1$, and $s_2$ basic first. To achieve this you can execute a sequence of pivots such that the matrix corresponding to these columns is the identity matrix $I$. This is the same process used in Gaussian elimination to produce a reduced row echelon form. After that continue with Phase II of the Simplex method.

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  • $\begingroup$ That makes sense. Thank you! $\endgroup$ Commented May 23, 2019 at 18:52

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