# How To Solve A Minimization Problem In Linear Programming When Objective Function Are All Non Negatives?

This is the Optimization Problem:

Minimize $C = x_4 + x_5$

Subject to constraints:

$x_1 + x_2 + x_4 = 2$

$2x_1 - x_3 + x_5 = 1$

I converted it to maximization: so my objective function becomes

Maximize $P = -x_4 - x_5$

=> $x_4 + x_5 + P = 0$

And my tableau becomes:

1 1 0 4 0 | 2 ~ constraint

2 0 -1 0 1 | 1 ~ constraint

0 0 0 1 1 | 0 ~ objective function

Basically, I'm not quite sure how to choose my pivot here, or if I set this up even correctly

I suppose that all variables are positive. Otherwise, the question wouldn't make sense.

Observe that $(x_4,x_5) = (2,1)$ is a basic feasible solution.

Note that the $x_4,x_4$-entry should be $1$ instead of $4$ as the coefficient of $x_4$ in the first constraint is one.
Before starting the simplex method, in the tableau, you have to make the columns representing the current basis ($x_4,x_5$-columns in this case) elementary, subtract the $x_4,x_5$-rows from the $z$-row.
The basic solution $(x_1,x_2) = (1/2,3/2)$ gives the minimum value $0$.