# Minimize using simplex method with mixed constraints

Using the simplex method, minimize $z= -2x_1+12x_2-3x_3-5x_4-x_5$ with $x_i \geq 0$ and subject to constraints:

• $x_1+4x_3-5x_4+2x_5 \leq 4$
• $4x_2-x_4 \leq 1$
• $5x_1-6x_2-3x_4 \leq 6$
• $6x_1-5x_2-x_3+2x_4 \geq 2$
• $3x_1+x_4-2x_5 \geq 5$

I want to double check I have the right idea about how to approach this. Here is what I have so far:

Reverse $z$ to get $z'=2x_1-12x_2+3x_3+5x_4+x_5$ which I will now seek to maximize.

Put constraints into standard form with slack variables:

• $x_1+4x_3-5x_4+2x_5 +s_1= 4$
• $4x_2-x_4 +s_2 = 1$
• $5x_1-6x_2-3x_4 +s_3 = 6$
• $6x_1-5x_2-x_3+2x_4 -s_4= 2$
• $3x_1+x_4-2x_5 -s_5 = 5$

Is that enough to begin with the tableau phase OR do I have to introduce reverse slack variables $R_1$ and $R_2$ such that:

• $x_1+4x_3-5x_4+2x_5 +s_1= 4$
• $4x_2-x_4 +s_2 = 1$
• $5x_1-6x_2-3x_4 +s_3 = 6$
• $6x_1-5x_2-x_3+2x_4 -s_4 + R_1 = 2$
• $3x_1+x_4-2x_5 -s_5 +R_2 = 5$

And then begin the two-phase simplex method and attempt to minimize $R= R_1 +R_2$?

I'm pretty stuck, any advice would be great!

• you have to use the two-phase method – Marcello Sammarra Nov 22 '17 at 18:37