# Max $Z=4x_1+5x_2$ Using two phase simplex

max $$z=4x_1+5x_2$$ s.t $$3x_1+x_2\leq27$$ $$x_1+x_2=12$$ $$3x_1+2x_2\geq 30$$ $$x_1,x_2\geq 0$$

To start the process we find an initial solution, so we add a slack variables:

$$3x_1+x_2+x_3 = 27$$ $$x_1+x_2+x_4=12$$ $$3x_1+2x_2-x_5= 30$$ $$x_1,x_2, x_3,x_4,x_5\geq 0$$

or should it be

$$3x_1+x_2+x_3 = 27$$ $$x_1+x_2+x_4=12$$ $$3x_1+2x_2-x_5+x_6= 30$$ $$x_1,x_2, x_3,x_4,x_5,x_6\geq 0$$

To set up an LP in augmented form, to prepare it for either the two-phase method or the big-$$M$$, method:
• Each $$\le$$ constraint gets a slack variable
• Each $$=$$ constraint gets an artificial variable
• Each $$\ge$$ constraint gets both a surplus variable and an artificial variable
• $$x_3$$ is a slack variable
• $$x_4$$ is an artificial variable
• $$x_5$$ is a surplus variable and $$x_6$$ is an artificial variable.