Standard Form With Equality Greater Less Constraints
Standard Form With Equality, Greater, and Less Constraints
Exam problems often mix all three constraint types in the same LP.
Worked example
Original:
\[\max x_1-5x_2+4x_3\]subject to
\[3x_1-2x_2+x_3=10,\] \[x_2+2x_3\le5,\] \[3x_1+5x_2+x_3\ge10,\] \[x_1,x_2,x_3\ge0.\]Step 1 — objective
Convert max to min:
\[\min -x_1+5x_2-4x_3.\]Step 2 — constraints
Equality stays:
\[3x_1-2x_2+x_3=10.\]Less-than row gets slack:
\[x_2+2x_3+s_1=5.\]Greater-than row gets surplus:
\[3x_1+5x_2+x_3-s_2=10.\]Final standard form
\[\min -x_1+5x_2-4x_3\]subject to
\[\begin{aligned} 3x_1-2x_2+x_3 &=10,\\ x_2+2x_3+s_1 &=5,\\ 3x_1+5x_2+x_3-s_2 &=10, \end{aligned}\] \[x_1,x_2,x_3,s_1,s_2\ge0.\]Exam checkpoint
For standard-form questions, use the course convention $\min c^Tx$ subject to $Ax=b$, $x\ge0$. Convert max objectives, add slack to $\le$, subtract surplus from $\ge$, and split free variables.