Given the following system : \begin{align*} \text{minimise } z = &2x_1 &+ 3x_2 &+ 3x_3 &+ x_4 &- 2x_5& \\ \end{align*} Subject to \begin{align*} & x_1 &+ 3x_2 & &+x_4 &+ x_5 &= 2 \\ & x_1 &+ 2x_2 & &- 3x_4 &+ x_5 &= 2 \\ - &x_1 &- 4x_2 & +x_3 & & &= 1 \\ \end{align*} with $x_1, x_2, x_3, x_4, x_5 \geq 0$ There should be Phase I and then Phase II of the simplex method. Q1 - how to explain why Phase I is required hereQ2 - how to know which rows should have artificial variables addedFor question 1, the objective function can be written as \begin{align*} -z + 2x_1 + 3x_2 + 3x_3 + x_4 - 2x_5 = 0\\ \end{align*} The way that the system is initially set up has basic variables $x_3$ and non-basic variables $x_1,x_2,x_4,x_5 = 0$. Meaning the objective function is \begin{align*} -z + 3x_3 = 0\\ \end{align*} Or \begin{align*} z = 3x_3 \end{align*} Why is this an issue? For Question 2 I'm not sure what to consider.
Date created: January 15, 2000 Date last modified: February 15, 2000 Authorised by: Moshe Sniedovich Maintained by: Moshe Sniedovich, Department of Mathematics and Statistics. Email: |