Chapter 4, Problem 35RE

In solving a linear programming problem, you are given the following initial tableau.

$\begin{array}{l}\left[\begin{array}{rrrrrrr}\hfill 4& \hfill 2& \hfill 3& \hfill 1& \hfill 0& \hfill 0& \hfill 9\\ \hfill 5& \hfill 4& \hfill 1& \hfill 0& \hfill 1& \hfill 0& \hfill 10\\ \hfill -6& \hfill -7& \hfill -5& \hfill 0& \hfill 0& \hfill 1& \hfill 0\end{array}\right]\hfill \end{array}$

1. (a) What is the problem being solved?
2. (b) If the 1 in row 1, column 4 were a −1 rather than a 1, how would it change your answer to part (a)?
3. (c) After several steps of the simplex algorithm, the following tableau results.

$\begin{array}{l}\left[\begin{array}{rrrrrrr}\hfill 3& \hfill 0& \hfill 5& \hfill 2& \hfill -1& \hfill 0& \hfill 8\\ \hfill 11& \hfill 10& \hfill 0& \hfill -1& \hfill 3& \hfill 0& \hfill 21\\ \hfill 47& \hfill 0& \hfill 0& \hfill 13& \hfill 11& \hfill 10& \hfill 227\end{array}\right]\hfill \end{array}$

What is the solution? (List only the values of the original variables and the objective function. Do not include slack or surplus variables.)

1. (d) What is the dual of the problem you found in part (a)?
2. (e) What is the solution of the dual you found in part (d)? (Do not perform any steps of the simplex algorithm; just examine the tableau given in part (c).)