dane502

Two days ago I posted a similar post in the "Calculus & Beyond Forum", but I guess that this forum is more appropriate - any admin should correct me if I am wrong..

1. The problem statement, all variables and given/known data

I am trying to solve the follwing linear program


[tex]
\max \qquad 4x_1+x_2+3x_3
[/tex]
[tex]
\text{s.t }\qquad x_1+4x_2\qquad\,\leq1
[/tex]
[tex]
\text{ }\qquad \quad3x_1-x_2+x_3\leq3
[/tex]

3. The attempt at a solution
Using the simplex method and a tableau (negated objective function in the last row, right-hand side of constraints in the last column)
[tex]

\begin{matrix}
\textcircled{1}&4&0&1&0&1\\
3&-1&1&0&1&3\\\hline
-4&-2&-3&0&0&0
\end{matrix}
\rightarrow
\begin{matrix}
1&4&0&1&0&1\\
0&-13&\textcircled{1}&-3&1&0\\\hline
0&14&-3&4&0&4
\end{matrix}
\rightarrow
\begin{matrix}
1&\textcircled{4}&0&1&0&1\\
0&-13&1&-3&1&0\\\hline
0&-25&0&-5&3&4
\end{matrix}
\rightarrow
\begin{matrix}
1/4&1&0&1/4&0&1/4\\
13/4&0&1&1/4&1&13/4\\\hline
25/4&0&0&5/4&3&41/4
\end{matrix}

[/tex]

From which I conclude that the optimal objective value is 41/4
and the optimal solution is (0,1/4,13/4).

Inserting the optimal solution in the objective function does NOT yield 41/4.
It yields 10. I know from the textbook that the correct answer is 10, so my solution is correct. Can anyone explain then why my objective value in the tableau is not?