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I have an exam coming up so I have been going over math questions in my textbook to practice the simplex method. I ran into an issue with questions like this one, and also any that have more constraints then variables. we haven't gone over the DUAL problem (if that would even help in these cases) so I would like some guidance in solving this problem, thanks.

The Problem

$$ Maximize\: P = 2x_1 - 3x_2 + 4x_3 $$
subject to
$$ x_1 + x_3 ≤ 4 $$ $$ x_2 + x_3 ≤ 3 $$ $$ x_1, x_2, x_3 ≥ 0 $$

The Correct Solution

My solution below is not correct as the book tells me the results should be

$$Max\: P = 17\: at\: x_1 = 4,\: x2 = 3,\: and\: x3 = 0$$

My Solution

My attempted Solution was

$$Max\: P = 14\: at\: x1 = 1,\: x2 = 0,\: x3 = 3$$

How I Tackled the Problem

The steps I took to solve the problem, step one forming the tableau and choosing the pivot point, the pivot point I chose is encircled in ()

$$ \begin{bmatrix} 1 & 0 & 1 & 1 & 0 & 0 & 4 \\ 0 & 1 & (1) & 0 & 1 & 0 & 3 \\ -2 & 3 & -4 & 0 & 0 & 1 & 0 \end{bmatrix} $$

I reduced the rows as I did not need to reduce the pivot point using these functions

-R2 + R1 -> R1
4R2 + R3 -> R3

And then I selected a new pivot point

$$ \begin{bmatrix} (1) & -1 & 0 & 1 & -1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 3 \\ -2 & 7 & 0 & 0 & 4 & 1 & 12 \end{bmatrix} $$

Next I reduced the rows based on the newly selected pivot point using this calculation

2R1 + R3 -> R3

$$ \begin{bmatrix} 1 & -1 & 0 & 1 & -1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 3 \\ 0 & 5 & 0 & 2 & 2 & 1 & 14 \end{bmatrix} $$

Any help clarifying what could be causing my issues or where I am going wrong would be greatly appreciated, and sorry for the formatting, this was the best I could do given that the html on the site is limited.

Thanks!

Latency
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1 Answers1

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I don´t see any wrong calculation. But if the optimal value in the textbook is 17, then objective function should be

$\texttt{Maximize} \ P = 2x_1 \color{red}{+} 3x_2 + 4x_3$

Inserting the optimal value for $x_1, x_2$ and $x_3$:

$2\cdot 4+3\cdot 3+4 \cdot 0=17$

There is, almost sure, a typo in the objective function.

callculus42
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    @Latency Your are welcome. A good evidence, that your calculation is not wrong is this one: You maximize the "wrong" objective function. All coefficients of the constraints are positive. An in addition you have only $\leq$-constraints. Thus the optimal value for $x_2$ has to be zero. Otherwise you would decrease the objective function and spend some ressources. This wouldn´t be target-aimed. – callculus42 May 25 '15 at 17:34
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    Thanks, that makes sense, i also found your technique for checking the equation helpful in checking my answer with these problems, as an added bonus. – Latency May 25 '15 at 20:42