Use the resulting simplex tableau to answer the following questions: (a) Is this the final tableau? (b) State the corresponding corner point for the tableau (x, y, z) =
Understand the Problem
The question requires analyzing the given simplex tableau to determine if it's the final tableau and, if so, to state the corresponding corner point (x, y, z) based on the tableau's values.
Answer
(a) No (b) $(0, 8, 0)$
Answer for screen readers
(a) No (b) $(x, y, z) = (0, 8, 0)$
Steps to Solve
- Check for negative values in the bottom row
To determine if the tableau is final, we examine the bottom row (the row corresponding to $P$). If there are any negative entries in the columns corresponding to the variables ($x, y, z, s_1, s_2, s_3$), then it is not the final tableau.
In our tableau, the bottom row is $[-6 \quad 0 \quad -14 \quad 0 \quad 9 \quad 0 \quad 1 \quad 0]$. We observe negative values $-6$ and $-14$ in the $x$ and $z$ columns, respectively.
- Determine if the tableau is final
Since the bottom row contains negative entries (specifically, $-6$ and $-14$), the tableau is not final.
- Identify basic variables and their values
The basic variables are those with a column containing a single 1 and the rest 0s. From the tableau, we can identify the basic variables as $s_1$, $y$, $s_3$, and $P$. Their values are on the right-hand side of the tableau. The non-basic variables are $x$ and $z$, which are 0.
- Find values of basic variables
$s_1 = 32$, $y = 8$, $s_3 = 16$. Since $x$ and $z$ are non-basic, $x = 0$ and $z = 0$.
- State the corresponding corner point
The corner point for the tableau is given by $(x, y, z)$. Substituting the values we found in steps 4, we get $(0, 8, 0)$.
(a) No (b) $(x, y, z) = (0, 8, 0)$
More Information
The given simplex tableau is not final because the bottom row contains negative entries in the x and z columns. A final tableau can be identified if the bottom row has no negative entries in the variable columns. The corresponding corner point represents a solution to the original system of equations at a specific vertex of the feasible region.
Tips
A common mistake is to assume the tableau is final if there are only a few negative numbers in the last row, or to ignore the negative numbers completely. Remember that all entries in the bottom row, excluding the $P$ column, must be non-negative for the tableau to be final. Another mistake is incorrectly reading the values from the tableau. Ensure you are looking at the right-hand side for the values of the basic variables.
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