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Problem 1

# Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$\begin{array}{ccccc|c} x & y & u & v & P & \text { Constant } \\ \hline 0 & 1 & \frac{5}{7} & -\frac{1}{7} & 0 & \frac{20}{7} \\ 1 & 0 & -\frac{3}{7} & \frac{2}{7} & 0 & \frac{30}{7} \\ \hline 0 & 0 & \frac{13}{7} & \frac{3}{7} & 1 & \frac{220}{7} \end{array}$$

Expert verified
The given simplex tableau is in its final form. The solution to the associated regular linear programming problem is x = $$\frac{30}{7}$$, y = $$\frac{20}{7}$$, and the objective function value (maximum value) P = $$\frac{220}{7}$$.
See the step by step solution

## Step 1: Check if the tableau is in the final form.

First, let's evaluate the last row values (ignoring the Constant column) to see if they are non-negative: $$\frac{13}{7}, \frac{3}{7}, \text{ and } 1$$. Since all of these values are non-negative, the tableau is indeed in the final form.

## Step 2: Identify the solution to the given linear programming problem.

Since the tableau is in its final form, we can read the values directly. The first two columns represent the variables 'x' and 'y' respectively. Thus, we have x = $$\frac{30}{7}$$ and y = $$\frac{20}{7}$$.

## Step 3: Find the objective function value.

We are given the objective function value at the bottom right corner of the tableau, which is $$\frac{220}{7}$$. So, the associated regular linear programming problem has a solution x = $$\frac{30}{7}$$, y = $$\frac{20}{7}$$, and the objective function value (maximum value) P = $$\frac{220}{7}$$.

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