Minimize the objective function P = 2x + 3y subject to the following constraints:\[ \begin{align*} x + y &\geq 3 \\ 2x + y &\leq 6 \\ x, y &\geq 0 \end{align*} \] using the Simplex Method.
Answer
Since all entries in the bottom row are now nonnegative, we have found the optimal solution. The values of the variables are given by the rightmost column: \(x = 3\), \(y = 0\), \(s_1 = 0\), and \(s_2 = 3\). Substituting these into the objective function gives \(P = 2(3) + 3(0) = 6\).
Step 1 ：First, convert the inequalities into equalities by introducing slack variables. For the first inequality, subtract a slack variable \(s_1\), and for the second, add a slack variable \(s_2\). This gives us:\[ \begin{align*} x + y - s_1 &= 3 \\ 2x + y + s_2 &= 6 \\ x, y, s_1, s_2 &\geq 0 \end{align*} \] The objective function is now \(P = 2x + 3y + 0s_1 + 0s_2\).
Step 2 ：Next, set up the initial simplex tableau:\[ \begin{array}{cccc|c} 1 & 1 & -1 & 0 & 3 \\ 2 & 1 & 0 & 1 & 6 \\ \hline -2 & -3 & 0 & 0 & 0 \end{array} \] The bottom row represents the negative of the objective function.
Step 3 ：The pivot column is the one with the most negative entry in the bottom row, which is the second column. The pivot row is the one where the ratio of the rightmost column to the pivot column is smallest, which is the first row. Therefore, the pivot element is the 1 in the first row and second column.
Step 4 ：Perform the pivot operation to make the pivot element 1 and all other elements in the pivot column 0. This gives us the new simplex tableau:\[ \begin{array}{cccc|c} 1 & 1 & -1 & 0 & 3 \\ 0 & 1 & 2 & 1 & 3 \\ \hline 0 & -1 & 2 & 0 & 6 \end{array} \]
Step 5 ：Since all entries in the bottom row are now nonnegative, we have found the optimal solution. The values of the variables are given by the rightmost column: \(x = 3\), \(y = 0\), \(s_1 = 0\), and \(s_2 = 3\). Substituting these into the objective function gives \(P = 2(3) + 3(0) = 6\).
