91Ó°ÊÓ

First, write down the given constraints as linear equations with non-negative slack variables: $$ \begin{array}{ll} 3x + y + s_1 = 24 \\ 2x + y + s_2 = 18 \\ x + 3y + s_3 = 24 \\ \end{array} $$ where \(s_1,s_2,s_3 \geq 0\). Now add the objective function, initially setting P = 0 and the objective function coefficients as negative: $$ \begin{array}{ll} P - 4x - 6y = 0 \end{array} $$

Create the initial simplex tableau by placing the coefficients of the variables x, y, the slack variables, and the objective function in a matrix form: $$ \begin{array}{ccccccc|c} P & x & y & s_1 & s_2 & s_3 & & \\ \hline 1 &-4 &-6 & 0 & 0 & 0 & = & 0 \\ 0 & 3 & 1 & 1 & 0 & 0 & = & 24 \\ 0 & 2 & 1 & 0 & 1 & 0 & = & 18 \\ 0 & 1 & 3 & 0 & 0 & 1 & = & 24 \end{array} $$

Identify the pivot column by looking at the most negative coefficient in the objective function row (P row). The pivot column is 'y' (column 3), because the most negative value is -6. Next, find the pivot row by dividing each right-hand side value by its corresponding pivot column entry (must be positive) and select the lowest ratio. In this case, the pivot row is the third row because the lowest ratio is 18. Now we perform the pivot operation by making the pivot element (1) equal to 1 and the other elements in the pivot column to be zero. New tableau after pivot: $$ \begin{array}{ccccccc|c} P & x & y & s_1 & s_2 & s_3 & & \\ \hline 1 & 2 & 0 & 0 & 6 & 0 & = & 108 \\ 0 & 1 & 0 & 1/3 & -1/3 & 0 & = & 6 \\ 0 & 0 & 1 & -2/3 & 1/3 & 0 & = & 6 \\ 0 & 1 & 0 & 1/3 & -1 & 1 & = & 18 \end{array} $$ There are no more negative coefficients in the objective function row, so the solution is optimal.

In the final tableau, we can determine the optimal solution by finding the basic variables, which are indicated by identity matrix columns: Optimal solution: \(x = 6, y = 6\) Objective function value: \(P = 4(6) + 6(6) = 24 + 36 = 60\) Thus, the optimal solution for this linear programming problem is \(x = 6\), \(y = 6\), and the maximum value of \(P = 60\).