Given the following maximum problem: Maximize: \( \quad P=8 x_{1}+4 x_{2} \) Subject to the following constraints: \[ \left\{\begin{array}{l} x_{1}+x_{2} \leq 2 \\ 2 x_{1}+5 x_{2} \leq 8 \\ x_{1} \geq 0 \\ x_{2} \geq 0 \end{array}\right. \] the Initial Simplex Tableau can be formatted in this form: \[ \left[\begin{array}{cccccc} P & x_{1} & x_{2} & s_{1} & s_{2} & R H S \\ a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ \hline & & & & & \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \end{array}\right] \] In each box below, enter the appropriate value that must replace each of the letters \( a_{11} \) through \( a_{36} \) in the above tableau.

To construct the Initial Simplex Tableau for the given linear programming problem, we need to first convert the constraints into equations by introducing slack variables. The original problem is: Maximize: \[ P = 8x_1 + 4x_2 \] Subject to: \[ \begin{align*} 1. & \quad x_1 + x_2 \leq 2 \\ 2. & \quad 2x_1 + 5x_2 \leq 8 \\ 3. & \quad x_1 \geq 0 \\ 4. & \quad x_2 \geq 0 \end{align*} \] ### Step 1: Introduce Slack Variables We introduce slack variables \( s_1 \) and \( s_2 \) to convert the inequalities into equalities: 1. \( x_1 + x_2 + s_1 = 2 \) (for the first constraint) 2. \( 2x_1 + 5x_2 + s_2 = 8 \) (for the second constraint) ### Step 2: Set Up the Tableau The tableau will have the following structure: - The first row will represent the objective function. - The subsequent rows will represent the constraints. The tableau will look like this: \[ \begin{array}{cccccc} P & x_1 & x_2 & s_1 & s_2 & R H S \\ 0 & -8 & -4 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 2 \\ 0 & 2 & 5 & 0 & 1 & 8 \\ \hline \end{array} \] ### Step 3: Fill in the Values Now we can fill in the values for \( a_{ij} \): - For the first row (objective function), we have: - \( a_{11} = 0 \) (since we are maximizing, we set \( P \) to 0) - \( a_{12} = -8 \) (coefficient of \( x_1 \)) - \( a_{13} = -4 \) (coefficient of \( x_2 \)) - \( a_{14} = 0 \) (coefficient of \( s_1 \)) - \( a_{15} = 0 \) (coefficient of \( s_2 \)) - \( a_{16} = 0 \) (right-hand side for the objective function) - For the second row (first constraint): - \( a_{21} = 0 \) (constant term) - \( a_{22} = 1 \) (coefficient of \( x_1 \)) - \( a_{23} = 1 \) (coefficient of \( x_2 \)) - \( a_{24} = 1 \) (coefficient of \( s_1 \)) - \( a_{25} = 0 \) (coefficient of \( s_2 \)) - \( a_{26} = 2 \) (right-hand side) - For the third row (second constraint): - \( a_{31} = 0 \) (constant term) - \( a_{32} = 2 \) (coefficient of \( x_1 \)) - \( a_{33} = 5 \) (coefficient of \( x_2 \)) - \( a_{34} = 0 \) (coefficient of \( s_1 \)) - \( a_{35} = 1 \) (coefficient of \( s_2 \)) - \( a_{36} = 8 \) (right-hand side) ### Final Tableau Thus, the values for \( a_{ij} \) are: \[ \begin{align*} a_{11} & = 0, & a_{12} & = -8, & a_{13} & = -4, & a_{14} & = 0, & a_{15} & = 0, & a_{16} & = 0 \\ a_{21} & = 0, & a_{22} & = 1, & a_{23} & = 1, & a_{24} & = 1, & a_{25} & = 0, & a_{26} & = 2 \\ a_{31} & = 0, & a_{32} & = 2, & a_{33} & = 5, & a_{34} & = 0, & a_{35} & = 1, & a_{36} & = 8 \end{align*} \] This completes the setup of the Initial Simplex Tableau.