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Sagot :
To convert the given linear programming problem into the standard form suitable for the simplex method, we start by introducing slack variables. Slack variables are added to convert the inequalities into equalities.
Given the problem:
Maximize [tex]\( z = x_1 + 6x_2 \)[/tex]
Subject to:
[tex]\[ x_1 + 2x_2 \leq 10 \][/tex]
[tex]\[ 8x_1 + x_2 \leq 9 \][/tex]
[tex]\[ 2x_1 + 2x_2 \leq 9 \][/tex]
[tex]\[ x_1 \geq 0, x_2 \geq 0 \][/tex]
### Introducing Slack Variables
To convert the inequalities to equalities by adding slack variables [tex]\( s_1, s_2, \)[/tex] and [tex]\( s_3 \)[/tex]:
1. [tex]\( x_1 + 2x_2 + s_1 = 10 \)[/tex]
2. [tex]\( 8x_1 + x_2 + s_2 = 9 \)[/tex]
3. [tex]\( 2x_1 + 2x_2 + s_3 = 9 \)[/tex]
Here, [tex]\( s_1, s_2,\)[/tex] and [tex]\( s_3 \)[/tex] are the slack variables, and they are all non-negative: [tex]\( s_1, s_2, s_3 \geq 0 \)[/tex].
### Form the Initial Simplex Tableau
To set up the initial Simplex tableau, write down the coefficients of [tex]\( x_1, x_2, s_1, s_2, s_3 \)[/tex], and the Right Hand Side (RHS) values for each constraint and the objective function. The tableau will look as follows:
[tex]\[ \begin{array}{cccccc} \hline x_1 & x_2 & s_1 & s_2 & s_3 & \text{RHS} \\ \hline 1 & 2 & 1 & 0 & 0 & 10 \\ 8 & 1 & 0 & 1 & 0 & 9 \\ 2 & 2 & 0 & 0 & 1 & 9 \\ \hline -1 & -6 & 0 & 0 & 0 & 0 \\ \hline \end{array} \][/tex]
Here's the breakdown of the tableau:
- The first row represents the constraint [tex]\( x_1 + 2x_2 + s_1 = 10 \)[/tex] with coefficients of [tex]\( x_1, x_2, \)[/tex] and [tex]\( s_1 \)[/tex] and the RHS value 10.
- The second row represents the constraint [tex]\( 8x_1 + x_2 + s_2 = 9 \)[/tex] with coefficients of [tex]\( x_1, x_2, \)[/tex] and [tex]\( s_2 \)[/tex] and the RHS value 9.
- The third row represents the constraint [tex]\( 2x_1 + 2x_2 + s_3 = 9 \)[/tex] with coefficients of [tex]\( x_1, x_2, \)[/tex] and [tex]\( s_3 \)[/tex] and the RHS value 9.
- The last row represents the negation of the objective function [tex]\( z = x_1 + 6x_2 \)[/tex], hence the coefficients of [tex]\( -1 \)[/tex] for [tex]\( x_1 \)[/tex] and [tex]\( -6 \)[/tex] for [tex]\( x_2 \)[/tex], and 0 for all slack variables and the RHS.
This tableau is now ready to be used in the Simplex method to find the optimal solution.
Given the problem:
Maximize [tex]\( z = x_1 + 6x_2 \)[/tex]
Subject to:
[tex]\[ x_1 + 2x_2 \leq 10 \][/tex]
[tex]\[ 8x_1 + x_2 \leq 9 \][/tex]
[tex]\[ 2x_1 + 2x_2 \leq 9 \][/tex]
[tex]\[ x_1 \geq 0, x_2 \geq 0 \][/tex]
### Introducing Slack Variables
To convert the inequalities to equalities by adding slack variables [tex]\( s_1, s_2, \)[/tex] and [tex]\( s_3 \)[/tex]:
1. [tex]\( x_1 + 2x_2 + s_1 = 10 \)[/tex]
2. [tex]\( 8x_1 + x_2 + s_2 = 9 \)[/tex]
3. [tex]\( 2x_1 + 2x_2 + s_3 = 9 \)[/tex]
Here, [tex]\( s_1, s_2,\)[/tex] and [tex]\( s_3 \)[/tex] are the slack variables, and they are all non-negative: [tex]\( s_1, s_2, s_3 \geq 0 \)[/tex].
### Form the Initial Simplex Tableau
To set up the initial Simplex tableau, write down the coefficients of [tex]\( x_1, x_2, s_1, s_2, s_3 \)[/tex], and the Right Hand Side (RHS) values for each constraint and the objective function. The tableau will look as follows:
[tex]\[ \begin{array}{cccccc} \hline x_1 & x_2 & s_1 & s_2 & s_3 & \text{RHS} \\ \hline 1 & 2 & 1 & 0 & 0 & 10 \\ 8 & 1 & 0 & 1 & 0 & 9 \\ 2 & 2 & 0 & 0 & 1 & 9 \\ \hline -1 & -6 & 0 & 0 & 0 & 0 \\ \hline \end{array} \][/tex]
Here's the breakdown of the tableau:
- The first row represents the constraint [tex]\( x_1 + 2x_2 + s_1 = 10 \)[/tex] with coefficients of [tex]\( x_1, x_2, \)[/tex] and [tex]\( s_1 \)[/tex] and the RHS value 10.
- The second row represents the constraint [tex]\( 8x_1 + x_2 + s_2 = 9 \)[/tex] with coefficients of [tex]\( x_1, x_2, \)[/tex] and [tex]\( s_2 \)[/tex] and the RHS value 9.
- The third row represents the constraint [tex]\( 2x_1 + 2x_2 + s_3 = 9 \)[/tex] with coefficients of [tex]\( x_1, x_2, \)[/tex] and [tex]\( s_3 \)[/tex] and the RHS value 9.
- The last row represents the negation of the objective function [tex]\( z = x_1 + 6x_2 \)[/tex], hence the coefficients of [tex]\( -1 \)[/tex] for [tex]\( x_1 \)[/tex] and [tex]\( -6 \)[/tex] for [tex]\( x_2 \)[/tex], and 0 for all slack variables and the RHS.
This tableau is now ready to be used in the Simplex method to find the optimal solution.
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