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To transform the given Linear Programming (LP) problem into its standard form, follow these steps:
### Step 1: Objective Function
Rewrite the objective function to a standard minimization form. The problem currently is:
[tex]\[ \text{Maximize } z = -2x_1 + 3x_2 - 2x_3 \][/tex]
We need to convert maximization into minimization. Maximizing [tex]\( z \)[/tex] is equivalent to minimizing [tex]\( -z \)[/tex]. Therefore, rewrite the objective function as:
[tex]\[ \text{Minimize } -z = 2x_1 - 3x_2 + 2x_3 \][/tex]
### Step 2: Constraints
The constraints are given as:
[tex]\[ 4x_1 - x_2 - 5x_3 = 10 \][/tex]
[tex]\[ 2x_1 + 3x_2 + 2x_3 = 12 \][/tex]
[tex]\[ x_1 \geq 0 \][/tex]
[tex]\[ x_2 \text{ unrestricted} \][/tex]
[tex]\[ x_3 \text{ unrestricted} \][/tex]
To convert the unrestricted variables [tex]\( x_2 \)[/tex] and [tex]\( x_3 \)[/tex] into non-negative variables, introduce new variables such that:
[tex]\[ x_2 = x_2' - x_2'' \quad \text{where } x_2', x_2'' \geq 0 \][/tex]
[tex]\[ x_3 = x_3' - x_3'' \quad \text{where } x_3', x_3'' \geq 0 \][/tex]
Substitute these new variables into the constraints.
### Step 3: Substitute and Rewrite Constraints
Substitute [tex]\( x_2 = x_2' - x_2'' \)[/tex] and [tex]\( x_3 = x_3' - x_3'' \)[/tex] into the constraints:
[tex]\[ 4x_1 - (x_2' - x_2'') - 5(x_3' - x_3'') = 10 \][/tex]
[tex]\[ 2x_1 + 3(x_2' - x_2'') + 2(x_3' - x_3'') = 12 \][/tex]
Simplify these equations:
[tex]\[ 4x_1 - x_2' + x_2'' - 5x_3' + 5x_3'' = 10 \][/tex]
[tex]\[ 2x_1 + 3x_2' - 3x_2'' + 2x_3' - 2x_3'' = 12 \][/tex]
### Step 4: Rewrite Objective Function
Rewriting the objective function in terms of [tex]\( x_2' \)[/tex], [tex]\( x_2'' \)[/tex], [tex]\( x_3' \)[/tex], and [tex]\( x_3'' \)[/tex]:
[tex]\[ -z = 2x_1 - 3x_2 + 2x_3 \][/tex]
[tex]\[ -z = 2x_1 - 3(x_2' - x_2'') + 2(x_3' - x_3'') \][/tex]
[tex]\[ -z = 2x_1 - 3x_2' + 3x_2'' + 2x_3' - 2x_3'' \][/tex]
So the objective function becomes:
[tex]\[ \text{Minimize } -z = 2x_1 - 3x_2' + 3x_2'' + 2x_3' - 2x_3'' \][/tex]
### Step 5: Standard Form
Now collect all parts together. The standard form of the LP problem is:
[tex]\[ \text{Minimize } -z = 2x_1 - 3x_2' + 3x_2'' + 2x_3' - 2x_3'' \][/tex]
subject to:
[tex]\[ 4x_1 - x_2' + x_2'' - 5x_3' + 5x_3'' = 10 \][/tex]
[tex]\[ 2x_1 + 3x_2' - 3x_2'' + 2x_3' - 2x_3'' = 12 \][/tex]
[tex]\[ x_1 \geq 0 \][/tex]
[tex]\[ x_2', x_2'', x_3', x_3'' \geq 0 \][/tex]
Thus, we have successfully converted the given LP problem into its standard form.
### Step 1: Objective Function
Rewrite the objective function to a standard minimization form. The problem currently is:
[tex]\[ \text{Maximize } z = -2x_1 + 3x_2 - 2x_3 \][/tex]
We need to convert maximization into minimization. Maximizing [tex]\( z \)[/tex] is equivalent to minimizing [tex]\( -z \)[/tex]. Therefore, rewrite the objective function as:
[tex]\[ \text{Minimize } -z = 2x_1 - 3x_2 + 2x_3 \][/tex]
### Step 2: Constraints
The constraints are given as:
[tex]\[ 4x_1 - x_2 - 5x_3 = 10 \][/tex]
[tex]\[ 2x_1 + 3x_2 + 2x_3 = 12 \][/tex]
[tex]\[ x_1 \geq 0 \][/tex]
[tex]\[ x_2 \text{ unrestricted} \][/tex]
[tex]\[ x_3 \text{ unrestricted} \][/tex]
To convert the unrestricted variables [tex]\( x_2 \)[/tex] and [tex]\( x_3 \)[/tex] into non-negative variables, introduce new variables such that:
[tex]\[ x_2 = x_2' - x_2'' \quad \text{where } x_2', x_2'' \geq 0 \][/tex]
[tex]\[ x_3 = x_3' - x_3'' \quad \text{where } x_3', x_3'' \geq 0 \][/tex]
Substitute these new variables into the constraints.
### Step 3: Substitute and Rewrite Constraints
Substitute [tex]\( x_2 = x_2' - x_2'' \)[/tex] and [tex]\( x_3 = x_3' - x_3'' \)[/tex] into the constraints:
[tex]\[ 4x_1 - (x_2' - x_2'') - 5(x_3' - x_3'') = 10 \][/tex]
[tex]\[ 2x_1 + 3(x_2' - x_2'') + 2(x_3' - x_3'') = 12 \][/tex]
Simplify these equations:
[tex]\[ 4x_1 - x_2' + x_2'' - 5x_3' + 5x_3'' = 10 \][/tex]
[tex]\[ 2x_1 + 3x_2' - 3x_2'' + 2x_3' - 2x_3'' = 12 \][/tex]
### Step 4: Rewrite Objective Function
Rewriting the objective function in terms of [tex]\( x_2' \)[/tex], [tex]\( x_2'' \)[/tex], [tex]\( x_3' \)[/tex], and [tex]\( x_3'' \)[/tex]:
[tex]\[ -z = 2x_1 - 3x_2 + 2x_3 \][/tex]
[tex]\[ -z = 2x_1 - 3(x_2' - x_2'') + 2(x_3' - x_3'') \][/tex]
[tex]\[ -z = 2x_1 - 3x_2' + 3x_2'' + 2x_3' - 2x_3'' \][/tex]
So the objective function becomes:
[tex]\[ \text{Minimize } -z = 2x_1 - 3x_2' + 3x_2'' + 2x_3' - 2x_3'' \][/tex]
### Step 5: Standard Form
Now collect all parts together. The standard form of the LP problem is:
[tex]\[ \text{Minimize } -z = 2x_1 - 3x_2' + 3x_2'' + 2x_3' - 2x_3'' \][/tex]
subject to:
[tex]\[ 4x_1 - x_2' + x_2'' - 5x_3' + 5x_3'' = 10 \][/tex]
[tex]\[ 2x_1 + 3x_2' - 3x_2'' + 2x_3' - 2x_3'' = 12 \][/tex]
[tex]\[ x_1 \geq 0 \][/tex]
[tex]\[ x_2', x_2'', x_3', x_3'' \geq 0 \][/tex]
Thus, we have successfully converted the given LP problem into its standard form.
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