To solve the given linear programming problem using the simplex method, we will summarize the steps and present the solution in a clear format.
The problem is:
[tex]\[
\begin{array}{lc}
\text { Maximize } & z=700 x_1+400 x_2+100 x_3 \\
\text { subject to } & x_1+x_2+x_3 \leq 110 \\
& 2 x_1+3 x_2+4 x_3 \leq 320 \\
& 2 x_1+x_2+x_3 \leq 200 \\
& x_1 \geq 0, x_2 \geq 0, x_3 \geq 0
\end{array}
\][/tex]
### Step-by-step solution:
1. Standard Form: Convert inequality constraints into equalities by introducing slack variables [tex]\( s_1, s_2, s_3 \)[/tex]:
[tex]\[
\begin{array}{lc}
\text{Maximize} & z = 700 x_1 + 400 x_2 + 100 x_3 \\
\text{subject to} & x_1 + x_2 + x_3 + s_1 = 110 \\
& 2 x_1 + 3 x_2 + 4 x_3 + s_2 = 320 \\
& 2 x_1 + x_2 + x_3 + s_3 = 200 \\
& x_1, x_2, x_3, s_1, s_2, s_3 \geq 0
\end{array}
\][/tex]
2. Initial Basic Feasible Solution: Assume slack variables are the initial basic variables, and decision variables [tex]\( x_1, x_2, x_3 \)[/tex] are non-basic:
[tex]\[
x_1 = 0, x_2 = 0, x_3 = 0, \, s_1 = 110, s_2 = 320, s_3 = 200
\][/tex]
3. Optimal Solution: By following the simplex algorithm iteratively to find the optimal values, we will get:
- Decision variables: [tex]\( x_1, x_2, x_3 \)[/tex]
- Optimal value of [tex]\( z \)[/tex]
- Slack variables: [tex]\( s_1, s_2, s_3 \)[/tex]
### Final Solution:
From the calculations, let's summarize the optimal solution as follows:
- The maximum value is [tex]\( 71,000 \)[/tex]
- [tex]\( x_1 = 90 \)[/tex]
- [tex]\( x_2 = 20 \)[/tex]
- [tex]\( x_3 = 0 \)[/tex]
- Slack variables:
- [tex]\( s_1 = 0 \)[/tex]
- [tex]\( s_2 = 80 \)[/tex]
- [tex]\( s_3 = 0 \)[/tex]
### Answer
A. Treating [tex]\( x_3 \)[/tex] as a nonbasic variable, the maximum is [tex]\( 71,000 \)[/tex] when:
- [tex]\( x_1 = 90 \)[/tex]
- [tex]\( x_2 = 20 \)[/tex]
- [tex]\( x_3 = 0 \)[/tex]
- [tex]\( s_1 = 0 \)[/tex]
- [tex]\( s_2 = 80 \)[/tex]
- [tex]\( s_3 = 0 \)[/tex]
