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Problem 396

# Find a basic feasible solution to the problem: Maximize $\quad \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3}+4 \mathrm{x}_{4}$ while satisfying the conditions \begin{aligned} &\mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4}=3 \\ &\mathrm{x}_{1}-\mathrm{x}_{2}+2 \mathrm{x}_{3}+\mathrm{x}_{4}=4 \\ &\mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}-\mathrm{x}_{4}=-1 \end{aligned} $$\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4} \geq 0$$

Expert verified
A basic feasible solution for this problem cannot be found using only the current tableau. Further analysis, such as using the dual simplex method, is necessary to find a feasible solution given the constraints. The current tableau gives the solution $$x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0, x_{5} = 3, x_{6} = 4, x_{7} = -1$$, but this is not feasible as $$x_{7} < 0$$.
See the step by step solution

## Step 1: Convert Constraints into Standard Form

To convert the given constraints into standard equality form, we need to turn inequalities into equalities by introducing slack variables ($$\mathrm{x}_{5}, \mathrm{x}_{6}, \mathrm{x}_{7}$$). The constraints become: \begin{aligned} \mathrm{x}_{1} + 2\mathrm{x}_{2} + \mathrm{x}_{3} + \mathrm{x}_{4} -\mathrm{x}_{5} = 3\\ \mathrm{x}_{1} - \mathrm{x}_{2} + 2\mathrm{x}_{3} + \mathrm{x}_{4} -\mathrm{x}_{6} = 4\\ \mathrm{x}_{1} + \mathrm{x}_{2} - \mathrm{x}_{3} - \mathrm{x}_{4} -\mathrm{x}_{7} = -1 \end{aligned} with $$\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}, \mathrm{x}_{5}, \mathrm{x}_{6}, \mathrm{x}_{7} \geq 0$$.

## Step 2: Set Up Tableau

We can now set up a tableau for the problem: $$\begin{array}{c|ccccccc|c} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & \\ \hline x_{5} & 1 & 2 & 1 & 1 & 1 & 0 & 0 & 3 \\ x_{6} & 1 &-1 & 2 & 1 & 0 & 1 & 0 & 4 \\ x_{7} & 1 & 1 & -1 & -1 & 0 & 0 & 1 & -1\\ \hline z & 1 & 2 & 3 & 4 & 0 & 0 & 0 & 0 \end{array}$$

## Step 3: Identify a Basic Feasible Solution

In the above tableau, columns with a single 1 and all other elements 0 represent basic variables. For this tableau, our basic variables are: $$x_{5}, x_{6}$$, and $$x_{7}$$. The basic feasible solution given by this tableau is: $$x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0, x_{5} = 3, x_{6} = 4, x_{7} = -1$$ However, since $$x_{7} = -1$$ and we require $$x_{7} \geq 0$$, this solution is not feasible for the given problem. At this point, we cannot find a basic feasible solution using only the current tableau. Further analysis, such as using the dual simplex method, is necessary to find a feasible solution given the constraints.

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