(Solved): Consider the following LP: maxs.t.z=3x1+4x22x1+5x24x1+3 ...

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Consider the following LP: $$\begin{array}{lrl}\max & z=3x_1+4x_2& \\ \text{ s.t. }& 2x_1+5x_2& ?20\\ & 4x_1+3x_2& ?24\\ & x_1+x_2& ?2\\ & x_1,x_2& ?0\end{array}$$ (a) (15) Solve the LP graphically. Also identify ALL corner points of the feasible region (i.e., give the $$x_1$$ and $$x_2$$ coordinates for each of them). Name the corner points by letters, e.g., $$\mathrm{A},\mathrm{B},\mathrm{C}$$, etc. (b) (30) Write the LP in standard form. How many basic solutions are there for this LP in standard form? Find the basic solutions corresponding to each corner point identified in part (a). You must clearly indicate the basis (i.e., the column indices that determine the basis; you could use the notation $$B(1),B(2),\dots )$$, and show that the basis is indeed a basis in each case (you could show that the basis matrix $$B$$ is invertible, or $$\det (B)?=0$$ ). Then you should find the solution to $$B{\mathbf{x}}_B=\mathbf{b}$$. You can use Matlab to solve each of these systems of linear equations. Finally, you must point out the values of $$x_1$$ and $$x_2$$ for each basic solution, showing the correspondence with the corner points identified in part (a). (c) (10) Now add the fourth constraint $$14x_1+7x_2?76$$. Is there a degenerate bfs now? If yes, identify the corner point on the graph of the feasible region. Then find the basic solution corresponding to this corner point. Demonstrate that there are more than $$n?m{x}_j$$ 's set at zero at this basic solution.