$1.(20)$ The following digraph depicts a partially solved minimum cost flow problem using the network simplex method with labels on the nodes indicating net demand and those on the arcs showing unit cost, capacity, and current flow.

$($a$)(4)$ Show the node$-$arc incidence matrix of the above network.

$($b$)(4)$ Suppose that node $1$ is taken as the root node such that we added an artifical variable $\backslash ($ x$\_\{$a$\}\backslash )$ to the flow conservation constraint for node $1.$ Assume that the current basis $($with artificial variable $\backslash ($ x$\_\{$a$\}\backslash ))$ is $\backslash (\backslash \{(1,2),(1,3),(2,4)\backslash \}\backslash ).$ Show that the current solution is a basic feasible solution.

$($c$)(4)$ Compute all simplex directions available at this basis.

$($d$)(4)$ Determine whether each of the simplex directions is improving.

$($e$)(4)$ Regardless of whether they are improving, determine the maximum feasible step $\backslash (\backslash$lambda $\backslash )$ that could be applied to each of the simplex directions.