$9\mathrm{.}2-13,$ part $($a$)$ to $($e$).$ Make sure that each unit cost contains both a purchasing and a hauling component. For example, the total cost of supplying a ton of gravel from the north pit to building site $1$ is $$400.$ In question $($e$),$ the adjustment to add to the reduced cost of variable xij is D$($cij $-$ ui $-$ vj$)=$cij $,$ since dual variables ui and vj are constant $($basic solution remains unchanged$).$ After you add the adjustment to the corresponding reduced cost, apply the optimality condition to obtain an inequality for $$cij $.$ Solve the inequality to obtain the range of $$cij and cij so that the current solution remains optimal. Note that to invoke an improvement over the current optimal solution, the unit cost has to decrease below the allowable range. Add two more questions to this problem: $($f$)$ Show that the Vogel's approximation method finds the optimal solution obtained in $($d$)$; $($g$)$ Formulate the transportation problem using LINGO'S modeling language: Define sets, such as sources, destinations and routes $($origin$-$destination pairs$),$ and solve. Verify the optimal solution by comparing it with the one obtained in $($d$).$ Generate the ranges report and verify the answer obtained in $($e$).$