Consider the following cost matrix for a transportation problem in which the objective is to minimize the total transportation cost.

Destination

Source $123$ Supply $185450$

$268920$

Demand $102040$

$($a$)$Write down the linear programming formulation for this problem.

$($b$)$Write down the dual problem of the linear program in part $($a$).$

$($c$)$Write down all of the complementary slackness conditions.

$($d$)$Set up the transportation tableau and use the northwest corner rule to find an initial basic feasible solution.

$($e$)$Solve the problem using the transportation algorithm. Give a primal optimal solution and a dual optimal solution.