SOLVE QUESTION $3$ ONLY. Show a step by step solution not an explanation on how to do it$.$

$2.$ A Corporation has decided to produce three new products. Five branch plants now have excess production capacity. The unit manufacturing cost of the first product would be $\backslash (\backslash$$ $31\backslash ),\backslash (\backslash$$ $29,\backslash$$ $32,\backslash$$ $28\backslash ),$ and $\backslash (\backslash$$ $29\backslash )$ in Plants $1,2,3,4,$ and $5,$ respectively. The unit manufacturing

cost of the second product would be $\backslash (\backslash$$ $45,\backslash$$ $41,\backslash$$ $46,\backslash$$ $42\backslash ),$ and $\backslash (\backslash$$ $43\backslash )$ in Plants $\backslash (1,2,3,4\backslash ),$ and $5,$ respectively. The unit manufacturing cost of the third product would be $\backslash (\backslash$$ $38,\backslash$$ $35\backslash ),$ and $\backslash (\backslash$$ $40\backslash )$ in Plants $1,2,$ and $3,$ respectively, whereas Plants $4$ and $5$ do not have the capability for producing this product. Sales forecasts indicate that $600,1000,$ and $800$ units of products $1,2,$ and $3,$ respectively, should be produced per day. Plants $\backslash (1,2,3,4\backslash ),$ and $5$ have the capacity to produce $\backslash (400,600,400,600\backslash ),$ and $1000$ units daily, respectively, regardless of the product or combination of products involved. Assume that any plant having the capability and capacity to produce them can produce any combination of the products in any quantity.

$($a$)($points: $9)$ Formulate this problem as a balanced transportation problem by constructing the representation table.

$($b$)($points: $4)$ Find an initial basic feasible solution by the northwest corner rule. Show the intermediate steps.

$3.$ Reconsider the scenario in Problem $2.$ Suppose that the sales forecasts have been revised downward to $240,400,$ and $320$ units per day of products $1,2,$ and $3,$ respectively, and that each plant now has the capacity to produce all that is required of any one product$-$except plants $4$ and $5$ which still cannot produce the third product. Therefore, management has decided that each new product should be assigned to only one plant and that no plant should be assigned more than one product $($so that three plants are each to be assigned one product, and two plants are to be assigned none$).$ The objective is to make these assignments so as to minimize the total cost of producing these amounts of the three products.

$($a$)($points: $9)$ Formulate this problem as an assignment problem by constructing the appropriate cost table. It should be a balanced assignment problem.

$($b$)($points: $4)$ Obtain an optimal solution using the Hungarian algorithm. Show the intermediate steps.