Plans need to be made for the energy systems for a new building. The three possible sources of energy are electricity, natural gas, and a solar heating unit. Energy needs in the building are for electricity, water heating, and space heating, where the daily requirements $($all measured in the same units$)$ are

The size of the roof limits the solar heater to $30$ units, but thereis no limit to the electricity and natural gas available. Electricity needs can be met by purchasing electricity $($at a cost of $\backslash (\backslash$$ $200\backslash )$ per unit$).$ Both other energy needs can be met by any source or combination of sources. The unit costs are

$($a$)$ Formulate this problem as a transportation problem by constructing the appropriate cost and requirement table.

$($b$)$ Use the northwest corner rule to obtain an initial basic feasible solution for the problem as formulated in $($a$).$

$($c$)$ Starting with the initial basic feasible solution from $($b$),$ use the transportation simplex method to obtain an optimal solution.

$($d$)$ Use Vogel's approximation method to obtain an initial basic feasible solution for the problem as formulated in $($a$).$

$($e$)$ Starting with the initial basic feasible solution from $($d$),$ use the transportation simplex method to obtain an optimal solution. Compare the number of iterations required by the transportation simplex method here and in $($c$).$

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