The demand for a perishable item over the next four months is $400,300,420,$ and $380$ tons, respectively. The supply capacities for the same months are $500,600,200,$ and $300$ tons. The purchase price per ton varies from month to month and is estimated at $$100,$ $$140,$ $$120,$ and $$150,$ respectively. Because the item is perishable, a current month$$s supply must be consumed within $3$ months $($starting with current month$).$ The storage cost per ton per month is $3$$$.$ The nature of the item does not allow back$-$ordering. Formulate the problem as a transportation model and find an initial solution by Northwest$-$corner method.The demand for a special small engine over the next five quarters is $200,150,300,250,$ and

$400$ units, respectively. The manufacturer supplying the engine has different production

capacities estimated at $180,230,430,300,$ and $300$ for the five quarters. Back$-$ordering is not

allowed, but the manufacturer may use overtime to fill the immediate demand, if necessary.

The overtime capacity for each period is half the regular capacity. The production costs

per unit for the five periods are $$100,$ $$96,$ $$116,$ $$102,$ and $$106,$ respectively. The overtime

production cost per engine is $50\%$ higher than the regular production cost. If an engine is

produced now for use in later periods, an additional storage cost of $$4$ per engine per period

is incurred. Formulate the problem as a transportation model. Determine the optimum

number of engines to be produced during regular time and overtime of each period.