A large Saskatchewan feed mill, B$.$ Swart Processing, prepares its sixty$-$month aggregate plan by forecasting demand for $50-$pound bags of cattle feed as follows: January $1000$ bags; February $1200$; March $1250$; April $1450$; May $1400$; and June $1400.$ The feed mill plans to begin the new year with no inventory left over from the previous year, and backorders are not permitted. It projects that capacity $($during regular hours$)$ for producing bags of feed will remain constant at $800$ until the end of April, and then increase to $1100$ bags per month when a planned expansion is completed on May $1.$ Overtime capacity is set at $300$ bags per month until the expansion, at which time it will increase to $400$ bags per month. A friendly competitor in Alberta is also available as a backup source to meet demand$$but can provide only $500$ bags total during the six$-$month period.

Develop a six$-$month production plan for the feed mill using the transportation method.

Cost data are as follows:

Regular$-$time cost per bag $($until April $30)$ $$12\mathrm{.}00$

Regular$-$time cost per bag $($after May $1)$ $$11\mathrm{.}00$

Overtime cost per bag $($during entire period$)$ $$16\mathrm{.}00$

Cost of outside purchase per bag $$18\mathrm{.}50$

Carrying cost per bag per month $ $1\mathrm{.}00$