$3\mathrm{.}1$ A large St$.$ Louis feed mill, Robert Orwig Processing, prepares its $6-$month aggregate plan by forecasting demand for $50-$pound bags of cattle feed as follows: January, $1,000$ bags; February, $1,200$; March, $1,250$; April, $1,450$; May, $1,400$; and June, $1,400.$ The feed mill plans to begin the new year with no inventory left over from the previous year, and backorders are not permit$-$ ted. 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 $1,100$ 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 Sioux City, Iowa, is also available as a backup source to meet demand$$but can provide only $500$ bags total during the $6-$month period. Develop a $6-$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 $ $1\mathrm{.}00$