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Part $1$

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$$ follows: January, $1$ comma $000$bags; February, $1$ comma $200$; $$March$,1$ comma $250$; $$April$,1$ comma $450$; $$May$,1$ comma $400$; and$$ June, $1$ comma $400.$

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 $1$ comma $100$ 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$$ City, Iowa, is also available as a backup source to meet demandlong dashbut can provide only $500$ bags total during the$6-$month period. Cost data are as$$ 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$