A company has hired you to develop an aggregate plan to determine the number of workers they will need for the next $9$ months $($February through October$).$ This company makes $8$ different products. The time $($in hours$)$ required to produce each product is as follows:

Product Number of hours to produce $1$ unit

Product A $1\mathrm{.}14$

Product B $2\mathrm{.}34$

Product C $2\mathrm{.}03$

Product D $2\mathrm{.}82$

Product E $1\mathrm{.}15$

Product F $2\mathrm{.}46$

Product G $2\mathrm{.}78$

Product H $0\mathrm{.}94$

Suppose the following demands for each product have been forecast for the next $9$ months:

Month # days per month Forecast Demand

Product A Product B Product C Product D Product E Product F Product G Product H

February $2898044252615363731332260182$

March $21167464498723875402511462311$

April $30109249066518944131832352227$

May $296152593781249213944178110$

June $2774231145011132411043180124$

July $2949819230292419866117281$

August $21152564795522735122153460256$

September $2141518923279116168411973$

October $24108144659415113611390281201$

When developing aggregate plans the company assumes workers regularly work $8$ hours each day; no overtime or subcontracting will be used. The firm would like to maintain a minimum buffer inventory $($i$.$e$.$ safety stock$)$ each month $($see table below$).$ The starting inventory level $($i$.$e$.$ inventory at end of January$),$ desired inventory at the end of the planning horizon $($in addition to monthly safety stock$),$ initial workforce level, hiring cost, firing cost, regular time cost, and inventory holding cost $($in $ per aggregate unit per month$)$ are shown in the table below. Based on the aggregate forecast demand and planning parameters $($in table below$),$ determine the number of workers this company will need over the next $9$ months using the following aggregate planning strategies.

Safety Stock $($in aggregate units$)700$

Beginning of planning horizon Inventory $($in aggregate units$)900$

End of planning horizon Inventory, additional to any SS $($in aggregate units$)0$

# workers employed at end of January $45$

Hiring Cost $($in $ per hired worker$)$ $$800$

Firing Cost $($in $ per fired worker$)$ $$600$

Regular Time Cost $($in $ per hour$)$ $$30\mathrm{.}00$

Inventory Holding Cost $($in $ per aggregate unit per month$)$ $$7\mathrm{.}25$

a$.$ Find the minimum constant workforce needed to meet demand assuming no shortages are allowed $($i$.$e$.$ no negative inventory in any month$).$ What is the cost associated with this constant workforce strategy? Note: use scheduled idle time to find the minimum cost solution.

b$.$ Find the number of workers needed each month using the zero inventory strategy $($no shortages allowed$).$ What is the cost associated with this zero inventory workforce strategy? Note: use scheduled idle time to find the minimum cost solution.

c$.$ Find the number of workers needed each month using a mixed strategy $($no shortages allowed$).$ What is the cost associated with this mixed strategy? Note: use scheduled idle time to find the minimum cost solution.