Supreme Cola is a supplier of fountain equipment to restaurants, bars and cafeterias. The fountain equipment is manufactured at their York PA plant site. A national distribution center $($DC$)$ for the fountain equipment is also maintained adjacent to the plant. Supreme has one common platform design to which they add various features and accessories to create $10$ different product options.

The company is interested in using the Order$-$up$-$to approach at their DC$.$ Depending upon the length of the review interval, the company is considering three alternatives:

$1.$ Alternative $1$: L$=4$; demand is normally distributed with mean $=100$ units and std$.$ dev. $=40$ units.

$2.$ Alternative $2$: L$=2$; demand is normally distributed with mean $=200$ units, and std dev. $=56$ units.

$3.$ Alternative $3$: L$=1$; demand is normally distributed with mean $=400$ units, and std$.$ dev. $=75$ units.

Answer the following questions.

i$.$ For each alternative, determine the optimal order$-$up$-$to level $($S$*)$ as to achieve: $$ an In$-$stock probability of $98\%.$ a Fill rate of $96\%.$

ii$.$ For an in$-$stock probability target of $97\%,$ determine the alternative which will minimize on$-$hand inventory.

iii. For a fill rate target of $98\%,$ determine the alternative which will minimize the expected on$-$hand inventory.