In the single$-$period inventory problem, the service level that it is optimal for a product to provide is based on the ratio of the

cost of under$-$supplying to the sum of the cost of under$-$ or over$-$supplying. This fraction represents the service level that it is

most adventageous to the seller to acheive, based on how they will be penalized for too much or too little.

a$.$ If we assume that the desired service level is $94\%,$ the per$-$item cost is $$22$ and the sale price is $$57,$ what would the

under$-$supply and over$-$supply penalties need to be in order to make sense with that service level? Can you please work out how to get the under$-$supply and over$-$supply penalties?

b$.$ If we are selling cakes, give all of the information $($both provided and calculated$)$ in $1-2$ sentences, making it make sense

in the context of the problem.