You are trying to help a high$-$end car dealership make decisions on inventory policy. The dealership

operates $5$ days a week, Monday through Friday, each Friday the dealership can place an order to the

manufacturer for more cars, which will arrive early Monday morning. Each car costs $$50\mathrm{.}000$ to buy from

the manufacturer and is sold for $$90\mathrm{.}000.$ Each time you place an order for more cars the dealership has to

pay a fixed cost of $$100\mathrm{.}000$ for transportation and processing fees, regardless of how many cars are ordered.

At the end of the week each unsold car generates an inventory holding cost of $$5\mathrm{.}000.$ The storage capacity

in the dealership makes it impossible to hold more than $20$ cars at any given time.

Let Dn be the $($random$)$ demand of week n$.$ Through some preliminary work you have determined that

$0<=$ Dn $<=20$ for all n and that demand every week is i$.$i$.$d$.$ Define qi $=$ P r$($Dn $=$ i$),$ that is qi is the

probability that on any week the demand is equal to i$.$

You have been hired to advise on the decision of the best inventory policy for the dealership. It has

long been known $($see The Optimality of $($S$,$s$)$ Policies in the Dynamic Inventory Problem by Herbert Scarf,

$1959),$ that under this scenario an $($S$,$ s$)$ inventory policy is optimal. This type of policy works as follows:

$1.$ at each revision $($Friday nights$)$ the inventory, call it x$,$ is observed, and if it is less than or equal to

some cutoff s then an order is placed for S $$ x units

$2.$ if x $>$ s then no order is placed that week

$3.$ recall that orders arrive before opening on Monday.

A preliminary study has also narrowed the potential $($S$,$ s$)$ policies to $5$ alternatives. These are