Problem $1.$ Daily random demand $D$ for newspapers at a newspaper stand has a Poisson distri$-$

bution with parameter $>0.$ That is$,$

$P(D=d)=\frac{e^{-}{}^d}{d!},$ for $d=0,1,2,$dots

A Newsvendor has observed that the daily demands for newspapers over the past $8$ days were

$10,0,14,8,9,13,17,9.$ The Newsvendor utilizes this data to develop an MLE for $,$ and then uses

this MLE to make order quantity decisions. Suppose the overage and underage costs are $h=15$

cents and $p=60$ cents. Decide the optimal order quantity $Q^{***}.$ Calculate an upper bound on the

probability that there will be a stock out, that is$,$ an upper bound on $P(D>Q^{***}).$

Problem $2.$ Weekly demand for a product is Binomial $(3,0\mathrm{.}6).$ Weekly overage and underage costs

are $h=3$$ and $p=6$$ per unit. The purchase cost is $c=1$$ per unit, and there is no fixed cost

of purchasing. The order$-$up$-$to level in each period must be in the set $\{0,1,2,3\}.$ The planning

horizon is $T=2$ weeks and the terminal cost at the end of this planning horizon is $u(x)=x^{+}+2x^{-}.$

We begin week $1$ with $x=2$ units of inventory. The weekly discount factor is $=0\mathrm{.}95.$ Draw a

portion of the probabilistic shortest path network for period $1$ starting with a node corresponding

to inventory $x=2.$ Apply DP to compute and report in a tabular format the optimal cost${}_t(x)$

and the optimal order$-$up$-$to levels $y_t(x)$ for each week $t$ and each inventory level $x.$

Problem $3.$ Consider an inventory system with an infinite planning horizon of days $t=1,2,$dots

and daily discount factor $=0\mathrm{.}9.$ Suppose the costs are as follows: $K=150$$ per order, $c=75$$

per unit, $h=40$$ per unit per day, and $p=125$$ per unit per day. Suppose the daily demand is

Poisson distributed with $=4.$ Use a spreadsheet to simulate an $(s,S)=(4,10)$ ordering policy

over $365$ days $($as an approximation$)$ for this inventory system, starting with inventory $x=0$ on

day $1.$ Generate $10$ independent copies of your simulation using different random number generator

seeds, and for each copy, calculate the total discounted cost over $365$ days. Average these costs

over your $10$ simulations. Report that average as the $($estimated$)$ cost of $(s,S)=(4,10)$ policy.