Problem $1.(35$ points$)$ Consider $an$ inventory system, where $at$ the beginning $of$ period $k,$

the inventory level $is{S}_k,$ and $we$ can order $x_k$ units $of$ goods. The available units $of$ goods are

then used $to$ serve a random demand $W_k,$ and the amount $of$ inventory carried over $to$ the

next period $is{S}_{k+1}=$max$\{0,S_k+x_k-W_k\}.We$ assume that $S_k,x_k,W_k$ are non$-$negative

integers, and that the random demand $W_k$ follows the probability distribution

$P(W_k=0)=0\mathrm{.}1,P(W_k=1)=0\mathrm{.}8-0\mathrm{.}2k,P(W_k=2)=0\mathrm{.}1+0\mathrm{.}2k,$ for all $k=1,2,3.$

$In$ each stage, the cost consists $of$ a ordering cost $x_k,$ a holding cost $of2$max$\{0,S_k+x_k-W_k\}$

and a backorder cost $of5$max$\{0,W_k-S_k-x_k\}.In$ sum, the cost function $is$ given $by$

$C_k(S_k,x_k,W_k)=x_k+2$max$\{0,S_k+x_k-W_k\}+5$max$\{0,W_k-S_k-x_k\}$

Let $J_k(S_k,x_k)be$ the cost function from period $kto$ period $3$ given that the inventory level

$at$ the beginning $of$ period $kis{S}_k$ and the ordering quantity $is{x}_k.$ The cost function upon

the optimal decision $is$ denoted $as{J}_k^{*}(S_k).$ The dynamic relation $is$ given $by$

$J_k(S_k,x_k)=E_{W_k}[C_k(S_k,x_k,W_k)+J_{k+1}^{*}(max\{0,S_k+x_k-W_k\})]$

Now, consider $a3-$period problem $.$ Our goal $isto$ find the optimal ordering quantities

$x_1,x_2,x_3$ that minimize the total expected cost over the $3$ periods.

$(a)$ Verify that

$J_k(S_k,x_k)=x_k+J_k(S_k+x_k,0).$

$(b)$ Calculate the optimal cost given $S_1=0.(Hint$: the result $in(a)$ can help $to$ simplify

the calculation.$)$ can you slove the problem