# Minimize global costs

Given the inventory process $$\{X_t\}$$ with dynamic $$X_{t+1}=X_t+A_t-\xi_{t+1}$$, where $$A_t=\#$$ of items produced and $$\xi_t\sim Poiss(5)$$, i want to find the process $$\{A_t\}$$ such that $$S=E\left[\sum_{t=0}^{59}0.3^t(10A_t+2\max\{0,X_{t+1}\}+\max\{0,-X_{t+1}\})\right]$$ is minimum.

First of all, note that trivially $$S=\sum_{t=0}^{59}0.3^tE[10A_t+2\max\{0,X_{t+1}\}+\max\{0,-X_{t+1}\}]$$. Now, $$X_{t+1}\ge0$$ iff $$0\ge-X_{t+1}$$, so $$2\max\{0,X_{t+1}\}+\max\{0,-X_{t+1}\}=2E[X_{t+1}]$$ if $$X_{t+1}\ge0$$ and $$-E[X_{t+1}]$$ if $$X_{t+1}<0$$, so I tried to use $$E[X]=E[E[X|Y]]$$, but i can’t get explicitely the pdf of $$X_{t}$$.\

On the other hand, i believe that $$\{A_t\}$$ should be, intuitively, $$\xi_{t+1}$$, since it represents the items produced and $$\xi_{t+1}$$ is the number of sold items.

How can i solve this problem?