The production process is as follows. Initially, the company has $0$ trucks in their

garage, which has a maximum capacity of $3$ trucks per season. Since the production is

time$-$consuming, the company can produce $$ in $\{0,1,2\}$ trucks each season before the

demand is revealed. The production has a fixed cost of $$30$k and a variable cost of

$$20$k per truck in any season. After the production occurs, the demand level $$

becomes known. Each available truck in the garage is sold for $$100$k units. All unsold

trucks in spring or summer are kept for the next season with $$10$k of holding cost per

truck per season. All extra trucks at the end of the fall are sold for only $$50$k$.$

a$)$ Use stochastic dynamic programming to solve the problem of maximizing the

company$$s expected profit $($revenue $-$ costs$)$ using DP tables. Clearly define the stages,

states, and actions. Write the state transition function as well as the recursive DP

functions. $(20$ points$)$

b$)$ What is the maximum expected total profit? What is the optimal policy for the

company regarding the production plan? Clearly state how they should plan their

production at each season. $(10$ points$)$