A dynamic programming solution to the unit commitment problem consists of $5$ stages

named A$,$ B$,$ C$,$ D$,$ and E$.$ Stage A means all units are shut down. Each stage B$,$ C$,$ and D

has $4$ possible states $($combinations of units$).$ The final stage $E$ has only two possible states.

a$.$ How many costs $($branches$)$ do you have to evaluate if you use the forward

method? Explain your reasoning for the answer.

b$.$ How many costs $($branches$)$ do you have to evaluate if you use the backward

method $($Bellman$\text{'}$s principle$)?$ Explain your reasoning for the answer.

c$.$ Calculate the percentage computational saving you obtained by using backward

method $($assume one computation $=$ evaluation of one branch$).$