There are $4$ generating units in a balancing area. The market operator is trying to find the optimal unit commitment option for the balancing area. Below two tables are given to the market operator to make decisions.Start up conditions are at the beginning of the first period units $1$ and $2$ are up$,$ units $3$ and $4$ are down and have been down for $8$ hrs$.$

$1)$ Use Forward dynamic programing and find the optimum unit commitment schedules covering the $8$ hr load cycle. The starting conditions are at the beginning of first period units $1$ and $2$ are up $,$ units $3$ and $4$ are down and have been down for $8$ h$.$