Problem $3(12$ points$).$ Consider the problem of scheduling n jobs with different processing times and weights on one machine to minimize the total weighted completion time. That is$,$ we need to try to minimize summation over all jobs jin$[$n$]$ the product of the weight and completion time of j$.($a$)(2$ points$)$ For example, we have $5$ jobs $1,2,3,4$ and $5,$ and their weights and processing times are as follows: For example, consider the following schedule: We process the jobs in the order $1,2,3,4,5.$ The completion times of the $5$ jobs are respectively $20,20+60=80,80+15=95,95+18=113$ and $113+8=121.$ The weighted completion time is $20\backslash$times $3+80\backslash$times $6+95\backslash$times $1+113\backslash$times $6+121\backslash$times $2=60+480+95+678+242=1555$ Your goal is to give the best schedule $($or equivalently, the best order$)$ for the $5$ jobs. Please provide the optimal schedule and compute its weighted completion time. $($b$)(10$ points$)$ State whether the following two strategies $($A$)$ and $($B$)$ in the first step is safe or not. If a strategy is safe, provide a proof; if it is not safe, provide a counterexample to demonstrate why it is not safe. Strategy $($A$)$: schedule the job with the smallest processing time p$\_($j$)$ to be executed first. Strategy $($B$)$: Schedule the job that has the biggest ratio of weight to length $(($w$\_($j$))/($p$\_($j$)))$ to be executed first.

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