Problem $1$

Greedy Scheduling

$3$ points

Scheduling is a problem where the goal is to permute a set of $n$ tasks, where each $i^{th}$ task is prescribed with a length $l_i($which represents the time taken to complete this task$)$ and a priority weight $w_i($where higher weights represent higher priority$).$

$($a$)$ Implement your own class called Tasks which represents a data structure that stores all the jobs $J,$ along with their respective lengths $l$ and weights $w.$

$($b$)$ Within the Tasks class, implement the GreedyRatio$($self$)$ subroutine in this class to find the optimal schedule that minimizes the sum of weighted completion times.

$($c$)$ Perform the empirical runtime analysis of your GreedyRatio$($self$)$ subroutine by simulating multiple Tasks objects with randomly generating $n$ tasks $($Let $n=5$:$5$:$101),$ each having a uniformly random length $l_i$ using the statement randrange $(5,51,5),$ and a uniformly random weight $w_i$ using the statement randrange $(1,n+1,1).$