factory manufactures custom widgets, each requiring a specific assembly time

and a painting time. The factory has multiple workers available for painting, but only one

assembly line, which can handle one widget at a time. Each widget Wi needs Ai minutes

for assembly and Pi minutes for painting. The painting of a widget can only begin after it

has been assembled, but multiple widgets can be painted in parallel. Write an algorithm

that, given a set of n widgets, determines a schedule Wi$1,$ Wi$2,,$ Win that such that all the

widgets are completely finished in the shortest overall time. Derive the runtime and prove

the correctness of your algorithm formally