$3.(30$ points$)$ Problem $5\mathrm{.}6($a$)$ A machine operators cost $$9\mathrm{.}50$ an hour. The idle cost to run the machines in the accompanying table is $$10,$ $$12,$ and $$15,$ respectively. It takes $30$ seconds to get from one machine to the next.

Assuming that the takt time is $25$ minutes, evaluate the machine$-$coupling alternatives $(1),(2)$ and $(3)$ below and recommend the best alternative. Show the worker$-$machine charts for each alternative.

$(1)3$ operators $/3$ machines

$(2)2$ Operators

AB $(2$ machines$/1$ operator$)$ C $(1$ machine$/1$ operator$)$

$(3)1$ operator

ABC $(3$ machines$/1$ operator$)$