$3.$ A manual assembly line is to be designed to make a small consumer product. The work elements, their times, and precedence constraints are given in the table below. The workers will operate the line for $400$ min per day and must produce $300$ products per day. Because of the variability in the time required to perform the assembly operations, it has been determined that the tolerance time should be $1\mathrm{.}5$ times the cycle time of the line.

$($a$)$ Determine the ideal minimum number of workers on the line.

$($b$)$ Use the Ranked Positional Weights method to balance the line.

$($c$)$ Compute the balance delay for your solution

Element

Time Te

Preceded by:

Element

Time Te

Preceded by:

$1$

$0\mathrm{.}4$ min

$6$

$0\mathrm{.}2$ min

$3$

$0\mathrm{.}7$ min

$1$

$7$

$0\mathrm{.}3$ min

$4$

$0\mathrm{.}5$ min

$1$

$8$

$0\mathrm{.}9$ min

$4,9$

$4$

$0\mathrm{.}8$ min

$2$

$9$

$0\mathrm{.}3$ min

$5,6$

$1\mathrm{.}0$ min

$2.3$

$10$

$0\mathrm{.}5$ min

$7,8$