A manual assembly line is to be designed to make a small consumer product. The work elements, their times, and the precedence constraints are given in the table below.

$\backslash$table$[[$Element$,$Time $(min),$Preceded by$],[1,0\mathrm{.}4,-],[2,0\mathrm{.}7,1],[3,0\mathrm{.}5,1],[4,0\mathrm{.}8,2],[5,1\mathrm{.}0,2,3],[6,0\mathrm{.}2,3],[7,0\mathrm{.}3,4],[8,0\mathrm{.}9,4,9],[9,0\mathrm{.}3,5,6],[10,0\mathrm{.}5,7,8]]$

$T_c=5\mathrm{.}6$

The workers will operate the line for $400$min per day, and must produce $300$ products per day. A mechanized belt, moving at a speed of $1\mathrm{.}25\frac{m}{m}in$ will transport the products between stations. 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 workstations on the line.

$($b$)($i$)$ Use the Largest Candidate Rule method to balance the line.

$($ii$)$ Use the Ranked Positional Weight method to balance the line.

$($c$)$ Calculate the balance delay for your solution in part $($b$)($i$)$ & $($b$)($ii$).$

$($d$)$Determine the spacing between assemblies on the line.

$($e$)$ the required length of each workstation to meet the specifications of the line.