$15\mathrm{.}33$ Two models, A and B$,$ are to be assembled on a mixed$-$model line. Hourly production rates for the two models are: A$,25$ units$/$hr; and B$,18$ units$/$hr$.$ The work elements, element times, and precedence requirements are given in the table below. Elements $6$ and $8$ are not required for model A$,$ and elements $4$ and $7$ are not required for model B $.$ Assume $\backslash ($ E$=1\mathrm{.}0\backslash ),\backslash ($ E$\_\{$r$\}=1\mathrm{.}0\backslash ),$ and $\backslash ($ M$\_\{$i$\}=1\backslash ).($a$)$ Construct the precedence diagram for each model and for both models combined into one diagram. $($b$)$ Find the theoretical minimum number of workstations required to achieve the required production rate. $($c$)$ Use the Kilbridge and Wester method to solve the line balancing problem. $($d$)$ Determine the balance efficiency for your solution in $($c$).$

$\backslash$begin$\{$tabular$\}\{$ccccc$\}$

$\backslash$hline Work Element $\backslash ($ k $\backslash )$ & $\backslash ($ T$\_\{$e A k$\}\backslash )$ & Preceded by & $\backslash ($ T$\_\{$e B k$\}\backslash )$ & Preceded by $\backslash \backslash$

$\backslash$hline $1$ & $0\mathrm{.}5$ min & $-$ & $0\mathrm{.}5$ min & $-\backslash \backslash$

$2$ & $0\mathrm{.}3$ min & $1$ & $0\mathrm{.}3$ min & $1\backslash \backslash$

$3$ & $0\mathrm{.}7$ min & $1$ & $0\mathrm{.}8$ min & $1\backslash \backslash$

$4$ & $0\mathrm{.}4$ min & $2$ & $-$ & $-\backslash \backslash$

$5$ & $1\mathrm{.}2$ min & $2,3$ & $1\mathrm{.}3$ min & $2,3\backslash \backslash$

$6$ & $-$ & $-$ & $0\mathrm{.}4$ min & $3\backslash \backslash$

$7$ & $0\mathrm{.}6$ min & $4,5$ & $-$ & $-\backslash \backslash$

$8$ & $-$ & $-$ & $0\mathrm{.}7$ min & $5,6\backslash \backslash$

$9$ & $0\mathrm{.}5$ min & $7$ & $0\mathrm{.}5$ min & $8\backslash \backslash$

$\backslash$hline$\backslash ($ T$\_\{$w c$\}\backslash )$ & $4\mathrm{.}2$ min & & $4\mathrm{.}5$ min &

$\backslash$end$\{$tabular$\}$