$19\mathrm{.}4($A$)$ An FMS has three machining stations with seven machine tools as indicated in the table. Other stations are load$/$unload $(2$ servers$),$ transport $(1$ shuttle cart$),$ and part storage $(20$ storage spaces$).$ Four part styles $($A$,$ B$,$ C$,$ and D$)$ are produced, all with part$-$mix fractions $\backslash (=0\mathrm{.}25\backslash ).$ The sequence of processing is in numerical order of stations, with cycle times listed. At the load$/$unload station, time to load and clamp a part unto a pallet fixture is listed to the left of the slash and the unload time is to the right. A linear pallet system is used to move parts on pallets. Mean part transport time is $1\mathrm{.}0$ min $.$ Move time empty is $0\mathrm{.}5$ min $.$ For every transport move, there is an empty move. The parts are transported on pallet fixtures from station $4$ to part$-$storage station $6,$ from which they are moved through their respective process routings, then back to storage, and finally to unloading as time permits at station $4.$ Determine $($a$)$ bottleneck station, $($b$)$ total hourly output of the system, $($c$)$ production rate of each product, $($d$)$ numbers of busy servers at each station, $($c$)$ utilization of each server,

Chap. $19/$ Multi$-$Station Manufacturing Systems: Automated for Flexibility

$($f$)$ average utilization of the processing stations, and $($ g $)$ average utilization of all stations except storage.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multicolumn$\{3\}\{|$c$|\}\{$Part style $($j$)\}$ & A & B & C & D $\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Stations $\backslash \backslash$

$($i$)$

$\backslash$end$\{$tabular$\}$ & Operation & Servers & $\backslash ($ T$\_\{\backslash$text $\{$CAA $\}\}\backslash )($min$)$ & $\backslash [$

$\backslash$begin$\{$array$\}\{$c$\}$

T$\_\{\backslash$mathrm$\{$CB$\}\}\backslash \backslash$

$(\backslash$mathrm$\{$~min$\})$

$\backslash$end$\{$array$\}$

$\backslash ]$ & $\backslash ($ T$\_\{$c c$\}\backslash )($min$)$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

$\backslash [$

T$\_\{$c D$\}$

$\backslash ]\backslash \backslash$

$($min$)$

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

$\backslash$hline $1$ & HMC & $2$ & $27$ & $15$ & & $18\backslash \backslash$

$\backslash$hline $2$ & VMC & $2$ & $16$ & $13$ & $25$ & $\backslash \backslash$

$\backslash$hline $3$ & MTM & $3$ & $20$ & $31$ & $40$ & $30\backslash \backslash$

$\backslash$hline $\backslash (4($n$+1)\backslash )$ & Load$/$unload & $2$ & $10/4$ & $\backslash (8/3\backslash )$ & $\backslash (11/4\backslash )$ & $9/3\backslash \backslash$

$\backslash$hline $\backslash (5($n$+2)\backslash )$ & Transport & $1$ & $1\mathrm{.}0/0\mathrm{.}5$ & $1\mathrm{.}0/0\mathrm{.}5$ & $1\mathrm{.}0/0\mathrm{.}5$ & $1\mathrm{.}0/0\mathrm{.}5\backslash \backslash$

$\backslash$hline

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

Key: HMC $=$ horizontal machining center, VMC $=$ vertical machining center, MTM $=$ multi$-$tasking machine.