Each aisle of a four aisle AS$/$RS contains $60$ storage compartments in the length and $12$ in the width direction. All compartments are the same size and accommodate pallets of dimendions $x=42in,y=48in$ and $z=36$ in; The allowances are $a=6$ in$,b=8in,c=10in$ in $x,y$ and $z$ directions respectively. The horizontal and vertical speeds of the $\frac{S}{R}$ machine are $200f\frac{t}{m}in$ and $75f\frac{t}{m}in$ respectively. The machine requires $20$ sec to do the P&D operation.

a$)$ Determine the dimensions of the AS$/$RS system. $(10$ points$)$

b$)$ Determine the single command and dual command cycle times per aisle. $(20$ points$)$

c$)$ Determine throughput $($number of transactions per hour$)$ per aisle if storage system utilization $=90\%$ and number of dual command cycles is equal to number of single command cycles. $(20$ points$)$