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Part $1$

A U$.$S$.$ Postal Service supervisor is looking for ways to reduce stress in the sorting department. With the existing$$ arrangement, stamped letters are$$ machine$-$canceled and loaded into tubs with $425$ letters per tub. The tubs are then pushed to postal$$ clerks, who read and key zip codes into an automated sorting machine at the rate of one tub per $425$ seconds. To overcome the stress caused when the stamp canceling machine outpaces the sorting$$ clerks, a pull system is proposed. When the clerks are ready to process another tub of$$ mail, they will pull the tub from the canceling machine area. How many tubs should circulate between the sorting clerks and the canceling machine if $85,000$ letters are to be sorted during a $12-$hour $$shift$,$ the safety stock policy$$ variable, $\backslash$alpha $,$ is $0\mathrm{.}25,$ and the average waiting time plus material handling time is $30$ minutes per$$ tub?

Part $2$

The average processing time per tub of$$ letters, p$,$ is $0\mathrm{.}00240\mathrm{.}0024$ day. $($Enter your response rounded to four decimal$$ places.$)$

Part $3$

The average waiting time per tub of$$ letters, w$,$ is $0\mathrm{.}02080\mathrm{.}0208$ day. $($Enter your response rounded to four decimal$$ places.$)$

Part $4$

$$Therefore$,$ the number of tubs that should circulate between the sorting clerks and the canceling machine is $$enter your response here tubs. $($Enter your response rounded up to the next whole$$ number.$)$