A coffee shop is considering adding automatic order kiosks for customer self$-$checkout. Currently,

baristas take orders manually and prepare drinks. Observations show that each customer orders an

average of $3$ items, and each item takes $15$ seconds for the barista to key in and confirm, followed

by $45$ seconds per order to prepare. It takes an additional $30$ seconds to deliver the order and

transition to the next customer. If automatic kiosks are introduced, the ordering time per customer

would drop to $8$ seconds per item, while the preparation and delivery times remain the same.

$($a$)$ What is the hourly throughput rate $($number of customers served per hour$)$ under the current

system?

$($b$)$ What would be the estimated hourly throughput rate if automatic kiosks were used?

$($c$)$ Determine the productivity index for the case in $($b$),$ using $($a$)$ as the basis of comparison and

hourly customers served per labor hour as the measure of productivity.