This problem involves a cost$-$benefit process analysis for laying out a facility. The process involves a $$Chipotle$-$style$$ line whereby customers arrive, order food, and move through the system.

Your objective is to staff the system with the optimal number of employees, given an estimate of how many customers will be arriving per hour. Each of your employees is paid $$11/$hour$,$ and each customer$$s food order averages $$9\mathrm{.}80,$ out of which $$1\mathrm{.}3$The starting point for analysis is when you staff the system with one employee $($@ $$11/$hour; we will treat revenue and costs as separate$).$ This system is depicted below: the single employee takes care of the entire transaction in $150$ seconds for each customer.

$150$ seconds

For now, assume that the demand rate $($i$.$e$.,$ the arrival rate in this system$)$ is at least as fast as the maximum rate at which the system can operate. To put it a little differently, we are assuming that the system is in steady$-$s$($f$.)$ The whole point of this exercise is to enable you, as managers of such a system, to plan for staffing the system dynamically, that is$,$ to staff the system differently as its characteristics change over time. So$,$ consider three different times of day $($e$.$g$.,$ morning, midday, and evening$)$ which have different customer arrival rates. Suppose that the arrival rate in the morning is $22$ customers per hour; that the midday arrival rate is $67/$hour; and that the evening arrival rate is $45$ per hour. For each of these three scenarios, and for the costs and profits originally provided, what are the optimal employee staffing levels?

tate.