Problem $4$: Stir$-$Fry Food Truck

The Stir$-$Fry food truck on campus serves customers during the lunch hour by taking orders and making fresh batches of stir$-$fry. Customers have only one choice during the lunch hour, since the objective is to maximize the number of customers served. Assume that each customer places just one lunch order, and all lunch orders are the same size: one unit of stir$-$fry.

The stir$-$fry cooking works in this manner. First, a batch of orders is cooked in a wok by a single cook. The cooking time depends upon the number of orders in the batch. The fixed $($setup$)$ time for any size batch is $2$ minutes, and the time to process each order $($time per unit$)$ is $0\mathrm{.}5$ minutes. Thus, time to cook just one order is $2\mathrm{.}5$ minutes, cooking two orders in a batch takes $3$ minutes, cooking three orders in a batch takes $3\mathrm{.}5$ minutes, and so on$.$ There is no limit to the maximum size of a $()$ at the cooking step.

After a batch finishes at the cooking step, a single cashier bags each meal and accepts payments, which takes a total of $40$ seconds per order $($there is no batching in this step$).$ The process flow diagram is given below:

$18.$ If Stir$-$Fry operates with cooking batch sizes of $6$ orders, what is the capacity of the cooking step $($in orders$/$min$)?$ If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$

$8$

$19.$ What is the capacity of the cashier $($in orders$/$min$)?$ If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$

$20.$ If Stir$-$Fry operates with cooking batch sizes of $6$ units, what is their system capacity $($i$.$e$.,$ the bottleneck capacity$)($in orders$/$min$)?$ If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$

$21.$ Find the batch size $($in number of orders$)$ for the cooking step that will make the capacity of this step equal to the capacity of the cashier step.

Assume that fractional batch size is allowed. If your answer is not an integer, provide at least three decimal places, e$.$g$.,7\mathrm{.}500.$