Consider the following two$-$step process:

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Step A has a processing time of $1$ minute per unit, but no setup is required. Step B has a processing time of $0\mathrm{.}1$ minute per unit, but a setup time of $9$ minutes is required per batch.

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$($a$)$ Suppose units are produced in batches of $5($i$.$e$.,$ after each set of $5$ units are produced, step B must incur a setup of $9$ minutes$).$ What is the capacity of the process $($in units per minute$)?($Round the answer to $2$ decimal places.$)$

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$($b$)$ Suppose they operate with a batch size of $15$ and with this batch size step A is the bottleneck. What would be the average inventory in step B$?($Round the answer to $2$ decimal places. Round other intermediate computations to one decimal place. $)$

c$.$What is the batch size that maximizes the flow rate of this process with minimal inventory? Assume there is ample demand.