$.)$ JCL Inc. is a major chip$-$manufacturing firm that sells its products to computer manufacturers like Dell, HP$,$ and others. In simplified terms, chip making at JCL Inc. involves three basic operations: depositing, patterning, and etching.

$$ Depositing: Using chemical vapor deposition $($CVD$)$ technology, an insulating material is deposited on the wafer surface, forming a thin layer of solid material on the chip.

$$ Patterning: Photolithography projects a microscopic circuit pattern on the wafer surface, which has a light$-$sensitive chemical like the emulsion on photographic film. It is repeated many times as each layer of the chip is built.

$$ Etching: Etching removes selected material from the chip surface to create the device structures.

The following table lists the required processing times and setup times at each of the steps. There is unlimited space for buffer inventory between these steps. Assume that the unit of production is a wafer, from which individual chips are cut at a later stage.

Note: A Setup can only begin once the batch has arrived at the machine.

Process Step $1$ Depositing $2$ Patterning $3$ Etching

Setup time $45$ min. $30$ min. $20$ min.

Processing time $0\mathrm{.}15$ min.$/$unit $0\mathrm{.}25$ min.$/$unit $0\mathrm{.}2$ min.$/$unit

a$.$ What is the process capacity in units per hour with a batch size of $100$ wafers?

b$.$ For the current batch size of $100$ wafers, how long would it take to produce $50$ wafers? Assume that the batch needs to stay together during deposition and patterning $($i$.$e$.,$ the firm does not work with transfer batches that are less than production batch$).$ However, the $50$ wafers can leave the process the moment all $50$ wafers have passed through the etching stage. Recall that a setup can only be started upon the arrival of the batch at the machine.

c$.$ For what batch size is step $3($etching$)$ the bottleneck?

d$.$ Suppose JCL Inc. came up with a new technology that eliminated the setup time for step $1($deposition$),$ but increased the processing time to $0\mathrm{.}45$ min.$/$unit$.$ What would be the batch size you would choose so as to maximize the overall capacity of the process, assuming all units of a batch stay together for the entire process?