Kinga Doll Company manufactures eight versions of its popular girl doll, Shari. The company operates on a $40-$hour workweek. The eight versions differ in doll skin, hair, and eye color, enabling most children to have a doll with a similar appearance to them. It currently sells an average of $3,500$ dolls $($spread equally among its eight versions$)$ per week to boutique toy retailers. In simplified terms, doll making at Kinga involves three basic operations: molding the body and hair, painting the face, and dressing the doll. Changing over between versions requires setup time at the molding and painting stations due to the different colors of plastic pellets, hair, and eye color paint required. The following table lists the setup and processing times for a batch and the activity times for each unit at each step.

$\backslash$table$[[$Process Step,Molding,Painting,Dressing$],[$Setup time $($minute$),18\mathrm{.}00,31\mathrm{.}00,$No setup$],[$Processing time $($minute$),0\mathrm{.}21,0\mathrm{.}12,0\mathrm{.}29]]$

a$.$ What is the process capacity in dolls per hour with a batch size of $400$ dolls?

Note: Do not round intermediate calculations. Round your final answer to nearest whole number.

Capacity

dolls per hour

b$.$ What is the utilization of molding in dolls per hour with a batch size of $850$ dolls?

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

Utilization

$\%$

c$.$ Referencing the molding station in Part b$,$ what batch size would minimize inventory without decreasing process capacity?

Note: Do not round intermediate calculations. Round your final answer to nearest whole number.

Batch size

dolls per batch