Question $4$

Suppose you are operating a bakery that makes croissants $($a type of roll with filling inside of it$)$ in

batches of $60$ croissants at a time.

Preparation Department: The dough and filling for the croissants are prepared separately in the same

department. However, since the dough and the filling preparation require different equipment and

labor, they can be prepared at the same time. Only enough dough and filling are prepared for one batch

at a time and due to space constraints, only one batch $($which contains both the dough and the filling$)$

can be in the department at a time. Thus, the batch is not done in the preparation department until

both the dough and the filling preparations have been completed. Preparing the dough involves three

sequential steps, mixing, proofing, and roll & cutting. Mixing takes $5$ minutes$/$batch$,$ proofing takes $10$

minutes per batch, and rolling and cutting takes a total of $5$ minutes$/$batch$.$ Preparation of the filling

requires a single step that takes $10$ minutes per batch. Once the batch leaves the preparation

department, it will go through the Fill &$.$ Fold Department, Baking Department, and a Packing

Department, in that order. Each department can only handle one batch at a time.

Fill & Fold Department takes $20$ seconds per croissant

Baking Department takes $23$ minutes per batch.

Packins Department takes $10$ minutes per batch

Queues: On average there are $0\mathrm{.}25$ batches of croissants between the Preparation Department and the

Fill & Fold Department and $1\mathrm{.}0$ batches between the Fill & Fold Department and the Baking

epartment.