Step $1$: Batching. Water, eggs, and the dry ingredients are weighed, measured, and added to the mixing bowl. This step takes $2$ minutes per bowl and there are enough employees and equipment to batch two bowls simultaneously. The bowls and mixers can accommodate batch sizes from $8$ to $16$ pounds, and the default batch size is $8$ pounds. The current $8$ pound batch size minimizes wear and tear on the machinery and requires less maintenance than larger batch sizes.

Step $2$: Mixing: The bowl containing the ingredients is moved to the mechanical mixer. The mixing process takes $2\mathrm{.}5$ minutes per batch per mixer, and C$4$ has two mixers which can run simultaneously.

Step $3$: Extruding. Once the batch of dough is mixed it is moved to the extruder, which has a container into which the batches of mixed dough are loaded. The extruder shapes the dough and then cuts the dough into single$-$size cookie pieces. The machine can produce cookies in range of sizes from $2-4$ ounces of dough, but the standard cookie size being produced on a daily basis is $2$ ounces. As for setting up the extruder for each batch, it takes $30$ seconds to load each batch of dough into the extruder's container. Cleaning the cutters is also part of the per$-$batch setup. The base machine has $3$ cutters and each cutter must be cleaned between batches for operational consistency and food safety. Cleaning takes $30-50$ seconds per cutter and can only be done one cutter at a time, so cleaning three cutters takes $90-150$ seconds. After cleaning, the machine can extrude between $22-32$ cookies per minute per cutter, for a total output of $66-96$ cookies per minute with three cutters. Additional cutters can be added to the machine for an additional cost, and each additional cutter will be able to work at the same rate $-$ adding $22-32$ cookies per minute to the total output, however, each added cutter will also add to the setup time because of the added time to clean the additional cutter.

Step $4$: Baking. After being shaped, the cookies go onto a conveyor belt and travel through an oven and then on to a cooling rack. The oven and cooling rack can accommodate as many cookies as the extruder can produce.

Step $5$: Boxing: Employees check the cooled cookies for defects $($and sometimes eat the broken ones!$)$ and then place them in boxes of $3,6,$ and $12$ for transport to local retailers. Cookies are also placed on decorative trays to be displayed and sold one$-$at$-$a$-$time at the retail counter of C$4\text{'}$s shop. The boxing process is easy to scale up or down by adding employees or taking them away from this process step, so there is never a hold$-$up due to boxing.

The shop keeps meticulous records and knows that the overall daily demand for the C$4$ Chocolate Chip Crunch Cookie is between $1,000$ and $1,400$ dozen. The production line runs from $9$am to $4$pm daily Monday through Saturday, although this can be changed. C$4$ has very high freshness and quality standards, so they only sell cookies made on the same day. The owner, Charlie Cooker, knows that he can sell more cookies if he can just make them. He has identified the bottleneck to be the extruding machine and he wants to analyze the impact of using four managerial levers he has at his disposal to increase the capacity of the extruder and therefore increase the daily production of C$4$ Chocolate Chip Crunch Cookies. Each of these levers is explained below.

Reducing setup time $($loading and cleaning time$)$ will make the machine work more efficiently with little to no additional investment.

Increasing batch size will increase the benefit of economies of scale, but may affect product quality and require more equipment maintenance.

Adding cutters to the extruder will speed up the rate of production at the machine, but will increase the setup time for each batch.

Extending the daily production shift length will increase daily cookie output, but comes with increased labor and overhead cost.

Before he starts to analyze the impact of these levers, he also needs to know the current actual Capacity $($what his process can produce in one day$)$ and the Takt time $($the cycle time he needs to operate at in order to meet the daily demand$).$ Takt time may be above or below the actual cycle time the system is capable of $-$ Takt time is a target that would mean the system is producing exactly the customer demand.

We know that a process is only as fast as its bottleneck. This means the speed or capacity of a bottleneck in a process really determines the overall output or capacity of the process. Charlie Cooker knows that his extruder is the bottleneck for the cookie shop, so to find the daily capacity of the shop, he simply needs to find the daily capacity of the extruder. The process is a batch process, meaning the product moves from step to step in batches, where batch size is determined by the weight of dough. Each batch must be completed by the extruder before a new batch can be started at the extruder. The time it takes for t