Helpful Hints: Here are some hints that will help you through the process:

$$ Little$$s Law: Inventory $=$ ThroughputRate $\backslash$times ThroughputTime

$$ ThroughputRate $=1/($CycleTime$)$

$$ Capacity of the process depends upon the bottleneck step.

$$ To identify the bottleneck step, first calculate the capacity of each step and then the step with the least capacity is the bottleneck step.

$1.$ Capacity Analysis $(65$ points$)$

Instant Pot produces a popular multicooker appliance called the Duo. The Duo is produced in a factory that operates $24$ hours per day $($in $3$ shifts of $8$ hours$),365$ days per year. The final product consists of the pot$/$control panel assembly and one lid. The net profit on each Duo, after considering all relevant costs, is $$20$ per unit. You can assume that demand for the Duo is so strong that Instant Pot can sell as many Duo devices as they can produce.

The production process that Instant Pot uses to manufacture the Duo consists of five steps:

$$ Pot Production: It takes $1$ person $10$ minutes to manufacture a pot. There are $8$ people working on pot production during each shift

$$ Lid Production: An injection molding machine produces lids in batches of $25.$ The machine can produce one batch every $20$ minutes, and is operated at all times.

$$ Control Panel Production: It takes one person $4$ minutes to produce a control panel, and there are $4$ people working on control panel production during each shift.

$$ Duo Assembly: During assembly, the pot and control panel of the Duo are integrated. Each shift has $4$ people working on Duo assembly, and it takes $1$ person $6$ minutes to assemble a Duo.

$$ Duo Test: Each Duo that is produced is tested on Instant Pot$$s single test machine. The machine tests the Duos in batches of $15.$ It takes $9$ minutes to set up the machine, and then $12$ seconds to test each Duo. $95\%$ of Duos pass inspection. When a Duo fails inspection, the pot$/$control panel assembly is sent to another division of Instant Pot for recycling $($i$.$e$.,$ neither the pot nor the control panel can be used to make a Duo$).$ The only reason a Duo fails inspection is due to a defect in the control panel. Lids are not impacted by the test process.

$$

$($a$)(30$ points$)$ How much profit does the factory generate in a day?