For the first part of your assignment:

$($i$)$ Here's a simplified flowchart depicting the passenger arrival process at the airport:

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Start

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V

Check$-$in & Luggage drop

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V

Airline security interview

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V

Pass through airport security

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V

Wait at gate

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V

Boarding pass authentication

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V

Boarding

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V

End

$($ii$)$ To improve this process and avoid delays, several measures can be implemented:

Implement online check$-$in and baggage drop services to reduce the time spent at the airport.

Utilize automated security procedures such as TSA PreCheck or facial recognition for faster screening.

Increase the efficiency of boarding pass authentication by employing self$-$service kiosks or mobile boarding passes.

Optimize luggage size restrictions and enforce them strictly to reduce delays caused by oversized carry$-$on bags.

Streamline the interview process by utilizing advanced security screening technologies to identify potential threats more efficiently.

For the second part:

$($ii$)$ The likely causes of Bed Bath & Beyond filing for Chapter $11$ bankruptcy protection could include:

Declining sales due to increased competition from online retailers and other brick$-$and$-$mortar stores.

Poor strategic decisions, such as expanding too quickly or investing in unprofitable ventures.

High levels of debt resulting from previous acquisitions or expansions.

Inefficient cost management and operational inefficiencies lead to decreased profitability.

Failure to adapt to changing consumer preferences and shopping trends.

Economic downturns or external factors impacting consumer spending habits.

Creating a cause$-$and$-$effect diagram $($also known as a fishbone or Ishikawa diagram$)$ can help visualize these potential causes and their interrelationships.

For the third part:

$($i$)$ To calculate the upper and lower control limits for an X$-$bar chart and R$-$chart with a sample size of four, you would typically use statistical formulas based on the data provided. The formulas for control limits involve calculating the mean, standard deviation, and other statistical parameters from the sample data.

$($ii$)$ To determine if corrective action is needed after observing a sample of four additional observations, you would compare these new data points to the control limits established earlier. If any data points fall outside the control limits or show a significant shift or trend, it may indicate a process variation that requires investigation and corrective action.