Question $5(15$ pts$.)$: Continuous Distribution $-$ PYTHON Simulations

A manufacturing system produces components that require assembly in a multi$-$stage

process. Each stage has different characteristics modeled by exponential, Erlang, and

gamma distributions. The task is to analyze and simulate the system's performance under

the following conditions:

Stage $1-$ Quality Inspection: The time between defective components arriving at the

inspection station follows an exponential distribution with a mean of $5$ minutes.

Stage $2-$ Assembly: The assembly process requires multiple sequential sub$-$tasks, with each

sub$-$task having an average time of $2$ minutes. The total time for this stage is modeled

using an Erlang distribution with three sub$-$tasks.

Stage $3-$ Quality Assurance: The total time to inspect and rectify defects $($if any$)$ follows

a Gamma distribution with a shape parameter alpha$=2$ and a rate parameter beta$=0\mathrm{.}4.$

Assumptions:

Each component must go through all three stages sequentially.

Assume independent processing times for different components.

Use random number generators to simulate the data for each distribution.

a$)(10$ Pts$.)$ Simulate the overall production process for $1,000$ components to evaluate:

The average and variance of the total production time per component.

The percentage of components taking more than $20$ minutes for the overall process.

b$)(5$ Pts$.)$ Perform sensitivity analysis: Vary the parameters of each distribution $($e$.$g$.,$

mean, shape, rate$)$ and analyze their impact on the overall production time. solve every question