A manufacturing facility has decided to install a transfer line to produce a new product category com$-$

prising several widgets. The line is made up of five processing stations arranged in series, without buffers

between the stations. Material is released $($from an endless supply$)$ to the first station, whenever the station

becomes available. Upon completing the required work in the first station, a job goes to the second station,

if there is available space there. Otherwise, it waits at the first station $($blocking it$)$ until room becomes

available in the next station, and so on until station $5.$ After work is completed in station $5$ it is shipped

to finished goods inventory $($which has infinite capacity$).$ Each station was designed to have an average

throughput rate of three $(3)$ parts per hour. However, actual processing times have a high degree of vari$-$

ability, because the line makes many different widgets. Additionally, each station has operation$-$dependent

failures that occur after a random number of cycles. Each time there is a failure, it takes a random amount

of time to bring the station back to work.

Management quickly realized that, without buffers, the stations were frequently starved or blocked be$-$

cause of the failures and the variability of processing and repair times. Therefore, the total throughput of the

line was far below the expected three $(3)$ parts per hour. They conducted a month$-$long experiment, where

they ran the line all the time. That is$,$ they ran each station constantly, whether there was material available

or not. They are sure that whether the station is working on real material or running empty will not affect

processing times, the failure rate, or repair times. This allowed them to collect a lot of detailed process,

failure, and repair data for each station independently $($thus excluding the effects of dependent events$).$

The attached file Case$1$Data.zip contains $5$ comma separated value $($csv$)$ files. Each file has the data for

one station, they are named M$1$data.csv$,$ etc. Each of these files has three columns of data. The first column

is the start time of the operation, the second is the completion time of the operation and the third column

takes the value TRUE is the operation was completed successfully or FALSE otherwise. If an operation is

completed successfully, then the next operation should start immediately. On the other hand, if an operation

is not successful, it means there was a station failure, which must be repaired, in this case, the next operation

begins as soon as the failure is fixed.

This company has hired you as a consultant to perform a buffer$-$capacity analysis of their widget produc$-$

tion line. They want you to recommend a buffer size for each station that will achieve a high throughput,

without having enormous buffers. After some preliminary work, you have settled on four specific tasks that

you are going to perform, to find suitable buffer sizes. Your job is to perform the four tasks and present a

report with your findings.

Tasks

$1.$ Use the data to estimate the process parameters. You can use Stat:Fit, MINITAB, R$,$ @Risk or

whatever software you prefer, to calculate the following:

$($a$)$ The best$-$fit distribution of the processing times for each station.

$($b$)$ The best$-$fit distribution of the repair times for each station.

$($c$)$ The breakdown probability for each station.

$2.$ Analyze the line assuming that stations never break down and that they are identical $($in distribution$).$

This is$,$ of course, not true, but these results may serve as an approximation.

$($a$)$ Calculate the average variance of the processing time for the $5$ stations and use it to calculate

an average coefficient of variation CV $.$ Use this CV and Figure $3\mathrm{.}6$ from A&S as described in

example $3\mathrm{.}7($also covered in class$)$ to estimate the relative output of the line, with respect to the

theoretical maximum. What is the expected throughput of the line without buffers?

$($b$)$ Use Figure $3\mathrm{.}7$ from A&S as described in example $3\mathrm{.}8($also covered in class$)$ to estimate the buffer

size required between each station to recover $80\%$ of the lost throughput. Call this number Z$1.$

What is the expected throughput of the line with buffers equal to Z$1?$

$($c$)$ Use formula $3\mathrm{.}18$ in A&S $($also covered in class$)$ to verify your results in parts $($a$)$ and $($b$).$

$3.$ Analyze the line assuming that stations have constant identical service times and random failures.

Again, this is not true, but these results may serve as a second approximation.

$($a$)$ Calculate the average variance of the repair times for the $5$ stations and use it to calculate an

average coefficient of variation cvR$.$ Calculate an average breakdown rate of all $5$ machines $($you

can just take an average of the $5$ rates$).$ Calculate E$\backslash$infty for the system. What is the theoretical

maximum throughput adjusted for failures?

$($b$)$ Estimate the throughput of the unbuffered line $($this is E$0\backslash$times $\backslash$lambda $),$ as in example $3\mathrm{.}9$ in A&S $($also

covered in class$).$

$($c$)$ Use Table $3\mathrm{.}3$