Part II: Practical case

Consider a production line producing some commodity with a high failure rate $($maybe

because of variance in materials, or difficulty of production$).$ To finish the product $2$

different steps need to be completed, step A and step $B.$ The overall structure of the

production line is described in the figure below:

In steps $A$ or $B$ the piece goes through the respective procedure. The control steps

are to determine if the quality is good enough to continue to the next step:

If quality is good enough, the piece is finished.

Otherwise, the piece is sent to be fixed.

In the attempt fix steps we try to fix the piece: if the issue is fixable or small enough,

then the piece is sent back to quality control otherwise the piece is trashed. The values

on the edges represent the probabilities to go from one state to another.

a$)$ Create the stochastic matrix describing the Markov chain corresponding to this

production line.

b$)$ What is the probability that a product starting in step A ends up in the trash?

What is the probability that it is finished?

c$)$ What is the average number of steps a piece starting in A goes through before

it is finished or trashed?

d$)$ If step A and $B$ each cost $4,$ both control steps each cost $1,$ and the attempted

fixes cost $2$ each, what is the average cost of producing $10$ finished products?

$($Products that are trashed incur no extra cost$).$

e$)$ Assume no started products are trashed. Instead, a more expensive fix costing

$4$ is attempted on those defective pieces. If each finished piece could be sold

for $25,$ would it still be profitable to finish these products? $($You can assume

that the extra time spent on these products is neglectable$).$