Question

Q3. Press shop A press line of a car manufacturing company consists of different equipment: A feeder that provides the metal to press shop, a press machine that shapes the metal using a hydraulic press, a punch press that is used to form holes, and a robot that removes the final product from the press line. Suppose the removal robot and the feeder are reliable and one-year overhaul maintenance is planned for them. However, the press machine and punch press break down occasionally. If one of them breaks down, the whole press line has to be shut down. A quality control engineer tries to write a maintenance plan for press machine and punch press in order to minimize the sudden breakdown of the press line. She collects data for several months and she calculated that on average the press machine break down once every week and the punch press breaks down on average once in every month. She believes the numbers of breakdown for these two presses follow Poisson distribution. Until she cannot figure out a suitable timing for maintenance, she does not order any maintenance (therefore the number of breakdowns in the two equipment remain the same). Using this information, help her to answer the following questions: (Let's call the press line "LINE", the press machine "PRESS", and the punch press "PUNCH confusion) in order to avoid What is the probability that the PRESS breaks down 1 time in a month? What is the probability a. that the PUNCH breaks down 1 time in a month? (3 points) b. What is the probability that the LINE shuts down once in a month either because of a breakdown in press machine or a breakdown in punch press? (hint: these two machines work independently)(2 points) What is the probability that she waits at least a month to see PRESS breaks down? What is the c. probability that she waits at least a month to see PUNCH breaks down?( 3 points) d. What is the probability that she waits at least a month to see the LINE shuts down either because of a breakdown in press machine or a breakdown in punch press? (hint: these two machines work independently) (2 points) e. What is the probability that she waits at least three months to see the LINE shuts down either because of a breakdown in press machine or a breakdown in punch press? (hint: these two machines work independently) (4 points) A clear definition of random variablefs), choice of model, and a brief statement about the results have points

Answer 1

Let variable P be used for Press and Pu be used for Punch.

We take 4 weeks in 1 month

On an avg Press breaks down once a week therefore on an avg it 4 times a month.

PPoi(1)per week

P Poi(4)per month

On an avg Punch breaks down once a month.

Pu Poi(1)per month

e P(X z!

(a) Probability of breaking down once in 1 month

= 0.07326

e 4 P(P 1) 1!

= 0.3679

P(Pu 1)e1 1!

(b) Line will shut if one of the machines breakdown once.

Line to shut down in one month one of the machine has to break down once in a ,month

P(line will shut) =

P(P = 1) P(Pu = 1)

P(Line will shut) = 0.07326 + 0.3679

Probability that the line will shut in a month is 0.4412.

(c) If she is waiting for at least a month for a break down that means that there will be no break downs in the first month.

= 0.01832

e 40 P(P = 0) 0!

= 0.36789

-11 P(Pu 0) =

(d) For line to shut after a month, there has to be no break downs in either of the machine.

P( line to not be shut for at least a month) =

P(P 0)P(Pu 0)

= 0.01832 + 0.36789

Probability that line will not shut down for at least a month is 0.3862.

(e) If the line shall not shut down for at least 3 months there has to be no break downs for 3 months in either of the machines

P~Poi(12)per 3 months

$Pu\sim Poi(3)per\;3\;months$

P( line to not be shut for at least 3 months) =

P(P 0)P(Pu 0)

=

12120 e + 30 e

= 0.00000614 + 0.049787

Probability that she will for at least 3 months for the line to be shut is 0.49793.