I am having a problem with a Markov chain transition matrix question. Can anyone help me?
I have 2 questions in relation to this topic:
1)In a certain production process, items are pass through two manufacturing stages. At the end of each stage, the items are either scrapped with a probability of 0.15, sent through the stage again for rework with a probability of 0.25, or passed on the the next step with a probability of 0.6. Describe this as a Markov chain and set up the transition matrix? What is the expected number of steps to absorption?
And:
2)Ball bearings in a machine have an increased probability of failure with increased service life. This machine has two bearings. Bearings cost $15 each plus $50 per unit installation cost. A bearing failure results in an additional cost of $200. Past history of failures are tabulated below. Note that no bearing has ever run more than 5000 hours. Management wishes to examine two possible maintenance procedures.
i) Replace as failures occur.
ii) Replace individual bearings on an optimum replacement life basis.
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|Service Life (1000 hours) | No. of times failure occurred |
--------------------------|---------------------------------
| 1 | 10 |
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| 2 | 15 |
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| 3 | 15 |
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| 4 | 30 |
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| 5 | 20 |
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Which replacement policy results in minimum cost? What is the average life of a bearing (from installation to replacement) under each policy?
If you could help me with these it would be very much appreciated.
Let the state of an item can be Scrapped, ToBeTested, Passed, then we may represent an item as a colum vector of the probabilities of it being in each state:
$\displaystyle P=\left[ \begin{array}{c}S\\T\\P \end{array} \right]$
where S, T and P are the probabilities that it is each of the states.
Then the transition matrix is:
$\displaystyle
A=\left[ \begin{array}{ccc}1 & 0.15 & 0\\0 & 0.25 & 0\\0&0.6&1 \end{array} \right]
$
which tells us that:
(first row) if the item is Scrapped, it remains Scrapped, if it is to be Tested it is Scrapped with probability 0.15, and if it has been Passed it cannot be Scrapped.
(second row) if an item is Scrapped it cannot be tested, if it was tested it will be reworked and returned for Testing with probability 0.25, and if it has been passed it cannot be Tested.
(third row) if an item is Scrapped it cannot be Passed, if it was tested it will be Passed with probability 0.6, and if it has been Passed it remains Passed.
Note the convention I am using is that if $\displaystyle P(n)$ is a column vector of state probabilities at epoc $\displaystyle n$, then the new state at epoc $\displaystyle n+1$ is $\displaystyle P(n+1)=AP(n) $
CB