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Math Help - stock problem

  1. #1
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    Question stock problem

    I have a little question.

    All the components of which I speak are identical. I have a system which is composed of 4 of those components. If one of them breaks down the entire system breaks down. I have 7 components in stock.
    If I hand in the broken components into repair it takes about three months to get them repaired, but unfortunately only about 1/3 of the pieces get repaired (the other ones are trash).
    The mean time between failure of one of the components is about three months.

    So:
    • 4 components
    • 7 stock
    • 3 months repair time
    • 1/3 repair ratio

    Now I want to know two things:
    • The repair time can be changed slightly (a matter of given priority); which is the optimal repair time and why?
    • How long will my system last (best case)? What’s the formula to determine that?

    I tried to figure those out by myself but didn’t manage to compose a formula for this problem because the components of which the system is composed can be changed (once they are repaired they can replace broken ones of the system, and the original can get repaired too…).

    Can someone help me to compose the formula?
    Where can I find the mathematical background?
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  2. #2
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    This is a wonderful problem for Simulation. have you met WinBUGS or @Risk? Delightfully useful programs.

    1) It should be obvious that your system is good until you have the 8th failure (assuming your stored replacements are in working order). It is only then that you rely on getting a component back from the repair shop.

    2) Unfortunately, you are running four components simultaneously, so your risk of present failure is nowhere near the three months for which you may be hoping.

    3) Are the failures independent? For example, old strings of lights used to go very bad very quickly right after the first light burned out. The mehanism was the spread of voltage of the burned out light over the remaining lights. This created increasingly increased voltage over remaining lights as subsequent lights burnd out.
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  3. #3
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    Thanks a lot. So this means there is no "fomula-solution"?

    I have no experience at all using simulation programs. Can you recomend me one of the two you named ? Which one is the easiest to learn?
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  4. #4
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    The question may be price, rather than ease of use.

    I am not saying there is no formulaic solutions. There may be.

    The only real question I see is the repair shop. Can they work on multiple parts at once or is it a "single-server" setup?

    Answer these questions.

    1) Are the failures independent? If they are independent, you can build the distribution of failure rather easily.

    P(any failure) = 1 - p(no failure) = 1 - [1-p(any failure)]^4

    And you can see the distribution of failures for the entire system.

    2) Can repairs be done simultaneously or does #2 have to wait foir #1? Waiting is easier, but clearly will kill the system sooner.

    3) One more thing about the repairs. How does one tell if the components are trash? Does it take three months in the repair shop to decide? That would be VERY discouraging. If you can decide BEFORE they get into the repair system, that would waste far less valuable repair time.

    The repair model is far more complicated than the failure model. If you can decide trash or repair up front, and if the repairs have to wait for each other, then you can bring to bear all the power of a Single-Server Queueing Model with Balking. If they can be repaired at the same time, it can be modelled as a Multi-Server model. There is a good body of literature on Queueing Theory.
    Last edited by TKHunny; August 30th 2008 at 08:31 PM.
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  5. #5
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    Hi,

    The repair shop can work on multiple parts simultaneausly.

    We only know when we pick up the repaired pieces how much of them are trash. So, it takes them three months to decide wether the pieces can be repaired (in fact, the once that can't be repaired get burned in the attempt to repair them - about 2/3)

    The failures are independant...

    Thanks a lot for your guidance so far !
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  6. #6
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    Quote Originally Posted by pliniusken View Post
    The repair shop can work on multiple parts simultaneausly.
    Okay. It's a multiple-server model with essentially unlimited servers. In other words, if you bring in a part, we'll start working on it.

    We only know when we pick up the repaired pieces how much of them are trash. So, it takes them three months to decide wether the pieces can be repaired (in fact, the once that can't be repaired get burned in the attempt to repair them - about 2/3)
    Blech!! That's horrible. I'd hire a guy just to figure out if there is a better way. Maybe the repair process is faulty. Anyway..

    The failures are independant...
    Another nice simplification.

    Well, not we're down the the actual application.

    The Arrival Process: I think we've established a distribution or that.

    The Server Process: This is simple enough. It takes 3 months. The 2/3 failure rate is troublesome. Does this mean we can just extend the EXPECTED time to repair to 9 months?

    It amy be that simple.

    Anyway, here's a whole book on the subject.

    An Introduction to Queueing Systems

    Kick through a few quick chapters and see if any of it makes sense. Keep your problem in mind.
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