{"id":5123,"date":"2015-08-29T14:12:00","date_gmt":"2015-08-29T19:12:00","guid":{"rendered":"http:\/\/www.livingreliability.com\/en\/?p=5123"},"modified":"2025-11-06T05:20:25","modified_gmt":"2025-11-06T10:20:25","slug":"ra-micro-day-to-day-decision-analysis","status":"publish","type":"post","link":"https:\/\/www.livingreliability.com\/en\/posts\/ra-micro-day-to-day-decision-analysis\/","title":{"rendered":"RA – Micro (day-to-day decision) analysis"},"content":{"rendered":"

Reliability Analysis: 2 dimensions (Statistical and probabilistic concepts)<\/h1>\n

Reliability analysis is counting (slides)<\/a><\/h2>\n
\"\"
Slide 1 Financial analysis is counting money<\/figcaption><\/figure>\n

All types of analysis are about counting in one way or another. The elemental units of information that we count vary with each area of endeavor. Financial analysis is, of course, counting money and categorizing it in a variety of accounts that form a balance sheet or a profit and loss statement.<\/p>\n

 <\/p>\n

Reliability analysis (RA) is counting instances of unreliability<\/h3>\n
\"\"
Slide 2 Counting instances of failure modes and converting the count to a probability density curve.<\/figcaption><\/figure>\n

Just as the penny is the basic unit of finance it is said that the failure mode is the basic unit of maintenance and reliability. We have seen that the failure mode is the lowest common denominator of RCM – the level at which the effects, consequences, and mitigating tasks are determined. At its most basic RA counts the number of instances of a failure mode that occurred within a calendar window. Dividing that number by the accumulated ages of items at\u00a0 failure approximates the average life or mean time to failure (MTTF). We can go a step further in our counting procedure and graph the relationship between reliability and an item’s age. For a given equipment or type of equipment, count up the number of failures that occurred within consecutive age groups (e.g. from 0-1 month old, 1- two months old, etc) . Divide each count by the number of items in the sample.\u00a0 This will approximate the probability density graph shown at the bottom left of Slide 2. In theory the EAM should contain all the needed data to perform reliability analysis.<\/p>\n

CBM “predictive” reliability analysis<\/h2>\n
\"\"
Slide 3 CBM is multi-dimensional reliability analysis<\/figcaption><\/figure>\n

We extend two dimensional RA to include other significant dimensions or condition indicators<\/em>. Slide 3 depicts correlating instances of of failed failure modes with features in an atomic emission spectrometer scan of an oil sample. By counting up the number of times a failure mode occurred and was preceded by a high value of say, the parts per million of iron, we may determine, if a strong correlation was found, a predictive algorithm for potential failure prediction.<\/p>\n

RA for confidence in CBM predictions<\/h3>\n
\"\"
Slide 4 RULE and standard deviation (confidence).<\/figcaption><\/figure>\n

CBM predictions provide little benefit to the maintenance organization without a prediction performance metric for the CBM mitigation task. The CBM task generates a remaining useful life estimate (RULE) which is also known as the conditional mean time to failure. The RULE is determined from the model developed by statistically analyzing past occurrences of failure and potential failure in relation to CBM monitored data. The model monitors its own predictive performance by reporting a standard deviation of the scatter about the RULE (mean). A maintenance department can improve CBM predictive performance (i.e. reduce the standard deviation) by correctly recording information on the EAM work order. This means consistent identification of the failure mode and distinguishing on the work order between instances of failed failure modes and suspended failure modes.<\/p>\n

The six RCM curves. What do they really mean?<\/a><\/h2>\n

\"RcmCurves\"<\/a>See “The RCM Curves (Slides 7-18)<\/a>”\u00a0 In this set of slides we learned that real world data has immense value. Understanding failure behavior depends on having acquired the ages of failure modes at failure and\u00a0 suspension.<\/p>\n

<\/h2>\n

Conditional probability of failure<\/a><\/h2>\n
\"\"
Slide 2 Conditional failure probability confers power.<\/figcaption><\/figure>\n

The most powerful\u00a0information sought by all maintenance engineers and managers boils down to\u00a0the conditional failure probability<\/em>. It is the probability of an item failing in an\u00a0upcoming period of interest given that it is currently in operation. If you knew that the conditional probability of failure of a part or component were unusually high you could channel your\u00a0manpower to intervene propitiously, thereby preempting\u00a0the consequences of a failure in service while\u00a0avoiding waste of\u00a0resources and unnecessary downtime on items where\u00a0failure is not<\/em> imminent.<\/p>\n

What is the conditional probability?<\/h3>\n
\"Slide<\/a>
Slide 3 What is conditional probability?<\/figcaption><\/figure>\n

A little further down we’ll describe how to calculate\u00a0the conditional probability of failure<\/em> of an item. Right now we\u2019ll discuss\u00a0Conditional Probability<\/em> itself and then we\u2019ll define the Conditional Probability of Failure<\/em>.<\/p>\n

Let\u2019s begin with a card experiment. A card is chosen at random from a standard\u00a0deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? The events are said to be dependent because the probability of the second depends on the first.<\/p>\n

A. P(queen on first pick) = 4\/52
\nB. P(jack on 2nd pick given queen on 1st pick) = 4\/51, a higher\u00a0probability than 4\/52<\/p>\n

Then the probability that both events occur P(queen first and then jack)= (4\/52)(4\/51)=4\/663<\/p>\n

The probability of choosing a jack on the second pick given that a queen was chosen on the first pick is called a conditional probability<\/em>.\u00a0The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A) [2<\/a>]<\/sup>.<\/p>\n

When two events, A and B, are dependent, the probability of both occurring (denoted by \u201cA\u2229B\u201d<\/em>)\u00a0will, according to the card experiment, be the product of their probabilities, that is:<\/p>\n

P(A\u2229B) = P(A) \u00b7 P(B|A) ,\u00a0<\/em>or \"P(B|A)=\\frac{P(A)\\cap<\/p>\n

When two events are dependent (the probability of one depends on the other\u2019s occurrence) their probability areas intersect in a Venne graphical representation.<\/p>\n

\"venne\"<\/a>What is the conditional probability of failure?<\/h3>\n
\"Slide<\/a>
Slide 4 What is conditional probability of failure?<\/figcaption><\/figure>\n

Suppose\u00a0the two dependent events were:<\/p>\n

    \n
  1. X>t<\/em>, an\u00a0item survives to time t, X being the time of failure, and<\/li>\n
  2. t\u2264X\u2264t+\u0394t<\/em>, the\u00a0item fails in the interval between\u00a0t and t+\u0394t given event 1.<\/li>\n<\/ol>\n

    As in\u00a0the card experiment the probability of the second event depends on the first. Then the Conditional Probability of Failure is:<\/p>\n

    \"H(t)=P(t\\leq<\/p>\n

    It is the probability of failure in the interval\u00a0between\u00a0t and t+\u0394t (analogous to selecting a\u00a0jack\u00a0on the second\u00a0pick) given that the item has survived to time t (analogous to selecting a\u00a0queen on the first pick).<\/p>\n

    Slide 5 below shows that the conditional failure probability is a special case of the conditional probability where the numerator\u00a0reduces\u00a0simply to P(t\u2264X\u2264t+\u0394t)<\/em>. So that:<\/p>\n

    \"H(t)=P(t\\leq<\/p>\n

    The conditional failure probability is a special case of the conditional probability<\/h3>\n
    \"The<\/a>
    Slide 5 The conditional failure probability is a special case of conditional failure<\/figcaption><\/figure>\n

    X is the failure time. By definition the denominator is the survival or reliability function at time t<\/em>, i.e.\u00a0P(X>t) =<\/em>\u00a0R(t). The Conditional Probability of Failure is a special case of conditional probability\u00a0wherein the numerator is the intersection of two event probabilities, the first being\u00a0entirely contained within the probability space of the second, as depicted in the Venne diagram of Slide 5.<\/p>\n

    Conditional failure probability and reliability<\/h2>\n
    \"Slide<\/a>
    Slide 6 Conditional failure probability and reliability<\/figcaption><\/figure>\n

    Therefore the numerator, which is the intersection of P(t\u2264X\u2264t+\u0394t) <\/em>and \u00a0P(X>t)<\/em>\u00a0reduces to\u00a0simply\u00a0\u00a0P(t\u2264X\u2264t+\u0394t).<\/em>\u00a0Also, by expressing\u00a0P(t\u2264X\u2264t+\u0394t) <\/em>as the difference between the Cumulative Failure Probabilities calculated at t<\/em> and t+\u0394t<\/em>\u00a0<\/em>\u00a0the numerator can be expressed as the change in\u00a0Reliability over the interval \u0394t<\/em>\u00a0as:<\/p>\n

    P(t\u2264X\u2264t+\u0394t) =\u00a0<\/em>F(t+\u0394t) \u2013 F(t) =\u00a0<\/em>1-R(t+\u0394t) \u2013 (1-R(t)) = R(t)-R(t+\u0394t)<\/em><\/p>\n

    where the Cumulative Failure Probability F(t) and the Reliability R(t) \u00a0are complements, i.e. F(t) = 1-R(t), so that<\/p>\n

    \"H(t)=\\frac{R(t)-R(t+\\Delta<\/p>\n

    What about the failure rate?<\/h2>\n
    \"Slide<\/a>
    Slide 7 What about the failure rate?<\/figcaption><\/figure>\n

    We define the failure rate[1<\/a>]<\/sup> h(t) as the limit of the ratio H(t)<\/em>\/\u0394t<\/em> as \u0394t<\/em>\u21920:<\/p>\n

    \"\"<\/a><\/p>\n

    Differentiating \"F(t)=1-R(t)\" we have the density function f(t)<\/em>:<\/p>\n

    \"f(t)=-dR(t)\/dt\":<\/p>\n

    Then<\/p>\n

    \"h(t)=\\frac{f(t)}{R(t)}\"<\/p>\n

    Random failure. Is it really “random”?<\/h2>\n
    \"Random<\/a>
    Random failure is exponential decay<\/figcaption><\/figure>\n

    When something decays or grows “exponentially” it means that it changes regularly by a constant factor. An\u00a0example of exponential growth is the principle in a compound interest bank account which increases\u00a0at regular intervals by a constant factor[1<\/a>]<\/sup>. <\/em><\/p>\n

    It’s not really “random” but rather “age independent”<\/h3>\n

    \"CarExample\"<\/a>Assume that you drive your car normally. You replace\u00a0tires whenever the tread depth falls below the manufacturer’s safety recommendation.\u00a0[2<\/a>]<\/sup>\u00a0Intuitively, we would agree that you’re no more likely to have a punctured\u00a0tire in any one year than in any other.<\/em>[3<\/a>]<\/sup><\/p>\n

    In other words the conditional probability of failure (a flat tire) is constant\u00a0or\u00a0age independent<\/em>. We call this failure pattern “random”.[4<\/a>]<\/sup> \u00a0Assume that the conditional failure probability of a puncture\u00a0in any year is a constant\u00a025%. \u00a0When\u00a0you drive the car off the dealer’s lot for the first time, at that moment the Reliability R1 is 100%. What is the Reliability R2 at the beginning\u00a0of the second\u00a0year? \u00a0In the article here<\/a> we showed the Conditional Probability of Failure to be:<\/p>\n

    \n

    (1) <\/span>   <\/span>\"\begin{align*}<\/p>\n<\/p>\n

    Rearranging and substituting<\/p>\n

    \n

    (2) <\/span>   <\/span>\"\begin{align*}<\/p>\n<\/p>\n

    The reliability at the beginning of year 3 is:<\/p>\n

    \n

    (3) <\/span>   <\/span>\"\begin{align*}<\/p>\n<\/p>\n

    Excel calculation and graph of Reliability and Conditional Failure Probability<\/h3>\n
    \"Slide<\/a>
    Slide 10 Excel calculation and graph of Reliability (Survival) and Conditional Failure Probability<\/figcaption><\/figure>\n

    Repeating the calculation in an Excel spreadsheet\u00a0for each subsequent year reveals\u00a0the exponentially decaying Reliability curve of Slide 10.<\/p>\n

     <\/p>\n

     <\/p>\n

     <\/p>\n

    Conclusion, so what?<\/h2>\n
    \"Slide<\/a>
    Slide 11 So what?<\/figcaption><\/figure>\n

    You have learned most of what you need to know in reliability theory. The real lesson is that the maintenance department, if it is going to leverage the principles of Reliability Analysis to develop optimal and verifiable decision rules, must capture the right<\/em> information in the course of its day-to-day activities. Of prime importance is the distinction between failure mode instances that ended in failure and those that ended in suspension. In the upcoming sections,<\/p>\n

      \n
    1. “Weibull Exercises” and<\/li>\n
    2. “Optimal Preventive Renewal Exercises”<\/li>\n
    3. “The PF interval?”<\/li>\n
    4. “EXAKT Analysis and Model Building”<\/li>\n
    5. “Defeating CBM (reporting suspensions as failures)”<\/li>\n<\/ol>\n

      we will see how good data fuels practical decision making.<\/p>\n\n

        \n\t
      1. [1]<\/sup><\/strong>The factor is 1 plus the interest rate↩<\/a><\/li>\n\t
      2. [2]<\/sup><\/strong>This is an example of applying a Preventive Maintenance strategy in order to gain the desired constant, low conditional failure probability pattern.<\/p>\n

        If we allow the treads to wear beyond the safety depth then tire\u00a0failure would become age related meaning that\u00a0the conditional probability of failure would increase with age\u00a0 conforming to\u00a0Nowlan and Heap’s pattern B<\/a>.↩<\/a><\/li>\n\t

      3. [3]<\/sup><\/strong>Nevertheless the probability of surviving to an\u00a0age t\u00a0decreases with increasing age because, obviously, if you keep driving the car, eventually you will have a flat. This does not contradict the fact (although it seems paradoxical) that the probability of getting a flat in any one year, if you ask the question at the start of the\u00a0year, remains constant.↩<\/a><\/li>\n\t
      4. [4]<\/sup><\/strong>The word “random” when used in the reliability sense differs from the often conjured image of throwing dice. In the latter situation it is impossible to predict the result of the next throw. To the Reliability Engineer, however, “random” means merely that the conditional failure probability in any interval is independent of the item’s age. Therefore, even when an item’s failure behavior is\u00a0random<\/em>, observed condition data can be used to\u00a0predict its failure.↩<\/a><\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

        Reliability Analysis: 2 dimensions (Statistical and probabilistic concepts) Reliability analysis is counting (slides) All types of analysis are about counting in one way or another. The elemental units of information that we count vary with each area of endeavor. Financial analysis is, of course, counting money and categorizing it in a variety of accounts that […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[89],"tags":[],"class_list":["post-5123","post","type-post","status-publish","format-standard","hentry","category-theory-and-definitions"],"_links":{"self":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/5123","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/comments?post=5123"}],"version-history":[{"count":2,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/5123\/revisions"}],"predecessor-version":[{"id":5883,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/5123\/revisions\/5883"}],"wp:attachment":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/media?parent=5123"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/categories?post=5123"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/tags?post=5123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}