The type of production that puts components together is referred to as a(n)

The study of component and process reliability is the basis of many efficiency evaluations in Operations Management discipline. For example, in the calculation of the Overall Equipment Effectiveness (OEE) introduced by Nakajima [1], it is necessary to estimate a crucial parameter called availability. This is strictly related to reliability. Still as an example, consider how, in the study of service level, it is important to know the availability of machines, which again depends on their reliability and maintainability.

Reliability is defined as the probability that a component (or an entire system) will perform its function for a specified period of time, when operating in its design environment. The elements necessary for the definition of reliability are, therefore, an unambiguous criterion for judging whether something is working or not and the exact definition of environmental conditions and usage. Then, reliability can be defined as the time dependent probability of correct operation if we assume that a component is used for its intended function in its design environment and if we clearly define what we mean with "failure". For this definition, any discussion on the reliability basics starts with the coverage of the key concepts of probability.

A broader definition of reliability is that "reliability is the science to predict, analyze, prevent and mitigate failures over time." It is a science, with its theoretical basis and principles. It also has sub-disciplines, all related - in some way - to the study and knowledge of faults. Reliability is closely related to mathematics, and especially to statistics, physics, chemistry, mechanics and electronics. In the end, given that the human element is almost always part of the systems, it often has to do with psychology and psychiatry.

In addition to the prediction of system durability, reliability also tries to give answers to other questions. Indeed, we can try to derive from reliability also the availability performance of a system. In fact, availability depends on the time between two consecutive failures and on how long it takes to restore the system. Reliability study can be also used to understand how faults can be avoided. You can try to prevent potential failures, acting on the design, materials and maintenance.

Reliability involves almost all aspects related to the possession of a property: cost management, customer satisfaction, the proper management of resources, passing through the ability to sell products or services, safety and quality of the product.

This chapter presents a discussion of reliability theory, supported by practical examples of interest in operations management. Basic elements of probability theory, as the sample space, random events and Bayes' theorem should be revised for a deeper understanding.

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2. Reliability basics

The period of regular operation of an equipment ends when any chemical-physical phenomenon, said fault, occurred in one or more of its parts, determines a variation of its nominal performances. This makes the behavior of the device unacceptable. The equipment passes from the state of operation to that of non-functioning.

In Table 1 faults are classified according to their origin. For each failure mode an extended description is given.

Failure causeDescriptionStress, shock, fatigueFunction of the temporal and spatial distribution of the load conditions and of the response of the material. The structural characteristics of the component play an important role, and should be assessed in the broadest form as possible, incorporating also possible design errors, embodiments, material defects, etc..TemperatureOperational variable that depends mainly on the specific characteristics of the material (thermal inertia), as well as the spatial and temporal distribution of heat sources.WearState of physical degradation of the component; it manifests itself as a result of aging phenomena that accompany the normal activities (friction between the materials, exposure to harmful agents, etc..)CorrosionPhenomenon that depends on the characteristics of the environment in which the component is operating. These conditions can lead to material degradation or chemical and physical processes that make the component no longer suitable.

Table 1.

Main causes of failure. The table shows the main cases of failure with a detailed description

To study reliability you need to transform reality into a model, which allows the analysis by applying laws and analyzing its behavior [2]. Reliability models can be divided into static and dynamic ones. Static modelsassume that a failure does not result in the occurrence of other faults. Dynamic reliability, instead, assumes that some failures, so-called primary failures, promote the emergence of secondary and tertiary faults, with a cascading effect. In this text we will only deal with static models of reliability.

In the traditional paradigm of static reliability, individual components have a binary status: either working or failed. Systems, in turn, are composed by an integer numbern of components, all mutually independent. Depending on how the components are configured in creating the system and according to the operation or failure of individual components, the system either works or does not work.

Let’s consider a genericXsystem consisting ofnelements. The static reliability modeling implies that the operating status of thei-thcomponent is represented by the state functionXidefined as:

Xi=1     if the  i-th component works 0     if the  i-th component fails  E1

The state of operation of the system is modeled by the state functionΦX

ΦX=1      if the system works0     if the system fails    E2

The most common configuration of the components is the series system. A series system works if and only if all components work. Therefore, the status of a series system is given by the state function:

ΦX=∏i=1nXi=mini∈1,2,…,n⁡XiE3

where the symbol∏indicates the product of the arguments.

System configurations are often represented graphically with Reliability Block Diagrams (RBDs) where each component is represented by a block and the connections between them express the configuration of the system. The operation of the system depends on the ability to cross the diagram from left to right only by passing through the elements in operation. Figure 1 contains the RBD of a four components series system.

Figure 1.

Reliability block diagram for a four components (1,2,3,4) series system.

The second most common configuration of the components is the parallel system. A parallel system works if and only if at least one component is working. A parallel system does not work if and only if all components do not work. So, ifΦ-Xis the function that represents the state of not functioning of the system andX-iindicates the non-functioning of thei-thelement, you can write:

Accordingly, the state of a parallel system is given by the state function:

ΦX=1-∏i=1n1-Xi=∐i=1nXi=maxi∈1,2,…,n⁡XiE5

where the symbol∐indicates the complement of the product of the complements of the arguments. Figure 2 contains a RBD for a system of four components arranged in parallel.

Figure 2.

Parallel system. The image represents the RBD of a system of four elements (1,2,3,4) arranged in a reliability parallel configuration.

Another common configuration of the components is the series-parallel systems. In these systems, components are configured using combinations in series and parallel configurations. An example of such a system is shown in Figure 3.

State functions for series-parallel systems are obtained by decomposition of the system. With this approach, the system is broken down into subsystems or configurations that are in series or in parallel. The state functions of the subsystems are then combined appropriately, depending on how they are configured. A schematic example is shown in Figure 4.

Figure 3.

Series-parallel system. The picture shows the RBD of a system due to the series-parallel model of 9 elementary units.

Figure 4.

Calculation of the state function of a series-parallel. Referring to the configuration ofFigure 3, the state function of the system is calculated by first making the state functions of the parallel of{1,2}, of{3,4,5}and of{6,7,8 , 9}. Then we evaluate the state function of the series of the three groups just obtained.

A particular component configuration, widely recognized and used, is the parallelkout ofn. A systemkout ofnworks if and only if at leastkof thencomponents works. Note that a series system can be seen as a systemnout ofnand a parallel system is a system 1 out ofn. The state function of a systemkout ofnis given by the following algebraic system:

ΦX=1      if ∑i=1nXi≥k0      otherwise      E6

The RBD for a systemkout ofnhas an appearance identical to the RBD schema of a parallel system ofncomponents with the addition of a label "kout ofn". For other more complex system configurations, such as the bridge configuration (see Figure 5), we may use more intricate techniques such as the minimal path set and the minimal cut set, to construct the system state function.

A Minimal Path Set - MPS is a subset of the components of the system such that the operation of all the components in the subset implies the operation of the system. The set is minimal because the removal of any element from the subset eliminates this property. An example is shown in Figure 5.

Figure 5.

Minimal Path Set. The system on the left contains the minimal path set indicated by the arrows and shown in the right part. Each of them represents a minimal subset of the components of the system such that the operation of all the components in the subset implies the operation of the system.

A Minimal Cut Set - MCS is a subset of the components of the system such that the failure of all components in the subset does not imply the operation of the system. Still, the set is called minimal because the removal of any component from the subset clears this property (see Figure 6).

Figure 6.

Minimal Cut Set. The system of the left contains the minimal cut set, indicated by the dashed lines, shown in the right part. Each of them represents a minimum subset of the components of the system such that the failure of all components in the subset does not imply the operation of the system.

MCS and MPS can be used to build equivalent configurations of more complex systems, not referable to the simple series-parallel model. The first equivalent configuration is based on the consideration that the operation of all the components, in at least a MPS, entails the operation of the system. This configuration is, therefore, constructed with the creation of a series subsystem for each path using only the minimum components of that set. Then, these subsystems are connected in parallel. An example of an equivalent system is shown in Figure 7.

Figure 7.

Equivalent configurations with MPS. You build a series subsystem for each MPS. Then such subsystems are connected in parallel.

The second equivalent configuration, is based on the logical principle that the failure of all the components of any MCS implies the fault of the system. This configuration is built with the creation of a parallel subsystem for each MCS using only the components of that group. Then, these subsystems are connected in series (see Figure 8).

Figure 8.

Equivalent configurations with MCS. You build a subsystem in parallel for each MCS. Then the subsystems are connected in series.

After examining the components and the status of the system, the next step in the static modeling of reliability is that of considering the probability of operation of the component and of the system.

The reliabilityRiof thei-thcomponent is defined by:

while the reliability of the systemRis defined as in equation 8:

The methodology used to calculate the reliability of the system depends on the configuration of the system itself. For a series system, the reliability of the system is given by the product of the individual reliability (law of Lusser, defined by German engineer Robert Lusser in the 50s):

R=∏i=1nRi     since     R=P⋂i=1nXi=1=∏i=1nPXi=1=∏i=1nRiE9

For an example, see Figure 9.

Figure 9.

serial system consisting of 4 elements with reliability equal to0.98, 0.99, 0.995and0.975. The reliability of the whole system is given by their product:R = 0.98 · 0.99 · 0.995 · 0.975 = 0.941

For a parallel system, reliability is:

In fact, from the definition of system reliability and by the properties of event probabilities, it follows:

R=P⋃i=1nXi=1=1-P⋂i=1nXi=0=1-∏i=1nPXi=0==1-∏i=1n1-PXi=1=1-∏i=1n1-Ri=∐i=1nRiE11

In many parallel systems, components are identical. In this case, the reliability of a parallel system withnelements is given by:

Figure 10.

A parallel system consisting of 4 elements with the same reliability of 0.85. The system reliability s given by their co-product:1-1-0.854=0.9995.

For a series-parallel system, system reliability is determined using the same approach of decomposition used to construct the state function for such systems. Consider, for instance, the system drawn in Figure 11, consisting of 9 elements with reliabilityR1=R2=0.9; R3=R4=R5=0.8andR6=R7=R8=R9=0.7. Let’s calculate the overall reliability of the system.

Figure 11.

The system consists of three groups of blocks arranged in series. Each block is, in turn, formed by elements in parallel. First we must calculateR1,2=1-1-0.82=0.99. So it is possible to estimatedR3,4,5=1-1-0.83=0.992. Then we must calculate the reliability of the last parallel blockR6,7,8,9=1-1-0.74=0.9919. Finally, we proceed to the series of the three blocks:R=R1,2∙R3,4,5∙R6,7,8,9=0.974.

To calculate the overall reliability, for all other types of systems which can’t be brought back to a series-parallel scheme, it must be adopted a more intensive calculation approach [3] that is normally done with the aid of special software.

Reliability functions of the system can also be used to calculate measures of reliability importance.

These measurements are used to assess which components of a system offer the greatest opportunity to improve the overall reliability. The most widely recognized definition of reliability importanceI'iof the components is the reliability marginal gain, in terms of overall system rise of functionality, obtained by a marginal increase of the component reliability:

For other system configurations, an alternative approach facilitates the calculation of reliability importance of the components. LetR1ibe the reliability of the system modified so thatRi=1andR0ibe the reliability of the system modified withRi=0, always keeping unchanged the other components. In this context, the reliability importanceIiis given by:

In a series system, this formulation is equivalent to writing:

Thus, the most important component (in terms of reliability) in a series system is the less reliable. For example, consider three elements of reliabilityR1=0.9,R2=0.8eR3=0.7. It is therefore:I1=0.8∙0.7=0.56,I2=0.9∙0.7=0.63andI3=0.9·0.8=0.72which is the higher value.

If the system is arranged in parallel, the reliability importance becomes as follows:

It follows that the most important component in a parallel system is the more reliable. With the same data as the previous example, this time having a parallel arrangement, we can verify Eq. 16 for the first item:I1=R11-R01=1-1-1·1-0.8∙1-0.7-1-1-0·1-0.8∙1-0.7=1-0-1+1-0.8∙1-0.7=1-0.8∙1-0.7.

For the calculation of the reliability importance of components belonging to complex systems, which are not attributable to the series-parallel simple scheme, reliability of different systems must be counted. For this reason the calculation is often done using automated algorithms.

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3. Fleet reliability

Suppose you have studied the reliability of a component, and found that it is 80% for a mission duration of 3 hours. Knowing that we have 5 identical items simultaneously active, we might be interested in knowing what the overall reliability of the group would be. In other words, we want to know what is the probability of having a certain number of items functioning at the end of the 3 hours of mission. This issue is best known as fleet reliability.

Consider a set ofmidentical and independent systems in a same instant, each having a reliabilityR. The group may represent a set of systems in use, independent and identical, or could represent a set of devices under test, independent and identical. A discrete random variable of great interest reliability isN, the number of functioning items. Under the assumptions specified,Nis a binomial random variable, which expresses the probability of a Bernoulli process. The corresponding probabilistic model is, therefore, the one that describes the extraction of balls from an urn filled with a known number of red and green balls. Suppose that the percentageRof green balls is coincident with the reliability after 3 hours. After each extraction from the urn, the ball is put back in the container. Extraction is repeatedmtimes, and we look for the probability of findingngreen. The sequence of random variables thus obtained is a Bernoulli process of which each extraction is a test. Since the probability of obtainingNsuccesses inmextractions from an urn, with restitution of the ball, follows the binomial distributionBm,RB, the probability mass function ofNis the well-known:

The expected value ofNis given by:EN=μN=m∙Rand the standard deviation is:σN=m∙R∙1-R.

Let’s consider, for example, a corporate fleet consisting of 100 independent and identical systems. All systems have the same mission, independent from the other missions. Each system has a reliability of mission equal to 90%. We want to calculate the average number of missions completed and also what is the probability that at least 95% of systems would complete their mission. This involves analyzing the distribution of the binomial random variable characterized byR = 0.90andm = 100. The expected value is given byEN=μN=100∙0.9=90.

The probability that at least 95% of the systems complete their mission can be calculated as the sum of the probabilities that complete their mission 95, 96, 97, 98, 99 and 100 elements of the fleet:

PN≥n=∑n=95100m!n!m-n!Rn1-Rm-n=0,058E18

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4. Time dependent reliability models

When reliability is expressed as a function of time, the continuous random variable, not negative, of interest isT, the instant of failure of the device. Letf(t)be the probability density function ofT, and letF(t)be the cumulative distribution function ofT.F(t)is also known as failure function or unreliability function [4].

In the context of reliability, two additional functions are often used: the reliabilityand the hazard function. Let’s define ReliabilityR(t)as the survival function:

The Mean Time To Failure - MTTFis defined as the expected value of the failure time:

Integrating by parts, we can prove the equivalent expression:

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5. Hazard function

Another very important function is the hazard function, denoted byλ(t), defined as the trend of the instantaneous failure rate at timetof an element that has survived up to that timet. The failure rate is the ratio between the instantaneous probability of failure in a neighborhood oft- conditioned to the fact that the element is healthy int- and the amplitude of the same neighborhood.

The hazard functionλ(t)[5] coincides with the intensity functionz(t)of a Poisson process. The hazard function is given by:

λt=limΔt→0⁡Pt≤T

Thanks to Bayes' theorem, it can be shown that the relationship between the hazard function, density of probability of failure and reliability is the following:

Thanks to the previous equation, with some simple mathematical manipulations, we obtain the following relation:

In fact, sincelnR0=ln1=0,we have:

Rt=ftλt=1λt∙dFtdt=-1λt∙dRtdt→1RtdRt=-λtdt→lnRt-lnR0=-∫0tλuduE25

From equation 24 derive the other two fundamental relations:

Ft=1-e-∫0tλu∙duft=λt∙e-∫0tλu∙duE26

The most popular conceptual model of the hazard function is the bathtub curve. According to this model, the failure rate of the device is relatively high and descending in the first part of the device life, due to the potential manufacturing defects, called early failures. They manifest themselves in the first phase of operation of the system and their causes are often linked to structural deficiencies, design or installation defects. In terms of reliability, a system that manifests infantile failures improves over the course of time.

Later, at the end of the life of the device, the failure rate increases due to wear phenomena. They are caused by alterations of the component for material and structural aging. The beginning of the period of wear is identified by an increase in the frequency of failures which continues as time goes by. The wear-out failuresoccur around the average age of operating; the only way to avoid this type of failure is to replace the population in advance.

Between the period of early failures and of wear-out, the failure rate is about constant: failures are due to random events and are called random failures. They occur in non-nominal operating conditions, which put a strain on the components, resulting in the inevitable changes and the consequent loss of operational capabilities. This type of failure occurs during the useful life of the system and corresponds to unpredictable situations. The central period with constant failure rate is called useful life. The juxtaposition of the three periods in a graph which represents the trend of the failure rate of the system, gives rise to a curve whose characteristic shape recalls the section of a bathtub, as shown in Figure 12.

Figure 12.

Bathtub curve. The hazard function shape allows us to identify three areas: the initial period of the early failures, the middle time of the useful life and the final area of wear-out.

The most common mathematical classifications of the hazard curve are the so called Constant Failure Rate - CFR, Increasing Failure Rate - IFRand Decreasing Failure Rate - DFR.

The CFR model is based on the assumption that the failure rate does not change over time. Mathematically, this model is the most simple and is based on the principle that the faults are purely random events. The IFR model is based on the assumption that the failure rate grows up over time. The model assumes that faults become more likely over time because of wear, as is frequently found in mechanical components. The DFR model is based on the assumption that the failure rate decreases over time. This model assumes that failures become less likely as time goes by, as it occurs in some electronic components.

Since the failure rate may change over time, one can define a reliability parameter that behaves as if there was a kind of counter that accumulates hours of operation. The residual reliabilityfunctionRt+t0|t0, in fact, measures the reliability of a given device which has already survived a determined timet0. The function is defined as follows:

Applying Bayes' theorem we have:

PT>t+t0|T>t0=PT>t0|T>t+t0∙PT>t+t0PT>t0E28

And, given thatPT>t0|T>t+t0=1, we obtain the final expression, which determines the residual reliability:

The residual Mean Time To Failure– residual MTTFmeasures the expected value of the residual life of a device that has already survived a timet0:

MTTFt0=ET-t0|T>t0=∫0∞Rt+t0|t0∙dtE30

For an IFR device, the residual reliability and the residual MTTF, decrease progressively as the device accumulates hours of operation. This behavior explains the use of preventive actions to avoid failures. For a DFR device, both the residual reliability and the residual MTTF increase while the device accumulates hours of operation. This behavior motivates the use of an intense running (burn-in) to avoid errors in the field.

The Mean Time To Failure–MTTF, measures the expected value of the life of a device and coincides with the residual time to failure, wheret0=0. In this case we have the following relationship:

MTTF=MTTF0=ET|T>0=∫0∞Rt∙dtE31

The characteristic lifeof a device is the timetCcorresponding to a reliabilityRtCequal to1e, that is the time for which the area under the hazard function is unitary:

RtC=e-1=0,368 →RtC=∫0tCλu∙du=1E32

Let us consider a CFR device with a constant failure rateλ. The time-to-failure is an exponential random variable. In fact, the probability density function of a failure, is typical of an exponential distribution:

ft=λt∙e-∫0tλu∙du=λe-λ∙tE33

The corresponding cumulative distribution functionF(t)is:

Ft=∫-∞tfzdz=∫-∞tλe-λ∙zdz=1-e-λ∙tE34

The reliability functionR(t)is the survival function:

For CFR items, the residual reliability and the residual MTTF both remain constant when the device accumulates hours of operation. In fact, from the definition of residual reliability,∀t0∈0,∞, we have:

Rt+t0|t0=Rt+t0Rt0=e-λ∙t+t0e-λ∙t0=e-λ∙t+t0+λ∙t0=e-λ∙t=RtE36

Similarly, for the residual MTTF, is true the invariance in time:

MTTFt0=∫0∞Rt+t0|t0∙dt=∫0∞Rt∙dt     ∀t0∈0,∞E37

This behavior implies that the actions of prevention and running are useless for CFR devices. Figure 13 shows the trend of the functionft=λ∙e-λ∙tand of the cumulative distribution functionFt=1-e-λ∙tfor a constant failure rateλ=1. In this case, sinceλ=1, the probability density function and the reliability function, overlap:ft=Rt=e-t.

Figure 13.

Probability density function and cumulative distribution of an exponential function. In the figure is seen the trend offt=λ∙e-λ∙tand offt=λ∙e-λ∙twithλ=1.

The probability of having a fault, not yet occurred at timet, in the nextdt, can be written as follows:

Recalling the Bayes' theorem, in which we consider the probability of an hypothesis H, being known the evidence E:

we can replace the evidence E with the fact that the fault has not yet taken place, from which we obtainP(E)→P(T>t). We also exchange the hypothesis H with the occurrence of the fault in the neighborhood oft, obtainingPH→ Pt

Ptt=PT>t|ttE40

SincePT>t|t

Ptt=Ptt=ftdte-λ∙t=λe-λ∙tdte-λ∙t=λ∙dtE41

As can be seen, this probability does not depend ont, i.e. it is not function of the life time already elapsed. It is as if the component does not have a memory of its own history and it is for this reason that the exponential distribution is called memoryless.

The use of the constant failure rate model, facilitates the calculation of the characteristic life of a device. In fact for a CFR item,tCis the reciprocal of the failure rate. In fact:

Therefore, the characteristic life, in addition to be calculated as the time valuetCfor which the reliability is 0.368, can more easily be evaluated as the reciprocal of the failure rate.

The definition of MTTF, in the CFR model, can be integrated by parts and give:

MTTF=∫0∞Rt∙dt=∫0∞e-λ∙t∙dt=-1λe-λ∙t∞0=-0λ+1λ=1λE43

In the CFR model, then, the MTTF and the characteristic life coincide and are equal to1λ.

Let us consider, for example, a component with constant failure rate equal toλ=0.0002failures per hour. We want to calculate the MTTF of the component and its reliability after10000hours of operation. We’ll calculate, then, what is the probability that the component survives other10000hours. Assuming, finally, that it has worked without failure for the first6000hours, we’ll calculate the expected value of the remaining life of the component.

From equation 43 we have:

MTTF=1λ=10.0002failuresh=5000 hE44

For the law of the reliabilityRt=e-λ∙t, you get the reliability at10000hours:

R10000=e-0.0002∙10000=0.135E45

The probability that the component survives other10000hours, is calculated with the residual reliability. Knowing that this, in the model CFR, is independent from time, we have:

Rt+t0|t0=Rt→R20000|10000=R10000=0.135E46

Suppose now that it has worked without failure for6000hours. The expected value of the residual life of the component is calculated using the residual MTTF, that is invariant. In fact:

MTTFt0=∫0∞Rt+t0|t0∙dt→MTTF6000=∫0∞Rt+6000|6000∙dt=∫0∞Rt∙dt=MTTF=5000hE47

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6. CFR in series

Let us considerndifferent elements, each with its own constant failure rateλiand reliabilityRi=e-λi∙t, arranged in series and let us evaluate the overall reliabilityRS. From equation 9 we have:

RS=∏i=1nRi=∏i=1ne-λi∙t=e-∑i=1nλi∙tE48

Since the reliability of the overall system will take the form of the typeRS=e-λs∙t, we can conclude that:

RS=e-∑i=1nλi∙t=e-λs∙t→λs=∑i=1nλiE49

In a system of CFR elements arranged in series, then, the failure rate of the system is equal to the sum of failure rates of the components. The MTTF can thus be calculated using the simple relation:

For example, let me show the following example. A system consists of a pump and a filter, used to separate two parts of a mixture: the concentrate and the squeezing. Knowing that the failure rate of the pump is constant and isλP=1,5∙10-4failures per hour and that the failure rate of the filter is also CFR and isλF=3∙10-5, let’s try to assess the failure rate of the system, the MTTF and the reliability after one year of continuous operation.

To begin, we compare the physical arrangement with the reliability one, as represented in the following figure:

Figure 14.

physical and reliability modeling of a pump and a filter producing orange juice.

As can be seen, it is a simple series, for which we can write:

λs=∑i=1nλi=λP+λF=1.8∙10-4failureshE51

MTTFis the reciprocal of the failure rate and can be written:

MTTF=1λs=11.8∙10-4=5,555hE52

As a year of continuous operation is24·365=8,760hours, the reliability after one year is:

RS=e-λs∙t=e-1.8∙10-4·8760=0.2066

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7. CFR in parallel

If two components arranged in parallel are similar and have constant failure rate λ, the reliability of the systemRPcan be calculated with equation 10, whereinRCis the reliability of the componentRC=e-λt:

RP=1-∏i=121-Ri=1-1-R12=2RC-RC2=2e-λ∙t-e-2λ∙tE53

The calculation of the MTTF leads toMTTF=32λ. In fact we have:

MTTF=∫0∞Rt∙dt=∫0∞2e-λ∙t-e-2λ∙t∙dt=-2λe-λt+12λe-2λt∞0=2λ0-1+12λ0-1=32λE54

Therefore, the MTTF increases compared to the single component CFR. The failure rate of the parallel systemλP, reciprocal of the MTTF, is:

As you can see, the failure rate is not halved, but was reduced by one third.

For example, let us consider a safety system which consists of two batteries and each one is able to compensate for the lack of electric power of the grid. The two generators are equal and have a constant failure rateλB=9∙10-6failures per hour. We’d like to calculate the failure rate of the system, the MTTF and reliability after one year of continuous operation.

As in the previous case, we start with a reliability block diagram of the problem, as visible in Figure 15.

Figure 15.

Physical and reliability modeling of an energy supply system.

It is a parallel arrangement, for which the following equation is applicable:

λP=23λ=239∙10-6=6∙10-6guastihE56

The MTTF is the reciprocal of the failure rate and is:

MTTF=1λp=16∙10-6=166,666hE57

As a year of continuous operation is24·365=8,760hours, the reliability after one year is:

RP=e-λp∙t=e-6∙10-6·8,760=0.9488E58

It is interesting to calculate the reliability of a system of identical elements arranged in a parallel configurationkout ofn. The system is partially redundant since a group ofkelements is able to withstand the load of the system. The reliability is:

Rk out of n=Pk≤j≤n=∑j=knnjRj∙1-Rn-jE59

Let us consider, for example, three electric generators, arranged in parallel and with failure rate λ=9·10-6. In order for the system to be active, it is sufficient that only two items are in operation. Let’s get the reliability after one year of operation.

We’ll have:n=3,k=2. So, after a year of operation (t=8760 h), reliability can be calculated as follows:

R2 out of 3=∑j=233jRj∙1-R3-j=32e-2λt1-e-λt3-2+33e-λt1-e-λt3-3==3!2!3-2!e-2λt∙1-e-λt3-2+3!3!3-3!e-λt∙1-e-λt3-3==3∙e-2λt∙1-e-λt3-2+1∙e-λt∙1-e-λt3-3=0.963

A particular arrangement of components is that of the so-called parallel with stand-by: the second component comes into operation only when the first fails. Otherwise, it is idle.

Figure 16.

RBD diagram of a parallel system with stand-by. When component 1 fails, the switch S activates component 2. For simplicity, it is assumed that S is not affected by faults.

If the components are similar, thenλ1=λ2. It’s possible to demonstrate that for the stand-by parallel system we have:

Thus, in parallel with stand-by, the MTTF is doubled.=

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8. Repairable systems

The devices for which it is possible to perform some operations that allow to reactivate the functionality, deserve special attention. A repairable system [6] is a system that, after the failure, can be restored to a functional condition from any action of maintenance, including replacement of the entire system. Maintenance actions performed on a repairable system can be classified into two groups: Corrective Maintenance - CMand Preventive Maintenance - PM. Corrective maintenance is performed in response to system errors and might correspond to a specific activity of both repair of replacement. Preventive maintenance actions, however, are not performed in response to the failure of the system to repair, but are intended to delay or prevent system failures. Note that the preventive activities are not necessarily cheaper or faster than the corrective actions.

As corrective actions, preventive activities may correspond to both repair and replacement activities. Finally, note that the actions of operational maintenance (servicing) such as, for example, put gas in a vehicle, are not considered PM [7].

Preventative maintenance can be divided into two subcategories: scheduledand on-condition. Scheduled maintenance (hard-time maintenance) consists of routine maintenance operations, scheduled on the basis of precise measures of elapsed operating time.

Condition-Based Maintenance - CBM[8] (also known as predictive maintenance) is one of the most widely used tools for monitoring of industrial plants and for the management of maintenance policies. The main aim of this approach is to optimize maintenance by reducing costs and increasing availability. In CBM it is necessary to identify, if it exists, a measurable parameter, which expresses, with accuracy, the conditions of degradation of the system. What is needed, therefore, is a physical system of sensors and transducers capable of monitoring the parameter and, thereby, the reliability performance of the plant. The choice of the monitored parameter is crucial, as is its time evolution that lets you know when maintenance action must be undertaken, whether corrective or preventive.

To adopt a CBM policy requires investment in instrumentation and prediction and control systems: you must run a thorough feasibility study to see if the cost of implementing the apparatus are truly sustainable in the system by reducing maintenance costs.

The CBM approach consists of the following steps:

• group the data from the sensors;

• diagnose the condition;

• estimate the Remaining Useful Life – RUL;

• decide whether to maintain or to continue to operate normally.

CBM schedule is modeled with algorithms aiming at high effectiveness, in terms of cost minimization, being subject to constraints such as, for example, the maximum time for the maintenance action, the periods of high production rate, the timing of supply of the pieces parts, the maximization of the availability and so on.

In support of the prognosis, it is now widespread the use of diagrams that do understand, even graphically, when the sensor outputs reach alarm levels. They also set out the alert thresholds that identify ranges of values for which maintenance action must arise [9].

Starting from a state of degradation, detected by a measurement at the timetk, we calculate the likelihood that the system will still be functioning within the next instant of inspectiontk+1. The choice to act with a preventive maintenance is based on the comparison of the expected value of the cost of unavailability, with the costs associated with the repair. Therefore, you create two scenarios:

• continue to operate: if we are in the area of not alarming values. It is also possible that being in the area of preventive maintenance, we opt for a postponement of maintenance because it has already been established replacement intervention within a short interval of time

• stop the task: if we are in the area of values above the threshold established for preventive maintenance of condition.

The modeling of repairable systems is commonly used to evaluate the performance of one or more repairable systems and of the related maintenance policies. The information can also be used in the initial phase of design of the systems themselves.

In the traditional paradigm of modeling, a repairable system can only be in one of two states: working (up) or inoperative (down). Note that a system may not be functioning not only for a fault, but also for preventive or corrective maintenance.

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9. Availability

Availability may be generically be defined as the percentage of time that a repairable system is in an operating condition. However, in the literature, there are four specific measures of repairable system availability. We consider only the limit availability, defined with the limit of the probabilityAtthat the system is working at timet, whenttends to infinity.

The limit availability just seen is also called intrinsic availability, to distinguish it from the technical availability, which also includes the logistics cycle times incidental to maintenance actions (such as waiting for the maintenance, waiting for spare parts, testing...), and from the operational availabilitythat encompasses all other factors that contribute to the unavailability of the system such as time of organization and preparation for action in complex and specific business context [10].

The models of the impact of preventive and corrective maintenance on the age of the component, distinguish in perfect, minimal and imperfect maintenance. Perfect maintenance(perfect repair) returns the system as good as newafter maintenance. The minimal repair, restores the system to a working condition, but does not reduce the actual age of the system, leaving it as bad as old. The imperfect maintenance refers to maintenance actions that have an intermediate impact between the perfect maintenance and minimal repair.

The average duration of maintenance activity is the expected value of the probability distribution of repair time and is called Mean Time To Repair - MTTRand is closely connected with the concept of maintainability. This consists in the probability of a system, in assigned operating conditions, to be reported in a state in which it can perform the required function.

Figure 17 shows the state functions of two repairable systems with increasing failure rate, maintained with perfect and minimal repair.

Figure 17.

perfect maintenance vs minimal repair. In figure are represented the state functions of two systems both with IFR.Y(t)is equal to 1 when the system wotks, otherwise it’s 0. The left system is subject to a policy of perfect repair and shows homogeneous durations of the periods of operation. The right system adopts the minimal repair for which the durations of the periods of operation are reducing as time goes by.

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10. The general substitution model

The general substitution model, states that the failure time of a repairable system is an unspecified random variable. The duration of corrective maintenance (perfect) is also a random variable. In this model it is assumed that preventive maintenance is not performed.

Let’s denote byTithe duration of thei-thinterval of operation of the repairable system. For the assumption of perfect maintenance (as good as new),T1,T2,…,Ti,…,Tnis a sequence of independent and identically distributed random variables.

Let us now designate withDithe duration of thei-thcorrective maintenance action and assume that these random variables are independent and identically distributed. Therefore, each cycle (whether it is an operating cycle or a corrective maintenance action) has an identical probabilistic behavior, and the completion of a maintenance action coincides with time when system state returns operating

Regardless of the probability distributions governingTiandDi, the fundamental result of the general pattern of substitution is as follows:

A=ETiETi+EDi=MTTFMTTF+MTTR=MTTFMTBFE62

11. The substitution model for CFR

Let us consider the special case of the general substitution model whereTiis an exponential random variable with constant failure rateλ. Let alsoDibe an exponential random variable with constant repair rateμ. Since the reparable system has a constant failure rate (CFR), we know that aging and the impact of corrective maintenance are irrelevant on reliability performance. For this system it can be shown that the limit availability is:

Let us analyze, for example, a repairable system, subject to a replacement policy, with failure and repair times distributed according to negative exponential distribution. MTTF=1000 hours and MTTR=10 hours.

Let’s calculate the limit availability of the system. The formulation of the limit availability in this system is given by eq. 63, so we have:

A=μλ+μ=11011000+110=0,10.101=0.990E64

This means that the system is available for 99% of the time.

12. General model of minimal repair

After examining the substitution model, we now want to consider a second model for repairable system: the general model of minimal repair. According to this model, the time of system failure is a random variable. Corrective maintenance is instantaneous, the repair is minimal, and not any preventive activity is performed.

The times of arrival of faults, in a repairable system corresponding to the general model of minimal repair, correspond to a process of random experiments, each of which is regulated by the same negative exponential distribution. As known, having neglected the repair time, the number of faults detected by timet,Nt, t≥0, is a non-homogeneous Poisson process, described by the Poisson distribution.

13. Minimal repair with CFR

A well-known special case of the general model of minimal repair, is obtained if the failure timeTis a random variable with exponential distribution, with failure rate λ.

In this case, the general model of minimal repair is simplified because the numberENtof faults that occur within the timet:Nt, t≥0is described by a homogeneous Poisson process with intensityzt=λ, and is:

ENt=μNt=Zt=∫0tzu∙du=∫0tλ∙du=λtE65

If, for example, we considerλ=0.1faults/hour, we obtain the following values at time 100, 1000 and 10000:E[N(100)]=0,1∙100=10; E[N(1000)]=0,1∙1000=100; E[N(10000)]=0,1∙10000=1000. It should be noted, as well, a linear trend of the expected number of failures given the width of the interval taken.

Finally, we can obtain the probability mass function ofN(t), being a Poisson distribution:

PNt=n=Ztnn!e-Zt=λtnn!e-λtE66

Also, the probability mass function ofNt+s-Ns, that is the number of faults in a range of amplitude tshifted forward of s, is identical:

Since the two values are equal, the conclusion is that in the homogeneous Poisson process (CFR), the number of faults in a given interval depends only on the range amplitude.

The behavior of a Poisson mass probability distribution, with rate equal to 5 faults each year, representing the probability of havingn∈Nfaults within a year, is shown in Figure 18.

Figure 18.

Poisson distribution. In the diagram you can see the probability of havingNfaults within a year, having a homogeneous Poisson process with a rate of 5 faults each year.

Since in the model of minimal repair with CFR, repair time is supposed to be zero (MTTR = 0), the following relation applies:

MTBF=MTTF+MTTR=MTTF=1λE68

Suppose that a system, subjected to a repair model of minimal repair, shows failures according to a homogeneous Poisson process with failure rateλ= 0.0025failures per hour. We’d like to estimate the average number of failures that the system will have during 5000 hours. Then, determine the probability of having not more than 15 faults in a operation period of 5000 hours.

The estimate of the average number of failures in 5000 hours, can be carried out with the expected value function:

ENt=λ∙t →  EN5000=0.0025failuresh∙5000h=12.5failuresE69

The probability of having not more than 15 faults in a period of 5000 hours of operation, is calculated with the sum of the probability mass function evaluated between 0 and 15:

PN5000≤15=∑n=015λtnn!e-λ∙t=∑n=01512.5nn!e-12.5=0.806E70

14. Minimal repair: Power law

A second special case of the general model of minimal repair, is obtained if the failure timeTis a random variable with a Weibull distribution, with shape parameterβand scale parameterα.

In this case the sequence of failure times is described by a Non-Homogeneous Poisson Process - NHPPwith intensityz(t)equal to the probability density function of the Weibull distribution:

Since the cumulative intensity of the process is defined by:

the cumulative function is:

Zt=∫0tβαβuβ-1∙du=βαβ∙uββ0t=tβαβ=tαβE73

As it can be seen, the average number of faults occurring within the timet≥0of this not homogeneous poissonian processENt=Zt, follows the so-called power law.

Ifβ>1, it means that the intensity functionz(t)increases and, being this latter the expression of the average number of failures, it means that faults tend to occur more frequently over time. Conversely, ifβ<1, faults decrease over time.

In fact, if we take  α=10hours (λ=0.1failures/h) andβ=2, we have:EN100=0.1∙1002=100=102;EN1000=0.1∙10002=10000=1002;EN10000=0.1∙100002=1000000=10002. We can observe a trend no longer linear but increasing according to a power law of a multiple of the time width considered.

The probability mass function ofN(t)thus becomes:

PNt=n=Ztnn!e-Zt=tαβ∙nn!e-tαβE74

For example, let us consider a system that fails, according to a power law, havingβ=2.2andα=1500hours. What is the average number of faults occurring during the first 1000 hours of operation? What is the probability of having two or more failures during the first 1000 hours of operation? Which is the average number of faults in the second 1000 hours of operation?

The average number of failures that occur during the first 1000 hours of operation, is calculated with the expected value of the distribution:

ENt=μNt=Zt=tαβ→EN1000=100015002.2=0.41E75

The probability of two or more failures during the first 1000 hours of operation can be calculated as complementary to the probability of having zero or one failure:

PN1000≥2=1-PN1000<2=1-∑n=01tαβ∙nn!e-tαβ=1-0.4100!e-0.41-0.4111!e-0.41=1-0.663-0.272=0.064E76

The average number of faults in the succeeding 1000 hours of operation is calculated using the equation:

that, in this case, is:

EN2000-N1000=Z2000-Z1000=1.47E78

15. Conclusion

After seeing the main definitions of reliability and maintenance, let's finally see how we can use reliability knowledge also to carry out an economic optimization of replacement activities.

Consider a process that follows the power law withβ>1. As time goes by, faults begin to take place more frequently and, at some point, it will be convenient to replace the system.

Let us define withτthe time when the replacement (here assumed instantaneous) takes place. We can build a cost model to determine the optimal preventive maintenance timeτ*which optimizes reliability costs.

Let’s denote byCfthe cost of a failure and withCrthe cost of replacing the repairable system.

If the repairable system is replaced everyτtime units, in that time we will have the replacement costsCrand so many costs of failureCfas how many are the expected number of faults in the time range(0;τ]. The latter quantity coincides with the expected value of the number of faultsE[N(τ)].

The average cost per unit of timec(τ), in the long term, can then be calculated using the following relationship:

Then follows:

Differentiatingc(τ)with respect toτand placing the differential equal to zero, we can find the relative minimum of costs, that is, the optimal timeτ*of preventive maintenance. Manipulating algebraically we obtain the following final result:

Consider, for example, a system that fails according to a Weibull distribution withβ=2.2andα=1500hours. Knowing that the system is subject to replacement instantaneous and that the cost of a faultCf=2500 €and the cost of replacingCr=18000 €, we want to evaluate the optimal interval of replacement.

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