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Study on Preventive Maintenance for Priority Standby Redundant System ⁎
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th Modelling, Management and 9th IFAC IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Management and Available online at www.sciencedirect.com Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, Germany, August August 28-30, 28-30, 2019 2019 Berlin, Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 183–188
Study on Preventive Maintenance for Study Study on on Preventive Preventive Maintenance Maintenance for for⋆ Priority Standby Redundant System Study on Preventive Maintenance for⋆⋆ Priority Standby Redundant System Priority Standby Redundant System Priority Standby Redundant System ⋆∗∗∗ ∗ ∗∗
Ryosuke Yasuhiko Takemoto ∗∗∗ Ryosuke Hirata Hirata ∗∗ Ikuo Ikuo Arizono Arizono ∗∗ ∗∗ Yasuhiko Takemoto ∗∗∗ ∗∗∗ Ryosuke Hirata ∗∗ Ikuo Arizono ∗∗ Yasuhiko Takemoto Ryosuke Hirata Ikuo Arizono ∗∗ Yasuhiko Takemoto ∗∗∗ ∗ ∗ Graduate School of Natural Science and Technology Okayama ∗ Graduate School of Natural Science and Technology Okayama ∗ University, Japan (e-mail: Science [email protected]). School of Natural and Technology Okayama ∗ Graduate University, Japan (e-mail: [email protected]). School of Natural Science and Technology Okayama ∗∗Graduate University, Japan (e-mail: [email protected]). School of Natural Science and ∗∗ Graduate School of (e-mail: Natural Science and Technology Technology Okayama Okayama ∗∗ University, Japan [email protected]). ∗∗ Graduate Graduate School of Natural Science and Technology Okayama University, Japan (e-mail: [email protected]) ∗∗ University, Japan (e-mail: [email protected]) ∗∗∗Graduate School of Natural Science and Technology Okayama of Engineering Kindai University, Japan and (e-mail: [email protected]) ∗∗∗ Faculty of Science Science and Engineering Kindai University, University, Japan Japan ∗∗∗ Faculty University, Japan [email protected]) (e-mail: [email protected]) ∗∗∗ (e-mail: Faculty of Science and Engineering Kindai University, Japan ∗∗∗ [email protected]) Faculty of(e-mail: Science and Engineering Kindai University, Japan (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Redundancy Redundancy architecture architecture is is known known as as an an important important and and effective effective method method for for Abstract: Abstract: Redundancy architecturea 2-component is known as standby an important and system effectiveis method for improving reliability. In particular, redundant one of the improving reliability. In particular, a 2-component standby redundant system is method one of the Abstract: Redundancy architecture is known as in anthe important and effective for most fundamental fundamental andInimportant important redundant models reliability theory. In improving reliability. particular, a 2-component standby redundant system is oneprevious of the most and redundant models in the reliability theory. In some some previous improving reliability. In particular, a 2-component standby redundant system is one of the investigations, 2-component standby redundant system with priority priority has In been considered most fundamental and important redundant modelssystem in the reliability theory. some previous investigations, aa 2-component standby redundant with has been considered most fundamental and important redundant models in of thefailure reliability theory. some previous investigations, a 2-component standby redundant system with priority has In been considered and then the mean time to failure (MTTF), variance time (VOFT) and reliability and then the mean time to failure (MTTF), variance of failure time (VOFT) andconsidered reliability investigations, a system 2-component standby redundant system with stochastic priority has been and then in thethe mean timehave to failure (MTTF), variance oflimited failure time (VOFT) and reliability function been evaluated based on information about function in system have been evaluated based onoflimited information about and then thethe mean time to failure (MTTF), variance failure stochastic time to (VOFT) and reliability each component. In this study, we address a preventive maintenance the system based on function in the system have been evaluated based on limited stochastic information about each component. In this have study,been we address a preventive maintenance to theinformation system based on function in thein system evaluated based on limited stochastic about the outcomes the previous investigations. In particular, the digital SC twin framework is each component. In this study, we address a preventive maintenance to the system based on the outcomes in the previous investigations. particular, the digitaltoSC framework is each component. In this study, we address a In preventive maintenance thetwin system based on the outcomes in the previous investigations. In particular, the digital SC twin framework is c presented. Copy-right ⃝2019 IFAC c presented. Copy-right ⃝2019 IFAC the outcomes in the previous investigations. In particular, the digital SC twin framework is c presented. Copy-right ⃝2019 IFAC c Federation presented. Copy-right ⃝2019 IFAC of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2019, IFAC (International Keywords: system system reliability, reliability, preventive preventive maintenance, maintenance, 2-component 2-component priority priority standby standby redundant redundant Keywords: Keywords: system reliability, preventiveentropy maintenance, 2-component priority standby redundant system, reliability function, maximum principle, method of Lagrange. system, reliability function, maximum principle, method of priority Lagrange. Keywords: system reliability, preventiveentropy maintenance, 2-component standby redundant system, reliability function, maximum entropy principle, method of Lagrange. system, reliability function, maximum entropy principle, method of Lagrange. 1. INTRODUCTION INTRODUCTION et 1. et al. al. (2017) (2017) have have proposed proposed the the evaluation evaluation method method of of the the 1. INTRODUCTION variance of failure time (VOFT) in the system. et al. (2017) have proposed the evaluation method of the variance of failure time (VOFT) in the system. 1. INTRODUCTION et al. (2017) have proposed the evaluation method of the of failure time (VOFT) in the system. The latest latest systems systems are are becoming becoming extremely extremely advanced advanced and and variance Through the above researches, we can evaluate MTTF The variance of failure time (VOFT) in the system. the above researches, we can evaluate MTTF The latest systems extremely advanced and Through complicated, and at atare thebecoming same time time the reliability reliability of syssysThrough the above researches,priority we can evaluate MTTF and in 2-component standby redundant complicated, and the same the of The latest systems are extremely advanced and and VOFT VOFTthe in the the 2-component priority standby redundant complicated, and at thebecoming sameRedundancy time the reliability of systems has become important. architecture is Through above researches, we evaluate can evaluate MTTF and VOFT in the 2-component priority standby redundant system quantitatively. Then, we can qualitatively tems has become important. Redundancy architecture is complicated, and at the and same time the reliability of syssystem quantitatively. Then, wepriority can evaluate qualitatively tems has become important. Redundancy architecture is and known as an important effective method for improvVOFT in the 2-component standby redundant the reliability of 2-component priority standby known as become an important and effective method for improvquantitatively. we can evaluate tems has important. Redundancy architecture is system the reliability of the the Then, 2-component priority qualitatively standby rereknown as an important and effective method for improving reliability. reliability. Further, 2-component standby redundant system quantitatively. Then, weand canVOFT evaluate qualitatively dundant system using MTTF in the system. ing Further, 2-component standby redundant the reliability of the 2-component priority standby reknown as an important and effective method for improvdundant system using MTTF and VOFT in the system. ing reliability. Further, 2-component standby redundant systems are are one one of of the the most most fundamental fundamental and and important important dundant the reliability ofusing the 2-component priority resystem MTTF and VOFT in standby the in system. In this study, we consider a reliability function order systems ing reliability. Further, 2-component standby redundant In this study, weusing consider a reliability function order systems aremodels one of in thethemost fundamental important redundant reliability theory.and In this this study, dundant system MTTF and VOFT in the in system. In this study, we consider a reliability function in order to evaluate quantitatively the system reliability in the redundant models in the reliability theory. In study, systems aremodels one of inthethemost fundamental important to this evaluate quantitatively system function reliabilityininorder the redundant reliability theory.and Insystem this study, we consider consider 2-component standby redundant con- In study, we consider athe reliability situation that only and variances about failure we aa 2-component standby redundant system conevaluate quantitatively reliability the redundant models in the reliability theory. In this study, situation that only averages averagesthe andsystem variances about in failure we consider a 2-component standby redundant system con- to sisting of prior and standby components, where the prior to evaluate quantitatively the system reliability in the and repair times in the respective components are provided sisting of prior and standbystandby components, where the prior situation that only averages and variances about failure we consider a 2-component redundant system conand repairthat timesonly in the respective components are provided sisting of prior and standby components, where the prior component is used whenever it is available. Until now, situation averages and variances about failure and repair times in the respective components are provided information. In of reliability component is used whenevercomponents, it is available. Until now, as sisting of prior and standby where the prior as limited limited information. In the the derivation derivation of the the component isonused whenever priority it is available. Until now, and some studies studies 2-component standby redundant repair times in thepriority respective components arereliability provided as limited information. In thestandby derivation of the reliability of the 2-component redundant system, some onused 2-component priority standby redundant component is whenever it is available. Until now, of the 2-component priority standby redundant system, some studies on 2-component priority standby redundant systems have been published. For example, Osaki (1970, as limited information. In the derivation of the reliability we consider developing the PDF about failure time in systems have been published. For example, Osaki (1970, of the 2-component priority standby redundant some studies on 2-component priority standby redundant we consider developing the PDF about failure time in the the systems have been published. For example, Osaki (1970, of the 2-component priority standby redundant system, 2002) and and Buzacott (1971) have have derived the the evaluation system, entire system based on MTTF and VOFT proposed by 2002) Buzacott (1971) derived evaluation we consider developing the PDF about failure time in the systems have been published. For example, Osaki (1970, entire system based on MTTF and VOFT proposed by 2002) andofBuzacott (1971) have derived the evaluation methods mean time to failure (MTTF) of the entire consider developing PDFand about failure time in the entire system based onthe MTTF and VOFT proposed by Takemoto and Arizono (2016) Oigawa et al. (2017). methods ofBuzacott mean time to failure (MTTF)the of evaluation the entire we 2002) and (1971) have derived Takemoto and based Arizono (2016) and Oigawa etproposed al. (2017). methods of the mean time to failure (MTTF) of the entire entire system for 2-component priority redundant systems. system on MTTF and VOFT by Takemoto and Arizono (2016) and Oigawa et al. (2017). Hence, we use the maximum entropy principle (MEP) to system for the 2-component priority redundant systems. methods mean time todensity failure (MTTF) of the entire Hence, we and use the maximum entropy principle (MEP) to system forof the 2-component priority redundant systems. Note that any probability functions (PDFs) and/or Takemoto Arizono (2016) and Oigawa et al. (2017). develop the PDF about failure time in the entire system. Note that any probability density functions (PDFs) and/or Hence, we use the maximum entropy principle (MEP) to system for the 2-component priority redundant systems. develop the PDF about failure time in the entire system. Note that any probability density functions (PDFs) and/or cumulative distribution distribution functions functions (CDFs) (CDFs) about about failure failure Hence, we use the maximum entropy principle (MEP) to The procedure based on MEP in the information theory cumulative develop the PDF about failure time in the entire system. Note that any probability density functions (PDFs) and/or The procedure based onfailure MEP time in theininformation theory cumulative distribution functionscomponents (CDFs) about failure develop and repair times in the respective are required the PDF about the entire system. The procedure based on MEP in the information theory known as approach for the and repair times in the respective components are required cumulative distribution functions (CDFs) about failure is is widely widely knownbased as an an elegant elegant approach for deriving deriving the and repair times in of thethe respective components are to evaluate evaluate MTTF entire system on using using therequired formuThe procedure MEP invalues the information theory is widely known as anonelegant approach for deriving PDF under the constraints of the of moments in the to MTTF of the entire system on the formuand repair times in the respective components are required PDF under the constraints of the values of moments in the to evaluate MTTF of the entire system on using the(1971). formu- is widely known as an elegant approach for deriving the lae derived by Osaki (1970, 2002) and Buzacott probability lae derived MTTF by Osaki (1970, 2002) andon Buzacott under distribution. the constraints of the values of moments in the to evaluate of the entire system using the(1971). formu- PDF probability lae derived by Osaki (1970, 2002) and Buzacott (1971). PDF under distribution. the constraints of the values of moments in the probability distribution. In recent years, Takemoto and Arizono (2016) have lae derived by Osaki (1970, 2002) and Buzacott (1971). Moreover, by using In recent years, Takemoto and Arizono (2016) have probability distribution. Moreover, by using the the reliability reliability function function of of 2-component 2-component In recent the years, Takemoto and Arizono proposed evaluation method of MTTF MTTF(2016) in the thehave 2- priority standby redundant system, we consider the proposed the evaluation method of in 2Moreover, by using the reliability function of 2-component In recent years, Takemoto and Arizono (2016) have priority standby redundant system, we consider the prepreproposed the evaluation method of MTTF in the 2component priority priority standby standby redundant redundant system system under under the the Moreover, by using the reliability function of 2-component ventive maintenance of 2-component priority standby recomponent priority standby redundant system, we consider the preproposed the evaluation method of MTTF in the 2ventive maintenance of 2-component priority standby recomponent priority that standby redundant system under the priority limited information only averages and variances about standby redundant system, we consider the ventive maintenance oftheory 2-component prioritymaintenance standbypreredundant system. The of preventive limited information that only redundant averages and variances about component priority standby system under the dundantmaintenance system. Theoftheory of preventive maintenance limited information that only averages and components variances about failure and repair times in the respective are ventive 2-component priority standby rewhich increase the system reliability is also important. In failure and repair times in the respective components are dundant system. The theory of preventive maintenance limited information thatunder only averages and variancesOigawa about increase theThe system reliability is also important. In failure and repair times in the respective components are which provided. Furthermore, the same condition, dundant system. theory of preventive maintenance concrete, based on the proposed reliability function, we provided. Furthermore, under the same condition, Oigawa which increase the system reliability is also important. In failure andFurthermore, repair times under in thethe respective components are concrete, based on the proposed reliability function, we provided. same condition, Oigawa which increase the system reliability is also important. In concrete, based on the proposed reliability function, we derive the optimal maintenance interval. provided. Furthermore, under the same condition, Oigawa ⋆ This work was supported by Japan Society for the Promotion of derive thebased optimal concrete, onmaintenance the proposedinterval. reliability function, we ⋆ This work was supported by Japan Society for the Promotion of derive the optimal maintenance interval. ⋆ ⋆ Science (JSPS) Number 18K04611: “Evaluation This work work wasKAKENHI supported Grant by Japan Japan Society for the the Promotion Promotion of derive the optimal maintenance interval. This was supported by Society for of Science (JSPS) KAKENHI Grant Number 18K04611: “Evaluation ⋆
This work wasKAKENHI supported by Japan under Society for the Promotion of of system performance and reliability incomplete information Science (JSPS) Grant 18K04611: “Evaluation Science (JSPS) KAKENHI Grant Number Number of system performance and reliability under 18K04611: incomplete “Evaluation information Science (JSPS) KAKENHI Grant Number environment”. of system system performance and reliability reliability under 18K04611: incomplete “Evaluation information of performance and under incomplete information environment”. of system performance and reliability under incomplete information environment”. environment”. environment”. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 188 Peer review responsibility of International Federation of Automatic Copyright © under 2019 IFAC 188 Control. Copyright © 2019 2019 IFAC IFAC 188 Copyright © 188 10.1016/j.ifacol.2019.11.173 Copyright © 2019 IFAC 188
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However, in some practical situations, it might be difficult to know exactly the probability distributions of failure and repair times. On the other hand, in many practical cases, the averages and variances about failure and repair times in respective components may be at least known based on historical information and so on. At the same time, it is impossible to evaluate the reliability of a system without fundamental information such as averages and variances on failure and repair times in each component. In recent years, Takemoto and Arizono (2016) have assumed the following situations in the 2-component priority standby redundant system illustrated in Fig. 1:
Fig. 1. 2-component priority standby redundant system 2. 2-COMPONENT PRIORITY STANDBY REDUNDANT SYSTEM We consider the 2-component standby redundant system composed of Component 1 having priority and Component 2 which is a standby component. In this case, the component with priority means a component which is always used whenever it is available. In contrast, the standby component means a component which stands by whenever the component with priority is available. The outline of behavior in this system is illustrated in Fig. 1. Initially, at time t = 0, Component 1 is working, and Component 2 is in the standby state. At this time, the standby time of the Component 2 is not included in its lifetime. When Component 1 fails, it starts to be repaired immediately and then Component 2 starts to work. If the repair of Component 1 has been completed and Component 2 is still working, then Component 1 starts to work immediately, and Component 2 gets into a standby state. However, if Component 2 fails during the repair of Component 1, the entire system goes down. The switching of components is reliably executed, and it is assumed that the switchover time is negligible. 3. OUTCOME OF PREVIOUS LITERATURE
i) The failure time distribution of Component 1 has the cumulative distribution function (CDF) F1 (t) with mean EF1 = 1/λ1 and variance VF1 . ii) The repair time of Component 1 obeys the probability distribution G1 (t)with mean EG1 and variance VG1 . iii) The failure time of Component 2 obeys the probability distribution F2 (t)with mean EF2 and variance VF 2 . Then, Takemoto and Arizono (2016) have developed a formula for evaluating MTTF in the 2-component priority standby redundant system illustrated in Fig. 1 by utilizing the approximation technique of Keizers et al. (2001). The approximation technique of Keizers et al. (2001) has been a procedure substituting a mixed Erlang distribution composed of two Erlang distributions for a distribution with known average and variance. Takemoto and Arizono have applied the approximation technique to the repair time distribution of Component 1 and the failure time distribution of Component 2. As an example, based on the approximation technique of Keizers et al. (2001), the PDF of the repair time distribution of Component 1 is expressed as follows: tn1 −1 −µ1 t e (n1 − 1)! tn1 −µ1 t e +(1 − p1 )µ1n1 +1 . (1) n1 ! where p1 , n1 and µ1 are parameters for the approximation. The approximation parameters are provided as follows: g1 (t) = p1 µn1 1
3.1 Derivation of MTTF For the 2-component priority standby redundant system of Fig. 1, Osaki (1970, 2002) has derived the formula for evaluating MTTF of the 2-component priority standby redundant system in Fig. 1 under the situation that the failure and repair time distributions of Component 1 are explicitly prescribed as individual specific probability distributions and Component 2 has an exponential failure time distribution. Further, Buzacott (1971) has assumed that the failure and repair time distributions of each component are individually prescribed as an explicit probability distribution. Under such an assumption, Buzacott (1971) has derived the formula for evaluating MTTF in the system. Remark that all the distribution functions of the failure and repair times in each component have to be explicitly specified for evaluating MTTF on using the theoretical formulae by Osaki (1970, 2002) and Buzacott (1971). 189
1 1 < φ21 ≤ , n1 + 1 n1 p1 =
(n1 + 1)φ21 − φ1 =
√ (n1 + 1)(1 − n1 φ21 ) , φ21 + 1
√ VG1 /EG1 ,
(2)
(3) (4)
n 1 + 1 − p1 . (5) EG1 Similarly, note that the approximation parameters with respect to the PDF f2 (t) of the failure time distribution F2 (t) are obtained as p2 , n2 , φ2 and λ2 using EF2 and VF2 . µ1 =
Then, the system has been reconstructed by four types of tentative systems based on the combinations of respective Erlang distributions in the mixed Erlang distribution for
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the repair time distribution of Component 1 and the failure time distribution of Component 2. The respective tentative systems have been analyzed using Markov process theory because the state transition in the respective tentative systems can be expressed by a state transition diagram based on an exponential distribution. MTTFs in the respective tentative systems are weightedly summed up using the ratios of mixture on the mixed Erlang distribution. As a consequence, Takemoto and Arizono (2016) have defined MTTF of the entire system as follows:
w ˜(n1 ) (s) =
+ (1 − p1 )(1 − p2 )M T T F(n1 +1,n2 +1) ,
(6)
R(t) = 1 −
n∑ 2 −1 a ˜(0) 1/λ1 d˜j(n1 ) (0) + 1 − ˜bn1 (0) j=0 1 − ˜bn1 (0) ) −1−j( n∑ 1 −1 n 2 −1 n2∑ ∑ i + k ˜i ck (0)d˜j(n1 ) (0). b (0)˜ × k i=0 j=0
=
(7)
k=0
˜ 1 (s) denotes the Laplace transformation for Further, R R1 (t) that is the reliability function of the entire system R1 (t) presented as R1 (t) = 1 − F1 (t). Then, we have d˜i(n1 ) (s)
i=0
−1−j( n∑ 1 −1 n 2 −1 n2∑ ∑ i=0
j=0
× ˜bi (s)˜ ck (s)d˜j(n1 ) (s),
e−st f (t)dt.
k=0
(17)
On the other hand, since the reliability function R(t) of the entire system is expressed as
˜ (n ,n ) (0) M T T F(n1 ,n2 ) = R 1 2
+a ˜(s)˜ v(n1 ) (s)
∫∞ 0
+ p1 (1 − p2 )M T T F(n1 ,n2 +1)
˜ (n ,n ) (s) = u R ˜(n1 ) (s) 1 2
(16)
Oigawa et al. (2017) have addressed VOFT of 2component priority standby redundant system. Denote the PDF of the failure time distribution of the entire system by f (t). Then, the Laplace transformation for f (t) is defined as: f˜(s) =
+ (1 − p1 )p2 M T T F(n1 +1,n2 )
n∑ 2 −1
˜ 1 (s)}˜bn1 (s) {1 − sR . ˜ 1 (s)}˜bn1 (s) 1 − {1 − sR
3.2 Derivation of VOFT
M T T F = p1 p2 M T T F(n1 ,n2 )
where
185
i+k k
) (8)
1 , a ˜(s) = s + µ1 + λ2
(9)
˜b(s) =
µ1 , s + µ1 + λ2
(10)
c˜(s) =
λ2 , s + µ1 + λ2
(11)
d˜0(n1 ) (s) = 1, d˜j(n1 ) (s) = w ˜(n1 ) (s) ( j−1 ∑ n1 − 1 + j − k ) c˜j−k (s)d˜k(n1 ) (s), × j−k
(12)
(13)
k=0
˜ 1 (s) R , ˜ 1 (s)}˜bn1 (s) 1 − {1 − sR
(14)
v˜(n1 ) (s) =
˜ 1 (s) 1 − sR , ˜ 1 (s)}˜bn1 (s) 1 − {1 − sR
(15) 190
f (τ )dτ,
(18)
0
the relationship between the Laplace transformations f˜(s) ˜ ˜ and R(s) can be given as f˜(s) = 1 − sR(s). Then, by applying the final-value theorem and the L’Hopital’s rule to the differential calculus forms for this relationship, the following equations can be shown: d ˜ f (s)|s=0 ≡ f˜(1) (0) = −E[t] = −M T T F, ds
(19)
d2 ˜ (20) f (s)|s=0 ≡ f˜(2) (0) = E[t2 ], ds2 where superscript (i) of the function f˜(i) (0) means i-th differentiation of the function f˜(s). Therefore, VOFT of 2-component priority standby redundant system can be obtained as
where
{ }2 V OF T = f˜(2) (0) − f˜(1) (0) ,
(21)
(2) f˜(2) (0) = p1 p2 f˜(n1 ,n2 ) (0)
(2) (2) + (1 − p1 )p2 f˜(n1 +1,n2 ) (0) + p1 (1 − p2 )f˜(n1 ,n2 +1) (0)
(2) + (1 − p1 )(1 − p2 )f˜(n1 +1,n2 +1) (0), (2) (1) u(n1 ) (0) f˜(n1 ,n2 ) (0) = −2˜
− 2˜ u(n1 ) (0)
n∑ 2 −1
n∑ 2 −1
(22)
d˜j(n1 ) (0)
j=0
(1) d˜j(n1 ) (0)
j=0
+ 2˜ a(0)˜ v(n1 ) (0)
u ˜(n1 ) (s) =
∫t
−1−j( n∑ 1 −1 n 2 −1 n2∑ ∑
{
i=0
j=0
k=0
i+k k
)
( i k ˜ ˜ c (0) dj(n1 ) (0)˜ a(0) 1 + i + k × b (0)˜ ) } h(n1 ,n2 ) (0) (1) ˜ − dj(n1 ) (0) , + a ˜(0)
(23)
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( )2 VF1 + λ11 (1) } u ˜(n1 ) (0) = − { 2 1 − ˜bn1 (0) { 1 ˜n1 ˜(0) + λ1 b (0) n1 a − { }2 1 − ˜bn1 (0) (1) (1) d˜j(n1 ) (0) = w ˜(n1 ) (0)
1 λ1
}
∫∞ (24)
,
j−k
× c˜j−k (0)d˜k(n1 ) (0) ) j−1 ( ∑ n1 − 1 + j − k −w ˜(n1 ) (0) j−k k=0
× (j − k)˜ a(0)˜ cj−k (0)d˜k(n1 ) (0) ) j−1 ( ∑ n1 − 1 + j − k +w ˜(n1 ) (0) j−k × c˜
j−k
k=0 (1) (0)d˜k(n1 ) (0),
(0) = −˜ vn1 (0)h(n1 ,n2 ) (0)˜bn1 (0) w ˜n(1) 1 − n1 a ˜(0)˜bn1 (0)˜ vn1 (0), ˜ (n ,n ) (0) h(n1 ,n2 ) (0) = R 1 2 ˜bn1 (0)R ˜ (n ,n ) (0) + n1 a ˜(0)˜bn1 (0) 1 2 , + 1 − ˜bn1 (0) (i) f˜(n1 ,n2 ) (0),
(25)
(1) u ˜(n1 ) (0), ˜(i)
(0).
Through adopting the variational method to the Lagrange function V , we have { } δV = − log f (x) + 1 δf
2
2
f (x) = e−1−α1 e−α2 x e−α3 (x−M T T F ) .
(33)
(34)
At the same instant, by partially differentiating V with α1 , α2 and α3 , we have equations (29)–(31). Therefore, based on equations (29)–(31) and (34), f (x) is rewritten as f (x) = Ce−
(x−m)2 2δ 2
(35)
,
where 2
C = e−1−α1 −α3 M T T F e
Based on MEP, the identification problem of f (t) can be formulated as follows: f (x) log f (x) dx,
0
(27)
Based on the limited information on MTTF and VOFT in the entire system, we evaluate the PDF f (t) of the failure time of the entire system. Then, it is widely known that MEP principle brings an elegant solution for deriving the PDF under the constraints of some values of moments. See Jain and Bhagat (2015) and so on. Therefore, we apply MEP to obtain the PDF f (t) of the failure time in the entire system.
f (t)
(31)
From equation (33), f (x) is expressed as
4. DERIVATION OF RELIABILITY FUNCTION
Maximize H (f ) = −
2
(x − M T T F ) f (x) dx = V OF T.
− α1 − α2 x − α3 (x − M T T F ) = 0.
(2) (2) Through the similar way, f˜(n1 +1,n2 ) (0)，f˜(n1 ,n2 +1) (0) and (2) f˜(n1 +1,n2 +1) (0) can be obtained.
∫∞
(30)
By using equations (28)–(31), a Lagrange function is defined with introducing Lagrange multipliers as ∞ ∫ V = H (f ) − α1 f (x) dx − 1 0 ∞ ∫ xf (x) dx − M T T F − α2 0 ∞ ∫ 2 (x − M T T F ) f (x) dx − V OF T . (32) − α3
(26)
where note that the notations of (1) (1) d˜ (0) and w ˜n1 (0) are similar to the notation of f j(n1 )
∫∞ 0
) j−1 ( ∑ n1 − 1 + j − k
k=0
xf (x) dx = M T T F,
0
m=
α3
( 2M T T F α3 −α2 )2 2α3
2M T T F α3 − α2 , 2α3
,
(36) (37)
1 . (38) 2α3 Accordingly, the derivation of f (x) is equivalent to a problem of finding C, m and δ 2 . δ2 =
By substituting equation (35) for equations (29), (30) and (31), the following equations can be shown:
(28) C
0
subject to
∫∞
e−
(x−m)2 2δ 2
dx = 1,
(39)
dx = M T T F,
(40)
0
∫∞
f (x) dx = 1,
(29)
0
C
∫∞ 0
191
xe−
(x−m)2 2δ 2
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C
∫∞ 0
Ryosuke Hirata et al. / IFAC PapersOnLine 52-13 (2019) 183–188
(x−m)2 − 2δ2
2
(x − M T T F ) e
dx = V OF T.
(41)
∫∞
e−
(x−m)2 2δ 2
(42)
dx.
0
Moreover, by multiplying both sides of equation (42) by √ 1/ 2πδ 2 , and through the variable transformation of t = (x − m)/δ, equation(42) is converted to √
1 1 1 1 = − erf 2πδ C 2 2
(
m −√ 2δ
)
,
(43)
where erf (z) denotes the following error function: 2 erf (z) = √ π
∫z
2
e−t dt.
(44)
0
Further, the error function and confluent hypergeometric function have the following relationship: 2z erf (z) = √ 1 F1 π
(
) 1 3 ; ; −z 2 . 2 2
(45)
Then, the confluent hypergeometric function is defined as 1 F1
(β; γ; z) =
∞ Γ(γ) ∑ Γ(β + n) z n . Γ(β) n=0 Γ(γ + n) n!
(46)
Furthermore, the following Kummer transformation can be applied to be confluent hypergeometric function: 1 F1
(β; γ; z) = e−z 1 F1 (γ − β; γ; −z) .
(47)
Consequently, based on equations (45) and (47), equation (43) is transformed as follows C=√
1 πδ 2 2
+ me
2
m − 2δ 2
( ). 3 m2 1; F ; 1 1 2 2δ 2
(48)
Further, as for equations (40) and (41), we have respectively the following equations: C=
MTTF − m m2
δ 2 e− 2δ2
,
(49)
M T T F 2 + V OF T − m2 − δ 2
. (50) m2 mδ 2 e− 2δ2 Then, based on equations (49) and (50), the following equations can be shown: C=
δ 2 = −M T T F m + M T T F 2 + V OF T .
(M T T F − m)
(51)
Accordingly, by substituting equation (51) for equation (48) and (49) (or (50)), we can obtain the following function for m: 192
π(−M T T F m + M T T F 2 + V OF T ) 2 m2
+me 2(−M T T F m+M T T F 2 +V OF T ) )} ( m2 3 × 1 F1 1, ; 2 2(−M T T F m + M T T F 2 + V OF T ) −
Then, from equation (39), we can derive as follows: 1 = C
{√
187
−(−M T T F m + M T T F 2 + σ 2 ) −
m2
×e 2(−M T T F m+M T T F 2 +σ2 ) = 0. (52) Then, we can obtain the value of m satisfying equation (52) numerically by appropriate search algorithms such as linear search. It is clear that we can get the values of δ 2 and C by calculated m. Moreover, by substituting equation (35) for equation (18), we can derive as follows. {√ πδ 2 R(t) = C − (t − m) 2 ( )} 2 (t−m)2 (t − m) 3 × e− 2δ2 1 F1 1; ; . (53) 2 2δ 2
Thus, the reliability of the entire system can be evaluated concretely by substituting calculated m, δ 2 and C for equation (53). 5. PREVENTIVE MAINTENANCE In this section, we propose the preventive maintenance of the 2-component priority standby redundant system based on the reliability function proposed in this study. In considering the preventive maintenance, we derive the optimal maintenance interval which maximizes the expected value of operating profit per unit time. At first, we define the time from the start of the system operation (t = 0) to the completion of the planned preventive maintenance or the completion of the repair of an unexpected failure as S(T ). Hence, we define T as maintenance interval, and S(T ) is derived as follows: T ∫ S(T ) = xf (x)dx + (1 − R(t)) Tf 0
(54) +R(T ) (T + Tp ) , where Tp and Tf are respectively the preventive maintenance time and the repair time of unexpected failure of the 2-component priority standby redundant system. For equation (54), the first term means the total time to complete the repair in case that the system fails before the planned preventive maintenance, and the second term means the time to complete the preventive maintenance without unexpected failure of the system. Thus, S(T ) means the time taken for one cycle. Further, the expected profit function P (t) is defined as: T ∫ P (T ) = G xf (x)dx − (1 − R(T )) Cf Tf 0
(55) +R(T ) (GT − Cp Tp ) , where G is the operating profit per unit time, Cp is the cost of preventive maintenance per unit time and Cf
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Fig. 2. Reliability function R(t). is the repair cost of unexpected failure per unit time. In equation (55), the first term means the total profit in case the system fails before the planned preventive maintenance, and the second term means the total profit without unexpected failure of the system. From equations (54) and (55), the profit per unit time is evaluated as P (T )/S(T ). Consequently, we define T maximizing P (T )/S(T ) as the optimal maintenance interval T ∗. 6. NUMERICAL ANALYSIS In this section, we investigate the validity of proposed preventive maintenance. As an example, supposed that EF1 = 1/λ1 = 100.0, VF1 = 20.02 , EG1 = 5.0, VG1 = 1.52 , EF2 = 50.0, and VF2 = 22.52 , respectively. Based on the above numerical values, the values of MTTF and VOFT for the failure times in the entire system are calculated as (M T T F, V OF T ) = (1106.089,( 239852.167). ) Further, m, δ 2 and C are obtained as m, δ 2 , C = (1057.804, 345871, 144, 0.000704). Thus, we can evaluate the reliability function in)equation (53) by applying the ( above values of m, δ 2 , C , and the result is shown Fig. 2. In addition, as an example of a comparison of the proposed reliability and simulation, the failure time distribution F1 (t) and the repair time distribution G1 (t) in Component 1 and the failure time distribution F2 (t) in Component 2 have been respectively supposed as the Weibull distribution with mean EF1 and variance VF1 , the log-normal distribution with mean EG1 and variance VG1 and the Weibull distribution with mean EF2 and variance VF2 . Then, note that the solid line shows the reliability function and the histogram shows the simulation result in Fig. 2. In addition, we set G = 100.0, Cf /Cp = 10.0 and Tf /Tp = 20 respectively. Based on the numerical values of (G, Cf /Cp , Tf /Tp ) and the reliability function, the profit per unit time is P ∗ (T ) is shown in Fig. 3. At this case, the optimal maintenance interval T ∗ and the reliability at that time are as (T ∗ , R(T ∗ )) = (290.777045, 0.954993). Because of the number of pages, we omit other results with different numerical values, but we have confirmed that as the values of Cf /Cp and Tf /Tp increase, the optimal maintenance interval gets shorter. Thus, we can get the results that the preventive maintenance in 2-component priority standby redundant system in this study is useful. 193
Fig. 3. P (t) in the case that G = 100.0, Cf /Cp = 10.0 and Tf /Tp = 20. 7. CONCLUSION In this study, we have addressed the evaluation method of the reliability function and preventive maintenance in the 2-component priority standby redundant system. Then, by utilizing the evaluation results for the mean time to failure (MTTF) by Takemoto and Arizono (2016) and the variance of the failure time (VOFT) by Oigawa et al. (2017), we have succeeded in deriving the evaluation method of the reliability function in the 2-component priority standby redundant system. In concrete, by using only the average and variance of the failure time in the entire system, the reliability function in the 2-component priority standby redundant system has been derived based on MEP. Further, by using the proposed reliability function, we can get the optimal maintenance interval in the 2-component priority standby redundant system. REFERENCES Buzacott, J.A. (1971). Availability of Priority Standby Redundant Systems. IEEE Transactions on Reliability, Vol. R-20, No. 2, pp. 60–63. Jain, M. and Bhagat, A. (2015). Analysis of Bulk Retrial Queue Using Maximum Entropy Principle. International Journal of Operational Research, Vol. 23, No. 4: 477–496. Keizers, J., Adan, I., and Van der Wal, J. (2001). A Queuing Model for Due Date Control in a Multiserver Repair Shop. Naval Res Logist, Vol. 48, No. 4, pp. 281– 292. Oigawa, S., Tomohiro, R., Arizono, I., and Takemoto, Y. (2017). Variance Evaluation of Time to Failure in 2Component Standby Redundancy System with Priority under Limited Information. Journal of Japan Industrial Management Association, Vol. 68, No. 2, pp. 120–123 in Japanese. Osaki, S. (1970). Reliability Analysis of a Two-unit Standby Redundant System with Priority Canadian Journal of Operations Research, Vol. 8, No. 1, pp. 60–62. Osaki, S. (2002). Stochastic Models in Reliability and Maintenance, Springer, New York. Takemoto, Y. and Arizono, I. (2016). A Study of MTTF in Two-unit Standby Redundant System with Priority under Limited Information about Failure and Repair Times. Journal of Risk and Reliability, Vol. 230, No. 1, pp. 67–74, 2016.
ScienceDirect
IFAC PapersOnLine 52-13 (2019) 183–188
Study on Preventive Maintenance for Study Study on on Preventive Preventive Maintenance Maintenance for for⋆ Priority Standby Redundant System Study on Preventive Maintenance for⋆⋆ Priority Standby Redundant System Priority Standby Redundant System Priority Standby Redundant System ⋆∗∗∗ ∗ ∗∗
Ryosuke Yasuhiko Takemoto ∗∗∗ Ryosuke Hirata Hirata ∗∗ Ikuo Ikuo Arizono Arizono ∗∗ ∗∗ Yasuhiko Takemoto ∗∗∗ ∗∗∗ Ryosuke Hirata ∗∗ Ikuo Arizono ∗∗ Yasuhiko Takemoto Ryosuke Hirata Ikuo Arizono ∗∗ Yasuhiko Takemoto ∗∗∗ ∗ ∗ Graduate School of Natural Science and Technology Okayama ∗ Graduate School of Natural Science and Technology Okayama ∗ University, Japan (e-mail: Science [email protected]). School of Natural and Technology Okayama ∗ Graduate University, Japan (e-mail: [email protected]). School of Natural Science and Technology Okayama ∗∗Graduate University, Japan (e-mail: [email protected]). School of Natural Science and ∗∗ Graduate School of (e-mail: Natural Science and Technology Technology Okayama Okayama ∗∗ University, Japan [email protected]). ∗∗ Graduate Graduate School of Natural Science and Technology Okayama University, Japan (e-mail: [email protected]) ∗∗ University, Japan (e-mail: [email protected]) ∗∗∗Graduate School of Natural Science and Technology Okayama of Engineering Kindai University, Japan and (e-mail: [email protected]) ∗∗∗ Faculty of Science Science and Engineering Kindai University, University, Japan Japan ∗∗∗ Faculty University, Japan [email protected]) (e-mail: [email protected]) ∗∗∗ (e-mail: Faculty of Science and Engineering Kindai University, Japan ∗∗∗ [email protected]) Faculty of(e-mail: Science and Engineering Kindai University, Japan (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Redundancy Redundancy architecture architecture is is known known as as an an important important and and effective effective method method for for Abstract: Abstract: Redundancy architecturea 2-component is known as standby an important and system effectiveis method for improving reliability. In particular, redundant one of the improving reliability. In particular, a 2-component standby redundant system is method one of the Abstract: Redundancy architecture is known as in anthe important and effective for most fundamental fundamental andInimportant important redundant models reliability theory. In improving reliability. particular, a 2-component standby redundant system is oneprevious of the most and redundant models in the reliability theory. In some some previous improving reliability. In particular, a 2-component standby redundant system is one of the investigations, 2-component standby redundant system with priority priority has In been considered most fundamental and important redundant modelssystem in the reliability theory. some previous investigations, aa 2-component standby redundant with has been considered most fundamental and important redundant models in of thefailure reliability theory. some previous investigations, a 2-component standby redundant system with priority has In been considered and then the mean time to failure (MTTF), variance time (VOFT) and reliability and then the mean time to failure (MTTF), variance of failure time (VOFT) andconsidered reliability investigations, a system 2-component standby redundant system with stochastic priority has been and then in thethe mean timehave to failure (MTTF), variance oflimited failure time (VOFT) and reliability function been evaluated based on information about function in system have been evaluated based onoflimited information about and then thethe mean time to failure (MTTF), variance failure stochastic time to (VOFT) and reliability each component. In this study, we address a preventive maintenance the system based on function in the system have been evaluated based on limited stochastic information about each component. In this have study,been we address a preventive maintenance to theinformation system based on function in thein system evaluated based on limited stochastic about the outcomes the previous investigations. In particular, the digital SC twin framework is each component. In this study, we address a preventive maintenance to the system based on the outcomes in the previous investigations. particular, the digitaltoSC framework is each component. In this study, we address a In preventive maintenance thetwin system based on the outcomes in the previous investigations. In particular, the digital SC twin framework is c presented. Copy-right ⃝2019 IFAC c presented. Copy-right ⃝2019 IFAC the outcomes in the previous investigations. In particular, the digital SC twin framework is c presented. Copy-right ⃝2019 IFAC c Federation presented. Copy-right ⃝2019 IFAC of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2019, IFAC (International Keywords: system system reliability, reliability, preventive preventive maintenance, maintenance, 2-component 2-component priority priority standby standby redundant redundant Keywords: Keywords: system reliability, preventiveentropy maintenance, 2-component priority standby redundant system, reliability function, maximum principle, method of Lagrange. system, reliability function, maximum principle, method of priority Lagrange. Keywords: system reliability, preventiveentropy maintenance, 2-component standby redundant system, reliability function, maximum entropy principle, method of Lagrange. system, reliability function, maximum entropy principle, method of Lagrange. 1. INTRODUCTION INTRODUCTION et 1. et al. al. (2017) (2017) have have proposed proposed the the evaluation evaluation method method of of the the 1. INTRODUCTION variance of failure time (VOFT) in the system. et al. (2017) have proposed the evaluation method of the variance of failure time (VOFT) in the system. 1. INTRODUCTION et al. (2017) have proposed the evaluation method of the of failure time (VOFT) in the system. The latest latest systems systems are are becoming becoming extremely extremely advanced advanced and and variance Through the above researches, we can evaluate MTTF The variance of failure time (VOFT) in the system. the above researches, we can evaluate MTTF The latest systems extremely advanced and Through complicated, and at atare thebecoming same time time the reliability reliability of syssysThrough the above researches,priority we can evaluate MTTF and in 2-component standby redundant complicated, and the same the of The latest systems are extremely advanced and and VOFT VOFTthe in the the 2-component priority standby redundant complicated, and at thebecoming sameRedundancy time the reliability of systems has become important. architecture is Through above researches, we evaluate can evaluate MTTF and VOFT in the 2-component priority standby redundant system quantitatively. Then, we can qualitatively tems has become important. Redundancy architecture is complicated, and at the and same time the reliability of syssystem quantitatively. Then, wepriority can evaluate qualitatively tems has become important. Redundancy architecture is and known as an important effective method for improvVOFT in the 2-component standby redundant the reliability of 2-component priority standby known as become an important and effective method for improvquantitatively. we can evaluate tems has important. Redundancy architecture is system the reliability of the the Then, 2-component priority qualitatively standby rereknown as an important and effective method for improving reliability. reliability. Further, 2-component standby redundant system quantitatively. Then, weand canVOFT evaluate qualitatively dundant system using MTTF in the system. ing Further, 2-component standby redundant the reliability of the 2-component priority standby reknown as an important and effective method for improvdundant system using MTTF and VOFT in the system. ing reliability. Further, 2-component standby redundant systems are are one one of of the the most most fundamental fundamental and and important important dundant the reliability ofusing the 2-component priority resystem MTTF and VOFT in standby the in system. In this study, we consider a reliability function order systems ing reliability. Further, 2-component standby redundant In this study, weusing consider a reliability function order systems aremodels one of in thethemost fundamental important redundant reliability theory.and In this this study, dundant system MTTF and VOFT in the in system. In this study, we consider a reliability function in order to evaluate quantitatively the system reliability in the redundant models in the reliability theory. In study, systems aremodels one of inthethemost fundamental important to this evaluate quantitatively system function reliabilityininorder the redundant reliability theory.and Insystem this study, we consider consider 2-component standby redundant con- In study, we consider athe reliability situation that only and variances about failure we aa 2-component standby redundant system conevaluate quantitatively reliability the redundant models in the reliability theory. In this study, situation that only averages averagesthe andsystem variances about in failure we consider a 2-component standby redundant system con- to sisting of prior and standby components, where the prior to evaluate quantitatively the system reliability in the and repair times in the respective components are provided sisting of prior and standbystandby components, where the prior situation that only averages and variances about failure we consider a 2-component redundant system conand repairthat timesonly in the respective components are provided sisting of prior and standby components, where the prior component is used whenever it is available. Until now, situation averages and variances about failure and repair times in the respective components are provided information. In of reliability component is used whenevercomponents, it is available. Until now, as sisting of prior and standby where the prior as limited limited information. In the the derivation derivation of the the component isonused whenever priority it is available. Until now, and some studies studies 2-component standby redundant repair times in thepriority respective components arereliability provided as limited information. In thestandby derivation of the reliability of the 2-component redundant system, some onused 2-component priority standby redundant component is whenever it is available. Until now, of the 2-component priority standby redundant system, some studies on 2-component priority standby redundant systems have been published. For example, Osaki (1970, as limited information. In the derivation of the reliability we consider developing the PDF about failure time in systems have been published. For example, Osaki (1970, of the 2-component priority standby redundant some studies on 2-component priority standby redundant we consider developing the PDF about failure time in the the systems have been published. For example, Osaki (1970, of the 2-component priority standby redundant system, 2002) and and Buzacott (1971) have have derived the the evaluation system, entire system based on MTTF and VOFT proposed by 2002) Buzacott (1971) derived evaluation we consider developing the PDF about failure time in the systems have been published. For example, Osaki (1970, entire system based on MTTF and VOFT proposed by 2002) andofBuzacott (1971) have derived the evaluation methods mean time to failure (MTTF) of the entire consider developing PDFand about failure time in the entire system based onthe MTTF and VOFT proposed by Takemoto and Arizono (2016) Oigawa et al. (2017). methods ofBuzacott mean time to failure (MTTF)the of evaluation the entire we 2002) and (1971) have derived Takemoto and based Arizono (2016) and Oigawa etproposed al. (2017). methods of the mean time to failure (MTTF) of the entire entire system for 2-component priority redundant systems. system on MTTF and VOFT by Takemoto and Arizono (2016) and Oigawa et al. (2017). Hence, we use the maximum entropy principle (MEP) to system for the 2-component priority redundant systems. methods mean time todensity failure (MTTF) of the entire Hence, we and use the maximum entropy principle (MEP) to system forof the 2-component priority redundant systems. Note that any probability functions (PDFs) and/or Takemoto Arizono (2016) and Oigawa et al. (2017). develop the PDF about failure time in the entire system. Note that any probability density functions (PDFs) and/or Hence, we use the maximum entropy principle (MEP) to system for the 2-component priority redundant systems. develop the PDF about failure time in the entire system. Note that any probability density functions (PDFs) and/or cumulative distribution distribution functions functions (CDFs) (CDFs) about about failure failure Hence, we use the maximum entropy principle (MEP) to The procedure based on MEP in the information theory cumulative develop the PDF about failure time in the entire system. Note that any probability density functions (PDFs) and/or The procedure based onfailure MEP time in theininformation theory cumulative distribution functionscomponents (CDFs) about failure develop and repair times in the respective are required the PDF about the entire system. The procedure based on MEP in the information theory known as approach for the and repair times in the respective components are required cumulative distribution functions (CDFs) about failure is is widely widely knownbased as an an elegant elegant approach for deriving deriving the and repair times in of thethe respective components are to evaluate evaluate MTTF entire system on using using therequired formuThe procedure MEP invalues the information theory is widely known as anonelegant approach for deriving PDF under the constraints of the of moments in the to MTTF of the entire system on the formuand repair times in the respective components are required PDF under the constraints of the values of moments in the to evaluate MTTF of the entire system on using the(1971). formu- is widely known as an elegant approach for deriving the lae derived by Osaki (1970, 2002) and Buzacott probability lae derived MTTF by Osaki (1970, 2002) andon Buzacott under distribution. the constraints of the values of moments in the to evaluate of the entire system using the(1971). formu- PDF probability lae derived by Osaki (1970, 2002) and Buzacott (1971). PDF under distribution. the constraints of the values of moments in the probability distribution. In recent years, Takemoto and Arizono (2016) have lae derived by Osaki (1970, 2002) and Buzacott (1971). Moreover, by using In recent years, Takemoto and Arizono (2016) have probability distribution. Moreover, by using the the reliability reliability function function of of 2-component 2-component In recent the years, Takemoto and Arizono proposed evaluation method of MTTF MTTF(2016) in the thehave 2- priority standby redundant system, we consider the proposed the evaluation method of in 2Moreover, by using the reliability function of 2-component In recent years, Takemoto and Arizono (2016) have priority standby redundant system, we consider the prepreproposed the evaluation method of MTTF in the 2component priority priority standby standby redundant redundant system system under under the the Moreover, by using the reliability function of 2-component ventive maintenance of 2-component priority standby recomponent priority standby redundant system, we consider the preproposed the evaluation method of MTTF in the 2ventive maintenance of 2-component priority standby recomponent priority that standby redundant system under the priority limited information only averages and variances about standby redundant system, we consider the ventive maintenance oftheory 2-component prioritymaintenance standbypreredundant system. The of preventive limited information that only redundant averages and variances about component priority standby system under the dundantmaintenance system. Theoftheory of preventive maintenance limited information that only averages and components variances about failure and repair times in the respective are ventive 2-component priority standby rewhich increase the system reliability is also important. In failure and repair times in the respective components are dundant system. The theory of preventive maintenance limited information thatunder only averages and variancesOigawa about increase theThe system reliability is also important. In failure and repair times in the respective components are which provided. Furthermore, the same condition, dundant system. theory of preventive maintenance concrete, based on the proposed reliability function, we provided. Furthermore, under the same condition, Oigawa which increase the system reliability is also important. In failure andFurthermore, repair times under in thethe respective components are concrete, based on the proposed reliability function, we provided. same condition, Oigawa which increase the system reliability is also important. In concrete, based on the proposed reliability function, we derive the optimal maintenance interval. provided. Furthermore, under the same condition, Oigawa ⋆ This work was supported by Japan Society for the Promotion of derive thebased optimal concrete, onmaintenance the proposedinterval. reliability function, we ⋆ This work was supported by Japan Society for the Promotion of derive the optimal maintenance interval. ⋆ ⋆ Science (JSPS) Number 18K04611: “Evaluation This work work wasKAKENHI supported Grant by Japan Japan Society for the the Promotion Promotion of derive the optimal maintenance interval. This was supported by Society for of Science (JSPS) KAKENHI Grant Number 18K04611: “Evaluation ⋆
This work wasKAKENHI supported by Japan under Society for the Promotion of of system performance and reliability incomplete information Science (JSPS) Grant 18K04611: “Evaluation Science (JSPS) KAKENHI Grant Number Number of system performance and reliability under 18K04611: incomplete “Evaluation information Science (JSPS) KAKENHI Grant Number environment”. of system system performance and reliability reliability under 18K04611: incomplete “Evaluation information of performance and under incomplete information environment”. of system performance and reliability under incomplete information environment”. environment”. environment”. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 188 Peer review responsibility of International Federation of Automatic Copyright © under 2019 IFAC 188 Control. Copyright © 2019 2019 IFAC IFAC 188 Copyright © 188 10.1016/j.ifacol.2019.11.173 Copyright © 2019 IFAC 188
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However, in some practical situations, it might be difficult to know exactly the probability distributions of failure and repair times. On the other hand, in many practical cases, the averages and variances about failure and repair times in respective components may be at least known based on historical information and so on. At the same time, it is impossible to evaluate the reliability of a system without fundamental information such as averages and variances on failure and repair times in each component. In recent years, Takemoto and Arizono (2016) have assumed the following situations in the 2-component priority standby redundant system illustrated in Fig. 1:
Fig. 1. 2-component priority standby redundant system 2. 2-COMPONENT PRIORITY STANDBY REDUNDANT SYSTEM We consider the 2-component standby redundant system composed of Component 1 having priority and Component 2 which is a standby component. In this case, the component with priority means a component which is always used whenever it is available. In contrast, the standby component means a component which stands by whenever the component with priority is available. The outline of behavior in this system is illustrated in Fig. 1. Initially, at time t = 0, Component 1 is working, and Component 2 is in the standby state. At this time, the standby time of the Component 2 is not included in its lifetime. When Component 1 fails, it starts to be repaired immediately and then Component 2 starts to work. If the repair of Component 1 has been completed and Component 2 is still working, then Component 1 starts to work immediately, and Component 2 gets into a standby state. However, if Component 2 fails during the repair of Component 1, the entire system goes down. The switching of components is reliably executed, and it is assumed that the switchover time is negligible. 3. OUTCOME OF PREVIOUS LITERATURE
i) The failure time distribution of Component 1 has the cumulative distribution function (CDF) F1 (t) with mean EF1 = 1/λ1 and variance VF1 . ii) The repair time of Component 1 obeys the probability distribution G1 (t)with mean EG1 and variance VG1 . iii) The failure time of Component 2 obeys the probability distribution F2 (t)with mean EF2 and variance VF 2 . Then, Takemoto and Arizono (2016) have developed a formula for evaluating MTTF in the 2-component priority standby redundant system illustrated in Fig. 1 by utilizing the approximation technique of Keizers et al. (2001). The approximation technique of Keizers et al. (2001) has been a procedure substituting a mixed Erlang distribution composed of two Erlang distributions for a distribution with known average and variance. Takemoto and Arizono have applied the approximation technique to the repair time distribution of Component 1 and the failure time distribution of Component 2. As an example, based on the approximation technique of Keizers et al. (2001), the PDF of the repair time distribution of Component 1 is expressed as follows: tn1 −1 −µ1 t e (n1 − 1)! tn1 −µ1 t e +(1 − p1 )µ1n1 +1 . (1) n1 ! where p1 , n1 and µ1 are parameters for the approximation. The approximation parameters are provided as follows: g1 (t) = p1 µn1 1
3.1 Derivation of MTTF For the 2-component priority standby redundant system of Fig. 1, Osaki (1970, 2002) has derived the formula for evaluating MTTF of the 2-component priority standby redundant system in Fig. 1 under the situation that the failure and repair time distributions of Component 1 are explicitly prescribed as individual specific probability distributions and Component 2 has an exponential failure time distribution. Further, Buzacott (1971) has assumed that the failure and repair time distributions of each component are individually prescribed as an explicit probability distribution. Under such an assumption, Buzacott (1971) has derived the formula for evaluating MTTF in the system. Remark that all the distribution functions of the failure and repair times in each component have to be explicitly specified for evaluating MTTF on using the theoretical formulae by Osaki (1970, 2002) and Buzacott (1971). 189
1 1 < φ21 ≤ , n1 + 1 n1 p1 =
(n1 + 1)φ21 − φ1 =
√ (n1 + 1)(1 − n1 φ21 ) , φ21 + 1
√ VG1 /EG1 ,
(2)
(3) (4)
n 1 + 1 − p1 . (5) EG1 Similarly, note that the approximation parameters with respect to the PDF f2 (t) of the failure time distribution F2 (t) are obtained as p2 , n2 , φ2 and λ2 using EF2 and VF2 . µ1 =
Then, the system has been reconstructed by four types of tentative systems based on the combinations of respective Erlang distributions in the mixed Erlang distribution for
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the repair time distribution of Component 1 and the failure time distribution of Component 2. The respective tentative systems have been analyzed using Markov process theory because the state transition in the respective tentative systems can be expressed by a state transition diagram based on an exponential distribution. MTTFs in the respective tentative systems are weightedly summed up using the ratios of mixture on the mixed Erlang distribution. As a consequence, Takemoto and Arizono (2016) have defined MTTF of the entire system as follows:
w ˜(n1 ) (s) =
+ (1 − p1 )(1 − p2 )M T T F(n1 +1,n2 +1) ,
(6)
R(t) = 1 −
n∑ 2 −1 a ˜(0) 1/λ1 d˜j(n1 ) (0) + 1 − ˜bn1 (0) j=0 1 − ˜bn1 (0) ) −1−j( n∑ 1 −1 n 2 −1 n2∑ ∑ i + k ˜i ck (0)d˜j(n1 ) (0). b (0)˜ × k i=0 j=0
=
(7)
k=0
˜ 1 (s) denotes the Laplace transformation for Further, R R1 (t) that is the reliability function of the entire system R1 (t) presented as R1 (t) = 1 − F1 (t). Then, we have d˜i(n1 ) (s)
i=0
−1−j( n∑ 1 −1 n 2 −1 n2∑ ∑ i=0
j=0
× ˜bi (s)˜ ck (s)d˜j(n1 ) (s),
e−st f (t)dt.
k=0
(17)
On the other hand, since the reliability function R(t) of the entire system is expressed as
˜ (n ,n ) (0) M T T F(n1 ,n2 ) = R 1 2
+a ˜(s)˜ v(n1 ) (s)
∫∞ 0
+ p1 (1 − p2 )M T T F(n1 ,n2 +1)
˜ (n ,n ) (s) = u R ˜(n1 ) (s) 1 2
(16)
Oigawa et al. (2017) have addressed VOFT of 2component priority standby redundant system. Denote the PDF of the failure time distribution of the entire system by f (t). Then, the Laplace transformation for f (t) is defined as: f˜(s) =
+ (1 − p1 )p2 M T T F(n1 +1,n2 )
n∑ 2 −1
˜ 1 (s)}˜bn1 (s) {1 − sR . ˜ 1 (s)}˜bn1 (s) 1 − {1 − sR
3.2 Derivation of VOFT
M T T F = p1 p2 M T T F(n1 ,n2 )
where
185
i+k k
) (8)
1 , a ˜(s) = s + µ1 + λ2
(9)
˜b(s) =
µ1 , s + µ1 + λ2
(10)
c˜(s) =
λ2 , s + µ1 + λ2
(11)
d˜0(n1 ) (s) = 1, d˜j(n1 ) (s) = w ˜(n1 ) (s) ( j−1 ∑ n1 − 1 + j − k ) c˜j−k (s)d˜k(n1 ) (s), × j−k
(12)
(13)
k=0
˜ 1 (s) R , ˜ 1 (s)}˜bn1 (s) 1 − {1 − sR
(14)
v˜(n1 ) (s) =
˜ 1 (s) 1 − sR , ˜ 1 (s)}˜bn1 (s) 1 − {1 − sR
(15) 190
f (τ )dτ,
(18)
0
the relationship between the Laplace transformations f˜(s) ˜ ˜ and R(s) can be given as f˜(s) = 1 − sR(s). Then, by applying the final-value theorem and the L’Hopital’s rule to the differential calculus forms for this relationship, the following equations can be shown: d ˜ f (s)|s=0 ≡ f˜(1) (0) = −E[t] = −M T T F, ds
(19)
d2 ˜ (20) f (s)|s=0 ≡ f˜(2) (0) = E[t2 ], ds2 where superscript (i) of the function f˜(i) (0) means i-th differentiation of the function f˜(s). Therefore, VOFT of 2-component priority standby redundant system can be obtained as
where
{ }2 V OF T = f˜(2) (0) − f˜(1) (0) ,
(21)
(2) f˜(2) (0) = p1 p2 f˜(n1 ,n2 ) (0)
(2) (2) + (1 − p1 )p2 f˜(n1 +1,n2 ) (0) + p1 (1 − p2 )f˜(n1 ,n2 +1) (0)
(2) + (1 − p1 )(1 − p2 )f˜(n1 +1,n2 +1) (0), (2) (1) u(n1 ) (0) f˜(n1 ,n2 ) (0) = −2˜
− 2˜ u(n1 ) (0)
n∑ 2 −1
n∑ 2 −1
(22)
d˜j(n1 ) (0)
j=0
(1) d˜j(n1 ) (0)
j=0
+ 2˜ a(0)˜ v(n1 ) (0)
u ˜(n1 ) (s) =
∫t
−1−j( n∑ 1 −1 n 2 −1 n2∑ ∑
{
i=0
j=0
k=0
i+k k
)
( i k ˜ ˜ c (0) dj(n1 ) (0)˜ a(0) 1 + i + k × b (0)˜ ) } h(n1 ,n2 ) (0) (1) ˜ − dj(n1 ) (0) , + a ˜(0)
(23)
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( )2 VF1 + λ11 (1) } u ˜(n1 ) (0) = − { 2 1 − ˜bn1 (0) { 1 ˜n1 ˜(0) + λ1 b (0) n1 a − { }2 1 − ˜bn1 (0) (1) (1) d˜j(n1 ) (0) = w ˜(n1 ) (0)
1 λ1
}
∫∞ (24)
,
j−k
× c˜j−k (0)d˜k(n1 ) (0) ) j−1 ( ∑ n1 − 1 + j − k −w ˜(n1 ) (0) j−k k=0
× (j − k)˜ a(0)˜ cj−k (0)d˜k(n1 ) (0) ) j−1 ( ∑ n1 − 1 + j − k +w ˜(n1 ) (0) j−k × c˜
j−k
k=0 (1) (0)d˜k(n1 ) (0),
(0) = −˜ vn1 (0)h(n1 ,n2 ) (0)˜bn1 (0) w ˜n(1) 1 − n1 a ˜(0)˜bn1 (0)˜ vn1 (0), ˜ (n ,n ) (0) h(n1 ,n2 ) (0) = R 1 2 ˜bn1 (0)R ˜ (n ,n ) (0) + n1 a ˜(0)˜bn1 (0) 1 2 , + 1 − ˜bn1 (0) (i) f˜(n1 ,n2 ) (0),
(25)
(1) u ˜(n1 ) (0), ˜(i)
(0).
Through adopting the variational method to the Lagrange function V , we have { } δV = − log f (x) + 1 δf
2
2
f (x) = e−1−α1 e−α2 x e−α3 (x−M T T F ) .
(33)
(34)
At the same instant, by partially differentiating V with α1 , α2 and α3 , we have equations (29)–(31). Therefore, based on equations (29)–(31) and (34), f (x) is rewritten as f (x) = Ce−
(x−m)2 2δ 2
(35)
,
where 2
C = e−1−α1 −α3 M T T F e
Based on MEP, the identification problem of f (t) can be formulated as follows: f (x) log f (x) dx,
0
(27)
Based on the limited information on MTTF and VOFT in the entire system, we evaluate the PDF f (t) of the failure time of the entire system. Then, it is widely known that MEP principle brings an elegant solution for deriving the PDF under the constraints of some values of moments. See Jain and Bhagat (2015) and so on. Therefore, we apply MEP to obtain the PDF f (t) of the failure time in the entire system.
f (t)
(31)
From equation (33), f (x) is expressed as
4. DERIVATION OF RELIABILITY FUNCTION
Maximize H (f ) = −
2
(x − M T T F ) f (x) dx = V OF T.
− α1 − α2 x − α3 (x − M T T F ) = 0.
(2) (2) Through the similar way, f˜(n1 +1,n2 ) (0)，f˜(n1 ,n2 +1) (0) and (2) f˜(n1 +1,n2 +1) (0) can be obtained.
∫∞
(30)
By using equations (28)–(31), a Lagrange function is defined with introducing Lagrange multipliers as ∞ ∫ V = H (f ) − α1 f (x) dx − 1 0 ∞ ∫ xf (x) dx − M T T F − α2 0 ∞ ∫ 2 (x − M T T F ) f (x) dx − V OF T . (32) − α3
(26)
where note that the notations of (1) (1) d˜ (0) and w ˜n1 (0) are similar to the notation of f j(n1 )
∫∞ 0
) j−1 ( ∑ n1 − 1 + j − k
k=0
xf (x) dx = M T T F,
0
m=
α3
( 2M T T F α3 −α2 )2 2α3
2M T T F α3 − α2 , 2α3
,
(36) (37)
1 . (38) 2α3 Accordingly, the derivation of f (x) is equivalent to a problem of finding C, m and δ 2 . δ2 =
By substituting equation (35) for equations (29), (30) and (31), the following equations can be shown:
(28) C
0
subject to
∫∞
e−
(x−m)2 2δ 2
dx = 1,
(39)
dx = M T T F,
(40)
0
∫∞
f (x) dx = 1,
(29)
0
C
∫∞ 0
191
xe−
(x−m)2 2δ 2
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C
∫∞ 0
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(x−m)2 − 2δ2
2
(x − M T T F ) e
dx = V OF T.
(41)
∫∞
e−
(x−m)2 2δ 2
(42)
dx.
0
Moreover, by multiplying both sides of equation (42) by √ 1/ 2πδ 2 , and through the variable transformation of t = (x − m)/δ, equation(42) is converted to √
1 1 1 1 = − erf 2πδ C 2 2
(
m −√ 2δ
)
,
(43)
where erf (z) denotes the following error function: 2 erf (z) = √ π
∫z
2
e−t dt.
(44)
0
Further, the error function and confluent hypergeometric function have the following relationship: 2z erf (z) = √ 1 F1 π
(
) 1 3 ; ; −z 2 . 2 2
(45)
Then, the confluent hypergeometric function is defined as 1 F1
(β; γ; z) =
∞ Γ(γ) ∑ Γ(β + n) z n . Γ(β) n=0 Γ(γ + n) n!
(46)
Furthermore, the following Kummer transformation can be applied to be confluent hypergeometric function: 1 F1
(β; γ; z) = e−z 1 F1 (γ − β; γ; −z) .
(47)
Consequently, based on equations (45) and (47), equation (43) is transformed as follows C=√
1 πδ 2 2
+ me
2
m − 2δ 2
( ). 3 m2 1; F ; 1 1 2 2δ 2
(48)
Further, as for equations (40) and (41), we have respectively the following equations: C=
MTTF − m m2
δ 2 e− 2δ2
,
(49)
M T T F 2 + V OF T − m2 − δ 2
. (50) m2 mδ 2 e− 2δ2 Then, based on equations (49) and (50), the following equations can be shown: C=
δ 2 = −M T T F m + M T T F 2 + V OF T .
(M T T F − m)
(51)
Accordingly, by substituting equation (51) for equation (48) and (49) (or (50)), we can obtain the following function for m: 192
π(−M T T F m + M T T F 2 + V OF T ) 2 m2
+me 2(−M T T F m+M T T F 2 +V OF T ) )} ( m2 3 × 1 F1 1, ; 2 2(−M T T F m + M T T F 2 + V OF T ) −
Then, from equation (39), we can derive as follows: 1 = C
{√
187
−(−M T T F m + M T T F 2 + σ 2 ) −
m2
×e 2(−M T T F m+M T T F 2 +σ2 ) = 0. (52) Then, we can obtain the value of m satisfying equation (52) numerically by appropriate search algorithms such as linear search. It is clear that we can get the values of δ 2 and C by calculated m. Moreover, by substituting equation (35) for equation (18), we can derive as follows. {√ πδ 2 R(t) = C − (t − m) 2 ( )} 2 (t−m)2 (t − m) 3 × e− 2δ2 1 F1 1; ; . (53) 2 2δ 2
Thus, the reliability of the entire system can be evaluated concretely by substituting calculated m, δ 2 and C for equation (53). 5. PREVENTIVE MAINTENANCE In this section, we propose the preventive maintenance of the 2-component priority standby redundant system based on the reliability function proposed in this study. In considering the preventive maintenance, we derive the optimal maintenance interval which maximizes the expected value of operating profit per unit time. At first, we define the time from the start of the system operation (t = 0) to the completion of the planned preventive maintenance or the completion of the repair of an unexpected failure as S(T ). Hence, we define T as maintenance interval, and S(T ) is derived as follows: T ∫ S(T ) = xf (x)dx + (1 − R(t)) Tf 0
(54) +R(T ) (T + Tp ) , where Tp and Tf are respectively the preventive maintenance time and the repair time of unexpected failure of the 2-component priority standby redundant system. For equation (54), the first term means the total time to complete the repair in case that the system fails before the planned preventive maintenance, and the second term means the time to complete the preventive maintenance without unexpected failure of the system. Thus, S(T ) means the time taken for one cycle. Further, the expected profit function P (t) is defined as: T ∫ P (T ) = G xf (x)dx − (1 − R(T )) Cf Tf 0
(55) +R(T ) (GT − Cp Tp ) , where G is the operating profit per unit time, Cp is the cost of preventive maintenance per unit time and Cf
2019 IFAC MIM 188 Germany, August 28-30, 2019 Berlin,
Ryosuke Hirata et al. / IFAC PapersOnLine 52-13 (2019) 183–188
Fig. 2. Reliability function R(t). is the repair cost of unexpected failure per unit time. In equation (55), the first term means the total profit in case the system fails before the planned preventive maintenance, and the second term means the total profit without unexpected failure of the system. From equations (54) and (55), the profit per unit time is evaluated as P (T )/S(T ). Consequently, we define T maximizing P (T )/S(T ) as the optimal maintenance interval T ∗. 6. NUMERICAL ANALYSIS In this section, we investigate the validity of proposed preventive maintenance. As an example, supposed that EF1 = 1/λ1 = 100.0, VF1 = 20.02 , EG1 = 5.0, VG1 = 1.52 , EF2 = 50.0, and VF2 = 22.52 , respectively. Based on the above numerical values, the values of MTTF and VOFT for the failure times in the entire system are calculated as (M T T F, V OF T ) = (1106.089,( 239852.167). ) Further, m, δ 2 and C are obtained as m, δ 2 , C = (1057.804, 345871, 144, 0.000704). Thus, we can evaluate the reliability function in)equation (53) by applying the ( above values of m, δ 2 , C , and the result is shown Fig. 2. In addition, as an example of a comparison of the proposed reliability and simulation, the failure time distribution F1 (t) and the repair time distribution G1 (t) in Component 1 and the failure time distribution F2 (t) in Component 2 have been respectively supposed as the Weibull distribution with mean EF1 and variance VF1 , the log-normal distribution with mean EG1 and variance VG1 and the Weibull distribution with mean EF2 and variance VF2 . Then, note that the solid line shows the reliability function and the histogram shows the simulation result in Fig. 2. In addition, we set G = 100.0, Cf /Cp = 10.0 and Tf /Tp = 20 respectively. Based on the numerical values of (G, Cf /Cp , Tf /Tp ) and the reliability function, the profit per unit time is P ∗ (T ) is shown in Fig. 3. At this case, the optimal maintenance interval T ∗ and the reliability at that time are as (T ∗ , R(T ∗ )) = (290.777045, 0.954993). Because of the number of pages, we omit other results with different numerical values, but we have confirmed that as the values of Cf /Cp and Tf /Tp increase, the optimal maintenance interval gets shorter. Thus, we can get the results that the preventive maintenance in 2-component priority standby redundant system in this study is useful. 193
Fig. 3. P (t) in the case that G = 100.0, Cf /Cp = 10.0 and Tf /Tp = 20. 7. CONCLUSION In this study, we have addressed the evaluation method of the reliability function and preventive maintenance in the 2-component priority standby redundant system. Then, by utilizing the evaluation results for the mean time to failure (MTTF) by Takemoto and Arizono (2016) and the variance of the failure time (VOFT) by Oigawa et al. (2017), we have succeeded in deriving the evaluation method of the reliability function in the 2-component priority standby redundant system. In concrete, by using only the average and variance of the failure time in the entire system, the reliability function in the 2-component priority standby redundant system has been derived based on MEP. Further, by using the proposed reliability function, we can get the optimal maintenance interval in the 2-component priority standby redundant system. REFERENCES Buzacott, J.A. (1971). Availability of Priority Standby Redundant Systems. IEEE Transactions on Reliability, Vol. R-20, No. 2, pp. 60–63. Jain, M. and Bhagat, A. (2015). Analysis of Bulk Retrial Queue Using Maximum Entropy Principle. International Journal of Operational Research, Vol. 23, No. 4: 477–496. Keizers, J., Adan, I., and Van der Wal, J. (2001). A Queuing Model for Due Date Control in a Multiserver Repair Shop. Naval Res Logist, Vol. 48, No. 4, pp. 281– 292. Oigawa, S., Tomohiro, R., Arizono, I., and Takemoto, Y. (2017). Variance Evaluation of Time to Failure in 2Component Standby Redundancy System with Priority under Limited Information. Journal of Japan Industrial Management Association, Vol. 68, No. 2, pp. 120–123 in Japanese. Osaki, S. (1970). Reliability Analysis of a Two-unit Standby Redundant System with Priority Canadian Journal of Operations Research, Vol. 8, No. 1, pp. 60–62. Osaki, S. (2002). Stochastic Models in Reliability and Maintenance, Springer, New York. Takemoto, Y. and Arizono, I. (2016). A Study of MTTF in Two-unit Standby Redundant System with Priority under Limited Information about Failure and Repair Times. Journal of Risk and Reliability, Vol. 230, No. 1, pp. 67–74, 2016.