- Email: [email protected]
Opportunistic maintenance scheduling with stochastic opportunities duration in a predictive maintenance strategy
Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Available online at www.sciencedirect.com Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Bergamo, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Information Control Problems in Bergamo, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Manufacturing Bergamo, Italy, June 11-13, 2018 Information Control in Manufacturing Bergamo, Italy, Italy, JuneProblems 11-13, 2018 2018 Bergamo, June 11-13, Bergamo, Italy, June 11-13, 2018
ScienceDirect
IFAC PapersOnLine 51-11 (2018) 453–458
Opportunistic maintenance scheduling with stochastic opportunities duration in aa Opportunistic maintenance scheduling with stochastic opportunities duration in Opportunistic maintenance scheduling with stochastic opportunities duration in aa Opportunistic maintenance scheduling with stochastic opportunities duration in predictive maintenance strategy Opportunistic scheduling with stochastic opportunities duration in aa predictive maintenance strategy Opportunistic maintenance maintenance scheduling with stochastic opportunities duration in predictive maintenance strategy predictive maintenance strategy predictive maintenance strategy predictive maintenance strategy C. Nzukam, A. Voisin, E. Levrat, D. Sauter, B. Iung
C. Nzukam, A. Voisin, E. Sauter, B. Iung C. Nzukam, A. Voisin, E. Levrat, D. D. Sauter, B. Iung Levrat, C. Nzukam, A. Voisin, E. D. Sauter, B. Iung Levrat, C. Nzukam, A. Voisin, E. Levrat, D. Sauter, B. Iung C. Nzukam, A. Voisin, E. Levrat, D. Sauter, B. Iung Lorraine University, CRAN, CNRS UMR 7039, Nancy, France Lorraine University, CRAN, CNRS UMR 7039, Nancy, France Lorraine University, CRAN, CNRS UMR 7039, Nancy, France collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr Lorraine University, CRAN, CNRS UMR 7039, Nancy, France Lorraine University, CRAN, CNRS UMR 7039, Nancy, France collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr Lorraine University, CRAN, CNRS UMR 7039, Nancy, France collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr Abstract: To guarantee a good level of facilities performance (reliability and availability), maintenance Abstract: To guarantee aa good level of facilities performance (reliability and availability), maintenance Abstract: To guarantee of facilities performance (reliability and availability), activities have to be planned andlevel scheduled efficiently. Maintenance scheduling decides, in a maintenance tactical way, Abstract: To guarantee aa good good level of facilities performance (reliability and availability), maintenance Abstract: To guarantee good level of facilities performance (reliability and availability), maintenance activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in a tactical way, activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in a tactical way, Abstract: To guarantee a good level of facilities performance (reliability and availability), maintenance when theses maintenance activities will be carried out according to the appearance of opportunities. activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in aaopportunities. tactical way, activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in tactical way, when theses maintenance activities will be carried out according to the appearance of when theses maintenance activities will be carried out according to the appearance of opportunities. activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in a tactical way, Numerous works on opportunistic maintenance have been proposed in order to take profit of stochastic, when theses maintenance activities will be carried out according to the appearance of opportunities. when theses maintenance activities will be carried out according to the appearance of opportunities. Numerous works on opportunistic maintenance have been proposed in order to take profit of stochastic, Numerous works on opportunistic maintenance have been proposed in order to take profit of stochastic, when theses maintenance activities will be carried out according to the appearance of opportunities. In the real context, the duration of an opportunity is not known structural and economic dependence. Numerousand works on opportunistic opportunistic maintenance have been proposed proposed in order order toopportunity take profit profit is of not stochastic, Numerous works on maintenance have been in take of stochastic, structural economic dependence. In the real context, the duration of anto known structural economic dependence. In real context, the duration of Numerous works on papers opportunistic maintenance been proposed in order take profit is of stochastic, accurately.and Very few take into account the stochastic nature of opportunities duration. Innot thisknown paper, structural and economic dependence. In the the the realhave context, the duration of an antoopportunity opportunity is not known structural and economic dependence. In the real context, the duration of an opportunity is not known accurately. Very few papers take into account stochastic nature of opportunities duration. In this paper, accurately. Very few papers take into account the nature of opportunities duration. this paper, structural dependence. In the realstochastic context, the duration of an opportunity isIn not known we presentand an economic opportunistic maintenance scheduling methodology considering the stochastic nature of accurately. Very few papers take into account the stochastic nature of opportunities duration. In this paper, accurately. Very few take into the nature of opportunities duration. In paper, we present an opportunistic maintenance scheduling methodology the stochastic nature of we present an opportunistic maintenance scheduling methodology considering the stochastic nature of accurately. Very few papers papers take into account account the stochastic stochastic nature of considering opportunities duration. In this this opportunities duration in a predictive maintenance strategy. The prognostic information is used to paper, select we present an opportunistic maintenance scheduling methodology considering the stochastic nature of we present opportunistic maintenance scheduling methodology considering the nature of opportunities duration in aa predictive maintenance strategy. The prognostic information is used to select opportunities duration in maintenance strategy. The prognostic information is used to select we present an ancoming opportunistic maintenance scheduling methodology considering the stochastic stochastic of before the failure. The proposed maintenance scheduling methodology is nature based on opportunities duration in aa predictive predictive maintenance strategy. The prognostic information is used to select opportunities duration in predictive maintenance strategy. The prognostic information is used to select coming before the failure. The proposed maintenance scheduling methodology is based on coming before the failure. The proposed maintenance scheduling methodology is based on opportunities duration in a predictive maintenance strategy. The prognostic information is used to select an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper is to opportunities coming before the failure. The proposed maintenance scheduling methodology is based on opportunities coming before the failure. The proposed maintenance scheduling methodology is based on an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper is to an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper is to opportunities coming before the failure. The proposed maintenance scheduling methodology is based on consider the stochastic nature of opportunities duration using a Monte-Carlo simulation. A numerical study an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper study is to an optimal stopping problem algorithm known as algorithm. The originality of this is consider the stochastic nature of opportunities duration using aa Monte-Carlo simulation. A numerical consider the stochastic nature of opportunities using Monte-Carlo simulation. A numerical an optimal stopping algorithm as Bruss Bruss The originality of this paper paper study is to to is finally presented to problem illustrate the use andknown theduration strengths of algorithm. the proposed strategy. consider the stochastic nature of opportunities duration using a Monte-Carlo simulation. A numerical study consider stochastic nature of opportunities aa Monte-Carlo simulation. A numerical study is finally the presented to illustrate illustrate the use and and the theduration strengthsusing of the the proposed strategy. strategy. is finally presented to the use strengths of proposed consider the stochastic nature of opportunities duration using Monte-Carlo simulation. A numerical study is finally presented to illustrate illustrate the use use and and theprognostics, strengths of opportunistic the proposed strategy. is presented to the the strengths of the proposed strategy. © finally 2018, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights Keywords: maintenance decision-making, maintenance, odds reserved. algorithm, is finally presented to illustrate the use of and theprognostics, strengths of opportunistic the proposed strategy. Keywords: maintenance decision-making, maintenance, odds algorithm, Keywords: maintenance decision-making, prognostics, opportunistic maintenance, odds algorithm, opportunity, optimal stopping. Keywords: maintenance maintenance decision-making, prognostics, prognostics, opportunistic opportunistic maintenance, maintenance, odds algorithm, Keywords: decision-making, odds algorithm, opportunity, optimal stopping. opportunity, optimal stopping. Keywords: maintenance decision-making, prognostics, opportunistic maintenance, odds algorithm, opportunity, optimal stopping. opportunity, optimal stopping. opportunity, optimal stopping. completion of a mission by a system. The failed/defective 1. INTRODUCTION completion of of aa mission mission by by aa system. system. The The failed/defective failed/defective completion components areareplaced immediately. (Wu etfailed/defective al. 2016a) and 1. INTRODUCTION completion of mission by aa system. The 1. INTRODUCTION completion of a mission by system. The failed/defective components are replaced immediately. (Wu et al. 2016a) 2016a) and and 1. INTRODUCTION are replaced immediately. (Wu et al. completion of a mission by a system. The failed/defective (Wu et al. 2016b) established a decision model and of Maintenance strategies have evolved over the last decades components 1. INTRODUCTION components are2016b) replacedestablished immediately.a (Wu (Wu et al. al. 2016a) 2016a) 1. INTRODUCTION components are replaced immediately. et and (Wu et al. decision model of Maintenance strategies strategies have evolved evolved over over the the last last decades decades (Wu et al. 2016b) established a decision model of Maintenance have components are replaced immediately. (Wu et al. 2016a) and opportunistic using several constraints. (Nguyen et al. 2017) from corrective, preventive, and risk-based, up to condition(Wu et et al. al. using 2016b) established a (Nguyen decision etmodel model of Maintenance strategies haveand evolved over the the lastconditiondecades (Wu 2016b) established a decision of opportunistic several constraints. al. 2017) Maintenance strategies have evolved over last decades from corrective, preventive, risk-based, up to opportunistic using several constraints. (Nguyen etmodel al. 2017) 2017) from preventive, risk-based, up to et aal.joint 2016b) established a (Nguyen decision of propose predictive maintenance and spare parts Maintenance strategies haveand evolved over the lastconditiondecades based and nowadays predictive maintenance in order to keep (Wu opportunistic using several constraints. et al. from corrective, corrective, preventive, and risk-based, up to conditionopportunistic using several constraints. (Nguyen et al. 2017) propose a joint predictive maintenance and spare parts from corrective, preventive, and risk-based, up to conditionbased and nowadays predictive maintenance in order to keep aaforjoint predictive maintenance and spare parts based and nowadays predictive in to keep opportunistic using severalcomponents constraints. (Nguyen et al. inventory a multiple system. (Lu & 2017) Zhou from corrective, preventive, andmaintenance risk-based, to aconditionsystem with required performance as wellup asorder level of propose propose joint predictive maintenance and spare parts based and nowadays predictive maintenance in order to keep propose aafor predictive maintenance and spare parts inventory forjoint multiple components system. (Lu & Zhou Zhou based and nowadays maintenance to system with requiredpredictive performance as well wellin asorder level of inventory aaa multiple components system. & system with required performance as aaa level of propose joint predictive maintenance and (Lu spare parts 2017) propose an opportunistic preventive scheduling based and nowadays predictive maintenance inas order to keep keep performance with a high level of reliability and availability. inventory for multiple components system. (Lu & Zhou system with required performance as well as level of inventory for a multiple components system. (Lu & Zhou 2017) propose an opportunistic preventive scheduling system with required performance as well as a level of performance with a high level of reliability and availability. 2017) propose an opportunistic preventive scheduling performance with a high level of reliability and availability. inventory for a multiple components system. (Lu & Zhou methodology for a serie-parallel multistage manufacturing system with required performance as well as a level of These maintenance strategies rely on maintenance stoppages 2017) propose opportunistic preventive scheduling performance with a strategies high level of reliability and 2017) propose an opportunistic preventive scheduling methodology for an serie-parallel multistage manufacturing performance with high level of and availability. availability. These maintenance rely on maintenance maintenance stoppages methodology for aaa serie-parallel multistage manufacturing These maintenance rely on stoppages 2017) propose an opportunistic preventive scheduling with multiple streams deterioration. (Xiao et al. 2016) performance with aa strategies high levelappearance of reliability reliability availability. performed according to the of and events such as systems methodology for serie-parallel multistage manufacturing These maintenance strategies rely on maintenance stoppages methodology for a serie-parallel multistage manufacturing systems with multiple streams deterioration. (Xiao et al. 2016) 2016) These maintenance strategies rely on maintenance stoppages performed according to the appearance of events such as systems with multiple streams deterioration. (Xiao et al. performed according to the appearance of events such as methodology for a serie-parallel multistage manufacturing develop a joint optimization model connecting group PM2016) with These maintenance strategies rely on maintenance stoppages failure, systematic time between maintenance or the crossing systems with multiple streams deterioration. (Xiao et al. performed according to the appearance of events such as systems with multiple streams deterioration. (Xiao et al. 2016) develop a joint optimization model connecting group PM with performed according to the appearance of events such as failure, systematic time between maintenance or the crossing develop a joint optimization model connecting group PM with failure, systematic time between maintenance or the crossing systems with multiple streams deterioration. (Xiao et al. 2016) production scheduling applied to an in-series-system where performed according to the appearance of events such as of threshold (e.g. age, degradation level). For multidevelop aa joint joint optimization model connecting group PM PMwhere with failure, systematic time between maintenance or the crossing develop optimization model connecting group with production scheduling applied to an in-series-system failure, systematic time between maintenance or the crossing of threshold (e.g. age, degradation level). For multiproduction scheduling applied to an in-series-system where of age, degradation Forcrossing multia joint optimization model connecting PMwhere with PM on any machine leads to unavailability ofgroup all machines. failure, systematic time between maintenance or the components or(e.g. complex systems, these level). stoppages may be develop production scheduling applied to an an in-series-system of threshold threshold (e.g. age, degradation level). For multiproduction scheduling applied to in-series-system where PM on any machine leads to unavailability of all machines. of threshold (e.g. age, degradation level). For multicomponents or complex systems, these stoppages may be PM on any machine leads to unavailability of all machines. components or complex systems, these stoppages may be production scheduling applied to an in-series-system where (Zhou et al. 2009) propose an opportunistic preventive of threshold (e.g. age, degradation level). For multiexploited in order to simultaneously maintain others PM on any machine leads to unavailability of all machines. componentsinor or order complex systems, these stoppages stoppages may be PM on any to of machines. (Zhou et al.machine 2009) leads propose an opportunistic opportunistic preventive components complex these be exploited to systems, simultaneously maintain may others 2009) propose an exploited to simultaneously maintain others PM on et anyal. machine leads to unavailability unavailability of all all preventive machines. maintenance scheduling algorithm for the multi-unit in-series componentsin or order complex systems, these stoppages may be (Zhou meanwhile. This strategy is named opportunistic (Zhou et al. 2009) propose an opportunistic preventive exploited in order to simultaneously maintain others (Zhou et al. 2009) propose an opportunistic preventive maintenance scheduling algorithm for the multi-unit in-series exploited in order to simultaneously maintain others components meanwhile. This strategy is named opportunistic maintenance scheduling algorithm for the multi-unit in-series components meanwhile. This strategy is named opportunistic (Zhou et al. 2009) propose an opportunistic preventive system based on dynamic programming with consideration of exploited in order to simultaneously maintain others maintenance meanwhile. strategy (OMS) sinceis the aforementioned maintenance scheduling algorithm for the multi-unit in-series components This strategy named opportunistic maintenance scheduling algorithm for the multi-unit in-series system based on dynamic programming with consideration of components meanwhile. This strategy is named opportunistic maintenance strategy (OMS) since the aforementioned system based on dynamic programming with consideration of maintenance strategy (OMS) since the aforementioned maintenance scheduling algorithm for the multi-unit in-series the imperfect maintenance actions. components meanwhile. This strategy is named opportunistic stoppages are considered as opportunities to carry out extra system based on dynamic programming with consideration of maintenance strategy (OMS) since the aforementioned system based on dynamic programming with consideration the imperfect maintenance actions. maintenance strategy (OMS) since the aforementioned stoppages are considered as opportunities to carry out extra imperfect actions. stoppages are considered(OMS) as opportunities to carry out out extra extra the system based maintenance on dynamic programming with consideration of of maintenance strategy theto aforementioned interventions than thesince planned one. the imperfect maintenance actions. stoppages are considered as opportunities carry the imperfect maintenance actions. stoppages are considered as opportunities carry out extra maintenance interventions than the planned planned to one. maintenance interventions than the one. the imperfect maintenance actions. stoppages are considered as opportunities to carry out extra In the literature, only few papers take into account the maintenance interventions than the planned one. maintenance interventions than planned one. In the the literature, literature, only only few few papers papers take take into into account account the the maintenance interventions thanonthe the planned stochastic nature ofonly opportunities. (Truong Ba et al. 2017) In literature, numerous works OMS haveone. been proposed in In In the literature, few papers take into account the In the literature, only few papers take into account the stochastic nature of opportunities. (Truong Ba et al. 2017) In literature, numerous works on OMS have been proposed in stochastic nature of opportunities. (Truong Ba et al. 2017) In literature, on OMS have been in In the literature, only few papers take into account the considers the stochastic nature of both the opportunity arrival order to takenumerous benefit ofworks stochastic, structural andproposed economic stochastic nature of opportunities. (Truong Ba et al. 2017) In literature, numerous works on OMS have been proposed in stochastic nature of opportunities. (Truong Ba et al. 2017) considers the stochastic nature of both the opportunity arrival In literature, numerous works on OMS have been proposed in order to take benefit of stochastic, structural and economic considers the stochastic nature of both the opportunity arrival order to take benefit of stochastic, structural and economic stochastic nature of opportunities. (Truong Ba et al. 2017) and duration. The authors propose an optimization procedure In literature, numerous works on OMS have been proposed in dependence (Nguyen et al. 2017). (Truong Ba et al. 2017) considers the stochastic stochastic nature of both both the opportunity opportunity arrival order to to take take(Nguyen benefit of of stochastic, structuralBaand and economic considers the nature of the arrival and duration. The authors authors propose an optimization optimization procedure order benefit stochastic, structural dependence et al. al. 2017). (Truong (Truong et economic al. 2017) 2017) and duration. The propose an procedure dependence (Nguyen et 2017). Ba et al. thedefine stochastic of both the opportunity arrival in order to the nature best maintenance time interval and order to take benefit of stochastic, structural economic develop a maintenance opportunistic model considering and duration. The authors propose an optimization procedure dependence (Nguyen et al. al. 2017). (Truong (Truong Baand et al. 2017) 2017) considers and duration. The authors propose an optimization procedure in order to define the best maintenance time interval and dependence (Nguyen et 2017). Ba et al. develop a maintenance opportunistic model considering order to define best maintenance time and develop aa maintenance opportunistic model considering and duration. The authors propose an optimization minimum duration athe priori in the design phase. Ininterval a procedure predictive dependence (Nguyen et al.arrivals 2017).and (Truong Ba et al. 2017) in homogeneous opportunity stochastic opportunity in order to define the best maintenance time interval and develop maintenance opportunistic model considering in order to define the best maintenance time interval and minimum duration a priori in the design phase. In a predictive develop a maintenance opportunistic model considering homogeneous opportunity arrivals and stochastic opportunity duration aathe priori in the design phase. aa predictive homogeneous opportunity and stochastic opportunity in order to define best maintenance timeIn interval and maintenance strategy, where prognostics gives RUL develop maintenance opportunistic model et considering durations.a(Yildirim et al. arrivals 2017) and (Kennedy al. 2014) minimum minimum duration priori in the design phase. In predictive homogeneous opportunity arrivals and stochastic opportunity minimum duration a priori in the design phase. In a predictive maintenance strategy, where prognostics gives RUL homogeneous opportunity arrivals and stochastic opportunity durations. (Yildirim et al. 2017) and (Kennedy et al. 2014) maintenance strategy, where prognostics gives RUL durations. (Yildirim et al. 2017) and (Kennedy et al. 2014) minimum duration a priori in the design phase. In a predictive information such a priori optimization is no longer the best homogeneous opportunity arrivals and stochastic opportunity propose each one, a unified framework in order to group wind maintenance strategy, prognostics gives durations. (Yildirim et al. 2017) and (Kennedy al. 2014) maintenance strategy, where prognostics gives RUL information such priori where optimization is no no longer longer theRUL best durations. (Yildirim et and (Kennedy et al. 2014) propose each one, aa unified unified framework in order order to to et group wind information such aaa priori optimization is best propose each one, framework in group wind maintenance strategy, where prognostics givesthe RUL solution. durations. (Yildirim et al. al. 2017) 2017) and et al. 2014) turbine maintenance together. (Shin & (Kennedy Yacout 2016) consider information such priori optimization is no longer the best propose each one, a unified framework in order to group wind information such a priori optimization is no longer the solution. propose each one, a unified framework in order to group wind turbine maintenance together. (Shin & Yacout 2016) consider turbine maintenance together. (Shin & Yacout 2016) consider information such a priori optimization is no longer the best best propose each one, maintenance a unified framework in order to group an opportunity strategy launched by wind the solution. solution. turbine maintenance together. (Shin & Yacout 2016) consider turbine maintenance together. (Shin & Yacout 2016) consider an opportunity maintenance strategy launched by the the solution. an opportunity maintenance strategy launched by solution. turbine maintenance together. (Shin & Yacout 2016) consider component failure or preventive maintenance. Others In this paper, we consider the stochastic nature of opportunities an opportunity maintenance strategy launched by the an opportunity maintenance strategy launched by the component failure or preventive preventive maintenance. Others In this this paper, paper, we we consider consider the the stochastic stochastic nature nature of of opportunities opportunities component failure or Others an opportunity launched time by the components are maintenance repaired whenstrategy theirmaintenance. remaining to In duration in a we predictive stochastic component failure or preventive maintenance. Others In this paper, paper, considermaintenance the stochastic stochasticstrategy. nature of ofThe opportunities component failure or preventive maintenance. Others In this we consider the nature opportunities components are repaired when their remaining time to duration in a predictive maintenance strategy. The stochastic components are repaired when their remaining time to duration in a predictive maintenance strategy. The stochastic component failure or preventive maintenance. Others In this paper, we consider the stochastic nature of opportunities preventive action is below a predetermined value. (Yang et al. nature of opportunity arrival is tackle by the agility of the componentsaction are isrepaired repaired when their remaining remaining timeet al. to duration durationofin inopportunity a predictive predictivearrival maintenance strategy. The stochastic components are when their time to a maintenance strategy. The stochastic preventive below a predetermined value. (Yang nature is tackle by the agility of the the preventive action is below a predetermined value. (Yang et al. nature of opportunity arrival is tackle by the agility of components are isrepaired when their remaining time to proposed durationofin a predictive maintenance strategy. The stochastic 2016) uses action inspection as opportunity for maintenance afteretthe algorithm: each time is a new opportunity appears, preventive below a predetermined value. (Yang al. nature opportunity arrival tackle by the agility of the preventive action is below a predetermined value. (Yang et al. nature of opportunity arrival is tackle by the agility of 2016) uses inspection as opportunity for maintenance after the proposed algorithm: each time a new opportunity appears, 2016) uses inspection as opportunity for maintenance after each time aa new opportunity appears, preventive action is below a predetermined value. (Yang etthe al. proposed nature of algorithm: opportunity arrival is tackle by the agility of the the 2016) uses inspection as opportunity for maintenance after the proposed algorithm: each time new opportunity appears, the 2016) 2016) uses uses inspection inspection as as opportunity opportunity for for maintenance maintenance after after the the proposed proposed algorithm: algorithm: each each time time aa new new opportunity opportunity appears, appears, the the
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 460Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 460 Peer review© of International Federation of Automatic Copyright ©under 2018 responsibility IFAC 460Control. Copyright © 460 10.1016/j.ifacol.2018.08.348 Copyright © 2018 2018 IFAC IFAC 460 Copyright © 2018 IFAC 460
IFAC INCOM 2018 454 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
algorithm is run in order to get the optimal opportunity. Since the algorithm do not requires optimization, the running time is acceptable. The prognostic information is used to select opportunities coming before the failure. The proposed methodology is based on Bruss algorithm (Bruss 2003) developed for optimal stopping problem and introduced in maintenance by Thomas in (Thomas et al. 2007). (Thomas et al. 2007) used this algorithm to find in a set of opportunities, the optimal one in order to carry out maintenance interventions. The originality of this paper is to consider the stochastic nature of opportunities duration using a MonteCarlo simulation.
To solve the aforementioned problem, Bruss theorem (Bruss 2000) also called odds theorem is used. The goal of Bruss theorem is to maximize the probability to stop at the last success of a sequence of independent events. The optimal decision will be considered to this criteria. (Bruss 2000) shown that the optimal decision rule to select the farthest success is to choose the success for which the sum of odds from the last event (stoppage in our case) is equal or exceeds 1: add for decreasing value of 𝑛𝑛 until the value of 𝑅𝑅𝑠𝑠 exceeds 1, where 𝑛𝑛
𝑠𝑠 ≔ 𝑠𝑠𝑠𝑠𝑠𝑠 (1; 𝑠𝑠𝑠𝑠𝑠𝑠 {𝑘𝑘 ∈ [1; 𝑛𝑛]|𝑅𝑅𝑠𝑠 = ∑ 𝑟𝑟𝑖𝑖 ≥ 1})
The rest of the paper is organized as follows: Section 2 gives a set of mathematical tools used to formalize the aforementioned problem. Section 3 provides results in the deterministic case. Section 4 studies the effects of stochastic duration on the choice of opportunity. A conclusion is provided in Section 5.
𝑗𝑗=𝑘𝑘
(3)
𝑃𝑃𝑖𝑖 ⁄1 − 𝑃𝑃 with 𝑃𝑃𝑖𝑖 is the probability of success of 𝑖𝑖 the event 𝑖𝑖 and 1 − 𝑃𝑃𝑖𝑖 the probability without success.
Where 𝑟𝑟𝑖𝑖 =
General assumptions:
The first stoppage where the sum of odds reaches or exceeds 1 is the optimal one. 𝑠𝑠 is the index corresponding to the optimal stoppage. In case where the sum of odds do not reach 1, the maintenance manager must choice the forst occuring event/stoppage/opportunity.
An existing prognostic model provides the remaining useful life of the component. As explained in (Nzukam et al. 2017), an alpha level corresponding to the decision maker choice is defined. This level allows setting a time, corresponding to the time where the cumulative density function associated with the RUL equals to alpha. The selection of opportunities will be based on this time noted 𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎ℎ𝑎𝑎 that represents the failure time of the system (Figure 1).
Its application in maintenance is described in (Thomas et al. 2007) and (Nzukam et al. 2017). It considers reliability (R) and maintainability (M) for the computation of an event probability. An opportunity is called a success if the system/component is alive at time 𝑆𝑆𝑆𝑆𝑖𝑖 and maintainable during 𝐿𝐿𝑖𝑖 . The probability 𝑃𝑃𝑖𝑖 associated to this success is computed as follow:
2. PROBLEM FORMALIZATION The purpose of this section is to present the formalization of the problem of opportunistic maintenance
𝑃𝑃𝑖𝑖 = 𝑅𝑅(𝑆𝑆𝑆𝑆𝑖𝑖 ). 𝑀𝑀(𝐿𝐿𝑖𝑖 )
(1)
Then, the odds 𝑟𝑟𝑖𝑖 , 1 ≤ 𝑖𝑖 ≤ 𝑛𝑛, are computed through the following equation (the ratio between the probability of success and the probability of failure):
2.1 System stoppages modelling
A stoppage is defined by the couple (𝑆𝑆𝑆𝑆, 𝐿𝐿) with 𝑆𝑆𝑆𝑆 the stoppage starting time and 𝐿𝐿 the stoppage length with 𝐿𝐿 > 0. We consider a set of 𝑛𝑛 stoppages, a maintenance manager has at his disposal (Figure 1).
𝑟𝑟𝑖𝑖 =
𝑅𝑅(𝑆𝑆𝑆𝑆𝑖𝑖 ). 𝑀𝑀(𝐿𝐿𝑖𝑖 ) ⁄1 − 𝑅𝑅(𝑆𝑆𝑆𝑆 ). 𝑀𝑀(𝐿𝐿 ) 𝑖𝑖 𝑖𝑖
(2)
In order to get an ordered list of opportunities, the next suboptimal opportunity is determined by re-computing the sum on the set of opportunities except the optimal one. The ordered list of success gives to the maintenance manager more flexibility to plan the maintenance actions (for example in the context of unavailability of spare parts or human resource). Figure 1: Stoppage modelling
The following section provides an application of odds algorithm on an academic example.
Thus, given a maintenance action, what is the most appropriate stoppage that allows performing the maintenance action before the failure time? The next section describes Bruss algorithm on which lays the choice of appropriate opportunity.
3. DETERMINISTIC CASE This part is dedicated to show results with deterministic opportunity duration. Data considered in this paper are the same used in (Thomas et al. 2006). Let’s consider 1500ℎ for the planning horizon and twelve planned system stoppages as indicated in Table 1.
2.2 Bruss theorem in maintenance
461
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
455
Table 1: Stoppages characteristics 𝑵𝑵𝒐𝒐 St (h) L (h) 𝑵𝑵𝒐𝒐 St (h) L (h) 1
200
3
7
800
4
2
310
2
8
910
2
3
400
4
9
980
3 Figure 2: Reliability curve
4
560
2
10 1100
7 The next section introduces stochastic opportunities duration.
5
620
1
11 1250
3
4. USING MONTE-CARLO SIMULATION FOR STOCHASTIC OPPORTUNITIES DURATION
6
690
4
12 1360
4
The same example is used but stochastic durations are considered. The uncertainty on stoppage duration is introduced in the experiment by the means of Gaussian distribution and is processed using a Monte-Carlo simulation. Monte-Carlo simulation is described in the following.
The reliability is assumed to be a Weibull distribution with the shape parameter 𝛽𝛽 = 1.5 and the scale parameter 𝜂𝜂 = 700 ℎ. Maintainability is supposed following an exponential law with parameter 𝜇𝜇 = 0.5 ℎ−1 .
4.1 Monte-Carlo approach and stochastic duration modelling
Table 2: Classification Results 𝑵𝑵𝒐𝒐 Odds (r)
𝑹𝑹𝒔𝒔
𝑵𝑵𝒐𝒐 Odds (r)
Monte-Carlo simulation is a computerized mathematical technique that takes into account risk in quantitative analysis and decision-making (Rubinstein & Kroese 2016). In the present work, the risk is associated to the quality of available knowledge on opportunities duration. The essential idea of Monte-Carlo simulation is to develop possible outcome models, considering a range of values described by a probability distribution) for any uncertainty-carrying factor. The advantage of Monte-Carlo simulation is to get the uncertainty on the outcome when no analytical solution is available.
𝑹𝑹𝒔𝒔
1
2.0016
2.9797
7
0.3420
0.9781
2
0.8895
1.8676
8
0.1676
0.6361
3
1.2798
2.2579
9
0.1740
0.4685
4
0.4473
1.4254
10
0.1564
0.2945
5
0.2062
1.1843
11
0.0769
0.1381
6
0.4814
1.4595 12
0.0612
0.0612
Considering the formalism introduced in section 2, the uncertainty is modeled using a Gaussian distribution for the length of the opportunities. The mean of the distribution will be 𝐿𝐿𝑖𝑖 and the standard deviation will vary in order to show the influence of the uncertainty level. Every length ℒ𝑖𝑖 is considered as a random variable 𝐿𝐿𝑖𝑖 such that: ℒ𝑖𝑖 = 𝒩𝒩(𝐿𝐿𝑖𝑖 , 𝜎𝜎)
Where 𝒩𝒩(𝜇𝜇, 𝜎𝜎) is the normal distribution with a mean equals to 𝜇𝜇 and a standard deviation equals to 𝜎𝜎.
Table 2 shows the values for every opportunity computed from (2) and the bold results shows the list of success. The last success, i.e. #6, is the optimal one. The stoppages 5, 4, 3, 2 𝑎𝑎𝑎𝑎𝑎𝑎 1 are considered as alternatives (for flexibility) in the case where it could not be possible to perform maintenance at stoppage #6.
4.2 Results of the Monte-Carlo simulations The Monte-Carlo simulation considers 10000 draws. The results are presented in Table 3 with range of values for 𝜎𝜎 from 0.1 to 2.5. Each figure must be read in relation (or compared)
Figure 2 shows the reliability curve associated to the considered component. For stoppage 6, the reliability is 0.38 at time 690ℎ. 462
IFAC INCOM 2018 456 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
𝜎𝜎 = 0.7
to the deterministic case which give stoppage #6 as optimal stoppage.
Table 3: Numerical results 𝜎𝜎 = 0.1
𝜎𝜎 = 1
𝜎𝜎 = 0.25 𝜎𝜎 = 1.5
𝜎𝜎 = 0.3
𝜎𝜎 = 2
𝜎𝜎 = 0.4
𝜎𝜎 = 2.5
463
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
On the whole tests, the most probable stoppage is the stoppage #6 as in the deterministic case. On one hand, only few cases lead to consider an optimal stoppage earlier than #6 and only in the tests where the uncertainty is the highest (𝜎𝜎 >1.5). This means that the sum of the odds is close enough from 1 at stoppage #7 and the odd of stoppage #6, even when the duration decrease, is big enough to allows the sum to reach 1. When stoppage duration shortens, the maintainability decreases and the odd decreases. On the contrary, when stoppage duration lengthens, the maintainability increases and the odd increases.
457
consumption and energy bill in the operational stage of existing non-residential buildings. (https://www.energyintime.eu/). REFERENCES Bruss, F.T., 2003. A note on bounds for the odds theorem of optimal stopping. Annals of Probability, 31(4), pp.1859–1861. Bruss, F.T., 2000. Sum the odds to one and stop. Annals of Probability, 28(3), pp.1384–1391.
On the other hand, stoppage #7 appears to be the second most probable optimal stoppage. As it allows the sum of the odds to be close from 1, i.e. 0,98, when its length increase, it increases its odd and allows the sum to reach 1.The sum of the odds at stoppage #8 is 0,64 and is not close enough from 1 to allows stoppage #8 to be the optimal stoppage.
Kennedy, K., Walsh, P. & Scully, T., 2014. Genetic Optimisation for a Stochastic Model for Opportunistic Maintenance Planning of Offshore Wind Farms. Lu, B. & Zhou, X., 2017. Opportunistic preventive maintenance scheduling for serial-parallel multistage manufacturing systems with multiple streams of deterioration. Reliability Engineering and System Safety, 168(May), pp.116–127. Available at: https://doi.org/10.1016/j.ress.2017.05.017.
From the decision maker point of view, i.e. maintenance manager, when the uncertainty becomes important (𝜎𝜎 =1, compared to the length duration mean equal to 2,6), the decision uncertainty allows to choose between stoppage 7 and stoppage 6. This means that the uncertainty allows to consider a stoppage further. Nevertheless, the decision is not easy since only binary results are shown. Indeed, decision maker would prefer to have a more detailed view of how the choice has been done, including uncertainty.
Nguyen, K.A., Do, P. & Grall, A., 2017. Joint predictive maintenance and inventory strategy for multicomponent systems using Birnbaum’s structural importance. Reliability Engineering and System Safety, 168(May), pp.249–261. Available at: https://doi.org/10.1016/j.ress.2017.05.034.
5. CONCLUSIONS Nzukam, C., Voisin, A., Levrat, E., Sauter, D., Iung, B., 2017. ScienceDirect A dynamic grouping costs savings Non-residential Non-residential buildings buildings costs savings in Non-residential. IFAC-PapersOnLine, 50(1), pp.13722–13727. Available at: https://doi.org/10.1016/j.ifacol.2017.08.2551.
In this paper, an opportunistic maintenance scheduling methodology considering the stochastic nature of opportunities duration is proposed in a predictive maintenance strategy. The proposed maintenance scheduling methodology used the prognostic information to select opportunities coming before the failure and is based on an optimal stopping problem algorithm known as Bruss algorithm. The Bruss algorithm is use since it has shows it usefulness on previous case. The uncertainty on opportunities duration is tackle with a MonteCarlo simulation as no analytical formulation exists. This makes this contribution original.
Rubinstein, R.Y. & Kroese, D.P., 2016. Simulation and the Monte Carlo method, John Wiley & Sons. Shin, H. & Yacout, S., 2016. Opportunistic Preventive Maintenance Strategy of a Multi-Component System with Hierarchical Structure by Simulation and Evaluation. Emerging Technologies and Factory Automation (ETFA), 2016 IEEE 21st International Conference on, pp.1–8.
Further studies will include both theoretical and practical aspects. From the theoretical aspect, the consideration of uncertainty in both length and starting date of stoppage will be considered and an analytical solution will be investigated in order to have the uncertainty on the sum of odds. From the practical aspect, two directions will be followed: (1) the consideration of the spare parts and human resources availability and (2) what information would be more relevant for the decision maker than a « binary » one (as in Table 3). Finally, the proposed approach has to be compared with a classic systematic maintenance scheduling in order to shows its benefit.
Thomas, E. , Levrat, E., Iung, B., & Monin, M., 2006. Odds algorithm-based opportunity-triggered preventive maintenance with production policy. In 6th IFAC Symposium. pp. 835–840. Thomas, E., Levrat, E. & Iung, B., 2007. L’algorithme de Bruss comme contribution à une maintenance préventive opportuniste. e-STA, Sciences et Technologies de l’Automatique, 3, pp.13–18. Available at: http://hal.archives-ouvertes.fr/hal-00149994/ [Accessed July 4, 2013].
The authors would like to acknowledge the financial support of the European Commission under the Seventh Framework Program titled Energy-In-Time that aims to develop Smart Energy Simulation Based Control method to reduce the energy 464
IFAC INCOM 2018 458 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
Truong Ba, H., Cholette, M.E., Borghesani, P., Zhou, Y., Ma, L., 2017. Opportunistic maintenance considering nonhomogenous opportunity arrivals and stochastic opportunity durations. Reliability Engineering and System Safety, 160(April 2016), pp.151–161. Wu, Q., Lv, C., Zhou, D., Wang, Y., Yu, Dequan., 2016a. Multi-event Maintenance Decision-making Model and Optimization Method Based on Opportunistic Maintenance Policy. 12th World Congress on Intelligent Control and Automation (WCICA), (June 12-15, 2016, Guilin, China), pp.1–5. Wu, Q., Lv, C., Zhou, D., Wang, Y., Yu, Dequan 2016b. Study on Multi-event Opportunistic Maintenance Decision-making Model Based on Condition. , pp.1–5. Xiao, L., Song, S., Chen, X., Coit, D.W., 2016. Joint optimization of production scheduling and machine group preventive maintenance. Reliability Engineering and System Safety, 146, pp.68–78. Available at: http://dx.doi.org/10.1016/j.ress.2015.10.013. Yang, L., Ma, X., Zhai, Q., Zhao, Y., 2016. A delay time model for a mission-based system subject to periodic and random inspection and postponed replacement. Reliability Engineering and System Safety, 150, pp.96– 104. Available at: http://dx.doi.org/10.1016/j.ress.2016.01.016. Yildirim, M., Gebraeel, N.Z. & Sun, X.A., 2017. Integrated Predictive Analytics and Optimization for Opportunistic Maintenance and Operations in Wind Farms. IEEE Transactions on Power Systems, 32(6), pp.4319–4328. Zhou, X., Xi, L. & Lee, J., 2009. Opportunistic preventive maintenance scheduling for a multi-unit series system based on dynamic programming. , 118, pp.361–366.
465
ScienceDirect
IFAC PapersOnLine 51-11 (2018) 453–458
Opportunistic maintenance scheduling with stochastic opportunities duration in aa Opportunistic maintenance scheduling with stochastic opportunities duration in Opportunistic maintenance scheduling with stochastic opportunities duration in aa Opportunistic maintenance scheduling with stochastic opportunities duration in predictive maintenance strategy Opportunistic scheduling with stochastic opportunities duration in aa predictive maintenance strategy Opportunistic maintenance maintenance scheduling with stochastic opportunities duration in predictive maintenance strategy predictive maintenance strategy predictive maintenance strategy predictive maintenance strategy C. Nzukam, A. Voisin, E. Levrat, D. Sauter, B. Iung
C. Nzukam, A. Voisin, E. Sauter, B. Iung C. Nzukam, A. Voisin, E. Levrat, D. D. Sauter, B. Iung Levrat, C. Nzukam, A. Voisin, E. D. Sauter, B. Iung Levrat, C. Nzukam, A. Voisin, E. Levrat, D. Sauter, B. Iung C. Nzukam, A. Voisin, E. Levrat, D. Sauter, B. Iung Lorraine University, CRAN, CNRS UMR 7039, Nancy, France Lorraine University, CRAN, CNRS UMR 7039, Nancy, France Lorraine University, CRAN, CNRS UMR 7039, Nancy, France collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr Lorraine University, CRAN, CNRS UMR 7039, Nancy, France Lorraine University, CRAN, CNRS UMR 7039, Nancy, France collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr Lorraine University, CRAN, CNRS UMR 7039, Nancy, France collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr collince-christian.nzukam, alexandre.voisin, eric.levrat, dominique.sauter, benoît.iung @univ-lorraine.fr Abstract: To guarantee a good level of facilities performance (reliability and availability), maintenance Abstract: To guarantee aa good level of facilities performance (reliability and availability), maintenance Abstract: To guarantee of facilities performance (reliability and availability), activities have to be planned andlevel scheduled efficiently. Maintenance scheduling decides, in a maintenance tactical way, Abstract: To guarantee aa good good level of facilities performance (reliability and availability), maintenance Abstract: To guarantee good level of facilities performance (reliability and availability), maintenance activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in a tactical way, activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in a tactical way, Abstract: To guarantee a good level of facilities performance (reliability and availability), maintenance when theses maintenance activities will be carried out according to the appearance of opportunities. activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in aaopportunities. tactical way, activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in tactical way, when theses maintenance activities will be carried out according to the appearance of when theses maintenance activities will be carried out according to the appearance of opportunities. activities have to be planned and scheduled efficiently. Maintenance scheduling decides, in a tactical way, Numerous works on opportunistic maintenance have been proposed in order to take profit of stochastic, when theses maintenance activities will be carried out according to the appearance of opportunities. when theses maintenance activities will be carried out according to the appearance of opportunities. Numerous works on opportunistic maintenance have been proposed in order to take profit of stochastic, Numerous works on opportunistic maintenance have been proposed in order to take profit of stochastic, when theses maintenance activities will be carried out according to the appearance of opportunities. In the real context, the duration of an opportunity is not known structural and economic dependence. Numerousand works on opportunistic opportunistic maintenance have been proposed proposed in order order toopportunity take profit profit is of not stochastic, Numerous works on maintenance have been in take of stochastic, structural economic dependence. In the real context, the duration of anto known structural economic dependence. In real context, the duration of Numerous works on papers opportunistic maintenance been proposed in order take profit is of stochastic, accurately.and Very few take into account the stochastic nature of opportunities duration. Innot thisknown paper, structural and economic dependence. In the the the realhave context, the duration of an antoopportunity opportunity is not known structural and economic dependence. In the real context, the duration of an opportunity is not known accurately. Very few papers take into account stochastic nature of opportunities duration. In this paper, accurately. Very few papers take into account the nature of opportunities duration. this paper, structural dependence. In the realstochastic context, the duration of an opportunity isIn not known we presentand an economic opportunistic maintenance scheduling methodology considering the stochastic nature of accurately. Very few papers take into account the stochastic nature of opportunities duration. In this paper, accurately. Very few take into the nature of opportunities duration. In paper, we present an opportunistic maintenance scheduling methodology the stochastic nature of we present an opportunistic maintenance scheduling methodology considering the stochastic nature of accurately. Very few papers papers take into account account the stochastic stochastic nature of considering opportunities duration. In this this opportunities duration in a predictive maintenance strategy. The prognostic information is used to paper, select we present an opportunistic maintenance scheduling methodology considering the stochastic nature of we present opportunistic maintenance scheduling methodology considering the nature of opportunities duration in aa predictive maintenance strategy. The prognostic information is used to select opportunities duration in maintenance strategy. The prognostic information is used to select we present an ancoming opportunistic maintenance scheduling methodology considering the stochastic stochastic of before the failure. The proposed maintenance scheduling methodology is nature based on opportunities duration in aa predictive predictive maintenance strategy. The prognostic information is used to select opportunities duration in predictive maintenance strategy. The prognostic information is used to select coming before the failure. The proposed maintenance scheduling methodology is based on coming before the failure. The proposed maintenance scheduling methodology is based on opportunities duration in a predictive maintenance strategy. The prognostic information is used to select an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper is to opportunities coming before the failure. The proposed maintenance scheduling methodology is based on opportunities coming before the failure. The proposed maintenance scheduling methodology is based on an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper is to an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper is to opportunities coming before the failure. The proposed maintenance scheduling methodology is based on consider the stochastic nature of opportunities duration using a Monte-Carlo simulation. A numerical study an optimal stopping problem algorithm known as Bruss algorithm. The originality of this paper study is to an optimal stopping problem algorithm known as algorithm. The originality of this is consider the stochastic nature of opportunities duration using aa Monte-Carlo simulation. A numerical consider the stochastic nature of opportunities using Monte-Carlo simulation. A numerical an optimal stopping algorithm as Bruss Bruss The originality of this paper paper study is to to is finally presented to problem illustrate the use andknown theduration strengths of algorithm. the proposed strategy. consider the stochastic nature of opportunities duration using a Monte-Carlo simulation. A numerical study consider stochastic nature of opportunities aa Monte-Carlo simulation. A numerical study is finally the presented to illustrate illustrate the use and and the theduration strengthsusing of the the proposed strategy. strategy. is finally presented to the use strengths of proposed consider the stochastic nature of opportunities duration using Monte-Carlo simulation. A numerical study is finally presented to illustrate illustrate the use use and and theprognostics, strengths of opportunistic the proposed strategy. is presented to the the strengths of the proposed strategy. © finally 2018, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights Keywords: maintenance decision-making, maintenance, odds reserved. algorithm, is finally presented to illustrate the use of and theprognostics, strengths of opportunistic the proposed strategy. Keywords: maintenance decision-making, maintenance, odds algorithm, Keywords: maintenance decision-making, prognostics, opportunistic maintenance, odds algorithm, opportunity, optimal stopping. Keywords: maintenance maintenance decision-making, prognostics, prognostics, opportunistic opportunistic maintenance, maintenance, odds algorithm, Keywords: decision-making, odds algorithm, opportunity, optimal stopping. opportunity, optimal stopping. Keywords: maintenance decision-making, prognostics, opportunistic maintenance, odds algorithm, opportunity, optimal stopping. opportunity, optimal stopping. opportunity, optimal stopping. completion of a mission by a system. The failed/defective 1. INTRODUCTION completion of of aa mission mission by by aa system. system. The The failed/defective failed/defective completion components areareplaced immediately. (Wu etfailed/defective al. 2016a) and 1. INTRODUCTION completion of mission by aa system. The 1. INTRODUCTION completion of a mission by system. The failed/defective components are replaced immediately. (Wu et al. 2016a) 2016a) and and 1. INTRODUCTION are replaced immediately. (Wu et al. completion of a mission by a system. The failed/defective (Wu et al. 2016b) established a decision model and of Maintenance strategies have evolved over the last decades components 1. INTRODUCTION components are2016b) replacedestablished immediately.a (Wu (Wu et al. al. 2016a) 2016a) 1. INTRODUCTION components are replaced immediately. et and (Wu et al. decision model of Maintenance strategies strategies have evolved evolved over over the the last last decades decades (Wu et al. 2016b) established a decision model of Maintenance have components are replaced immediately. (Wu et al. 2016a) and opportunistic using several constraints. (Nguyen et al. 2017) from corrective, preventive, and risk-based, up to condition(Wu et et al. al. using 2016b) established a (Nguyen decision etmodel model of Maintenance strategies haveand evolved over the the lastconditiondecades (Wu 2016b) established a decision of opportunistic several constraints. al. 2017) Maintenance strategies have evolved over last decades from corrective, preventive, risk-based, up to opportunistic using several constraints. (Nguyen etmodel al. 2017) 2017) from preventive, risk-based, up to et aal.joint 2016b) established a (Nguyen decision of propose predictive maintenance and spare parts Maintenance strategies haveand evolved over the lastconditiondecades based and nowadays predictive maintenance in order to keep (Wu opportunistic using several constraints. et al. from corrective, corrective, preventive, and risk-based, up to conditionopportunistic using several constraints. (Nguyen et al. 2017) propose a joint predictive maintenance and spare parts from corrective, preventive, and risk-based, up to conditionbased and nowadays predictive maintenance in order to keep aaforjoint predictive maintenance and spare parts based and nowadays predictive in to keep opportunistic using severalcomponents constraints. (Nguyen et al. inventory a multiple system. (Lu & 2017) Zhou from corrective, preventive, andmaintenance risk-based, to aconditionsystem with required performance as wellup asorder level of propose propose joint predictive maintenance and spare parts based and nowadays predictive maintenance in order to keep propose aafor predictive maintenance and spare parts inventory forjoint multiple components system. (Lu & Zhou Zhou based and nowadays maintenance to system with requiredpredictive performance as well wellin asorder level of inventory aaa multiple components system. & system with required performance as aaa level of propose joint predictive maintenance and (Lu spare parts 2017) propose an opportunistic preventive scheduling based and nowadays predictive maintenance inas order to keep keep performance with a high level of reliability and availability. inventory for multiple components system. (Lu & Zhou system with required performance as well as level of inventory for a multiple components system. (Lu & Zhou 2017) propose an opportunistic preventive scheduling system with required performance as well as a level of performance with a high level of reliability and availability. 2017) propose an opportunistic preventive scheduling performance with a high level of reliability and availability. inventory for a multiple components system. (Lu & Zhou methodology for a serie-parallel multistage manufacturing system with required performance as well as a level of These maintenance strategies rely on maintenance stoppages 2017) propose opportunistic preventive scheduling performance with a strategies high level of reliability and 2017) propose an opportunistic preventive scheduling methodology for an serie-parallel multistage manufacturing performance with high level of and availability. availability. These maintenance rely on maintenance maintenance stoppages methodology for aaa serie-parallel multistage manufacturing These maintenance rely on stoppages 2017) propose an opportunistic preventive scheduling with multiple streams deterioration. (Xiao et al. 2016) performance with aa strategies high levelappearance of reliability reliability availability. performed according to the of and events such as systems methodology for serie-parallel multistage manufacturing These maintenance strategies rely on maintenance stoppages methodology for a serie-parallel multistage manufacturing systems with multiple streams deterioration. (Xiao et al. 2016) 2016) These maintenance strategies rely on maintenance stoppages performed according to the appearance of events such as systems with multiple streams deterioration. (Xiao et al. performed according to the appearance of events such as methodology for a serie-parallel multistage manufacturing develop a joint optimization model connecting group PM2016) with These maintenance strategies rely on maintenance stoppages failure, systematic time between maintenance or the crossing systems with multiple streams deterioration. (Xiao et al. performed according to the appearance of events such as systems with multiple streams deterioration. (Xiao et al. 2016) develop a joint optimization model connecting group PM with performed according to the appearance of events such as failure, systematic time between maintenance or the crossing develop a joint optimization model connecting group PM with failure, systematic time between maintenance or the crossing systems with multiple streams deterioration. (Xiao et al. 2016) production scheduling applied to an in-series-system where performed according to the appearance of events such as of threshold (e.g. age, degradation level). For multidevelop aa joint joint optimization model connecting group PM PMwhere with failure, systematic time between maintenance or the crossing develop optimization model connecting group with production scheduling applied to an in-series-system failure, systematic time between maintenance or the crossing of threshold (e.g. age, degradation level). For multiproduction scheduling applied to an in-series-system where of age, degradation Forcrossing multia joint optimization model connecting PMwhere with PM on any machine leads to unavailability ofgroup all machines. failure, systematic time between maintenance or the components or(e.g. complex systems, these level). stoppages may be develop production scheduling applied to an an in-series-system of threshold threshold (e.g. age, degradation level). For multiproduction scheduling applied to in-series-system where PM on any machine leads to unavailability of all machines. of threshold (e.g. age, degradation level). For multicomponents or complex systems, these stoppages may be PM on any machine leads to unavailability of all machines. components or complex systems, these stoppages may be production scheduling applied to an in-series-system where (Zhou et al. 2009) propose an opportunistic preventive of threshold (e.g. age, degradation level). For multiexploited in order to simultaneously maintain others PM on any machine leads to unavailability of all machines. componentsinor or order complex systems, these stoppages stoppages may be PM on any to of machines. (Zhou et al.machine 2009) leads propose an opportunistic opportunistic preventive components complex these be exploited to systems, simultaneously maintain may others 2009) propose an exploited to simultaneously maintain others PM on et anyal. machine leads to unavailability unavailability of all all preventive machines. maintenance scheduling algorithm for the multi-unit in-series componentsin or order complex systems, these stoppages may be (Zhou meanwhile. This strategy is named opportunistic (Zhou et al. 2009) propose an opportunistic preventive exploited in order to simultaneously maintain others (Zhou et al. 2009) propose an opportunistic preventive maintenance scheduling algorithm for the multi-unit in-series exploited in order to simultaneously maintain others components meanwhile. This strategy is named opportunistic maintenance scheduling algorithm for the multi-unit in-series components meanwhile. This strategy is named opportunistic (Zhou et al. 2009) propose an opportunistic preventive system based on dynamic programming with consideration of exploited in order to simultaneously maintain others maintenance meanwhile. strategy (OMS) sinceis the aforementioned maintenance scheduling algorithm for the multi-unit in-series components This strategy named opportunistic maintenance scheduling algorithm for the multi-unit in-series system based on dynamic programming with consideration of components meanwhile. This strategy is named opportunistic maintenance strategy (OMS) since the aforementioned system based on dynamic programming with consideration of maintenance strategy (OMS) since the aforementioned maintenance scheduling algorithm for the multi-unit in-series the imperfect maintenance actions. components meanwhile. This strategy is named opportunistic stoppages are considered as opportunities to carry out extra system based on dynamic programming with consideration of maintenance strategy (OMS) since the aforementioned system based on dynamic programming with consideration the imperfect maintenance actions. maintenance strategy (OMS) since the aforementioned stoppages are considered as opportunities to carry out extra imperfect actions. stoppages are considered(OMS) as opportunities to carry out out extra extra the system based maintenance on dynamic programming with consideration of of maintenance strategy theto aforementioned interventions than thesince planned one. the imperfect maintenance actions. stoppages are considered as opportunities carry the imperfect maintenance actions. stoppages are considered as opportunities carry out extra maintenance interventions than the planned planned to one. maintenance interventions than the one. the imperfect maintenance actions. stoppages are considered as opportunities to carry out extra In the literature, only few papers take into account the maintenance interventions than the planned one. maintenance interventions than planned one. In the the literature, literature, only only few few papers papers take take into into account account the the maintenance interventions thanonthe the planned stochastic nature ofonly opportunities. (Truong Ba et al. 2017) In literature, numerous works OMS haveone. been proposed in In In the literature, few papers take into account the In the literature, only few papers take into account the stochastic nature of opportunities. (Truong Ba et al. 2017) In literature, numerous works on OMS have been proposed in stochastic nature of opportunities. (Truong Ba et al. 2017) In literature, on OMS have been in In the literature, only few papers take into account the considers the stochastic nature of both the opportunity arrival order to takenumerous benefit ofworks stochastic, structural andproposed economic stochastic nature of opportunities. (Truong Ba et al. 2017) In literature, numerous works on OMS have been proposed in stochastic nature of opportunities. (Truong Ba et al. 2017) considers the stochastic nature of both the opportunity arrival In literature, numerous works on OMS have been proposed in order to take benefit of stochastic, structural and economic considers the stochastic nature of both the opportunity arrival order to take benefit of stochastic, structural and economic stochastic nature of opportunities. (Truong Ba et al. 2017) and duration. The authors propose an optimization procedure In literature, numerous works on OMS have been proposed in dependence (Nguyen et al. 2017). (Truong Ba et al. 2017) considers the stochastic stochastic nature of both both the opportunity opportunity arrival order to to take take(Nguyen benefit of of stochastic, structuralBaand and economic considers the nature of the arrival and duration. The authors authors propose an optimization optimization procedure order benefit stochastic, structural dependence et al. al. 2017). (Truong (Truong et economic al. 2017) 2017) and duration. The propose an procedure dependence (Nguyen et 2017). Ba et al. thedefine stochastic of both the opportunity arrival in order to the nature best maintenance time interval and order to take benefit of stochastic, structural economic develop a maintenance opportunistic model considering and duration. The authors propose an optimization procedure dependence (Nguyen et al. al. 2017). (Truong (Truong Baand et al. 2017) 2017) considers and duration. The authors propose an optimization procedure in order to define the best maintenance time interval and dependence (Nguyen et 2017). Ba et al. develop a maintenance opportunistic model considering order to define best maintenance time and develop aa maintenance opportunistic model considering and duration. The authors propose an optimization minimum duration athe priori in the design phase. Ininterval a procedure predictive dependence (Nguyen et al.arrivals 2017).and (Truong Ba et al. 2017) in homogeneous opportunity stochastic opportunity in order to define the best maintenance time interval and develop maintenance opportunistic model considering in order to define the best maintenance time interval and minimum duration a priori in the design phase. In a predictive develop a maintenance opportunistic model considering homogeneous opportunity arrivals and stochastic opportunity duration aathe priori in the design phase. aa predictive homogeneous opportunity and stochastic opportunity in order to define best maintenance timeIn interval and maintenance strategy, where prognostics gives RUL develop maintenance opportunistic model et considering durations.a(Yildirim et al. arrivals 2017) and (Kennedy al. 2014) minimum minimum duration priori in the design phase. In predictive homogeneous opportunity arrivals and stochastic opportunity minimum duration a priori in the design phase. In a predictive maintenance strategy, where prognostics gives RUL homogeneous opportunity arrivals and stochastic opportunity durations. (Yildirim et al. 2017) and (Kennedy et al. 2014) maintenance strategy, where prognostics gives RUL durations. (Yildirim et al. 2017) and (Kennedy et al. 2014) minimum duration a priori in the design phase. In a predictive information such a priori optimization is no longer the best homogeneous opportunity arrivals and stochastic opportunity propose each one, a unified framework in order to group wind maintenance strategy, prognostics gives durations. (Yildirim et al. 2017) and (Kennedy al. 2014) maintenance strategy, where prognostics gives RUL information such priori where optimization is no no longer longer theRUL best durations. (Yildirim et and (Kennedy et al. 2014) propose each one, aa unified unified framework in order order to to et group wind information such aaa priori optimization is best propose each one, framework in group wind maintenance strategy, where prognostics givesthe RUL solution. durations. (Yildirim et al. al. 2017) 2017) and et al. 2014) turbine maintenance together. (Shin & (Kennedy Yacout 2016) consider information such priori optimization is no longer the best propose each one, a unified framework in order to group wind information such a priori optimization is no longer the solution. propose each one, a unified framework in order to group wind turbine maintenance together. (Shin & Yacout 2016) consider turbine maintenance together. (Shin & Yacout 2016) consider information such a priori optimization is no longer the best best propose each one, maintenance a unified framework in order to group an opportunity strategy launched by wind the solution. solution. turbine maintenance together. (Shin & Yacout 2016) consider turbine maintenance together. (Shin & Yacout 2016) consider an opportunity maintenance strategy launched by the the solution. an opportunity maintenance strategy launched by solution. turbine maintenance together. (Shin & Yacout 2016) consider component failure or preventive maintenance. Others In this paper, we consider the stochastic nature of opportunities an opportunity maintenance strategy launched by the an opportunity maintenance strategy launched by the component failure or preventive preventive maintenance. Others In this this paper, paper, we we consider consider the the stochastic stochastic nature nature of of opportunities opportunities component failure or Others an opportunity launched time by the components are maintenance repaired whenstrategy theirmaintenance. remaining to In duration in a we predictive stochastic component failure or preventive maintenance. Others In this paper, paper, considermaintenance the stochastic stochasticstrategy. nature of ofThe opportunities component failure or preventive maintenance. Others In this we consider the nature opportunities components are repaired when their remaining time to duration in a predictive maintenance strategy. The stochastic components are repaired when their remaining time to duration in a predictive maintenance strategy. The stochastic component failure or preventive maintenance. Others In this paper, we consider the stochastic nature of opportunities preventive action is below a predetermined value. (Yang et al. nature of opportunity arrival is tackle by the agility of the componentsaction are isrepaired repaired when their remaining remaining timeet al. to duration durationofin inopportunity a predictive predictivearrival maintenance strategy. The stochastic components are when their time to a maintenance strategy. The stochastic preventive below a predetermined value. (Yang nature is tackle by the agility of the the preventive action is below a predetermined value. (Yang et al. nature of opportunity arrival is tackle by the agility of components are isrepaired when their remaining time to proposed durationofin a predictive maintenance strategy. The stochastic 2016) uses action inspection as opportunity for maintenance afteretthe algorithm: each time is a new opportunity appears, preventive below a predetermined value. (Yang al. nature opportunity arrival tackle by the agility of the preventive action is below a predetermined value. (Yang et al. nature of opportunity arrival is tackle by the agility of 2016) uses inspection as opportunity for maintenance after the proposed algorithm: each time a new opportunity appears, 2016) uses inspection as opportunity for maintenance after each time aa new opportunity appears, preventive action is below a predetermined value. (Yang etthe al. proposed nature of algorithm: opportunity arrival is tackle by the agility of the the 2016) uses inspection as opportunity for maintenance after the proposed algorithm: each time new opportunity appears, the 2016) 2016) uses uses inspection inspection as as opportunity opportunity for for maintenance maintenance after after the the proposed proposed algorithm: algorithm: each each time time aa new new opportunity opportunity appears, appears, the the
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 460Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 460 Peer review© of International Federation of Automatic Copyright ©under 2018 responsibility IFAC 460Control. Copyright © 460 10.1016/j.ifacol.2018.08.348 Copyright © 2018 2018 IFAC IFAC 460 Copyright © 2018 IFAC 460
IFAC INCOM 2018 454 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
algorithm is run in order to get the optimal opportunity. Since the algorithm do not requires optimization, the running time is acceptable. The prognostic information is used to select opportunities coming before the failure. The proposed methodology is based on Bruss algorithm (Bruss 2003) developed for optimal stopping problem and introduced in maintenance by Thomas in (Thomas et al. 2007). (Thomas et al. 2007) used this algorithm to find in a set of opportunities, the optimal one in order to carry out maintenance interventions. The originality of this paper is to consider the stochastic nature of opportunities duration using a MonteCarlo simulation.
To solve the aforementioned problem, Bruss theorem (Bruss 2000) also called odds theorem is used. The goal of Bruss theorem is to maximize the probability to stop at the last success of a sequence of independent events. The optimal decision will be considered to this criteria. (Bruss 2000) shown that the optimal decision rule to select the farthest success is to choose the success for which the sum of odds from the last event (stoppage in our case) is equal or exceeds 1: add for decreasing value of 𝑛𝑛 until the value of 𝑅𝑅𝑠𝑠 exceeds 1, where 𝑛𝑛
𝑠𝑠 ≔ 𝑠𝑠𝑠𝑠𝑠𝑠 (1; 𝑠𝑠𝑠𝑠𝑠𝑠 {𝑘𝑘 ∈ [1; 𝑛𝑛]|𝑅𝑅𝑠𝑠 = ∑ 𝑟𝑟𝑖𝑖 ≥ 1})
The rest of the paper is organized as follows: Section 2 gives a set of mathematical tools used to formalize the aforementioned problem. Section 3 provides results in the deterministic case. Section 4 studies the effects of stochastic duration on the choice of opportunity. A conclusion is provided in Section 5.
𝑗𝑗=𝑘𝑘
(3)
𝑃𝑃𝑖𝑖 ⁄1 − 𝑃𝑃 with 𝑃𝑃𝑖𝑖 is the probability of success of 𝑖𝑖 the event 𝑖𝑖 and 1 − 𝑃𝑃𝑖𝑖 the probability without success.
Where 𝑟𝑟𝑖𝑖 =
General assumptions:
The first stoppage where the sum of odds reaches or exceeds 1 is the optimal one. 𝑠𝑠 is the index corresponding to the optimal stoppage. In case where the sum of odds do not reach 1, the maintenance manager must choice the forst occuring event/stoppage/opportunity.
An existing prognostic model provides the remaining useful life of the component. As explained in (Nzukam et al. 2017), an alpha level corresponding to the decision maker choice is defined. This level allows setting a time, corresponding to the time where the cumulative density function associated with the RUL equals to alpha. The selection of opportunities will be based on this time noted 𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎ℎ𝑎𝑎 that represents the failure time of the system (Figure 1).
Its application in maintenance is described in (Thomas et al. 2007) and (Nzukam et al. 2017). It considers reliability (R) and maintainability (M) for the computation of an event probability. An opportunity is called a success if the system/component is alive at time 𝑆𝑆𝑆𝑆𝑖𝑖 and maintainable during 𝐿𝐿𝑖𝑖 . The probability 𝑃𝑃𝑖𝑖 associated to this success is computed as follow:
2. PROBLEM FORMALIZATION The purpose of this section is to present the formalization of the problem of opportunistic maintenance
𝑃𝑃𝑖𝑖 = 𝑅𝑅(𝑆𝑆𝑆𝑆𝑖𝑖 ). 𝑀𝑀(𝐿𝐿𝑖𝑖 )
(1)
Then, the odds 𝑟𝑟𝑖𝑖 , 1 ≤ 𝑖𝑖 ≤ 𝑛𝑛, are computed through the following equation (the ratio between the probability of success and the probability of failure):
2.1 System stoppages modelling
A stoppage is defined by the couple (𝑆𝑆𝑆𝑆, 𝐿𝐿) with 𝑆𝑆𝑆𝑆 the stoppage starting time and 𝐿𝐿 the stoppage length with 𝐿𝐿 > 0. We consider a set of 𝑛𝑛 stoppages, a maintenance manager has at his disposal (Figure 1).
𝑟𝑟𝑖𝑖 =
𝑅𝑅(𝑆𝑆𝑆𝑆𝑖𝑖 ). 𝑀𝑀(𝐿𝐿𝑖𝑖 ) ⁄1 − 𝑅𝑅(𝑆𝑆𝑆𝑆 ). 𝑀𝑀(𝐿𝐿 ) 𝑖𝑖 𝑖𝑖
(2)
In order to get an ordered list of opportunities, the next suboptimal opportunity is determined by re-computing the sum on the set of opportunities except the optimal one. The ordered list of success gives to the maintenance manager more flexibility to plan the maintenance actions (for example in the context of unavailability of spare parts or human resource). Figure 1: Stoppage modelling
The following section provides an application of odds algorithm on an academic example.
Thus, given a maintenance action, what is the most appropriate stoppage that allows performing the maintenance action before the failure time? The next section describes Bruss algorithm on which lays the choice of appropriate opportunity.
3. DETERMINISTIC CASE This part is dedicated to show results with deterministic opportunity duration. Data considered in this paper are the same used in (Thomas et al. 2006). Let’s consider 1500ℎ for the planning horizon and twelve planned system stoppages as indicated in Table 1.
2.2 Bruss theorem in maintenance
461
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
455
Table 1: Stoppages characteristics 𝑵𝑵𝒐𝒐 St (h) L (h) 𝑵𝑵𝒐𝒐 St (h) L (h) 1
200
3
7
800
4
2
310
2
8
910
2
3
400
4
9
980
3 Figure 2: Reliability curve
4
560
2
10 1100
7 The next section introduces stochastic opportunities duration.
5
620
1
11 1250
3
4. USING MONTE-CARLO SIMULATION FOR STOCHASTIC OPPORTUNITIES DURATION
6
690
4
12 1360
4
The same example is used but stochastic durations are considered. The uncertainty on stoppage duration is introduced in the experiment by the means of Gaussian distribution and is processed using a Monte-Carlo simulation. Monte-Carlo simulation is described in the following.
The reliability is assumed to be a Weibull distribution with the shape parameter 𝛽𝛽 = 1.5 and the scale parameter 𝜂𝜂 = 700 ℎ. Maintainability is supposed following an exponential law with parameter 𝜇𝜇 = 0.5 ℎ−1 .
4.1 Monte-Carlo approach and stochastic duration modelling
Table 2: Classification Results 𝑵𝑵𝒐𝒐 Odds (r)
𝑹𝑹𝒔𝒔
𝑵𝑵𝒐𝒐 Odds (r)
Monte-Carlo simulation is a computerized mathematical technique that takes into account risk in quantitative analysis and decision-making (Rubinstein & Kroese 2016). In the present work, the risk is associated to the quality of available knowledge on opportunities duration. The essential idea of Monte-Carlo simulation is to develop possible outcome models, considering a range of values described by a probability distribution) for any uncertainty-carrying factor. The advantage of Monte-Carlo simulation is to get the uncertainty on the outcome when no analytical solution is available.
𝑹𝑹𝒔𝒔
1
2.0016
2.9797
7
0.3420
0.9781
2
0.8895
1.8676
8
0.1676
0.6361
3
1.2798
2.2579
9
0.1740
0.4685
4
0.4473
1.4254
10
0.1564
0.2945
5
0.2062
1.1843
11
0.0769
0.1381
6
0.4814
1.4595 12
0.0612
0.0612
Considering the formalism introduced in section 2, the uncertainty is modeled using a Gaussian distribution for the length of the opportunities. The mean of the distribution will be 𝐿𝐿𝑖𝑖 and the standard deviation will vary in order to show the influence of the uncertainty level. Every length ℒ𝑖𝑖 is considered as a random variable 𝐿𝐿𝑖𝑖 such that: ℒ𝑖𝑖 = 𝒩𝒩(𝐿𝐿𝑖𝑖 , 𝜎𝜎)
Where 𝒩𝒩(𝜇𝜇, 𝜎𝜎) is the normal distribution with a mean equals to 𝜇𝜇 and a standard deviation equals to 𝜎𝜎.
Table 2 shows the values for every opportunity computed from (2) and the bold results shows the list of success. The last success, i.e. #6, is the optimal one. The stoppages 5, 4, 3, 2 𝑎𝑎𝑎𝑎𝑎𝑎 1 are considered as alternatives (for flexibility) in the case where it could not be possible to perform maintenance at stoppage #6.
4.2 Results of the Monte-Carlo simulations The Monte-Carlo simulation considers 10000 draws. The results are presented in Table 3 with range of values for 𝜎𝜎 from 0.1 to 2.5. Each figure must be read in relation (or compared)
Figure 2 shows the reliability curve associated to the considered component. For stoppage 6, the reliability is 0.38 at time 690ℎ. 462
IFAC INCOM 2018 456 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
𝜎𝜎 = 0.7
to the deterministic case which give stoppage #6 as optimal stoppage.
Table 3: Numerical results 𝜎𝜎 = 0.1
𝜎𝜎 = 1
𝜎𝜎 = 0.25 𝜎𝜎 = 1.5
𝜎𝜎 = 0.3
𝜎𝜎 = 2
𝜎𝜎 = 0.4
𝜎𝜎 = 2.5
463
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
On the whole tests, the most probable stoppage is the stoppage #6 as in the deterministic case. On one hand, only few cases lead to consider an optimal stoppage earlier than #6 and only in the tests where the uncertainty is the highest (𝜎𝜎 >1.5). This means that the sum of the odds is close enough from 1 at stoppage #7 and the odd of stoppage #6, even when the duration decrease, is big enough to allows the sum to reach 1. When stoppage duration shortens, the maintainability decreases and the odd decreases. On the contrary, when stoppage duration lengthens, the maintainability increases and the odd increases.
457
consumption and energy bill in the operational stage of existing non-residential buildings. (https://www.energyintime.eu/). REFERENCES Bruss, F.T., 2003. A note on bounds for the odds theorem of optimal stopping. Annals of Probability, 31(4), pp.1859–1861. Bruss, F.T., 2000. Sum the odds to one and stop. Annals of Probability, 28(3), pp.1384–1391.
On the other hand, stoppage #7 appears to be the second most probable optimal stoppage. As it allows the sum of the odds to be close from 1, i.e. 0,98, when its length increase, it increases its odd and allows the sum to reach 1.The sum of the odds at stoppage #8 is 0,64 and is not close enough from 1 to allows stoppage #8 to be the optimal stoppage.
Kennedy, K., Walsh, P. & Scully, T., 2014. Genetic Optimisation for a Stochastic Model for Opportunistic Maintenance Planning of Offshore Wind Farms. Lu, B. & Zhou, X., 2017. Opportunistic preventive maintenance scheduling for serial-parallel multistage manufacturing systems with multiple streams of deterioration. Reliability Engineering and System Safety, 168(May), pp.116–127. Available at: https://doi.org/10.1016/j.ress.2017.05.017.
From the decision maker point of view, i.e. maintenance manager, when the uncertainty becomes important (𝜎𝜎 =1, compared to the length duration mean equal to 2,6), the decision uncertainty allows to choose between stoppage 7 and stoppage 6. This means that the uncertainty allows to consider a stoppage further. Nevertheless, the decision is not easy since only binary results are shown. Indeed, decision maker would prefer to have a more detailed view of how the choice has been done, including uncertainty.
Nguyen, K.A., Do, P. & Grall, A., 2017. Joint predictive maintenance and inventory strategy for multicomponent systems using Birnbaum’s structural importance. Reliability Engineering and System Safety, 168(May), pp.249–261. Available at: https://doi.org/10.1016/j.ress.2017.05.034.
5. CONCLUSIONS Nzukam, C., Voisin, A., Levrat, E., Sauter, D., Iung, B., 2017. ScienceDirect A dynamic grouping costs savings Non-residential Non-residential buildings buildings costs savings in Non-residential. IFAC-PapersOnLine, 50(1), pp.13722–13727. Available at: https://doi.org/10.1016/j.ifacol.2017.08.2551.
In this paper, an opportunistic maintenance scheduling methodology considering the stochastic nature of opportunities duration is proposed in a predictive maintenance strategy. The proposed maintenance scheduling methodology used the prognostic information to select opportunities coming before the failure and is based on an optimal stopping problem algorithm known as Bruss algorithm. The Bruss algorithm is use since it has shows it usefulness on previous case. The uncertainty on opportunities duration is tackle with a MonteCarlo simulation as no analytical formulation exists. This makes this contribution original.
Rubinstein, R.Y. & Kroese, D.P., 2016. Simulation and the Monte Carlo method, John Wiley & Sons. Shin, H. & Yacout, S., 2016. Opportunistic Preventive Maintenance Strategy of a Multi-Component System with Hierarchical Structure by Simulation and Evaluation. Emerging Technologies and Factory Automation (ETFA), 2016 IEEE 21st International Conference on, pp.1–8.
Further studies will include both theoretical and practical aspects. From the theoretical aspect, the consideration of uncertainty in both length and starting date of stoppage will be considered and an analytical solution will be investigated in order to have the uncertainty on the sum of odds. From the practical aspect, two directions will be followed: (1) the consideration of the spare parts and human resources availability and (2) what information would be more relevant for the decision maker than a « binary » one (as in Table 3). Finally, the proposed approach has to be compared with a classic systematic maintenance scheduling in order to shows its benefit.
Thomas, E. , Levrat, E., Iung, B., & Monin, M., 2006. Odds algorithm-based opportunity-triggered preventive maintenance with production policy. In 6th IFAC Symposium. pp. 835–840. Thomas, E., Levrat, E. & Iung, B., 2007. L’algorithme de Bruss comme contribution à une maintenance préventive opportuniste. e-STA, Sciences et Technologies de l’Automatique, 3, pp.13–18. Available at: http://hal.archives-ouvertes.fr/hal-00149994/ [Accessed July 4, 2013].
The authors would like to acknowledge the financial support of the European Commission under the Seventh Framework Program titled Energy-In-Time that aims to develop Smart Energy Simulation Based Control method to reduce the energy 464
IFAC INCOM 2018 458 Bergamo, Italy, June 11-13, 2018
C. Nzukam et al. / IFAC PapersOnLine 51-11 (2018) 453–458
Truong Ba, H., Cholette, M.E., Borghesani, P., Zhou, Y., Ma, L., 2017. Opportunistic maintenance considering nonhomogenous opportunity arrivals and stochastic opportunity durations. Reliability Engineering and System Safety, 160(April 2016), pp.151–161. Wu, Q., Lv, C., Zhou, D., Wang, Y., Yu, Dequan., 2016a. Multi-event Maintenance Decision-making Model and Optimization Method Based on Opportunistic Maintenance Policy. 12th World Congress on Intelligent Control and Automation (WCICA), (June 12-15, 2016, Guilin, China), pp.1–5. Wu, Q., Lv, C., Zhou, D., Wang, Y., Yu, Dequan 2016b. Study on Multi-event Opportunistic Maintenance Decision-making Model Based on Condition. , pp.1–5. Xiao, L., Song, S., Chen, X., Coit, D.W., 2016. Joint optimization of production scheduling and machine group preventive maintenance. Reliability Engineering and System Safety, 146, pp.68–78. Available at: http://dx.doi.org/10.1016/j.ress.2015.10.013. Yang, L., Ma, X., Zhai, Q., Zhao, Y., 2016. A delay time model for a mission-based system subject to periodic and random inspection and postponed replacement. Reliability Engineering and System Safety, 150, pp.96– 104. Available at: http://dx.doi.org/10.1016/j.ress.2016.01.016. Yildirim, M., Gebraeel, N.Z. & Sun, X.A., 2017. Integrated Predictive Analytics and Optimization for Opportunistic Maintenance and Operations in Wind Farms. IEEE Transactions on Power Systems, 32(6), pp.4319–4328. Zhou, X., Xi, L. & Lee, J., 2009. Opportunistic preventive maintenance scheduling for a multi-unit series system based on dynamic programming. , 118, pp.361–366.
465