- Email: [email protected]
Outsourcing selective maintenance problem in failure prone multi-component systems
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 Information Control Problems in Manufacturing Available online at www.sciencedirect.com Bergamo, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Manufacturing Proceedings,16th IFAC Symposium on Information Control Problems in Bergamo, Italy, June 11-13, 2018 Bergamo, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Bergamo, Italy, June 11-13, 2018
ScienceDirect
IFAC PapersOnLine 51-11 (2018) 525–530
Outsourcing selective maintenance problem Outsourcing Outsourcing selective selective maintenance maintenance problem problem Outsourcing selective maintenance problem in failure prone multi-component systems Outsourcing selective maintenance problem in failure prone multi-component systems in in failure failure prone prone multi-component multi-component systems systems inChaabane failure ∗∗prone multi-component ∗∗ ∗∗∗ systems ∗∗∗∗ K. A. Khatab ∗∗ E.H. Aghezzaf ∗∗∗ C. Diallo ∗∗∗∗ K. K. K. K.
Chaabane E.H. Aghezzaf C. Diallo ∗∗∗∗ ∗ A. Khatab ∗∗ ∗∗∗ † ∗∗ E.H. Aghezzaf ∗∗∗ C. Diallo ∗∗∗∗ Chaabane U. † Chaabane ∗∗ A. A. Khatab Khatab E.H. Aghezzaf C. Diallo ∗∗∗∗ U. Venkatadri Venkatadri ∗∗ ∗∗∗ † † Chaabane A. Khatab E.H. Aghezzaf C. Diallo U. U. Venkatadri Venkatadri † ∗ U. Venkatadri e de Lorraine, LGIPM, F-57000 Metz, France ∗ Universit´ de ∗ Universit´ ∗ Universit´eee([email protected]). de Lorraine, Lorraine, LGIPM, LGIPM, F-57000 F-57000 Metz, Metz, France France Universit´ de Lorraine, LGIPM, F-57000 Metz, France ∗ ∗∗ e([email protected]). de Lorraine, LGIPM, F-57000 Metz, France ([email protected]). Universit´ e de Lorraine, LGIPM, F-57000 Metz, France ∗∗Universit´ ([email protected]). Universit´ ee de Lorraine, LGIPM, F-57000 Metz, France ∗∗ ∗∗ Universit´ ([email protected]). de Lorraine, LGIPM, F-57000 Metz, France ([email protected]) Universit´ e de Lorraine, LGIPM, F-57000 Metz, France ([email protected]) ∗∗ ∗∗∗ Universit´ e de Lorraine, LGIPM, F-57000 Metz, FranceDesign, ([email protected]) of Industrial Systems Engineering and Product ∗∗∗ Department ([email protected]) Department of Industrial Systems Engineering and Product Design, ∗∗∗ ∗∗∗ Department of([email protected]) Industrialand Systems Engineering and Product and Design, Faculty of Engineering Architecture, Ghent University Department of Industrial Systems Engineering and Product Design, ∗∗∗ Faculty of Engineering and Architecture, Ghent University and Department of Industrial Systems Engineering and Product and Design, Faculty of Engineering and Architecture, Ghent University Flanders Make, Belgium. ([email protected]) Faculty of Engineering and Architecture, Ghent University and Flanders Make, Belgium. ([email protected]) ∗∗∗∗ Faculty of Engineering and Architecture, Ghent University and Flanders Make, Belgium. ([email protected]) Department of Industrial Engineering, Dalhousie University, ∗∗∗∗ Flanders Make,ofBelgium. ([email protected]) Department Industrial Engineering, Dalhousie University, ∗∗∗∗ ∗∗∗∗ Flanders Make,of Belgium. ([email protected]) Department Industrial Engineering, Canada ([email protected]) Department of Industrial Engineering, Dalhousie Dalhousie University, University, Canada ([email protected]) † ∗∗∗∗ Department of Industrial Engineering, Dalhousie University, Canada ([email protected]) Engineering, Dalhousie University, Canada † Department of Industrial Canada ([email protected]) Department of Industrial Engineering, Dalhousie University, Canada † † Department of Industrial Canada ([email protected]) Engineering, Dalhousie University, Canada ([email protected]) Department of Industrial Engineering, Dalhousie University, Canada ([email protected]) † Department of Industrial Engineering, Dalhousie University, Canada ([email protected]) ([email protected]) ([email protected]) R´eesum´ sum´eeIn In many many industrial industrial settings, settings, there there are are systems systems designed designed to to perform perform consecutive consecutive missions missions R´ R´ e sum´ e In many industrial settings, there are systems designed to perform consecutive missions interspersed with finite breaks during which only a set of component repairs can be carried R´ esum´eIn many settings, there are systems designed to perform consecutive interspersed withindustrial finite breaks during which only a set of component repairs can bemissions carried R´ esum´ eIntomany settings, there are systems designed to perform consecutive interspersed withindustrial finite breaks during which only a decision set of component component repairs can bemissions carried out due limited time, budget, or resources. The maker then has to decide which interspersed with finite breaks during which only a set of repairs can be carried out due to limited time, budget, or resources. The decision maker then has to decide which interspersed with finite breaks during which aonly a decision set of component repairs can be carried out due to limited time, budget, or resources. The maker then has to decide which components to repair in order to guarantee given performance level. This is known as the out due to limited time, budget, or resources. The decision maker then has to decide which components to repair in order to guarantee a given performance level. This is known as the out due to limited time, budget, or resources. The introduces decision maker then has is to decide which components to repair in order to guarantee aapaper given performance level. This known as the selective maintenance problem (SMP). This a new variant of the SMP by components to repair in order to guarantee given performance level. This is known as the selective maintenance problem (SMP). This paper introduces a new variant of the SMP by components to repair in order to guarantee a given performance level. This is known as the selective maintenance problem (SMP). This paper introduces a new variant of the SMP by specifically taking into account the maintenance outsourcing alternative. A novel integrated selective maintenance This paperoutsourcing introduces alternative. a new variant of theintegrated SMP by specifically taking intoproblem account (SMP). the maintenance A novel selective (SMP). This both paperoutsourcing introduces aand newoutsourcing variant of the SMP by specifically taking into intoproblem account the maintenance maintenance alternative. A novel novel integrated non-linearmaintenance programming formulation where the in-housealternative. maintenance specifically taking account the A integrated non-linear programming formulation where both outsourcing the in-house and outsourcing maintenance specifically taking into account the maintenance outsourcing alternative. Aofnovel integrated non-linear programming formulation where both the in-house and outsourcing maintenance alternatives are accounted for is developed and optimally solved. The effect the outsourcing non-linear programming formulation where both the in-house and outsourcing maintenance alternatives are accounted for is developed and optimally solved. The effect of the outsourcing non-linear both the in-house and alternatives are accounted for is and optimally solved. The effect outsourcing alternative programming on maintenance decisions iswhere investigated through numerical experiments. The overall overall alternatives aremaintenance accountedformulation for is developed developed and optimally solved. Theoutsourcing effect of of the themaintenance outsourcing alternative on decisions is investigated through numerical experiments. The alternatives are accounted for is developed and optimally solved. The effect of the outsourcing alternative on maintenance decisions is investigated through numerical experiments. The overall results obtained demonstrate the validity of the proposed approach. alternative on maintenance decisions is investigated through numerical experiments. The overall results obtained demonstrate the validity of the proposed approach. alternative on maintenance decisions is investigated through numerical experiments. The overall results obtained obtained demonstrate the validity validity of the the proposed proposed approach. results demonstrate the of approach. © 2018, obtained IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. results demonstrate the validity of the proposed approach. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. 1. INTRODUCTION INTRODUCTION varying from from 22 to to 88 hours hours (Diallo (Diallo et et al., al., 2017). 2017). To To prepare prepare 1. varying 1. INTRODUCTION INTRODUCTION varying fromto22 successfully to 88 hours hours (Diallo (Diallo et al., al., 2017). To prepare prepare the system complete its next mission, its 1. varying from to et 2017). To the system to successfully complete its next mission, its 1. INTRODUCTION varying from 2 to 8 hours (Diallo et al., 2017). To prepare the system to successfully complete its next mission, its components must be properly maintained during the schesystem to successfully complete its next mission, its components must be properly maintained during the scheAs organisations organisations strive strive to to reduce reduce operating operating costs costs and and enen- the As the system to successfully complete mission, its components must bebreak. properly maintained during theredunscheduled intermission Because ofits thenext built-in components must be properly maintained during the scheAs organisations strive to reduce operating costs and enduled intermission break. Because of the built-in redunsure lean operations, they usually resort to alternative As strivethey to reduce operating costs and en- components sureorganisations lean operations, usually resort to alternative must be properly maintained during the scheduled intermission break. Because of the built-in redundancies, and due to the limited duration of the scheduled duled intermission break. Because of the built-in redunAs organisations strive to reduce operating costs and ensure lean operations, operations, they usually resort resort to alternative alternative approaches to manage manage they their operations, operations, maintenance, and dancies, and due to the limited duration of the scheduled sure lean usually to approaches to their maintenance, and duled break. Because of theof redundancies, andscarce due to tomaintenance the limited limited duration ofbuilt-in the scheduled scheduled breaksintermission and resources, only limited and due the duration the sure lean operations, they usuallythese resort to alternative approaches to manage manage their operations, maintenance, and breaks and scarce maintenance resources, only aa limited asset management needs. Among alternative is the dancies, approaches to their operations, maintenance, and asset management needs. Among these alternative is the dancies, and due to the limited duration of the scheduled breaks and scarce maintenance resources, only a limited set of components have and can be maintained during the and scarce have maintenance resources, onlyduring a limited approaches to manage their operations, maintenance, asset management needs. Among these alternative is and the breaks set of components and can be maintained the outsourcing which is supply chain solution that enables enables asset management needs. Among these alternative is the outsourcing which is aa supply chain solution that breaks and scarce maintenance resources, only a limited set of components have and can be maintained during the breaks. It is therefore necessary to identify an optimal subset of components have and can be maintained during the asset management needs. Among these alternative isthat, the breaks. It is therefore necessary to identify an optimal suboutsourcing which is supply chain solution that so enables an organisation organisation to is outsource some of its needs outsourcing which aa supply chain solution that enables an to outsource some of its needs so that, set of components have and canto maintained duringsubthe breaks. It is is therefore therefore necessary to identify optimal to maintain maintain meet thean predetermined It necessary tobemeet identify an optimal suboutsourcing which a supply chain of solution that enables an organisation to isoutsource outsource some of its needs so that, breaks. set of components to to the predetermined for example, to face market demand or to keep core coman organisation to some its needs so that, for example, to face market demand or to keep core com- breaks. It is therefore necessary to identify an optimal subset of components to maintain to meet the predetermined reliability level required for the next mission. of components to maintain meet the predetermined an organisation to outsource some et of its needs so that, for example, to face market(Haoues demand oral., to keep keep core comreliability level required for theto next mission. petition (see to forface example 2016)core andcomthe set for example, market demand to petition (see for example (Haoues etoral., 2016) and the set components to maintain meet the was predetermined reliability levelmaintenance required forproblem theto next mission. Theofselective selective (SMP) first introintroreliability level required for the next mission. for example, to face market demand or to keep core competition (see for example (Haoues et al., 2016) and the The maintenance problem (SMP) was first reference therein). petition for example (Haoues et al., 2016) and the reliability reference(see therein). level required for the next mission. The selective maintenance problem (SMP) was first introduced by Rice et al. (1998) and applied in a series-parallel The selective maintenance problem (SMP) was first intropetition (see for example (Haoues al., 2016) and the duced by Rice et al. (1998) and applied in a series-parallel reference therein). To maintain maintain their production assets,et many many organisations reference therein). To their production assets, organisations The selective maintenance (SMP) first introduced byinRice Rice etsubsystems al. (1998) (1998) problem and applied in independent series-parallel system with composed of an by et al. and applied in aawas series-parallel reference therein). To maintain their production assets, many many organisations system in with subsystems composed of independent an outsource some of their maintenance activities by using duced To maintain their production assets, organisations outsource some of their maintenance activities by using duced by Rice et al. (1998) and applied in a series-parallel system in with subsystems composed of independent an identically exponentially distributed components. In this this system in with subsystems composed of independent an To maintain their production assets, many organisations outsource some their maintenance activities by using identically exponentially distributed components. In a combination of in-house and contract maintenance. outsource some of their maintenance activities by using system a combination in-house and contract maintenance. in with composed independent an identically exponentially distributed components. In this this work, the the only subsystems maintenance option components. isof the the replacement exponentially distributed In outsource some maintenance activities by using identically combination of their in-house and contract maintenance. work, only maintenance option is replacement Contractors are generally generally hired for contract specialist maintenance aa combination of in-house and maintenance. Contractors are hired for specialist maintenance identically exponentially distributed components. In this work, the only maintenance option is the replacement at failure. Cassady et al. (2001) extended the original work, the only maintenance option is the replacement aContractors combination of in-house and contract are generally generally hired for in-house. specialist maintenance. maintenance at failure. Cassady et al. (2001) extended the original where the expertise expertise is not not held Maintenance Contractors are hired for specialist maintenance where the is held in-house. Maintenance work, only option iswhere the replacement at failure. Cassady et al. for (2001) extended the original work ofthe (Rice et maintenance al., et 1998) systems lifetimes of failure. Cassady al. (2001) extended original Contractors are generally hired for specialist maintenance where theresolve expertise isproblems not held in-house. Maintenance work of (Rice et al., 1998) for systems wherethe lifetimes of tasks to basic are generally performed at where the expertise is not held in-house. Maintenance tasks to resolve basic problems are generally performed at failure. Cassady et al. (2001) extended the original work of (Rice et al., 1998) for systems where lifetimes of components are Weibull distributed. Three maintenance work of (Rice et al., 1998) for systems where lifetimes of where the expertise is not held in-house. Maintenance tasks to resolve basic problems are generally performed components are Weibull distributed. Three maintenance in-house. tasks to resolve basic problems are generally performed work in-house. of are (Rice et Weibull al., :1998) for systems wheremaintenance lifetimes of components are Weibull distributed. Three maintenance actions allowed minimal repair, Three corrective replaceare distributed. tasks to resolve basic problemsthe areselective generally performed components in-house. actions are allowed : minimal repair, corrective replaceThe present present paper investigates maintenance in-house. The paper investigates the selective maintenance components are Weibull distributed. Three maintenance actions are allowed : minimal repair, corrective replacement of failed components, and a preventive replacement actions are allowed : minimal repair, corrective replacein-house. The present paper paper investigates the the selective maintenance optimisation of multi-component multi-component systems in maintenance the outsouroutsour- ment of failed components, and a preventive replacement The present investigates selective optimisation of systems in the actions allowed : minimal repair, corrective replacement of are failed components, and preventive replacement of working working components. Theand resulting SMP is replacement solved using of failed components, aa preventive The present paper investigates the selective maintenance optimisation of multi-component multi-component systems in the outsouroutsourof components. The resulting SMP is solved using cing setting. The selective maintenance is appropriate in ment optimisation of systems in the cing setting. The selective maintenance is appropriate in ment of failed components, and a preventive replacement of working components. The resulting SMP is solved using an enumeration method. To deal with the combinatorial of working components. The resulting SMP is solved using optimisation of multi-component systems in the outsourcing setting. The selective maintenance is appropriate in an enumeration method. To deal with the combinatorial systems designed to perform consecutive missions intercing setting. The selective maintenance appropriate in of systems designed to perform consecutiveis missions interworking components. SMP solved using an enumeration method. To resulting deal with with the is combinatorial complexity arising from The large-size systems, four improved enumeration method. To deal the combinatorial cing setting. The maintenance appropriate in systems designed tobreaks perform consecutive missions intercomplexity arising from large-size systems, four improved spersed with finiteselective during whichisaamissions limited set set of an systems designed to perform consecutive interspersed with finite breaks during which limited of an enumeration method. To deal with the combinatorial complexity arising from large-size systems, four improved enumeration procedures are proposed in (Rajagopalan and complexity arising from large-size systems, four improved systems designed consecutive missions interspersed with finitetobreaks breaks during which limited set of enumeration procedures are proposed in (Rajagopalan and component repairs orperform replacements can aabe be carriedset out spersed with finite during which limited of component repairs or replacements can carried out complexity arising from are large-size systems, four improved enumeration procedures are proposed in (Rajagopalan (Rajagopalan and Cassady, 2006) to reduce the computation times. An exact enumeration procedures proposed in and spersed with finite breaks during which a limited set of component repairs or replacements can be carried out Cassady, 2006) to reduce the computation times. An exact due to to limited limited time,orbudget, budget, or resources. resources. Incarried the airline airline component repairs replacements can beIn out enumeration due time, or the procedures are in (Rajagopalan and Cassady, 2006) to reduce theproposed computation times. An exact exact method based based on reduce the branch-and-bound branch-and-bound procedure 2006) to the computation times. An component replacements can beIn carried out Cassady, due to limited limited time,orbudget, budget, or resources. Into the airline method on the procedure and industry for repairs example, aircraft areresources. subjected overnight due to time, or airline industry for example, aircraft are subjected tothe overnight Cassady, 2006) to reduce the computation times. An exact method based on the branch-and-bound procedure and a Tabu search based algorithm are proposed in (Lust method based on the branch-and-bound procedure and due to limited time, budget, or resources. In the airline industry for example, aircraft are subjected to overnight a Tabu search based algorithm are proposed in (Lust maintenance at night during a break interval typically industry for example, subjected to overnight maintenance at night aircraft during are a break interval typically method based on the branch-and-bound procedure and a Tabu search based algorithm are proposed in (Lust a Tabu search based algorithm are proposed in (Lust industry for example, aircraft are subjected to overnight maintenance at at night night during during aa break break interval interval typically typically maintenance a Tabu search based algorithm are proposed in (Lust maintenance atIFAC night during a Federation break interval typically 2405-8963 © (International of Automatic Control) Copyright © 2018, 2018 IFAC 532 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 532Control. Peer review© under of International Federation of Automatic Copyright © 532 Copyright © 2018 2018 IFAC IFAC 532 10.1016/j.ifacol.2018.08.372 Copyright © 2018 IFAC 532
IFAC INCOM 2018 526 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
et al., 2009). Khatab et al. (2007) proposed two heuristic methods, adapted from those used to solve the redundancy allocation problem in (Aggrawal, 1976; Gopal et al., 1978). The development of SMP models has since increased and dealt with extensions to include imperfect maintenance, system’s components dependencies, and mission and break profiles. Imperfect maintenance in the selective maintenance setting is addressed inLiu and Huang (2010), where the age reduction coefficient approach (Malik, 1979) is used to model imperfect maintenance. An imperfect selective maintenance model was also developed by (Zhu et al., 2011) and applied to a machining line system. Panday et al. (2013) also studied the selective maintenance problem for binary systems under imperfect maintenance using the hybrid hazard rate approach introduced in (Lin et al., 2000). In (Liu and Huang, 2010) and (Panday et al., 2013), a set of maintenance levels, ranging from minimal repair to replacement, are used to improve the reliability of a system component. (Khatab and Aghezzaf, 2016) studied the selective maintenance problem when the quality of imperfect maintenance is stochastic. A nonlinear and stochastic optimization problem was proposed and solved for a series-parallel system. In (Khatab et al., 2017) the SMP is addressed in systems operating missions and breaks both of stochastic durations. Schneider and Cassady (Schneider and Cassady, 2015) solve the selective maintenance problem for a fleet composed of a set of independent and identical systems. The fleet is required to execute a sequence of missions and return to a common base where maintenance actions may be performed on selected components. The selective maintenance problem has also been addressed in multi-state systems (MSS) settings. The most recent work appeared in Dao and Zuo (2017) where the authors investigated a MSS with structural relationships between its components. All papers surveyed above do not consider the very common case where multiple repair-persons are available to carry out the maintenance actions. Only few papers attempts to some extent to overcome this restrictive assumption. (Iyoub et al., 2006) and (Maillart et al., 2009) investigated the selective maintenance problem taking into account maintenance resources allocation. For its repair, each system’s component is assumed to consume an amount of a given maintenance resource. The total amount consumed from each maintenance resource must be less than or equal to the total amount of that resource allotted to perform maintenance on failed components. These approaches are however limited to systems with components of constant failure rates. In a more recent paper, Diallo et al. (2017) studied the SMP with multiple repair-persons. An integrated non-linear programming formulation was then developed and optimally solved. Numerical experiments conducted show the benefits of jointly carrying out the assignment of the tasks to repair-persons and the selection of the components to be repaired. Nonetheless, their work assume that component’s replacement as the only one available maintenance decision. In the present paper, the selective maintenance problem is investigated for systems with multiple repair-persons in outsourcing setting. The system must operate a sequence of alternating missions and scheduled breaks. Each component has a list of eligible maintenance actions ranging from minimal repair, through intermediate imperfect 533
maintenance actions, to replacement. It is assumed that some maintenance levels are performed in-house, while the others are outsourced. To maximise the probability of the system to successfully operate the next mission, the maintenance activities are performed on the system’s components during the break. Due to limited break duration, maintenance budget and repair crews, not all components are likely to be maintained. The resulting integrated optimisation model is a non-linear integer optimization model which is optimally solved and fully discussed. Numericals experiments are also provided to demonstrate the accuracy and the validity of the proposed approach. The remainder of the paper is organized as follows. Acronyms, notation and the main working assumptions are given in Section 2. System’s description and the imperfect maintenance model used are presented in Section 3. The selective maintenance model is developed in Section 4 where the new integrated selective maintenance problem is also formulated. Numerical experiments are provided in Section 5. Conclusion and future research extensions are drawn in Section 6. 2. ACRONYM, NOTATION AND MAIN WORKING ASSUMPTIONS In this section, acronyms, the list of notation and the main working assumptions made in this paper are provided in what follows. Acronyms CM (PM) Corrective (Preventive) maintenance SMP Selective maintenance problem OSMOP Outsourcing SM optimisation problem Notation n Number of subsystems i Index of subsystems ni Number of components in subsystem Si j Index of component in subsystem Si , Cij The j th component of subsystem Si L The highest maintenance level available l Maintenance level l ∈ {0, · · · , L} tcijl Time of CM of level l performed on Cij tpijl Time of PM of level l performed on Cij ch Cost rate of an in-house repair-person co Cost rate of an outsourcing repair-person Aij Age of Cij at the start of a break Bij Age of Cij at the end of a break Xij Cij status at the start of a break Yij Cij status at the end of a break C0 Maximum maintenance budget available D Limited break duration U Next mission duration hij (t) Failure rate of Cij (t) c Rij U |Bij Conditional reliability of component Cij R System reliability for the next mission Assumptions (1) The system consists of multiple, repairable binary components (the components and the system are either functioning or failed). (2) During the break, system components do not age, i.e. the age of a component is operation-dependent.
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
(3) No maintenance activity is allowed during the mission. Maintenance activities are allowed only during the break. (4) Multiple components can be worked on simultaneously without repair-persons colliding. 3. SYSTEM’S DESCRIPTION AND MAINTENANCE MODEL 3.1 System’s description Without loss of generality, the SMP studied in the present work considers a series-parallel system composed of n series subsystems Si (i = 1, ..., n) each of which is composed of ni independent and possibly non-identical components Cij (j = 1, ..., ni ) arranged in parallel. The system is assumed to have just finished the current mission, and then turned off during the scheduled break of finite length D for possible maintenance of their components. Thereafter, the system is used to operate the next mission of duration U . At the end of the current mission, a system’s component can be either in a functioning or in a failed state. Two state variables Xij and Yij are used to describe the status of component Cij , respectively, at the end of current mission and at the beginning of the next mission : 1 if Cij is working at the end of the current mission Xij = 0 otherwise. (1) 1 if Cij is working at the beginning of the next mission Yij = 0 otherwise. (2) Each component Cij is also characterized by its age Aij at the end of the current mission (i.e. at the start of te break), and its age Bij at the beginning of the next mission (i.e. at the end of the break). If the component is still functioning at the end of the current mission (i.e. Xij = 1), its corresponding probability to successfully operating the next mission is given by its conditional reliability R U |Bij such that : Bij +U hij (t)dt , (3) Rc U |Bij = exp Bij
Since the system’s reliability block diagram is seriesparallel, its reliability is :
R=
n
i=1
1 −
1 − R U |Bij · Y ij
ni
j=1
(4)
3.2 Imperfect maintenance model
the component Cij at the end of the current mission. Both maintenance types are modelled according to the age reduction imperfect maintenance model of (Malik, 1979). Without loss of generality, a common list {1, . . . , l, . . . , L} of L maintenance levels is available for all system’s components. To each maintenance level 1 ≤ l ≤ L corresponds an age reduction coefficient αl ∈ [0, 1]. If a maintenance of level l is carried out on Cij , its age Aij is reduced and becomes αl Aij . Two particular values of maintenance level can then be observed. The first corresponds to the maintenance level l = 1 whose age reduction is equal to 1 and stands then for minimal repair. The second corresponds to the highest maintenance level L whose age reduction is 0 which is equivalent to component’s renewal. It is worth noticing that minimal repair, as a maintenance level, is eligible only for failed components. To deal with outsourcing within SMP, the maintenance list is partitioned on two subsets of maintenance levels. The first k maintenance levels {1, . . . , k} are the set of maintenance actions that are performed in-house, while the remaining maintenance levels {k + 1, . . . , L} are assumed to require more skilled repair-persons and then outsourced. To alleviate the notation, we simply write k = k + 1. 4. SELECTIVE MAINTENANCE MODEL To develop the new integrated selective maintenance optimisation in the outsourcing setting, the following decision variable zijl is defined :
zijl
534
1 if Cij is selected for maintenance and maintenance l is performed = 0 otherwise.
(5)
According to the age reduction imperfect maintenance model, when a maintenance action of level l is performed on a component Cij , then the effective age Bij of Cij is evaluated as : Bij = [(αijl · zijl ) + (1 − zijl )] · Aij .
(6)
In what follows, we compute the total time spent on components’ maintenance and the total cost induced by the selected maintenance actions. To do so, let us denote by tpijl and tcijl , the time consumed by, respectively, a preventive and corrective maintenance level l when preformed on component Cij . We also denote by ch and co the cost rate (i.e. cost per unit time) induced by an in-house and outsourcing maintenance. It is reasonable to assume that outsourcing cost rate co is greater than the in-house cost rate ch . The total time spent by maintenance actions performed in-house is computed as : k ni n k p c tijl .Xij .zijl + tijl .(1 − Xij ).zijl . (7) i=1 j=1
Two types of maintenance are considered either preventive or corrective maintenance, depending on the status of
527
l=2
l=1
Similarly, the total time consumed by outsourced maintenance actions is computed as :
IFAC INCOM 2018 528 Bergamo, Italy, June 11-13, 2018
ni n i=1 j=1
L
tpijl .Xij .zijl
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
+
l=k
L
tcijl .(1
l=k
− Xij ).zijl
. (8)
The total cost induced by both in-house and outsourced maintenance actions is computed as : k ni n L p tijl .ch .Xij .zijl + tpijl .co .Xij .zijl + i=1 j=1 k
l=k
l=2
tcijl .ch .(1
l=1
− Xij ).zijl +
L
tcijl .co .(1
l=k
− Xij ).zijl
(9)
4.1 Problem formulation In this section, the outsourcing selective maintenance optimization problem (OSMOP) is developed. This model deals with the case where the objective of the maintenance decision-maker consists on maximizing the system reliability to successfully achieve the next mission, taking into consideration the pre-specified maintenance budget C0 . The optimization model is drawn subsequently. OSMOP : n
Max R =
i=1
Subject to : k ni n i=1 j=1 k l=1
i=1 j=1
ni n i=1 j=1
1 −
ni
j=1
k
tpijl .ch .Xij .zijl +
L
(10)
tpijl .co .Xij .zijl +
l=k
− Xij ).zijl +
L
tpijl .Xij .zijl +
k l=1
tpijl .Xij .zijl +
l=k
tcijl .co .(1
l=k
l=2
L
c 1 − Rij U |Aij · Yij
l=2
tcijl .ch .(1
ni n
− Xij ).zijl
tcijl .(1 − Xij ).zijl
L
l=k
tcijl .(1 − Xij ).zijl
≤ C0 (11)
≤D (12) ≤D (13)
Lij
zijl ≤ 1
(14)
zij1 ≤ (1 − Xij )
(15)
l=1
Lij
Bij =
l=1
[αijl · zijl + 1 − zijl )] · Aij
Yij = Xij +
L l=1
(1 − Xij ) · zijl
i = 1, . . . , n; j = 1, . . . , ni ; l = 1, . . . , L Yij , Xij ∈ {0, 1}, zijl ∈ {0, 1}.
(16) (17) (18) (19)
In the above optimization model, Equations (11) is the maintenance budget constraint. Equations (12) and (13) are the maintenance break duration constraints, respectively, for the in-house and outsourcing maintenance ac535
Figure 1. Reliability block diagram of the series-parallel system tivities. For each component Cij , Equations (14) states that only one maintenance level can be selected if the component is to be maintained by an in-house or an outsourcing repair-person. The constraint (15) states that minimal repair is eligible only on a failed component. The constraint (16) updates the effective age of components at the end of the break. The constraint (17) allows to update the operating state of components. 5. NUMERICAL EXPERIMENTS In this section, two numerical experiments are conducted. We are given a series-parallel system composed of n = 2 subsystems Si (i = 1, 2), its corresponding reliability block diagram is depicted in Figure (1). Subsystem S1 is composed of n1 = 2 components in parallel, while subsystem S2 contains n2 = 3 components arranged in parallel. It is assumed that the lifetimes of each component Cij follows a Weibull distribution with shape parameter β ij and scale parameter η ij . These parameters are reported in Table (1). This table gives also the age Aij and the value of the state variable Xij corresponding to component Cij at the end of the current mission. According to Table (1), only components C11 and C22 are failed to survive the current mission, while the other components are still functioning. We assume that L = 6 maintenance levels are available each of which the corresponding age reduction coefficient is given in Table (2). Times required to perform a corrective or a preventive maintenance of level l on components are reported, respectively, in Tables (3) and (4). In the all subsequent experiments, the in-house and the outsourcing cost rates are, respectively, set to ch = 5 and co = 7. The system has just finished the current mission and made available for maintenance. The next mission duration is set to U = 70 time units. Table 1. Components lifetimes parameters, ages and status at the start of break Cij C11 C12 C21 C22 C23
β ij 3 1.5 2.7 2 2.5
η ij 120 200 150 150 170
Aij 110 175 175 150 100
Xij 0 1 1 0 1
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
be realistic and one may be forced to resort to maintenance outsourcing contracts. Indeed, on one hand, not all organisations have highly-skilled maintenance operators, and, on the other hand, the maintenance budget is usually restricted. The maintenance manager should therefore strive to find an appropriate solution with mixed in-house and outsourcing maintenance activities. The following experiment investigates the effect of such combined solutions on maintenance decisions.
Table 2. Age reduction coefficients values l αl
1 1
2 0.8
3 0.6
4 0.3
5 0.2
6 0
Table 3. Corrective maintenance times tcijl l
1
2
3
4
5
6
C11 C12 C21 C22 C23
0.5 0.45 0.6 0.4 0.5
1.5 1.2 1.1 0.8 1
2.2 2.3 1.3 1.1 1.3
2.5 2.5 1.8 1.5 1.7
2.8 2.7 2.3 1.8 2
3 2.8 2.5 2.2 2.5
5.2 Experiment #2 This experiment investigates the effects of the outsourcing on the selective maintenance decisions. Using the optimisation model OSMOP, the maximum achievable system reliability will be determined for a given break duration and maintenance cost when the number k of in-house maintenance levels is taken variable. It is conjectured that the flexibility provided by the cohort with mixed in-house and outsourcing repair-persons will allow the model to reach the same or better solution than when maintenance activities are completely either in-house or outsourced.
Table 4. Preventive maintenance times tpijl l C11 C12 C21 C22 C23
2 1 0.9 0.8 0.7 0.8
3 1.9 1.7 1 0.9 1
4 2.3 2 1.2 1.2 1.1
5 2.7 2.2 1.4 1.4 1.7
529
6 2.8 2.5 1.9 2 2.2
5.1 Experiment#1 This experiments investigates the two extreme cases where the maintenance activities is completely outsourced or completely performed in-house. We then solve the selective maintenance problem for two particular values of the number k of in-house maintenance levels. In the case of complete maintenance outsourcing k = 0, while in the complete in-house case the value of k is set to k = L. In this experiment the duration of the break is set to D = 4, and the maintenance budget is set to C0 = 25. In complete outsourcing case, the optimal maintenance decisions suggests the maintenance plan according to which component C11 and C22 are both replaced (corrective replacement). The resulting maximal reliability to operate the next mission is evaluated to R = 63.09%. This maintenance plan induces a total cost of 23.8. In the case of complete in-house maintenance, the optimal maintenance decisions derived are such that three components are selected to be maintained. The optimal maintenance plan suggests to perform a preventive replacement on component C12 , a preventive maintenance of level l = 4 on component C23 , and a minimal repair on the failed component C22 . The resulting system’s reliability is estimated to R = 70.22%. This maintenance plan induces a total cost of only 20. From the above results, it follows that the maximal achievable system’s reliability when dealing with complete inhouse maintenance case is better that that obtained when dealing with the case of complete maintenance outsourcing. Furthermore the optimal in-house maintenance plan is less expensive than the optimal outsourcing maintenance plan. This result comes from the fact that hiring an outsourcing repair-person is more expensive than using an in-house repair-person. Consequently, using the complete outsourcing solution limits the number of components to be maintained. However, these two extreme cases may not 536
The same system and parameters used in the previous experiments are used. The optimisation problem OSMOP is solved for different values of the allowable maintenance budget C0 . The results obtained are depicted in Figure (2). From these results, it is clearly shown as conjectured that an appropriate combination of an in-house and an outsourcing cohort is able to achieve equivalent or better reliability than the uniform cohort (i.e. either complete in-house or complete outsourcing). For example, if we consider the case where the maintenance budget is limited to C0 = 25, it follows that the maximal achievable system’s reliability when dealing with a combined in-house and outsourcing solution is obtained when the first four maintenance levels are performed in-house (k = 4) while the highest maintenance levels l = 5 and l = 6 are outsourced. If the maintenance budget is increased to C0 = 30, the maximal achievable system’s reliability when dealing with a combined in-house and outsourcing solution is obtained when only the highest maintenance level l = 6 is outsourced. The same conclusion can also be made from the results obtained for the other values of the maintenance budget. In general, when the maintenance budget available decreases, the maximal reliability achieved decreases. When the maintenance budget permits, the highly skilled outsourcing repair-persons can be used in an adequate combination with the in-house repair-persons. As the budget decreases, the model suggests to use the in-house persons instead of resorting to outsourcing. 6. CONCLUSION This paper introduced a novel variant of the selective maintenance problem in multi-component system by specifically combining the in-house and the outsourcing maintenance alternatives. A novel integrated non-linear programming formulation was then proposed and optimally solved. Numerical experiments show the benefits of jointly carrying out : (1) the assignment of the maintenance tasks
IFAC INCOM 2018 530 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
Figure 2. Maximal achievable reliability versus the number k : case of Experiment#2 to the in-house and outsourcing repair-persons, and (2) the selection of the components to be repaired. Future extensions that the authors are currently working on, include the development of efficient solution methods to deal with large sized problems. Another important issue consist to investigate the selective maintenance problem where more than one objective is to be optimized. Other possible extensions may also be the generalization of the present work to consider multi-state systems and k-out-ofn systems. ´ ERENCES ´ REF Aggrawal, K.K. (1976). redundancy optimization in general systems. IEEE Transactions on Reliability, R-25(25), 330–332. Cassady, C.R., Murdock, W.P., and Pohl, E.A. (2001). Selective maintenance for support equipement involving multiple maintenance actions. European Journal of Operational Research, 129, 252–258. Dao, C.D. and Zuo, M.J. (2017). Selective maintenance of multi-state systems with structural dependence. Reliability Engineering and System Safety, 159, 184–195. Diallo, C., Khatab, A., Venkatadri, U., and Aghezzaf, E.H. (2017). A joint selective maintenance and multiple repair–person assignment problem. In 7th Industrial Engineering and Systems Management (IESM) Conference, Saarbrucken, Germany. Gopal, K., Aggrawal, K.K., and Gupta, J.S. (1978). An improved algorithm for reliability optimization. IEEE Transactions on Reliability, R-27(27), 325–328. Haoues, M., Dahane, M., and Mouss, N.K. (2016). Outsourcing optimization in two-echelon supply chain network under integrated production-maintenance constraints. Journal of Intelligent Manufacturing. doi :10.1007/s10845-016-1273-3. Iyoub, I., Cassady, C.R., and Pohl, E.A. (2006). Establishing maintenance resource levels using selective maintenance. Engineering Economist, 51(2), 99–114. Khatab, A. and Aghezzaf, E.H. (2016). Selective maintenance optimization when quality of imperfect mainte537
nance actions are stochastic. Reliability Engineering & System Safety,, 150, 182–189. Khatab, A., Aghezzaf, E.H., Djelloul, I., and Sari, Z. (2017). Selective maintenance optimization for series– parallel systems alternating missions and scheduled breaks with stochastic durations. International Journal of Production Research, 55(10), 168–177. Khatab, A., Ait-Kadi, D., and Nourelfath, M. (2007). Heuristic-based methods for solving the selective maintenance problem for series-prallel systems. In International Conference on Industrial Engineering and Systems Management, Beijing, China. Lin, D., Zuo, M.J., and Yam, R.C.M. (2000). General sequential imperfect preventive maintenance models. International Journal of Reliability, Quality and Safety Engineering, 7(3), 253–266. Liu, Y. and Huang, H.Z. (2010). Optimal selective maintenance strategy for multi-state systems under imperfect maintenance. IEEE Transactions on Reliability, 59(2), 356–367. Lust, T., Roux, O., and Riane, F. (2009). Exact and heuristic methods for the selective maintenance problem. European Journal of Operational Research, 197, 1166– 1177. Maillart, L.M., Cassady, C.R., Rainwater, C., and Schneide, K. (2009). Selective maintenance decisionmaking over extended planing horizons. IEEE Transactions on Reliability, 58(3), 462–469. Malik, M. (1979). Reliable preventive maintenance scheduling. AIIE transactions, 11(3), 221–228. Panday, M., Zuo, M.J., Moghaddass, R., and Tiwari, M.K. (2013). Selective maintenance for binary systems under imperfect repair. Reliability Engineering and System Safety, 113, 42–51. Rajagopalan, R. and Cassady, C.R. (2006). An improved selective maintenance solution approach. Journal of Quality in Maintenance Engineering, 12(2), 172–185. Rice, W.F., Cassady, C.R., and Nachlas, J. (1998). Optimal maintenance plans under limited maintenance time. In Proceedings of Industrial Engineering Conference, Banff, BC, Canada. Schneider, K. and Cassady, C.R. (2015). Evaluation and comparison of alternative fleet-level selective maintenance models. Reliability Engineering and System Safety, 134, 178–187. Zhu, H., Liu, F., Shao, X., Liu, Q., and Deng, Y. (2011). A cost-based selective maintenance decision-making method for machining line. Quality and Reliability Engineering International, 27, 191–201.
ScienceDirect
IFAC PapersOnLine 51-11 (2018) 525–530
Outsourcing selective maintenance problem Outsourcing Outsourcing selective selective maintenance maintenance problem problem Outsourcing selective maintenance problem in failure prone multi-component systems Outsourcing selective maintenance problem in failure prone multi-component systems in in failure failure prone prone multi-component multi-component systems systems inChaabane failure ∗∗prone multi-component ∗∗ ∗∗∗ systems ∗∗∗∗ K. A. Khatab ∗∗ E.H. Aghezzaf ∗∗∗ C. Diallo ∗∗∗∗ K. K. K. K.
Chaabane E.H. Aghezzaf C. Diallo ∗∗∗∗ ∗ A. Khatab ∗∗ ∗∗∗ † ∗∗ E.H. Aghezzaf ∗∗∗ C. Diallo ∗∗∗∗ Chaabane U. † Chaabane ∗∗ A. A. Khatab Khatab E.H. Aghezzaf C. Diallo ∗∗∗∗ U. Venkatadri Venkatadri ∗∗ ∗∗∗ † † Chaabane A. Khatab E.H. Aghezzaf C. Diallo U. U. Venkatadri Venkatadri † ∗ U. Venkatadri e de Lorraine, LGIPM, F-57000 Metz, France ∗ Universit´ de ∗ Universit´ ∗ Universit´eee([email protected]). de Lorraine, Lorraine, LGIPM, LGIPM, F-57000 F-57000 Metz, Metz, France France Universit´ de Lorraine, LGIPM, F-57000 Metz, France ∗ ∗∗ e([email protected]). de Lorraine, LGIPM, F-57000 Metz, France ([email protected]). Universit´ e de Lorraine, LGIPM, F-57000 Metz, France ∗∗Universit´ ([email protected]). Universit´ ee de Lorraine, LGIPM, F-57000 Metz, France ∗∗ ∗∗ Universit´ ([email protected]). de Lorraine, LGIPM, F-57000 Metz, France ([email protected]) Universit´ e de Lorraine, LGIPM, F-57000 Metz, France ([email protected]) ∗∗ ∗∗∗ Universit´ e de Lorraine, LGIPM, F-57000 Metz, FranceDesign, ([email protected]) of Industrial Systems Engineering and Product ∗∗∗ Department ([email protected]) Department of Industrial Systems Engineering and Product Design, ∗∗∗ ∗∗∗ Department of([email protected]) Industrialand Systems Engineering and Product and Design, Faculty of Engineering Architecture, Ghent University Department of Industrial Systems Engineering and Product Design, ∗∗∗ Faculty of Engineering and Architecture, Ghent University and Department of Industrial Systems Engineering and Product and Design, Faculty of Engineering and Architecture, Ghent University Flanders Make, Belgium. ([email protected]) Faculty of Engineering and Architecture, Ghent University and Flanders Make, Belgium. ([email protected]) ∗∗∗∗ Faculty of Engineering and Architecture, Ghent University and Flanders Make, Belgium. ([email protected]) Department of Industrial Engineering, Dalhousie University, ∗∗∗∗ Flanders Make,ofBelgium. ([email protected]) Department Industrial Engineering, Dalhousie University, ∗∗∗∗ ∗∗∗∗ Flanders Make,of Belgium. ([email protected]) Department Industrial Engineering, Canada ([email protected]) Department of Industrial Engineering, Dalhousie Dalhousie University, University, Canada ([email protected]) † ∗∗∗∗ Department of Industrial Engineering, Dalhousie University, Canada ([email protected]) Engineering, Dalhousie University, Canada † Department of Industrial Canada ([email protected]) Department of Industrial Engineering, Dalhousie University, Canada † † Department of Industrial Canada ([email protected]) Engineering, Dalhousie University, Canada ([email protected]) Department of Industrial Engineering, Dalhousie University, Canada ([email protected]) † Department of Industrial Engineering, Dalhousie University, Canada ([email protected]) ([email protected]) ([email protected]) R´eesum´ sum´eeIn In many many industrial industrial settings, settings, there there are are systems systems designed designed to to perform perform consecutive consecutive missions missions R´ R´ e sum´ e In many industrial settings, there are systems designed to perform consecutive missions interspersed with finite breaks during which only a set of component repairs can be carried R´ esum´eIn many settings, there are systems designed to perform consecutive interspersed withindustrial finite breaks during which only a set of component repairs can bemissions carried R´ esum´ eIntomany settings, there are systems designed to perform consecutive interspersed withindustrial finite breaks during which only a decision set of component component repairs can bemissions carried out due limited time, budget, or resources. The maker then has to decide which interspersed with finite breaks during which only a set of repairs can be carried out due to limited time, budget, or resources. The decision maker then has to decide which interspersed with finite breaks during which aonly a decision set of component repairs can be carried out due to limited time, budget, or resources. The maker then has to decide which components to repair in order to guarantee given performance level. This is known as the out due to limited time, budget, or resources. The decision maker then has to decide which components to repair in order to guarantee a given performance level. This is known as the out due to limited time, budget, or resources. The introduces decision maker then has is to decide which components to repair in order to guarantee aapaper given performance level. This known as the selective maintenance problem (SMP). This a new variant of the SMP by components to repair in order to guarantee given performance level. This is known as the selective maintenance problem (SMP). This paper introduces a new variant of the SMP by components to repair in order to guarantee a given performance level. This is known as the selective maintenance problem (SMP). This paper introduces a new variant of the SMP by specifically taking into account the maintenance outsourcing alternative. A novel integrated selective maintenance This paperoutsourcing introduces alternative. a new variant of theintegrated SMP by specifically taking intoproblem account (SMP). the maintenance A novel selective (SMP). This both paperoutsourcing introduces aand newoutsourcing variant of the SMP by specifically taking into intoproblem account the maintenance maintenance alternative. A novel novel integrated non-linearmaintenance programming formulation where the in-housealternative. maintenance specifically taking account the A integrated non-linear programming formulation where both outsourcing the in-house and outsourcing maintenance specifically taking into account the maintenance outsourcing alternative. Aofnovel integrated non-linear programming formulation where both the in-house and outsourcing maintenance alternatives are accounted for is developed and optimally solved. The effect the outsourcing non-linear programming formulation where both the in-house and outsourcing maintenance alternatives are accounted for is developed and optimally solved. The effect of the outsourcing non-linear both the in-house and alternatives are accounted for is and optimally solved. The effect outsourcing alternative programming on maintenance decisions iswhere investigated through numerical experiments. The overall overall alternatives aremaintenance accountedformulation for is developed developed and optimally solved. Theoutsourcing effect of of the themaintenance outsourcing alternative on decisions is investigated through numerical experiments. The alternatives are accounted for is developed and optimally solved. The effect of the outsourcing alternative on maintenance decisions is investigated through numerical experiments. The overall results obtained demonstrate the validity of the proposed approach. alternative on maintenance decisions is investigated through numerical experiments. The overall results obtained demonstrate the validity of the proposed approach. alternative on maintenance decisions is investigated through numerical experiments. The overall results obtained obtained demonstrate the validity validity of the the proposed proposed approach. results demonstrate the of approach. © 2018, obtained IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. results demonstrate the validity of the proposed approach. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. Keywords: Selective maintenance, Reliability, Imperfect maintenance, Outsourcing. 1. INTRODUCTION INTRODUCTION varying from from 22 to to 88 hours hours (Diallo (Diallo et et al., al., 2017). 2017). To To prepare prepare 1. varying 1. INTRODUCTION INTRODUCTION varying fromto22 successfully to 88 hours hours (Diallo (Diallo et al., al., 2017). To prepare prepare the system complete its next mission, its 1. varying from to et 2017). To the system to successfully complete its next mission, its 1. INTRODUCTION varying from 2 to 8 hours (Diallo et al., 2017). To prepare the system to successfully complete its next mission, its components must be properly maintained during the schesystem to successfully complete its next mission, its components must be properly maintained during the scheAs organisations organisations strive strive to to reduce reduce operating operating costs costs and and enen- the As the system to successfully complete mission, its components must bebreak. properly maintained during theredunscheduled intermission Because ofits thenext built-in components must be properly maintained during the scheAs organisations strive to reduce operating costs and enduled intermission break. Because of the built-in redunsure lean operations, they usually resort to alternative As strivethey to reduce operating costs and en- components sureorganisations lean operations, usually resort to alternative must be properly maintained during the scheduled intermission break. Because of the built-in redundancies, and due to the limited duration of the scheduled duled intermission break. Because of the built-in redunAs organisations strive to reduce operating costs and ensure lean operations, operations, they usually resort resort to alternative alternative approaches to manage manage they their operations, operations, maintenance, and dancies, and due to the limited duration of the scheduled sure lean usually to approaches to their maintenance, and duled break. Because of theof redundancies, andscarce due to tomaintenance the limited limited duration ofbuilt-in the scheduled scheduled breaksintermission and resources, only limited and due the duration the sure lean operations, they usuallythese resort to alternative approaches to manage manage their operations, maintenance, and breaks and scarce maintenance resources, only aa limited asset management needs. Among alternative is the dancies, approaches to their operations, maintenance, and asset management needs. Among these alternative is the dancies, and due to the limited duration of the scheduled breaks and scarce maintenance resources, only a limited set of components have and can be maintained during the and scarce have maintenance resources, onlyduring a limited approaches to manage their operations, maintenance, asset management needs. Among these alternative is and the breaks set of components and can be maintained the outsourcing which is supply chain solution that enables enables asset management needs. Among these alternative is the outsourcing which is aa supply chain solution that breaks and scarce maintenance resources, only a limited set of components have and can be maintained during the breaks. It is therefore necessary to identify an optimal subset of components have and can be maintained during the asset management needs. Among these alternative isthat, the breaks. It is therefore necessary to identify an optimal suboutsourcing which is supply chain solution that so enables an organisation organisation to is outsource some of its needs outsourcing which aa supply chain solution that enables an to outsource some of its needs so that, set of components have and canto maintained duringsubthe breaks. It is is therefore therefore necessary to identify optimal to maintain maintain meet thean predetermined It necessary tobemeet identify an optimal suboutsourcing which a supply chain of solution that enables an organisation to isoutsource outsource some of its needs so that, breaks. set of components to to the predetermined for example, to face market demand or to keep core coman organisation to some its needs so that, for example, to face market demand or to keep core com- breaks. It is therefore necessary to identify an optimal subset of components to maintain to meet the predetermined reliability level required for the next mission. of components to maintain meet the predetermined an organisation to outsource some et of its needs so that, for example, to face market(Haoues demand oral., to keep keep core comreliability level required for theto next mission. petition (see to forface example 2016)core andcomthe set for example, market demand to petition (see for example (Haoues etoral., 2016) and the set components to maintain meet the was predetermined reliability levelmaintenance required forproblem theto next mission. Theofselective selective (SMP) first introintroreliability level required for the next mission. for example, to face market demand or to keep core competition (see for example (Haoues et al., 2016) and the The maintenance problem (SMP) was first reference therein). petition for example (Haoues et al., 2016) and the reliability reference(see therein). level required for the next mission. The selective maintenance problem (SMP) was first introduced by Rice et al. (1998) and applied in a series-parallel The selective maintenance problem (SMP) was first intropetition (see for example (Haoues al., 2016) and the duced by Rice et al. (1998) and applied in a series-parallel reference therein). To maintain maintain their production assets,et many many organisations reference therein). To their production assets, organisations The selective maintenance (SMP) first introduced byinRice Rice etsubsystems al. (1998) (1998) problem and applied in independent series-parallel system with composed of an by et al. and applied in aawas series-parallel reference therein). To maintain their production assets, many many organisations system in with subsystems composed of independent an outsource some of their maintenance activities by using duced To maintain their production assets, organisations outsource some of their maintenance activities by using duced by Rice et al. (1998) and applied in a series-parallel system in with subsystems composed of independent an identically exponentially distributed components. In this this system in with subsystems composed of independent an To maintain their production assets, many organisations outsource some their maintenance activities by using identically exponentially distributed components. In a combination of in-house and contract maintenance. outsource some of their maintenance activities by using system a combination in-house and contract maintenance. in with composed independent an identically exponentially distributed components. In this this work, the the only subsystems maintenance option components. isof the the replacement exponentially distributed In outsource some maintenance activities by using identically combination of their in-house and contract maintenance. work, only maintenance option is replacement Contractors are generally generally hired for contract specialist maintenance aa combination of in-house and maintenance. Contractors are hired for specialist maintenance identically exponentially distributed components. In this work, the only maintenance option is the replacement at failure. Cassady et al. (2001) extended the original work, the only maintenance option is the replacement aContractors combination of in-house and contract are generally generally hired for in-house. specialist maintenance. maintenance at failure. Cassady et al. (2001) extended the original where the expertise expertise is not not held Maintenance Contractors are hired for specialist maintenance where the is held in-house. Maintenance work, only option iswhere the replacement at failure. Cassady et al. for (2001) extended the original work ofthe (Rice et maintenance al., et 1998) systems lifetimes of failure. Cassady al. (2001) extended original Contractors are generally hired for specialist maintenance where theresolve expertise isproblems not held in-house. Maintenance work of (Rice et al., 1998) for systems wherethe lifetimes of tasks to basic are generally performed at where the expertise is not held in-house. Maintenance tasks to resolve basic problems are generally performed at failure. Cassady et al. (2001) extended the original work of (Rice et al., 1998) for systems where lifetimes of components are Weibull distributed. Three maintenance work of (Rice et al., 1998) for systems where lifetimes of where the expertise is not held in-house. Maintenance tasks to resolve basic problems are generally performed components are Weibull distributed. Three maintenance in-house. tasks to resolve basic problems are generally performed work in-house. of are (Rice et Weibull al., :1998) for systems wheremaintenance lifetimes of components are Weibull distributed. Three maintenance actions allowed minimal repair, Three corrective replaceare distributed. tasks to resolve basic problemsthe areselective generally performed components in-house. actions are allowed : minimal repair, corrective replaceThe present present paper investigates maintenance in-house. The paper investigates the selective maintenance components are Weibull distributed. Three maintenance actions are allowed : minimal repair, corrective replacement of failed components, and a preventive replacement actions are allowed : minimal repair, corrective replacein-house. The present paper paper investigates the the selective maintenance optimisation of multi-component multi-component systems in maintenance the outsouroutsour- ment of failed components, and a preventive replacement The present investigates selective optimisation of systems in the actions allowed : minimal repair, corrective replacement of are failed components, and preventive replacement of working working components. Theand resulting SMP is replacement solved using of failed components, aa preventive The present paper investigates the selective maintenance optimisation of multi-component multi-component systems in the outsouroutsourof components. The resulting SMP is solved using cing setting. The selective maintenance is appropriate in ment optimisation of systems in the cing setting. The selective maintenance is appropriate in ment of failed components, and a preventive replacement of working components. The resulting SMP is solved using an enumeration method. To deal with the combinatorial of working components. The resulting SMP is solved using optimisation of multi-component systems in the outsourcing setting. The selective maintenance is appropriate in an enumeration method. To deal with the combinatorial systems designed to perform consecutive missions intercing setting. The selective maintenance appropriate in of systems designed to perform consecutiveis missions interworking components. SMP solved using an enumeration method. To resulting deal with with the is combinatorial complexity arising from The large-size systems, four improved enumeration method. To deal the combinatorial cing setting. The maintenance appropriate in systems designed tobreaks perform consecutive missions intercomplexity arising from large-size systems, four improved spersed with finiteselective during whichisaamissions limited set set of an systems designed to perform consecutive interspersed with finite breaks during which limited of an enumeration method. To deal with the combinatorial complexity arising from large-size systems, four improved enumeration procedures are proposed in (Rajagopalan and complexity arising from large-size systems, four improved systems designed consecutive missions interspersed with finitetobreaks breaks during which limited set of enumeration procedures are proposed in (Rajagopalan and component repairs orperform replacements can aabe be carriedset out spersed with finite during which limited of component repairs or replacements can carried out complexity arising from are large-size systems, four improved enumeration procedures are proposed in (Rajagopalan (Rajagopalan and Cassady, 2006) to reduce the computation times. An exact enumeration procedures proposed in and spersed with finite breaks during which a limited set of component repairs or replacements can be carried out Cassady, 2006) to reduce the computation times. An exact due to to limited limited time,orbudget, budget, or resources. resources. Incarried the airline airline component repairs replacements can beIn out enumeration due time, or the procedures are in (Rajagopalan and Cassady, 2006) to reduce theproposed computation times. An exact exact method based based on reduce the branch-and-bound branch-and-bound procedure 2006) to the computation times. An component replacements can beIn carried out Cassady, due to limited limited time,orbudget, budget, or resources. Into the airline method on the procedure and industry for repairs example, aircraft areresources. subjected overnight due to time, or airline industry for example, aircraft are subjected tothe overnight Cassady, 2006) to reduce the computation times. An exact method based on the branch-and-bound procedure and a Tabu search based algorithm are proposed in (Lust method based on the branch-and-bound procedure and due to limited time, budget, or resources. In the airline industry for example, aircraft are subjected to overnight a Tabu search based algorithm are proposed in (Lust maintenance at night during a break interval typically industry for example, subjected to overnight maintenance at night aircraft during are a break interval typically method based on the branch-and-bound procedure and a Tabu search based algorithm are proposed in (Lust a Tabu search based algorithm are proposed in (Lust industry for example, aircraft are subjected to overnight maintenance at at night night during during aa break break interval interval typically typically maintenance a Tabu search based algorithm are proposed in (Lust maintenance atIFAC night during a Federation break interval typically 2405-8963 © (International of Automatic Control) Copyright © 2018, 2018 IFAC 532 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 532Control. Peer review© under of International Federation of Automatic Copyright © 532 Copyright © 2018 2018 IFAC IFAC 532 10.1016/j.ifacol.2018.08.372 Copyright © 2018 IFAC 532
IFAC INCOM 2018 526 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
et al., 2009). Khatab et al. (2007) proposed two heuristic methods, adapted from those used to solve the redundancy allocation problem in (Aggrawal, 1976; Gopal et al., 1978). The development of SMP models has since increased and dealt with extensions to include imperfect maintenance, system’s components dependencies, and mission and break profiles. Imperfect maintenance in the selective maintenance setting is addressed inLiu and Huang (2010), where the age reduction coefficient approach (Malik, 1979) is used to model imperfect maintenance. An imperfect selective maintenance model was also developed by (Zhu et al., 2011) and applied to a machining line system. Panday et al. (2013) also studied the selective maintenance problem for binary systems under imperfect maintenance using the hybrid hazard rate approach introduced in (Lin et al., 2000). In (Liu and Huang, 2010) and (Panday et al., 2013), a set of maintenance levels, ranging from minimal repair to replacement, are used to improve the reliability of a system component. (Khatab and Aghezzaf, 2016) studied the selective maintenance problem when the quality of imperfect maintenance is stochastic. A nonlinear and stochastic optimization problem was proposed and solved for a series-parallel system. In (Khatab et al., 2017) the SMP is addressed in systems operating missions and breaks both of stochastic durations. Schneider and Cassady (Schneider and Cassady, 2015) solve the selective maintenance problem for a fleet composed of a set of independent and identical systems. The fleet is required to execute a sequence of missions and return to a common base where maintenance actions may be performed on selected components. The selective maintenance problem has also been addressed in multi-state systems (MSS) settings. The most recent work appeared in Dao and Zuo (2017) where the authors investigated a MSS with structural relationships between its components. All papers surveyed above do not consider the very common case where multiple repair-persons are available to carry out the maintenance actions. Only few papers attempts to some extent to overcome this restrictive assumption. (Iyoub et al., 2006) and (Maillart et al., 2009) investigated the selective maintenance problem taking into account maintenance resources allocation. For its repair, each system’s component is assumed to consume an amount of a given maintenance resource. The total amount consumed from each maintenance resource must be less than or equal to the total amount of that resource allotted to perform maintenance on failed components. These approaches are however limited to systems with components of constant failure rates. In a more recent paper, Diallo et al. (2017) studied the SMP with multiple repair-persons. An integrated non-linear programming formulation was then developed and optimally solved. Numerical experiments conducted show the benefits of jointly carrying out the assignment of the tasks to repair-persons and the selection of the components to be repaired. Nonetheless, their work assume that component’s replacement as the only one available maintenance decision. In the present paper, the selective maintenance problem is investigated for systems with multiple repair-persons in outsourcing setting. The system must operate a sequence of alternating missions and scheduled breaks. Each component has a list of eligible maintenance actions ranging from minimal repair, through intermediate imperfect 533
maintenance actions, to replacement. It is assumed that some maintenance levels are performed in-house, while the others are outsourced. To maximise the probability of the system to successfully operate the next mission, the maintenance activities are performed on the system’s components during the break. Due to limited break duration, maintenance budget and repair crews, not all components are likely to be maintained. The resulting integrated optimisation model is a non-linear integer optimization model which is optimally solved and fully discussed. Numericals experiments are also provided to demonstrate the accuracy and the validity of the proposed approach. The remainder of the paper is organized as follows. Acronyms, notation and the main working assumptions are given in Section 2. System’s description and the imperfect maintenance model used are presented in Section 3. The selective maintenance model is developed in Section 4 where the new integrated selective maintenance problem is also formulated. Numerical experiments are provided in Section 5. Conclusion and future research extensions are drawn in Section 6. 2. ACRONYM, NOTATION AND MAIN WORKING ASSUMPTIONS In this section, acronyms, the list of notation and the main working assumptions made in this paper are provided in what follows. Acronyms CM (PM) Corrective (Preventive) maintenance SMP Selective maintenance problem OSMOP Outsourcing SM optimisation problem Notation n Number of subsystems i Index of subsystems ni Number of components in subsystem Si j Index of component in subsystem Si , Cij The j th component of subsystem Si L The highest maintenance level available l Maintenance level l ∈ {0, · · · , L} tcijl Time of CM of level l performed on Cij tpijl Time of PM of level l performed on Cij ch Cost rate of an in-house repair-person co Cost rate of an outsourcing repair-person Aij Age of Cij at the start of a break Bij Age of Cij at the end of a break Xij Cij status at the start of a break Yij Cij status at the end of a break C0 Maximum maintenance budget available D Limited break duration U Next mission duration hij (t) Failure rate of Cij (t) c Rij U |Bij Conditional reliability of component Cij R System reliability for the next mission Assumptions (1) The system consists of multiple, repairable binary components (the components and the system are either functioning or failed). (2) During the break, system components do not age, i.e. the age of a component is operation-dependent.
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
(3) No maintenance activity is allowed during the mission. Maintenance activities are allowed only during the break. (4) Multiple components can be worked on simultaneously without repair-persons colliding. 3. SYSTEM’S DESCRIPTION AND MAINTENANCE MODEL 3.1 System’s description Without loss of generality, the SMP studied in the present work considers a series-parallel system composed of n series subsystems Si (i = 1, ..., n) each of which is composed of ni independent and possibly non-identical components Cij (j = 1, ..., ni ) arranged in parallel. The system is assumed to have just finished the current mission, and then turned off during the scheduled break of finite length D for possible maintenance of their components. Thereafter, the system is used to operate the next mission of duration U . At the end of the current mission, a system’s component can be either in a functioning or in a failed state. Two state variables Xij and Yij are used to describe the status of component Cij , respectively, at the end of current mission and at the beginning of the next mission : 1 if Cij is working at the end of the current mission Xij = 0 otherwise. (1) 1 if Cij is working at the beginning of the next mission Yij = 0 otherwise. (2) Each component Cij is also characterized by its age Aij at the end of the current mission (i.e. at the start of te break), and its age Bij at the beginning of the next mission (i.e. at the end of the break). If the component is still functioning at the end of the current mission (i.e. Xij = 1), its corresponding probability to successfully operating the next mission is given by its conditional reliability R U |Bij such that : Bij +U hij (t)dt , (3) Rc U |Bij = exp Bij
Since the system’s reliability block diagram is seriesparallel, its reliability is :
R=
n
i=1
1 −
1 − R U |Bij · Y ij
ni
j=1
(4)
3.2 Imperfect maintenance model
the component Cij at the end of the current mission. Both maintenance types are modelled according to the age reduction imperfect maintenance model of (Malik, 1979). Without loss of generality, a common list {1, . . . , l, . . . , L} of L maintenance levels is available for all system’s components. To each maintenance level 1 ≤ l ≤ L corresponds an age reduction coefficient αl ∈ [0, 1]. If a maintenance of level l is carried out on Cij , its age Aij is reduced and becomes αl Aij . Two particular values of maintenance level can then be observed. The first corresponds to the maintenance level l = 1 whose age reduction is equal to 1 and stands then for minimal repair. The second corresponds to the highest maintenance level L whose age reduction is 0 which is equivalent to component’s renewal. It is worth noticing that minimal repair, as a maintenance level, is eligible only for failed components. To deal with outsourcing within SMP, the maintenance list is partitioned on two subsets of maintenance levels. The first k maintenance levels {1, . . . , k} are the set of maintenance actions that are performed in-house, while the remaining maintenance levels {k + 1, . . . , L} are assumed to require more skilled repair-persons and then outsourced. To alleviate the notation, we simply write k = k + 1. 4. SELECTIVE MAINTENANCE MODEL To develop the new integrated selective maintenance optimisation in the outsourcing setting, the following decision variable zijl is defined :
zijl
534
1 if Cij is selected for maintenance and maintenance l is performed = 0 otherwise.
(5)
According to the age reduction imperfect maintenance model, when a maintenance action of level l is performed on a component Cij , then the effective age Bij of Cij is evaluated as : Bij = [(αijl · zijl ) + (1 − zijl )] · Aij .
(6)
In what follows, we compute the total time spent on components’ maintenance and the total cost induced by the selected maintenance actions. To do so, let us denote by tpijl and tcijl , the time consumed by, respectively, a preventive and corrective maintenance level l when preformed on component Cij . We also denote by ch and co the cost rate (i.e. cost per unit time) induced by an in-house and outsourcing maintenance. It is reasonable to assume that outsourcing cost rate co is greater than the in-house cost rate ch . The total time spent by maintenance actions performed in-house is computed as : k ni n k p c tijl .Xij .zijl + tijl .(1 − Xij ).zijl . (7) i=1 j=1
Two types of maintenance are considered either preventive or corrective maintenance, depending on the status of
527
l=2
l=1
Similarly, the total time consumed by outsourced maintenance actions is computed as :
IFAC INCOM 2018 528 Bergamo, Italy, June 11-13, 2018
ni n i=1 j=1
L
tpijl .Xij .zijl
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
+
l=k
L
tcijl .(1
l=k
− Xij ).zijl
. (8)
The total cost induced by both in-house and outsourced maintenance actions is computed as : k ni n L p tijl .ch .Xij .zijl + tpijl .co .Xij .zijl + i=1 j=1 k
l=k
l=2
tcijl .ch .(1
l=1
− Xij ).zijl +
L
tcijl .co .(1
l=k
− Xij ).zijl
(9)
4.1 Problem formulation In this section, the outsourcing selective maintenance optimization problem (OSMOP) is developed. This model deals with the case where the objective of the maintenance decision-maker consists on maximizing the system reliability to successfully achieve the next mission, taking into consideration the pre-specified maintenance budget C0 . The optimization model is drawn subsequently. OSMOP : n
Max R =
i=1
Subject to : k ni n i=1 j=1 k l=1
i=1 j=1
ni n i=1 j=1
1 −
ni
j=1
k
tpijl .ch .Xij .zijl +
L
(10)
tpijl .co .Xij .zijl +
l=k
− Xij ).zijl +
L
tpijl .Xij .zijl +
k l=1
tpijl .Xij .zijl +
l=k
tcijl .co .(1
l=k
l=2
L
c 1 − Rij U |Aij · Yij
l=2
tcijl .ch .(1
ni n
− Xij ).zijl
tcijl .(1 − Xij ).zijl
L
l=k
tcijl .(1 − Xij ).zijl
≤ C0 (11)
≤D (12) ≤D (13)
Lij
zijl ≤ 1
(14)
zij1 ≤ (1 − Xij )
(15)
l=1
Lij
Bij =
l=1
[αijl · zijl + 1 − zijl )] · Aij
Yij = Xij +
L l=1
(1 − Xij ) · zijl
i = 1, . . . , n; j = 1, . . . , ni ; l = 1, . . . , L Yij , Xij ∈ {0, 1}, zijl ∈ {0, 1}.
(16) (17) (18) (19)
In the above optimization model, Equations (11) is the maintenance budget constraint. Equations (12) and (13) are the maintenance break duration constraints, respectively, for the in-house and outsourcing maintenance ac535
Figure 1. Reliability block diagram of the series-parallel system tivities. For each component Cij , Equations (14) states that only one maintenance level can be selected if the component is to be maintained by an in-house or an outsourcing repair-person. The constraint (15) states that minimal repair is eligible only on a failed component. The constraint (16) updates the effective age of components at the end of the break. The constraint (17) allows to update the operating state of components. 5. NUMERICAL EXPERIMENTS In this section, two numerical experiments are conducted. We are given a series-parallel system composed of n = 2 subsystems Si (i = 1, 2), its corresponding reliability block diagram is depicted in Figure (1). Subsystem S1 is composed of n1 = 2 components in parallel, while subsystem S2 contains n2 = 3 components arranged in parallel. It is assumed that the lifetimes of each component Cij follows a Weibull distribution with shape parameter β ij and scale parameter η ij . These parameters are reported in Table (1). This table gives also the age Aij and the value of the state variable Xij corresponding to component Cij at the end of the current mission. According to Table (1), only components C11 and C22 are failed to survive the current mission, while the other components are still functioning. We assume that L = 6 maintenance levels are available each of which the corresponding age reduction coefficient is given in Table (2). Times required to perform a corrective or a preventive maintenance of level l on components are reported, respectively, in Tables (3) and (4). In the all subsequent experiments, the in-house and the outsourcing cost rates are, respectively, set to ch = 5 and co = 7. The system has just finished the current mission and made available for maintenance. The next mission duration is set to U = 70 time units. Table 1. Components lifetimes parameters, ages and status at the start of break Cij C11 C12 C21 C22 C23
β ij 3 1.5 2.7 2 2.5
η ij 120 200 150 150 170
Aij 110 175 175 150 100
Xij 0 1 1 0 1
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
be realistic and one may be forced to resort to maintenance outsourcing contracts. Indeed, on one hand, not all organisations have highly-skilled maintenance operators, and, on the other hand, the maintenance budget is usually restricted. The maintenance manager should therefore strive to find an appropriate solution with mixed in-house and outsourcing maintenance activities. The following experiment investigates the effect of such combined solutions on maintenance decisions.
Table 2. Age reduction coefficients values l αl
1 1
2 0.8
3 0.6
4 0.3
5 0.2
6 0
Table 3. Corrective maintenance times tcijl l
1
2
3
4
5
6
C11 C12 C21 C22 C23
0.5 0.45 0.6 0.4 0.5
1.5 1.2 1.1 0.8 1
2.2 2.3 1.3 1.1 1.3
2.5 2.5 1.8 1.5 1.7
2.8 2.7 2.3 1.8 2
3 2.8 2.5 2.2 2.5
5.2 Experiment #2 This experiment investigates the effects of the outsourcing on the selective maintenance decisions. Using the optimisation model OSMOP, the maximum achievable system reliability will be determined for a given break duration and maintenance cost when the number k of in-house maintenance levels is taken variable. It is conjectured that the flexibility provided by the cohort with mixed in-house and outsourcing repair-persons will allow the model to reach the same or better solution than when maintenance activities are completely either in-house or outsourced.
Table 4. Preventive maintenance times tpijl l C11 C12 C21 C22 C23
2 1 0.9 0.8 0.7 0.8
3 1.9 1.7 1 0.9 1
4 2.3 2 1.2 1.2 1.1
5 2.7 2.2 1.4 1.4 1.7
529
6 2.8 2.5 1.9 2 2.2
5.1 Experiment#1 This experiments investigates the two extreme cases where the maintenance activities is completely outsourced or completely performed in-house. We then solve the selective maintenance problem for two particular values of the number k of in-house maintenance levels. In the case of complete maintenance outsourcing k = 0, while in the complete in-house case the value of k is set to k = L. In this experiment the duration of the break is set to D = 4, and the maintenance budget is set to C0 = 25. In complete outsourcing case, the optimal maintenance decisions suggests the maintenance plan according to which component C11 and C22 are both replaced (corrective replacement). The resulting maximal reliability to operate the next mission is evaluated to R = 63.09%. This maintenance plan induces a total cost of 23.8. In the case of complete in-house maintenance, the optimal maintenance decisions derived are such that three components are selected to be maintained. The optimal maintenance plan suggests to perform a preventive replacement on component C12 , a preventive maintenance of level l = 4 on component C23 , and a minimal repair on the failed component C22 . The resulting system’s reliability is estimated to R = 70.22%. This maintenance plan induces a total cost of only 20. From the above results, it follows that the maximal achievable system’s reliability when dealing with complete inhouse maintenance case is better that that obtained when dealing with the case of complete maintenance outsourcing. Furthermore the optimal in-house maintenance plan is less expensive than the optimal outsourcing maintenance plan. This result comes from the fact that hiring an outsourcing repair-person is more expensive than using an in-house repair-person. Consequently, using the complete outsourcing solution limits the number of components to be maintained. However, these two extreme cases may not 536
The same system and parameters used in the previous experiments are used. The optimisation problem OSMOP is solved for different values of the allowable maintenance budget C0 . The results obtained are depicted in Figure (2). From these results, it is clearly shown as conjectured that an appropriate combination of an in-house and an outsourcing cohort is able to achieve equivalent or better reliability than the uniform cohort (i.e. either complete in-house or complete outsourcing). For example, if we consider the case where the maintenance budget is limited to C0 = 25, it follows that the maximal achievable system’s reliability when dealing with a combined in-house and outsourcing solution is obtained when the first four maintenance levels are performed in-house (k = 4) while the highest maintenance levels l = 5 and l = 6 are outsourced. If the maintenance budget is increased to C0 = 30, the maximal achievable system’s reliability when dealing with a combined in-house and outsourcing solution is obtained when only the highest maintenance level l = 6 is outsourced. The same conclusion can also be made from the results obtained for the other values of the maintenance budget. In general, when the maintenance budget available decreases, the maximal reliability achieved decreases. When the maintenance budget permits, the highly skilled outsourcing repair-persons can be used in an adequate combination with the in-house repair-persons. As the budget decreases, the model suggests to use the in-house persons instead of resorting to outsourcing. 6. CONCLUSION This paper introduced a novel variant of the selective maintenance problem in multi-component system by specifically combining the in-house and the outsourcing maintenance alternatives. A novel integrated non-linear programming formulation was then proposed and optimally solved. Numerical experiments show the benefits of jointly carrying out : (1) the assignment of the maintenance tasks
IFAC INCOM 2018 530 Bergamo, Italy, June 11-13, 2018
K. Chaabane et al. / IFAC PapersOnLine 51-11 (2018) 525–530
Figure 2. Maximal achievable reliability versus the number k : case of Experiment#2 to the in-house and outsourcing repair-persons, and (2) the selection of the components to be repaired. Future extensions that the authors are currently working on, include the development of efficient solution methods to deal with large sized problems. Another important issue consist to investigate the selective maintenance problem where more than one objective is to be optimized. Other possible extensions may also be the generalization of the present work to consider multi-state systems and k-out-ofn systems. ´ ERENCES ´ REF Aggrawal, K.K. (1976). redundancy optimization in general systems. IEEE Transactions on Reliability, R-25(25), 330–332. Cassady, C.R., Murdock, W.P., and Pohl, E.A. (2001). Selective maintenance for support equipement involving multiple maintenance actions. European Journal of Operational Research, 129, 252–258. Dao, C.D. and Zuo, M.J. (2017). Selective maintenance of multi-state systems with structural dependence. Reliability Engineering and System Safety, 159, 184–195. Diallo, C., Khatab, A., Venkatadri, U., and Aghezzaf, E.H. (2017). A joint selective maintenance and multiple repair–person assignment problem. In 7th Industrial Engineering and Systems Management (IESM) Conference, Saarbrucken, Germany. Gopal, K., Aggrawal, K.K., and Gupta, J.S. (1978). An improved algorithm for reliability optimization. IEEE Transactions on Reliability, R-27(27), 325–328. Haoues, M., Dahane, M., and Mouss, N.K. (2016). Outsourcing optimization in two-echelon supply chain network under integrated production-maintenance constraints. Journal of Intelligent Manufacturing. doi :10.1007/s10845-016-1273-3. Iyoub, I., Cassady, C.R., and Pohl, E.A. (2006). Establishing maintenance resource levels using selective maintenance. Engineering Economist, 51(2), 99–114. Khatab, A. and Aghezzaf, E.H. (2016). Selective maintenance optimization when quality of imperfect mainte537
nance actions are stochastic. Reliability Engineering & System Safety,, 150, 182–189. Khatab, A., Aghezzaf, E.H., Djelloul, I., and Sari, Z. (2017). Selective maintenance optimization for series– parallel systems alternating missions and scheduled breaks with stochastic durations. International Journal of Production Research, 55(10), 168–177. Khatab, A., Ait-Kadi, D., and Nourelfath, M. (2007). Heuristic-based methods for solving the selective maintenance problem for series-prallel systems. In International Conference on Industrial Engineering and Systems Management, Beijing, China. Lin, D., Zuo, M.J., and Yam, R.C.M. (2000). General sequential imperfect preventive maintenance models. International Journal of Reliability, Quality and Safety Engineering, 7(3), 253–266. Liu, Y. and Huang, H.Z. (2010). Optimal selective maintenance strategy for multi-state systems under imperfect maintenance. IEEE Transactions on Reliability, 59(2), 356–367. Lust, T., Roux, O., and Riane, F. (2009). Exact and heuristic methods for the selective maintenance problem. European Journal of Operational Research, 197, 1166– 1177. Maillart, L.M., Cassady, C.R., Rainwater, C., and Schneide, K. (2009). Selective maintenance decisionmaking over extended planing horizons. IEEE Transactions on Reliability, 58(3), 462–469. Malik, M. (1979). Reliable preventive maintenance scheduling. AIIE transactions, 11(3), 221–228. Panday, M., Zuo, M.J., Moghaddass, R., and Tiwari, M.K. (2013). Selective maintenance for binary systems under imperfect repair. Reliability Engineering and System Safety, 113, 42–51. Rajagopalan, R. and Cassady, C.R. (2006). An improved selective maintenance solution approach. Journal of Quality in Maintenance Engineering, 12(2), 172–185. Rice, W.F., Cassady, C.R., and Nachlas, J. (1998). Optimal maintenance plans under limited maintenance time. In Proceedings of Industrial Engineering Conference, Banff, BC, Canada. Schneider, K. and Cassady, C.R. (2015). Evaluation and comparison of alternative fleet-level selective maintenance models. Reliability Engineering and System Safety, 134, 178–187. Zhu, H., Liu, F., Shao, X., Liu, Q., and Deng, Y. (2011). A cost-based selective maintenance decision-making method for machining line. Quality and Reliability Engineering International, 27, 191–201.