U Aghezzaf

Venkatadri

K Chaabane
email: kenza.chaabane@univ-lorraine.fr

A Khatab

C Diallo

E.-H Aghezzaf

U Venkatadri
email: uday.venkatadri@dal.ca

Integrated imperfect

Keywords: multimission selective maintenance and repairpersons assignment problem Multimission selective maintenance, Reliability, Preventive maintenance, Repairpersons assignment, Optimization

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In many manufacturing and service industries such as maritime, production, military and transportation, there are complex multicomponent systems designed to run multiple sequences of missions with intermission breaks of finite duration. To improve the system performance, usually reliability, during the missions, its components are maintained during the breaks. For example, maintenance activities are carried out on production system between two production runs, military equipments are maintained between two combat missions, and boats and fishing gear are repaired between two fishing expeditions. However, given the limited intermission break duration and the scarce maintenance resources, it is often impossible to perform top level maintenance activities on all components of such systems. Therefore, the maintenance decision-maker is faced with the problem of selecting the appropriate components to maintain and the level of maintenance actions to be carried out on these selected components. In the literature, this is known as the selective maintenance problem (SMP).

The original work on selective maintenance (SM) is reported in [START_REF] Rice | Optimal maintenance plans under limited maintenance time[END_REF]. Since then, the SM has been extensively investigated in different ways and research issues addressed have enriched the maintenance literature [START_REF] Cao | A literature review on selective maintenance for multi-unit systems[END_REF][START_REF] Xu | Recent advances in selective maintenance from 1998 to 2014[END_REF]. Nevertheless, most papers dealing with the SMP are still based on the restrictive assumption of one break and only one subsequent mission. However, this is not the case in practice and real industrial applications require SMP decisions for multiple missions rather than a single one. Solving the multimission case gives the decision maker an optimized global plan for the maintenance activities and their assignment to repairpersons. From an optimization point of view, this optimal global plan covering all missions at once is better than sequentially planning multiple single missions (joint versus sequential optimization). This global planning allows to better manage and to ensure an optimal distribution of the overall maintenance resources during the overall multimission cycle. In such real situations, there are clear and strong trade-offs between the maintenance decisions and long-term resources management. To help the maintenance decision maker solve such more realistic SMP, the present paper develops a novel variant of the classical SMP to deal with the joint optimization of multimission maintenance decisions and maintenance resource assignment.

The remainder of this paper is structured as follows. Section 2 presents a brief literature review on SMP and its various extensions. In this section, the present paper is then positioned in the literature and its contributions highlighted. In Section 3, the notation and main working assumptions are presented. Section 3 also gives the description of the system under consideration and the formulation of its reliability function. Section 4 presents the imperfect maintenance model and computes the total cost and duration of the maintenance activities during the breaks. In section 5, the new integrated formulation of the SMP is developed and presented as a nonlinear binary program. The solution method used is also detailed in Section 5. Numerical experiments are investigated and discussed in Section 6 to demonstrate the validity and the accuracy of the proposed model and solution approach. Conclusions and future research directions are provided in Section 7.

Literature review and shortcommings & paper's contributions

Literature review

The original SM work [START_REF] Rice | Optimal maintenance plans under limited maintenance time[END_REF] considers a series-parallel system where the subsystems are composed of independent and identically components with exponentially distributed lifetimes. In [START_REF] Cassady | Selective maintenance modeling for industrial systems[END_REF], the authors extend the original SMP to deal with non-identical components. Cassady et al. [START_REF] Cassady | Selective maintenance for support equipment involving multiple maintenance actions[END_REF] study the SMP in a system where components' lifetimes are Weibull distributed with three maintenance actions: minimal repair (MR), corrective maintenance (CM) of failed components and preventive maintenance (PM) of functioning components. An enumeration method is used to solve the resulting SM optimization problem (SMOP).

Since its introduction, the SMP has been studied and extended into multiple directions such as improving the computational efficiency of solution methods, accounting for maintenance quality, more complex system configurations, scales, and missions profiles. Extensive literature reviews of the SMP can be found in [START_REF] Cao | A literature review on selective maintenance for multi-unit systems[END_REF][START_REF] Xu | Recent advances in selective maintenance from 1998 to 2014[END_REF]. The authors in [START_REF] Xu | Recent advances in selective maintenance from 1998 to 2014[END_REF] review 70 papers on SMP published between 1998 and 2014 and classify them according to the problem type (basic, multistate, multimission, fleet-level) and solution methodology (enumeration, heuristic). They then highlight the literature shortcomings and discuss future research issues.

To deal with the combinatorial complexity arising from large size systems while reducing computation times, Cassady et al. [START_REF] Rajagopalan | An improved selective maintenance solution approach[END_REF] propose four improved enumeration procedures. A branch-and-bound (B&B) procedure and a Tabu search based algorithms are proposed in [START_REF] Lust | Exact and heuristic methods for the selective maintenance problem[END_REF] to solve the SMP problem for the series-parallel problem in [START_REF] Cassady | Selective maintenance for support equipment involving multiple maintenance actions[END_REF]. As pointed out in [START_REF] Lust | Exact and heuristic methods for the selective maintenance problem[END_REF], the authors resort to the Tabu search heuristic when the number of components reaches approximately 20, for which the B&B method becomes computationally intensive. Inspired by [START_REF]redundancy optimization in general systems[END_REF][START_REF] Gopal | An improved algorithm for reliability optimization[END_REF][START_REF] Sharma | redundancy optimization in general systems[END_REF], two heuristic-based methods are proposed in [START_REF] Khatab | Heuristic-based methods for solving the selective maintenance problem for series-prallel systems[END_REF]. Pandey et al. [START_REF] Pandey | Selective maintenance for binary systems under imperfect repair[END_REF] propose a solution approach based on the differential evolution (DE) algorithm. Several other studies [START_REF] Liu | Optimal selective maintenance strategy for multi-state systems under imperfect maintenance[END_REF][START_REF] Zhu | A cost-based selective maintenance decision-making method for machining line[END_REF][START_REF] Khatab | Genetic algorithm for selective maintenance optimization of multi-mission oriented systems[END_REF][START_REF] Dao | Selective maintenance for multi-state series systems with s-dependent components[END_REF] use the genetic algorithm as a solution approach to handle SMP for large size systems.

The initial SMP has also been extended in binary states systems to include imperfect maintenance (IM) models. Zhu et al. [START_REF] Zhu | A cost-based selective maintenance decision-making method for machining line[END_REF] use the age reduction coefficient approach [START_REF] Malik | Reliable preventive maintenance scheduling[END_REF] to model IM. The resulting SMOP is applied to a machining line in the automobile industry. To model the imperfect PM actions, Pandey et al. [START_REF] Pandey | Selective maintenance for binary systems under imperfect repair[END_REF] use the hybrid hazard rate approach of Lin et al. [START_REF] Lin | General sequential imperfect preventive maintenance models[END_REF]. Different maintenance levels are then allowed and relationships linking the level of IM to the amount of resources consumed by the maintenance tasks are presented. In [START_REF] Maaroufi | Optimal selective renewal for systems subject to propoagated failures with global effect and failure isolation phenomena[END_REF], the authors study the SMP under failure propagation. Khatab et al. [START_REF] Khatab | Selective maintenance optimization when quality of imperfect maintenance actions are stochastic[END_REF] study the SMP when the quality of the IM is stochastic. A nonlinear and stochastic optimization was then proposed and solved for a series-parallel system. Diallo et al. [START_REF] Diallo | Optimal selective maintenance decisions for large serial k-out-of-n: G systems under imperfect maintenance[END_REF] propose a novel two-phase formulation to deal with complex reliability structures such as k-out-of-n systems connected in series. The authors in [START_REF] Khatab | Selective maintenance optimization for series-parallel systems alternating missions and scheduled breaks with stochastic durations[END_REF][START_REF] Khatab | Selective maintenance optimization for systems operating missions and scheduled breaks with stochastic durations[END_REF] extend the SMP to deal with cases where mission and break durations are stochastic with known distributions. Liu et al. [START_REF] Liu | On sequence planning for selective maintenance of multi-state systems under stochastic maintenance durations[END_REF] extend the work in [START_REF] Khatab | Selective maintenance optimization for series-parallel systems alternating missions and scheduled breaks with stochastic durations[END_REF][START_REF] Khatab | Selective maintenance optimization for systems operating missions and scheduled breaks with stochastic durations[END_REF] and propose a sequence planning model under stochastic maintenance durations. They conclude that the sequencing of the maintenance actions can significantly affect the achieved reliability especially when the break duration is uncertain. The condition-based SMP was proposed in [START_REF] Khatab | Condition-based selective maintenance for stochastically degrading multi-component systems under periodic inspection and imperfect maintenance[END_REF] for a system where components degrade according to a stationary stochastic gamma process. In a recent paper [START_REF] Jiang | Robust selective maintenance strategy under imperfect observations: A multi-objective perspective[END_REF], Jiang and Lin proposed a SM approach for a system under imperfect observations with uncertainties related to both component states and effective ages. A robust multi-objective SM optimization model is then developed to jointly maximize the system reliability during the next mission and minimize its variability.

Schneider et al. [START_REF] Schneider | Evaluation and comparison of alternative fleet-level selective maintenance models[END_REF] study the SMP in a fleet composed of multiple independent and identical systems. The fleet is required to operate a set of sequential missions and return to a common base where maintenance actions may be performed on some selected components. A nonlinear cost optimization model and its linearized version are proposed and solved. Yang et al. [START_REF] Yang | Fleet-level selective maintenance problem under a phased mission scheme with short breaks: A heuristic sequential game approach[END_REF] address the SMP for a fleet of systems required to perform phased missions with short scheduled breaks. A sequential game-based algorithm with state backtracking is proposed to reduce the maintenance frequency and costs under a minimum required reliability constraint.

SMPs have also been investigated for multi-state systems (MSS). In [START_REF] Cheng Chen | Selective maintenance optimization for multi-state systems[END_REF], the authors consider a series-parallel system where each subsystem consists of identical components arranged in parallel. Khatab et al. [START_REF] Khatab | Selective maintenance policy for multi-mission multi-state series-parallel systems[END_REF] develop a SMP optimization model for MSS as defined in [START_REF] Lisnianski | Multi-state systems reliability: Assesment, optimization and applications[END_REF][START_REF] Levitin | Universal generating function in reliability analysis and optimization[END_REF] and operating more than one mission. The resulting optimization program is solved using a simulated annealing algorithm. Liu and Huang [START_REF] Liu | Optimal selective maintenance strategy for multi-state systems under imperfect maintenance[END_REF] consider a MSS where components have two operating states and are subjected to IM. Pandey et al. [START_REF] Pandey | Selective maintenance modeling for multistate system with multistate components under imperfect maintenance[END_REF] investigate the SMP for a MSS where the functioning of each component is modelled as a continuoustime Markov chain with more than two states. Dao et al. [START_REF] Dao | Selective maintenance for multi-state series-parallel systems under economic dependence[END_REF][START_REF] Dao | Selective maintenance for multi-state series systems with s-dependent components[END_REF] study the SMP in MSS components under economic and stochastic dependencies. In [START_REF] Zhou | An effective approach to reducing strategy space for maintenance optimisation of multistate seriesparallel systems[END_REF], several levels of IM are considered and the system is composed of multi-state components modeled as in [START_REF] Pandey | Selective maintenance modeling for multistate system with multistate components under imperfect maintenance[END_REF]. The authors in [START_REF] Khatab | Selective maintenance for failure-prone multistate systems when the durations of missions and scheduled breaks are stochastic[END_REF] solve the SMP in a MSS while considering both random break and mission durations. Dao and Zuo [START_REF] Dao | Selective maintenance of multi-state systems with structural dependence[END_REF] investigate the SMP in a MSS with structural dependence between components. An optimal load distribution model for MSS subjected to SM was proposed by Chen et al. [START_REF] Chen | Optimal load distribution for multi-state systems under selective maintenance strategy[END_REF] where the objective is to maximize the system reliability by optimally distributing the loads among all components. Dao et al. [START_REF] Dao | Optimal selective maintenance for multi-state systems in variable loading conditions[END_REF] also study the SMP for series MSS working under variable loading conditions. They propose a load-dependent degradation model for multi-state components. Cao et al. [START_REF] Cao | Selective maintenance optimization for fuzzy multi-state systems[END_REF] investigate the SMP in a fuzzy MSS where components capacities and states transition rates, in addition to both the break and the next mission durations are considered as fuzzy values. The goal is to maximize the fuzzy system reliability to successfully complete the subsequent mission.

Several other papers in the literature study the SMP and the allocation of maintenance resources. In [START_REF] Iyoub | Establishing maintenance resource levels using selective maintenance[END_REF], the repair of each system 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. A similar approach is also used by [START_REF] Maillart | Selective maintenance decision-making over extended planning horizons[END_REF][START_REF] Maillart | Selective Maintenance decision-Making over extended Planing Horizons[END_REF]) when dealing with series-parallel systems with constant failure rate components. Furthermore, only perfect replacement is available as maintenance action. In more recent works [START_REF] Diallo | A joint selective maintenance and multiple repair-person assignment problem[END_REF][START_REF] Diallo | Optimal joint selective imperfect maintenance and multiple repairpersons assignment strategy for complex multicomponent systems[END_REF][START_REF] Khatab | Optimization of the joint selective maintenance and repairperson assignment problem under imperfect maintenance[END_REF], the authors extend the previous SMP models to systems where components lifetimes are generally distributed, IM is accounted for, and the assignment of the repair tasks to multiple repairpersons is explicitly modelled. The resulting integrated SM optimization models are then formulated so that decisions on maintenance tasks and their assignment to repairpersons are jointly made.

Literature shortcomings

The main common shortcoming in all of the above mentioned papers is that they all deal with a single mission. The one-mission based optimal plans obtained are no longer optimal when the system lifecycle covers multiple missions. According to our literature review, only few papers attempt to some extent to overcome this drawback. Maillart et al. [START_REF] Maillart | Selective maintenance decision-making over extended planning horizons[END_REF] formulate the finite-horizon, and infinite-horizon multimission SMP as stochastic dynamic programs. They conclude that these policies rarely differ and that the difference in long-run performance is minimal. Zhao and Zeng [START_REF] Zhao | Maintenance strategy for stochastic selective maintenance of a two-state system[END_REF] propose a model for the multimission SMP where the break duration is exponentially distributed. Khatab et al. [START_REF] Khatab | Selective maintenance policy for multi-mission multi-state series-parallel systems[END_REF] develop a SMP optimization model for MSS with multiple subsequent missions. The resulting optimization program is solved using a simulated annealing algorithm. In [START_REF] Khatab | Genetic algorithm for selective maintenance optimization of multi-mission oriented systems[END_REF], an approach based on the genetic algorithm is proposed to solve the SMP in a multimission setting. Pandey et al. [START_REF] Pandey | Selective maintenance scheduling over a finite planning horizon[END_REF] develop a maintenance scheduling model under IM for the finite planning horizon SMP using the hybrid imperfect maintenance model. The work in [START_REF] Pandey | Selective maintenance scheduling over a finite planning horizon[END_REF] is extended by Zhang et al. [START_REF] Zhang | multi-mission selective maintenance modelling for multistate systems over finite time horizon[END_REF] to a MSS setting. The authors in [START_REF] Zhang | multi-mission selective maintenance modelling for multistate systems over finite time horizon[END_REF] propose a SM model in a MSS whose components deteriorate according to an homogeneous time Markov chain. By taking the system reliability requirement as a constraint, the authors develop a SM optimization model to determine the maintenance times and options for each system's component to minimize the total maintenance cost. To solve the resulting SMOP, a differential evolution algorithm is implemented. Shahraki et al. [START_REF] Shahraki | Selective maintenance optimization for multi-state systems considering stochastically dependent components and stochastic imperfect maintenance actions[END_REF] extend the results of Dao and Zuo [START_REF] Dao | Selective maintenance for multi-state series systems with s-dependent components[END_REF] to deal with SMP in MSS where components are s-dependent and subjected to random IM. The s-dependency between components is represented by two types of interactions as a function of the system performance rate in addition to the number of components impacted. Similarly as in [START_REF] Jiang | Robust selective maintenance strategy under imperfect observations: A multi-objective perspective[END_REF], a multi-objective SMOP is then proposed to maximize the system reliability requirement and simultaneously to minimize its corresponding variability. Jiang and Lin [START_REF] Jiang | Selective maintenance strategy for systems executing multiple consecutive missions with uncertainty[END_REF] extend the results in [START_REF] Khatab | Selective maintenance optimization for series-parallel systems alternating missions and scheduled breaks with stochastic durations[END_REF][START_REF] Khatab | Selective maintenance optimization for systems operating missions and scheduled breaks with stochastic durations[END_REF] to deal with SMP in systems operating multiple missions and breaks with both random durations. The resulting SMOP is formulated as a max-min programming model and solved by a simulated annealing-based genetic algorithm. However, the above papers are still based on the restrictive assumption that only one repairperson or repair channel is available to perform maintenance actions during scheduled breaks. In many real industrial situations, there are multiple repairpersons or crews available to carry out inspections and maintenance activities. Furthermore, as maintenance workers are needed to carry out the maintenance operations, there is therefore a clear and strong dependence between the assignment of repairpersons to the maintenance tasks and the final performance of the maintenance activities. [START_REF] Diallo | A joint selective maintenance and multiple repair-person assignment problem[END_REF][START_REF] Khatab | Optimization of the joint selective maintenance and repairperson assignment problem under imperfect maintenance[END_REF].

Paper's contributions

The present paper develops a new approach where the SMP in [START_REF] Khatab | Optimization of the joint selective maintenance and repairperson assignment problem under imperfect maintenance[END_REF] is comprehensively extended to help maintenance decision makers resolve real industry occurrences of the SMP including the assignment of the maintenance actions to the repair crews in a multimission setting. The joint maintenance and the repairpersons assignment decisions is formulated over a multimission finite horizon for a multicomponent system running a sequence of alternating missions and scheduled breaks with possibly different durations. The lifetimes of the components are allowed to follow very general distributions. A list of IM actions including replacement is available for each component to bring the system reliability back up enough to successfully fulfill the subsequent missions. To avoid unplanned interruptions due to components failures during a mission, minimal repair are performed to restore operation. Crews are available to perform maintenance actions during breaks when necessary. To meet a pre-specified required performance level for the next mission, the maintenance activities are performed on the system components during the break. Due to the limited break durations, maintenance budget and repair crews, not all components are likely to be maintained.

The new integrated SMP is multidimensional and is developed to jointly make five decisions: selecting the components to be maintained, selecting the maintenance levels to be performed on the selected components, determining when (at which break) to perform the maintenance actions, determining the number of repairpersons to hire/utilize, and assigning them to components/maintenance levels. The objective is to find the optimal decisions minimizing the total maintenance and labour costs while ensuring a minimum reliability level during the missions. Because of the resulting combinatorial complexity of the optimization model, a solution method based on the genetic algorithm (GA) is developed for large scale problems.

The present SM approach is more suitable for systems with components that can be minimally repaired at failure during missions. This strategy is particularly interesting for series-parallel manufacturing systems. The function of the system is successfully accomplished whenever all subsystems operate successfully. It should be noted that it is not uncommon to find series-parallel systems where a subsystem is found to be composed of only one component or machine. Zhu et al. [START_REF] Zhu | A cost-based selective maintenance decision-making method for machining line[END_REF] describe such a system: the machining line of the automobile engine connecting rod. The machining line is composed of 10 subsystems in series. The first subsystem is a single machine whose failure would stop the whole line and cause production stoppage, economic losses and backlogs. To avoid such adverse consequences, minimal repair should be implemented to deal with failure during the missions while extensive maintenance alternatives are carried out during breaks. PM actions are performed to reduce random failures and to provide sufficient system reliability to accomplish the next mission.

System description and reliability computation

Notation and main working assumptions

We first present the notation system used and discuss the main working assumptions.

Notation

N

Number of subsystems in the multicomponent system i Index of subsystems

S i , i = 1, . . . , N N i Number of components in subsystem S i j Index of components in subsystem S i , j = 1, . . . , N i E ij
The j th component of subsystem

Duration of break m L ij

Highest maintenance level available for component

E ij l Index of maintenance level l = 0, • • • , L ij γ ijl Age reduction coefficient of maintenance level l performed on E ij t ijkl Duration of maintenance level l performed on E ij by repairperson k c r ij Cost of minimal repair on E ij c f k Fixed cost of hiring/using repairperson k c v k Variable cost rate of using repairperson k X ijm Age of component E ij at the start of break m, m = 1, . . . , M -1 Y ijm Age of component E ij at the end of break m, m = 1, . . . , M -1 R 0m Required minimum system reliability during mission m h ij (t) Failure rate of component E ij R c ij U m | Yijm Conditional reliability of component E ij during mission m given an initial age Y ijm R m
Overall system reliability during mission m Assumptions:

1. The system consists of multiple, s-independent repairable binary components (components and 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.

3. No maintenance activity other than minimal repair is allowed during the mission. When a minimal repair is performed on a failed component, its failure rate remains undisturbed.

4. Time to perform minimal repair is negligible compared to mission durations.

5. All required limited resources (budget, repairpersons, tools) are available when needed. If a resource is not available then the corresponding parameter should be penalized/reduced to reflect the shortage. The model would still remain valid.

6. Multiple components can be worked on simultaneously without repairpersons colliding.

System description

Without loss of generality, the SMP addressed in this work considers a binary series-parallel system S composed of n subsystems S i (i = 1, . . . , N ) in series, each of which is composed of s-independent components E ij (j = 1, . . . , N i ) arranged in parallel. Figure 1 depicts a 2 by 2 series-parallel system.

The system under consideration is tasked with running a series of alternating missions and scheduled breaks. To avoid any confusion while simplifying the notation, the present paper assumes that the system has just completed a mission and is entering the first break of a new sequence of multiple missions. It will run its first mission of the new sequence, after which it will be shutdown to start the second break, and so on. There are M missions and as many scheduled breaks, all indexed from m = 1 to M .

At the end of mission m of duration U m , the system is switched off for the (m + 1) th scheduled break during which maintenance activities can be carried out. After the break, the system will be operated during the following mission of duration U m+1 . X ijm and Y ijm denote the effective ages of component E ij at the beginning and at the end of m th break respectively. All components are subjected to perfect inspections that reveal their effective ages X ij1 at the beginning of the first break. The duration of these inspections are not included in the break duration. Following assumption (2), it is worth noticing that if no maintenance is carried out during break m, components' ages remain unchanged, i.e. X ijm = Y ijm . Given that only minimal repairs are carried out during the missions, the following recursive formula establishes the relationship between X ijm and Y ijm :

X ij,m+1 = Y ijm + U m , ∀m = 1, • • • , M -1.
(1)

System reliability during a mission

The system reliability for mission m is defined as the probability of the system to successfully complete the mission. To compute this reliability, we first compute the reliability for each component

E ij . Let R c ij U m | Y ij(m)
denote the conditional probability of a functioning component E ij at the end of the break m to survive the mission of length U m given that its age at the start of the mission is Y ijm . This conditional reliability depends on both the effective age Y ijm at the start of mission m and the mission duration U m . Let T ij be the random variable representing the lifetimes of component E ij . It follows that the

conditional reliability R c ij U m | Yijm during mission m is given by R c ij U m | Yijm = Pr T ij > Y ijm + U m | Tij >Yijm (2) = Pr (T ij > Y ijm + U m ) Pr (T ij > Y ijm ) ,
which equivalently gives:

R c ij U m | Yijm = R ij (U m + Y ijm ) R ij (Y ijm ) (3)
where R ij (t) is the unconditional reliability function of component E ij .

The reliability R m of the whole series-parallel system during mission m is given by:

R m = N i=1   1 - Ni j=1 1 -R c ij U m | Yijm   . (4)
4. Imperfect Maintenance: repair levels, total maintenance cost and duration

IM is an important topic in the field of engineering asset management and has gained much attention in reliability and maintenance theory. As a result, a multitude of IM models have been reported in the literature. Among these models are the age reduction model of Malik [START_REF] Malik | Reliable preventive maintenance scheduling[END_REF], the adjustment coefficient (hazard rate increased) model of Nakagawa [START_REF] Nakagawa | Sequential imperfect preventive maintenance policies[END_REF] (see also [START_REF] Nakagawa | A summary of maintenance policies for a finite interval[END_REF][START_REF] Nakagawa | Advanced reliability models and maintenance policies[END_REF]), the Kijima type I and type II models [START_REF] Kijima | Periodical replacement problem without assuming minimal repair[END_REF][START_REF] Kijima | Some results for repairable systems with general repair[END_REF], the quasi-renewal process model [START_REF] Park | Altered quasi-renewal concepts for modeling renewable warranty costs with imperfect repairs[END_REF][START_REF] Wang | Some maintenance models and availability with imperfect maintenance in production systems[END_REF][START_REF] Wang | Reliability and optimal maintenance[END_REF], the geometric process model [START_REF] Lam | Geometric processes and replacement problem[END_REF], the hybrid hazard rate model of Lin et al. [START_REF] Lin | General sequential imperfect preventive maintenance models[END_REF]. Recently, [START_REF] Zhang | An ameliorated improvement factor model for imperfect maintenance and its goodness of fit[END_REF] developed an ameliorated improvement factor model for IM and its goodness of fit to give practical grounds to the proposed model. Reviews of IM models appeared in [START_REF] Pham | Imperfect maintenance[END_REF][START_REF] Wang | A survey of maintenance policies of deteriorating systems[END_REF], and a more recent review is provided by [START_REF] Zhao | Age replacement models: A summary with new perspectives and methods[END_REF].

In this paper, the age reduction coefficient is used to model the effect of IM [START_REF] Malik | Reliable preventive maintenance scheduling[END_REF]. The age reduction model is based on the concept of system virtual age according to which a system becomes younger whenever it undergoes a PM. After an imperfect PM, the age t of the system is reduced to γ t where γ is the age reduction coefficient (0 ≤ γ ≤ 1). Accordingly, the system becomes as good as new if its age is reset to zero (γ = 0) while it becomes as bad as old if the age reduction coefficient γ = 1.

Imperfect maintenance model

During the break, each component E ij can potentially be subjected to a list of L ij + 1 maintenance levels l, l ∈ {0, 1, • • • , L ij }. The lowest maintenance level l = 0 corresponds to the "Do nothing" case, while the highest level l = L ij corresponds to the perfect replacement or as good as new case. Intermediate values of 0 < l < L ij represent IM actions. According to the age reduction IM model [START_REF] Malik | Reliable preventive maintenance scheduling[END_REF], when a maintenance level l is performed on E ij by a repairperson k, its age is reduced by a factor γ ijl (0 ≤ γ ijl ≤ 1) and it takes t ijkl time units.

In the present work, it is assumed that a maintenance action, whatever its level, can be assigned to any available repairperson regardless of their skills. All repairpersons are qualified to carry out any maintenance level. However, more skilled repairpersons incur higher labor cost while their repair times are shorter.

The following two binary decision variables z ijklm and w km are introduced.

z ijklm =      1, if maintenance level l is performed on E ij by repairperson k during break m, 0, otherwise.
(5)

w km = 1, if repairperson k is hired during break m, 0, otherwise. (6)
A component, say E ij , may be selected during break m for maintenance. Its effective age Y ijm at the end of the break m (i.e. at the beginning of the next mission m) can be expressed as a function of its age X ijm at the start of the break m and the selected maintenance level l to be carried out on it. Using Equation ( 5), the effective age Y ijm is given by:

Y ijm = X ijm   1 - K k=1 Lij l=0 z ijklm   + X ijm K k=1 Lij l=0 γ ijl • z ijklm (7)

Total maintenance cost

Component E ij with initial effective age Y ijm undergoes a maintenance action of level l including the "Do-Nothing" option during break m and also undergoes a number of minimal repairs during the subsequent mission of length U m .

The total expected maintenance cost T M C is then the sum of the cost of the maintenance action during the break and the cost for the expected number of minimal repairs carried out during the subsequent mission of length U m .

T M C = N i=1 Ni j=1 K k=1 Lij l=0 M m=1 c v k • t ijkl • z ijklm + Yijm+Um Yijm c r ij h ij (t)dt (8)
The total fixed cost for hiring the repairpersons denoted by HRC is:

HRC = M m=1 K k=1 c f k • w km (9)
The grand total expected cost C is the sum of the maintenance and hiring costs:

C = T M C + HRC. ( 10
)
Given that there is no risk of repairpersons colliding during the repairs (see assumption 6), the total time T km spent by each repairperson k to carry out their maintenance duties during break m is given by:

T km = N i=1 Ni j=1 Lij l=1 t ijkl • z ijklm (11)
5. Mathematical programming model and solution method

Mathematical programming model

The multimission selective maintenance problem under consideration jointly determines five decisions which minimize the grand total cost C subject to the limited break duration D m and the minimum required reliability during the missions R 0m . This joint multimission selective maintenance and repairperson assignment optimization problem (MMSMRAOP) is formulated as a mixed integer nonlinear model as follows.

MMSMRAOP:

Minimize C = N i=1 Ni j=1 K k=1 Lij l=0 M m=1 c f k • w km + c v k • t ijkl • z ijklm + c r ij Yijm+Um Yijm h ij (t)dt (12)
Subject to:

N i=1   1 - Ni j=1 1 -R c ij U m | Yijm   ≥ R 0m , ∀m (13)
N i=1 Ni j=1 Lij l=0 t ijkl • z ijklm ≤ D m • w km , ∀k, m (14)
N i=1 Ni j=1 Lij l=0 z ijlklm ≥ w km , ∀k, m (15)
K k Lij l=0 z ijklm ≤ 1, ∀i, j, m (16)
Y ijm = X ijm   1 - K k=1 Lij l=1 z ijklm   + X ijm   K k=1 Lij l=0 γ ijl • z ijklm   , ∀i, j, m (17)
X ij,m+1 = Y ijm + U m ∀i, j, m (18)
R c ij U m | Yijm = R ij (Y ijm + U m ) R ij (Y ijm ) , ∀i, j, m (19)
z ijklm , w km ∈ {0, 1}, 0 ≤ R c ij U m | Yijm ≤ 1, ∀i, j, k, l, m (20)
The objective function in Equation ( 12) minimizes the grand total cost. Equations ( 13) and ( 14) are the required minimum reliability during mission m and the break duration constraints respectively. Constraints [START_REF] Khatab | Genetic algorithm for selective maintenance optimization of multi-mission oriented systems[END_REF] ensure that a repairperson is hired before they can carry out any maintenance operation. For each component E ij , Equations ( 16) state that only one maintenance level can be performed if the component is to be maintained by a repairperson within a given break. Constraint [START_REF] Malik | Reliable preventive maintenance scheduling[END_REF] updates the effective age of the components at the end of the break. Equality constraint ( 18) is a recurrence computing the age at the end of each mission. Equality constraint [START_REF] Maaroufi | Optimal selective renewal for systems subject to propoagated failures with global effect and failure isolation phenomena[END_REF] is the conditional reliability of component E ij during mission m.

Solution method

The integrated MMSMRAOP formulated by Eq. ( 12)-( 19), is a mixed integer nonlinear program whose combinatorial complexity is higher than that of the classical SMOP. Indeed, the proposed SMOP covers several missions and incorporates imperfect preventive maintenance model in addition to repairpersons assignment to maintenance tasks. An exhaustive examination of all possible solutions is therefore not realistic due to the computational time limitations.

Meta-heuristic algorithms such as genetic algorithm (GA), Tabu search (TS), simulated annealing (SA), and ant colony optimization (ACO) are well known approaches for their efficiency and effectiveness in searching and finding optimal or a near optimal solutions of complex combinatorial and non-linear programming problems in a reasonable computation time. These heuristics do not require gradient information to determine the next direction of the search and easily adapt well to other problems [START_REF] Talbi | Metaheuristics: from design to implementation[END_REF]. GA is one of the most popular search metaheuristics that has proven to be suitable for a wide range of combinatorial optimization problems [START_REF] Talbi | Metaheuristics: from design to implementation[END_REF]. In this paper, a genetic algorithm (GA) is developed to solve the proposed integrated MMSMRAOP.

Solution encoding is crucial for the efficiency and effectiveness of any metaheuristic and constitutes an essential step in designing metaheuristics [START_REF] Talbi | Metaheuristics: from design to implementation[END_REF]. The design of the GA developed to solve the proposed MMSMRAOP is presented in the following subsections.

Solution representation, initial population and fitness determination

To simplify the solution encoding and representation, the components of the system are now considered as lexicographically numbered starting with components in the first subsystem, followed by those of the second subsystem and so on. Let us denote by T N = N i=1 N i the total number of components in the series-parallel system. A solution in the population is then represented by a pair of two matrices, here after denoted as M LS and RP S, each of which is composed of T N rows by M columns; their respective dimensions are T N × M . The first matrix refers to the maintenance levels selected for the system components, and the second matrix lists the repairpersons selected to perform those selected maintenance levels. Roughly speaking, each gene M LS im (i = 1, • • • , T N , and m = 1, • • • , M ) takes its values from the list {0, • • • , L i } of the available maintenance options and means that the i th component is selected to receive a maintenance level l ∈ {0, • • • , L i }, during the m th scheduled break. A gene RP S ik takes its values from the list {0, • • • , K} and means that repairperson k ∈ {0, • • • , K} is selected to perform the maintenance action M LS im during break m. It should be noted that, if M LS im = 0 (i.e. component i is not selected for maintenance during break m), the corresponding gene is set to RP S im = 0.

The initial population is generated by assigning random values to elements of both matrices M LS and RP S. The initial population size is denoted by N p . For each individual (chromosome) in this population, its corresponding fitness value is computed according to the cost function in Eq. [START_REF] Pandey | Selective maintenance for binary systems under imperfect repair[END_REF]. These fitness values are then used in the selection process. Each individual of the initial population is then characterized by a probability of surviving to the next generation.

Selection, Crossover and Mutation operators

Selection is a random process driven by the fitness values of individuals in the population that determines which solutions make it to the next generation. In this work, the elitism ranking procedure is used. It ranks individuals in the population based on an increasing order of their fitness values. Then, the best N s solutions out of N p are selected as candidates for the next population. The ratio N s /N p is randomly chosen and denoted by ξ s .

The partial mapping crossover strategy is implemented and two blocks of the parents genomes are exchanged to produce two children. A set of N c new solutions resulting from the crossover operation is then injected into the next population. The crossover rate is defined as ξ c = N c /N p .

The mutation operation allows the GA to avoid convergence toward any local optima. Mutation also ensures genetic diversity from one generation to the next. Here, the mutation operation is performed by modifying one gene chosen randomly among the genes of a parent which in turn is chosen randomly from a previous population. As a result, the number of new solutions generated is increased. When applying the mutation operator, the ratio ξ m represents the proportion of new solutions included in the population. To increase the diversity effect, additional solutions are randomly generated and accounted for as candidates for the next population. The proportion of these extra-solutions is given by the ration ξ e . Hence, the number of extra-solutions is ξ e × N p .

All above ratios are chosen such that their sum is equal to 100%, (i.e. ξ s + ξ c + ξ m + ξ e = 100%). The overall optimization search algorithm terminates either when a given number of generations N g is reached or when no solution improvement is achieved after a given number of successive generations N gi . The pseudo-code of the proposed GA-based solution method is presented in Algorithm 1 below.

Algorithm 1 Pseudo-code of the GA-based solution method

1: -Input data: N , N i , M , K, L ij , c f k , c v k , c r ij , t ijkl , X ij1 , β ij , η ij , γijl, D m , U m , R 0m (i = 1, • • • , N ; j = 1, • • • , N i ; k = 1, • • • , K; m = 1, • • • , M ) 2:
-Input GA data: N p , N g , N gi , ξ s , ξ c , ξ e , ξ m 3: -Randomly generate an initial population of N p individuals (i.e, randomly generate N p matrices M LS and SP S as described in Section 5.2.1.) 4: -For each individual, compute the corresponding fitness (i.e., cost function given by Eq. ( 12) ) 5: while Stopping condition is False do 6:

-Generate a new population of N p individuals by performing GA operators on the previous population as described in Section 5.2.2.

7:

-For each individual, compute the corresponding fitness (i.e., cost function value) according to Eq. ( 12)

8:

-Check and update Stopping condition (N g and N gi ) 9: end while 10: -Return best solution (i.e. best joint SM plan and repaircrew assignment)

Numerical examples

In this section, numerical experiments are conducted to demonstrate the validity and the accuracy of the proposed joint selective maintenance and re-pairpersons assignment approach in multimission setting. The first and second experiments consider a small size system to show the impact of considering multiple repairpersons and their composition when modeling the SMP in a multimission setting. The third experiment investigates a moderately large problem instance. The genetic algorithm was implemented in MATLAB R2018a and all experiments were performed on an Intel i7 2.5GHz laptop with 8 GB of RAM running Windows 10 TM . The performance in terms of efficiency and robustness of our heuristic will be measured by evaluating both the standard deviation (σ) and the coefficient of variation (C v ) over ten trials. In Experiment #1, the optimal solution from a full enumeration algorithm (FEA) is compared to the solution of our GA-based heuristic and it shows that the GA finds good quality and robust solutions in reasonable CPU times (CP U t ).

Numerical values of all parameters are arbitrarily chosen but maintain proportionality and satisfy practical maintenance principles. Thus, the lifetimes parameter values are chosen to ensure increasing failure rate of the components to justify PM. Usually, PM takes less time and cost less than CM. Lower level IM takes less time and cost less but has less age reduction effect than a higher level IM. All times and costs data are given in generic time and monetary units (e.g., hours and $).

Experiment #1: illustrative example

A 2-by-2 series-parallel system as depicted in Figure 1 is considered. Lifetimes of component E ij (i, j = 1, 2) are assumed to be governed by a Weibull distribution whose respective scale and shape parameters η ij and β ij are given in Table 1 along with corresponding minimal repair costs c r ij and ages X ij1 at the start of the first break (at the end of the current mission). A list of 5 maintenance levels including the "Do-nothing (DN)" case is available. Their respective age reduction coefficients are displayed in Table 2. Two repairpersons (K = 2) of equal skills and costs are available: c f k = 15 and c v k = 12.5. The maintenance level durations are given in Table 1. The system will operate M missions of equal durations U m = 60 (m = 1, • • • , M ). The scheduled break are also of identical length and set to D m = 10. The system is required to run all missions with a required minimum reliability In what follows, results are derived for a varying number of missions to be operated by the system. To avoid GA parameter over-fitting, preliminary numerical tests were performed to set parameter values while using randomly generated data. The appropriate values of these parameters are presented in Table 3. Recall that the solution search process terminates when N g generations are reached or when no solution improvement is achieved after N gi successive generations. The case of a single repairperson was examined first with the parameter values given above and it showed no solution improvement even for a simple instance with only two consecutive missions. However, when several repairpersons are available, the MMSMOP is able to provide joint maintenance and repairpersons assignment plans. In this case, for each problem instance, ten trials were carried out and the overall results obtained are reported in Tables 4 and5. The best integrated SM and repairpersons assignment plans are reported in Table 4. In this table, the SM plan is given as a list of component E ij followed by the maintenance level l performed on the component in parentheses: E ij (l). The general trend observed from the results in Table 4 is that as the number of missions to be performed increases, more maintenance activities with high levels including component replacement take place.

Table 5 gives the best, the average, and the worst costs, in addition to the average CPU time (CP U t ). Furthermore, this table presents the standard deviations (σ), and the coefficients of variation (C v ) of the ten solutions obtained.

From these results one may observe that, for each problem instance, the standard deviation as well as the the coefficient of variation are low which means that the proposed solution approach is robust. Table 5 also summarizes the performance of the proposed GA-based heuristic and the Full Enumeration Algorithm (FEA) in terms of solution quality and computational time. For the first instance of the problem (M = 2) in Table 5, the proposed heuristic provides the optimal solution after an average time of 8.2 seconds over ten runs. However, the full enumeration (FEA) finds the optimal solution in 13 seconds. The value of the proposed heuristic is highlighted when the number of mission is increased. For M = 3, the FEA takes 7,653 seconds (i.e. more than 2 hours) to find the optimal solution. However, the proposed heuristic finds the exact optimal solution after an average of 26.9 seconds. For M = 4, the proposed GA provides near optimal solutions in 112.3 seconds on average. In contrast, the FEA is unable to provide the optimal solution after 3 hours. From the results shown in Table 5, one may also observe that the values of the standard deviation as well as those of the coefficient of variation are low. Thus, the proposed heuristic is robust in addition of being able to quickly provide solutions. Now that the proposed SMP jointly dealing with maintenance and repairpersons assignment decisions in multicomponent systems performing multiple missions has been shown to make valid decisions, the following numerical experiments will consider multiple repairpersons with different skills to investigate the impact of repair crew composition on maintenance decisions.

Experiments set #2: multiple uniform and mixed skills repairpersons

Two repair crew compositions are considered: one homogeneous and the other heterogeneous. Three types of repairpersons are considered: low skill or trainee (T), skilled or standard (S) and highly skilled or pro (P).

The same system and parameters used in the previous experiments are used except that the system must accomplish M = 5 missions and scheduled breaks. Durations of each maintenance level for each repairperson skill level are listed in Table 6.2 along with their respective fixed and variable costs.

Trainee (T)

Standard (S) Pro (P) One experiment is run with a homogeneous or uniform cohort of three highly skilled (P) repairpersons for different values of the required minimum reliability level R 0m (m = 1, • • • , 5). Calibration/tuning tests are run to determine the best parameters to be used to run the GA as depicted in Table 7. For each problem instance, ten trials are carried out. The results obtained are reported in Tables 8 and9. The results in these tables show that the number of components to be maintained and the number of the repairpersons to hire and utilize and the grand total cost increase as the required minimum reliability increases. For R m0 = 75%, the best SM plan calls for maintenance during all breaks except the first one where no maintenance is needed since the system reliability is sufficient to operate the first mission with the required reliability level. A total of 6 maintenance including 3 component renewals are carried out through out the remaining breaks. Only one repairperson is hired and utilized to perform the selective maintenance plan suggested. When R m0 = 85%, the best solution suggests to hire one additional repairperson to be used from the third break. Furthermore, all components undergo maintenance for a total of 15 maintenance actions including 9 replacements. For each SM and repairpersons assignment plans in Table 8, the resulting best, average, and worst grand total costs obtained over the ten trials are summarized in Table 9 along with their standard deviations (σ) and coefficients of variation (C v ). The low values of σ and C v indicate that the proposed method is robust. Note also that, as expected, the best grand total cost decreases when the required minimum reliability decreases. The next experiment uses a mixed cohort of repairpersons composed of 1 pro (P), 1 standard (S), and 1 trainee (T). The tuned GA parameters are listed in Table 10. All other parameters remain the same. For each problem instance, ten trials were carried out. The results obtained are reported in Tables 11121314. The general trend observed with the uniform cohort also holds in the mixed cohort case. Results from Tables 11-14, show that as the required minimum reliability decreases, fewer repairpersons are hired, leading to a decrease in the grand total cost. For high values of the required minimum reliability level, the highly skilled repairperson is used in combination with the standard repairperson. As the required minimum reliability level decreases, the model resorts to the standard person in combination with the trainee. In fact, when the reliability level R 0m = 85%, the highly skilled and standard repairpersons are more frequently hired and utilized than the trainee. In this case, a total of 15 maintenance actions including 10 replacements are performed. When the reliability level is lowered down to R 0m = 75%, fewer maintenance actions are performed and only the standard repairperson is hired and utilized. When the reliability level is lowered further down to R 0m = 70%, the trainee is hired once during the second break, while the standard repairman is hired and utilized in all the subsequent breaks. For R 0m ∈ {70%, 75%}, the requirements are so low that there is no maintenance performed during the first break.

c f k =15, c v k =12.5 c f k =20, c v k =15 c f k =25, c v k =20 l 1 2
Table 7: Tuned GA parameters: case of uniform cohort (3 Pro) in Experiment #2 Parameters R 0m (%) N p ξ s (%) ξ c (%) ξ m (%) ξ e (%) N g N gi
Table 10: Tuned GA parameters: case of mixed cohort in Experiment #2 Rel. level Parameters R 0m (%) N p ξ s (%) ξ c (%) ξ m (%) ξ e (%) N g N gi
For each problem instance in Tables 11-14, the resulting best, average, and worst grand total costs obtained over the ten trial are reported in Table 15. The low values obtained for σ and C v confirm the conclusion already made about the robustness of the proposed method. Tables 16-19 compare the results obtained for both the uniform and mixed compositions of repair crews. The values in bold in are the best reliability, the best grand total cost, and the best average CPU time obtained. On all problem instances considered, the mixed cohort outperforms the uniform cohort policy in both solutions quality and CPU time. The mixed cohort is more cost-effective than the uniform one. Furthermore, in addition of being cost-effective, the mixed-skills cohort is also able to achieve equivalent or even better reliability than the uniform cohort on a majority of missions. For example, when R 0m = 85%, the maximal achievable system reliability when dealing with uniform cohort is indeed lower than that obtained with the mixed cohort for the second mission. From the results of Tables 16171819, one may also observe that the mixed skills cohort needs more repairpersons than the uniform cohort to perform maintenance actions. This is mainly due to the fact that the trainee and standard repairpersons take more time than the highly skilled repairperson to perform their duties. In this experiment, the coal transportation system in a plant supplying a boiler presented by Liu and Huang [START_REF] Liu | Optimal selective maintenance strategy for multi-state systems under imperfect maintenance[END_REF] is adapted to fit our setting. As shown in Figure 2, the system reliability block diagram is composed of five subsystems: 2 conveyors, 2 feeders and a stacker-reclaimer. The first feeder loads coal from the bin to the first conveyor which subsequently directs it to the stackerreclaimer, whose role is to lift the coal up to the burner level. Then after, the second feeder loads the second conveyor which supplies the burner feeding system of the boiler. The coal transportation system is composed of 14 components E ij each with Weibull distributed lifetimes with shape and scale parameters denoted by β ij and η ij respectively. Values of these lifetimes and other relevant cost and age parameters are given in Table 20 The same list of maintenance options defined in previous experiments is also used here. A mixed cohort of five repairpersons composed of 1 trainee (T), 2 standard (S) and 2 highly-skilled (Pro) repairpersons is available. The fixed cost and variable cost rates used for all maintenance workers in addition to their repair times are provided in Table 21.

When there is no maintenance carried out, the system reliabilities during the ten missions are presented in Table 22. The reliability of the system naturally decreases as the number of missions increases. The system survives the first mission with a probability of 79.36%. However, its chances to survive the last mission plummets to a very low value less than 1% meaning that there is a high risk of not completing the sequence of missions. It is therefore necessary to develop a multimission SM plan. Ten trials are run for each instance and the results obtained are reported in Tables 2425262728. The low values obtained for σ and C v confirm the conclusion already made about the robustness of the proposed method.

c f k =15, c v k =12.5 c f k =20, c v k =15 c f k =25, c v k =20 l 1 2
The results in Tables [START_REF] Liu | On sequence planning for selective maintenance of multi-state systems under stochastic maintenance durations[END_REF][START_REF] Khatab | Condition-based selective maintenance for stochastically degrading multi-component systems under periodic inspection and imperfect maintenance[END_REF][START_REF] Jiang | Robust selective maintenance strategy under imperfect observations: A multi-objective perspective[END_REF][START_REF] Schneider | Evaluation and comparison of alternative fleet-level selective maintenance models[END_REF] show that as the required minimum reliability increases, the number of components to be replaced increases and more repairpersons with increasing skills are needed to carry out all the operations in the break window available. When the reliability level is set to high values, the highly skilled repairperson (P) is frequently used along side the standard repairperson (S) to perform the highest number of operations. Indeed, when the required minimum reliability level is set to R 0m = 85%, 1 pro and the 2 standard S 1 and S 2 maintenance workers are hired and utilized 7, 10 and 8 times respectively. The trainee is however hired and utilized only 2 times. A total of 43 maintenance actions are performed and only 2 are carried out by the trainee, while the remaining actions are distributed between the pro and the standard repairpersons. When the required minimum reliability level drops to 75%, the highly-skilled, the two standard and the trainee repairpersons are still hired but used less often (i.e., 3, 7, 2 and 1 times respectively). In this case, the number of maintenance actions executed drops to 31 among which 1 and 9 actions are, respectively, assigned to the trainee and the pro. Now if the required minimum level is set to 70%, the pro worker is hired only once during the 8 th break to perform 2 maintenance actions among a total of 29. In this case, two standard workers are selected to carry out the remaining 27 maintenance tasks, while the trainee is not hired.

Comparing the CPU times in Table 28 to those previously obtained in Experiments #2 and reported in Tables 16171819, one can observe that these CPU times have increased when the problem instances changed from the case with 4 components, 5 missions, and 3 repairpersons, to the case with 14 components, 10 missions, and 5 maintenance workers.

Conclusions

This paper investigated a new variant of the selective maintenance optimization problem for a multicomponent system where more than one repairperson (or repair-channel) is assumed. The system carries out several missions separated by scheduled breaks during which maintenance of its components takes place. A component can receive different maintenance actions each characterized by a reliability improvement level. In contrast to the current SM models, the proposed new SMOP jointly include five rather than the two classical SM decisions, namely (1) the selection of the components set to be maintained, [START_REF] Cao | A literature review on selective maintenance for multi-unit systems[END_REF] the maintenance levels to be performed on the selected components, (3) the break during which the maintenance actions selected should be performed, (4) the number of repairpersons to hire/utilize within each break, and (5) the maintenance actions to be assigned to each of the hired repairpersons.

To help the maintenance decision maker solve such practical SMP, a novel integrated non-linear programming formulation was proposed. A solution method is proposed based on the genetic algorithm. Several numerical experiments are conducted and discussed to illustrate the model behaviour. The results obtained demonstrate the validity and the accuracy of the proposed SM modelling approach. They also successfully show the benefits of jointly carrying out the assignment of the tasks to repairpersons and the selection of the components to be repaired. Thus, it is demonstrated that the integrated approach of SM and repairpersons assignment problem provides more effective solutions in terms of cost and system reliability. The impact of the composition of the repair crews on the performance achieved is also investigated. The results showed that in general, the mixed cohort performed equally or better than the uniform cohorts, especially when differences, in terms of costs and required maintenance times, are sufficiently large between maintenance workers. The results obtained show that the proposed SMOP model can be used to deal with a moderately large series-parallel systems using the GA-based heuristic proposed. The results obtained also showed the robustness of the proposed solution heuristic.

Future research work would be to develop optimization methods to efficiently solve the new integrated selective maintenance optimization model presented in this paper. The initial analysis carried out indicates that heuristics will probably become necessary if the problem is be solved for large scale multicomponent systems. A standard GA was used in this paper. A natural extension would be to consider Tabu search, Bee and Particle Swarm algorithms, and other neighbourhood search methods. It would also be interesting to consider balancing the workload of the repairpersons as the problem size and number of missions increase. Taking into account the uncertainty related to spare parts and repairpersons availability is also a very interesting and challenging research issue that deserves to be added to the SMP. Finally, the present work could be extended to study the trade-offs between several objectives such as reliability, total costs, workload while considering economic and structural dependencies between com-

  S i K Number of repairpersons k Index of repairpersons, k = 1, . . . , K M Number of missions and breaks m Index of missions and breaks, m = 1, . . . , M U m Duration of mission m D m

Figure 1 :

 1 Figure 1: Series-parallel system structure

  . The system must run a series of M = 10 alternating missions and breaks. Missions and breaks have equal durations: U m = 100 and D m = 12 (m = 1, • • • , M ).

E

Figure 2 :

 2 Figure 2: RBD of the series-parallel system of Experiment #3

Table 1 :

 1 Components lifetimes parameters and maintenance times: case of Experiment #1

	E ij	β ij	η ij X ij1 t ijk1 t ijk2 t ijk3 t ijk4 c r ij
	E 11 1.32 145	60	7	8	9	10	25
	E 12 1.40 160	70	4	5	6	8	15
	E 21 1.35 145	50	6	7	8	9	35
	E 22 1.45 150	65	3	4	5	7	25

Table 2 :

 2 Age reduction coefficients for maintenance levels: case of Experiment #1 R 0m = 80%. When components are not subjected to any maintenance plan, the system reliabilities are evaluated to 75.04%, 67.14%, 61.17%, and 56.35% for missions 1, 2, 3, and 4 respectively. All these reliability values are lower than the required minimum reliability level. To improve the performance of the system, the MMSMOP must then be solved.

	l	0	1	2	3	4
	γ ijl 1 0.40 0.20 0.10 0

Table 3 :

 3 Values of the GA parameters: case of Experiment #1

	# of Missions			Parameters
	M N p ξ 2 200	5	70	20	5	750 150
	3	200	10	80	5	5	750 150
	4	200	10	70	10	10	750 150

s (%) ξ c (%) ξ m (%) ξ e (%) N g N gi

Table 4 :

 4 Best SM and repairpersons assignment results: case of Experiment #1

	M Break m k * Repairperson: SM plan R * m (%) CP Ut (s)
	2	1	1	Rep.1: E12(1), E22(2)	80.16	8.2
		2	2	Rep.1: E12(3)	80.54
				Rep.2: E22(4)
	3	1	1	Rep1: E12(1), E22(2)	80.16	26.9
		2	2	Rep.1: E12(3)	80.30
				Rep.2: E21(4)
		3	2	Rep.1: E22(4)	80.22
				Rep.2: E12(2)
	4	1	1	Rep1: E12(1), E22(2)	80.16	112.3
		2	2	Rep.1: E11(4)	80.16
				Rep.2: E22(3)
		3	2	Rep.1: E12(2)	80.16
				Rep.2: E21(4)
		4	2	Rep.1: E12(3)	80.09
				Rep.2: E22(3)

Table 5 :

 5 Comparative results: GA vs FEA

		FEA			GA
		C *	CP U t	Best	Mean	Worst	σ	C v CP U t
	M		(s)	cost	cost	cost		(%)	(s)
	2	392.25	13	392.25 392.25 392.25 0.0	0.0	8.2
	3	637.23	7653	637.23 637.23 637.23 0.0	0.0	26.9
	4	-	>10,800	869.37 878.34 895.52 9.2 1.04	112.3

Table 6 :

 6 Maintenance durations and costs: case of Experiment #2

Table 8 :

 8 Best results obtained: case of the uniform cohort (3 Pro) in Experiment #2

	R0m(%) Break m k * Repairperson: SM plan R * m (%) CP Ut (s)
	85	1	1	E12(4), E21(2) E22(4)	85.40	186.30
		2	1	E11(4), E12(3), E22(4)	85.06
		3	2	Rep.1: E12(3), E21(4)	85.30
				Rep.2: E22(4)
		4	2	Rep.1: E11(4), E22(4)	85.26
				Rep.2: E12(3)
		5	2	Rep.1: E12(3), E21(4)	85.09
				Rep.2: E22(2)
	80	1	1	E12(1), E22(2)	80.16	15.20
		2	1	E12(4), E22(3)	80.57
		3	1	E11(4), E22(2)	80.02
		4	1	E12(2), E21(4)	80.52
		5	1	E12(3), E22(3)	80.15
	75	1	1	DN	75.04	16.70
		2	1	E21(4), E22(1)	75.07
		3	1	E21(4)	75.26
		4	1	E22(4)	75.10
		5	1	E12(3), E22(2)	75.65
	70	1	1	DN	75.04	3.90
		2	1	E12(3)	72.17
		3	1	E22(3)	72.41
		4	1	E12(3)	70.22
		5	1	E22(3)	70.03

Table 9 :

 9 Performance of the GA: case of uniform cohort (3 Pro) in Experiment #2

	Rm0 (%) Best Cost Mean cost Worst cost	σ	Cv(%)
	85	1839.80	1902.90	1937.00	30.048	1.58
	80	1147.80	1196.90	1250.90	34.302	2.87
	75	842.78	871.22	886.46	14.493	1.66
	70	681.39	684.00	693.34	4.4963	0.66

Table 11 :

 11 Best solutions obtained: mixed cohort with R 0m = 85% in Experiment #2

	Break Repairperson SM plan	R * m (%) CP Ut
	m	T	S	P		(s)
	1			E21(4)	85.02	62.60
				E12(4), E22(2)
	2			E11(4)	85.30
				E12(3), E22(4)
	3			E12(3), E22(4)	85.30
				E21(4)
	4			E11(4)	85.26
				E12(3), E22(4)
	5			E12(3)	85.26
				E21(4), E22(4)

Table 12 :

 12 Best solutions obtained: mixed cohort with R 0m = 80% in Experiment #2

	Break Repairperson SM plan	R * m (%) CP Ut
	m	T	S	P		(s)
	1			E12(3) E22(1)	80.66	6.20
	2			E12(2), E22(4)	80.11
	3			E21(4)	80.09
				E12(3)
	4			E11(4), E22(2)	80.34
	5			E12(4), E22(4)	80.34

Table 13 :

 13 Best solutions obtained: mixed cohort with R 0m = 75% in Experiment #2

	Break Repairperson SM plan	R * m (%) CP Ut
	m	T S	P		(s)
	1		DN	75.04	8.10
	2		E12(3), E22(3)	75.56
	3		E21(4)	75.24
	4		E12(1), E22(3)	75.18
	5		E12(3), E22(1)	75.10

Table 14 :

 14 Best solutions obtained: mixed cohort with R 0m = 70% in Experiment #2

	Break Repairperson SM plan R * m (%) CP Ut
	m	T	S	P		(s)
	1			DN	75.04	3.60
	2			E22(1)	70.36
	3			E21(4)	70.46
	4			E12(3)	70.45
	5			E22(3)	70.74

Table 15 :

 15 Performance of the GA: case of mixed cohort in Experiment #2

	R0m (%) Best Cost Mean cost Worst cost	σ	Cv(%)
	85	1727.20	1820.90	1897.80	57.19	3.14
	80	1068.80	1108.80	1135.60	28.75	2.59
	75	745.31	770.39	797.62	16.33	2.12
	70	593.51	598.49	603.62	4.08	0.68

Table 16 :

 16 Comparative results: uniform versus mixed cohorts (R 0m = 85%)

			Uniform cohort (3P)		Mixed cohort
	m R * m (%) K * C *	CP U t (s)	R * m (%) K *	C *	CP U t (s)
	1	85.40	1	1839.80 186.30	85.02	2(T,S) 1727.20 62.60
	2	85.06	1			85.30	2(T,S)
	3	85.30	2			85.30	2(S,P)
	4	85.26	2			85.20	2(S,P)
	5	85.09	2			85.26	2(S,P)

Table 17 :

 17 Comparative results: uniform versus mixed cohorts (R 0m = 80%)

			Uniform cohort (3P)		Mixed cohort
	m R * m (%) K * C *	CP U t (s)	R * m (%) K *	C *	CP U t (s)
	1	80.16	1	1147.80 15.20	80.66	1(S)	1068.80 6.20
	2	80.57	1			80.11	1(S)
	3	80.02	1			80.09	2(T,S)
	4	80.52	1			80.34	1(P)
	5	80.15	1			80.34	1(S)

Table 18 :

 18 Comparative results: uniform versus mixed cohorts (R 0m = 75%)

			Uniform cohort (3P)		Mixed cohort
	m R * m (%) K * C *	CP U t (s)	R * m (%) K *	C *	CP U t (s)
	1	75.04	1	842.78 16.70	75.04	0	745.31 8.10
	2	75.07	1			75.56	1(S)
	3	75.26	1			75.24	1(S)
	4	75.10	1			75.18	1(P)
	5	75.65	1			75.10	1(S)

Table 19 :

 19 Comparative results: uniform versus mixed cohorts (R 0m = 70%)

			Uniform cohort (3P)		Mixed cohort
	m R * m (%) K * C *	CP U t (s)	R * m (%) K *	C *	CP U t (s)
	1	75.04	1	681.39 3.90	75.04	0	593.51 3.60
	2	72.17	1			70.36	1(T)
	3	72.41	1			70.46	1(S)
	4	70.22	1			70.45	1(S)
	5	70.03	1			70.74	1(S)

Table 20 :

 20 Reliability data for Experiment #3

	Eij β ij	η ij	Xij1 c r ij
	E11 1.5 250	110	15
	E12 2.4 380	150	20
	E13 1.6 280	170	15
	E21 2.6 400	120	25
	E22 1.5 280	180	10
	E31 2.4 340	100	15
	E32 2.5 260	130	30
	E33 2.0 280	170	25
	E41 1.2 260	150	15
	E42 1.4 350	120	30
	E51 2.8 400	180	35
	E52 1.5 350	130	20
	E53 2.4 300	100	30
	E54 2.2 450	150	15

Table 21 :

 21 Repairpersons data for Experiment #3

	Trainee (T)	Standard (S)	Pro (P)

Table 22 :

 22 System reliability with no maintenance.In this experiment, the proposed optimization model (Equations 12-20) is also solved for different values of the required minimum reliability level R 0m (%) ∈ {70, 75, 80, 85} (m = 1, • • • , 10). The best tuned GA parameters are presented in

			3 4	1	2	3	4	1	2	3	4
		E11 7 8 9 10	5.3 6	6.8 7.5	3.5 4	4.5 5
		E12 4 5 6 8	3	3.8 4.5 6	2	2.5 3	4
		E13 6 7 8 9	4.5 5.3 6	6.8	3	3.5 4	4.5
		E21 3 4 5 7	2.3 3	3.8 5.3	1.5 2	2.5 3.5
		E22 7 8 9 10	5.3 6	6.8 7.5	3.5 4	4.5 5
		E31 4 5 6 8	3	3.8 4.5 6	2	2.5 3	4
		E32 6 7 8 9	4.5 5.3 6	6.8	3	3.5 4	4.5
		E33 3 4 5 7	2.3 3	3.8 5.3	1.5 2	2.5 3.5
		E41 7 8 9 10	5.3 6	6.8 7.5	3.5 4	4.5 5
		E42 4 5 6 8	3	3.8 4.5 6	2	2.5 3	4
		E51 6 7 8 9	4.5 5.3 6	6.8	3	3.5 4	4.5
		E52 3 4 5 7	2.3 3	3.8 5.3	1.5 2	2.5 3.5
		E53 7 8 9 10	5.3 6	6.8 7.5	3.5 4	4.5 5
		E54 4 5 6 8	3	3.8 4.5 6	2	2.5 3	4
	m	1	2	3	4	5	6	7	8	9	10
	Rm(%) 79.36 60.63 40.96 25.11 14.45 8.05 4.44 2.47 1.39 0.80

Table 23 .

 23

Table 23 :

 23 Tuned GA parameters used for Experiment #3 Parameters R 0m (%) N p ξ s (%) ξ c (%) ξ m (%) ξ e (%)

	N g	N gi

Table 24 :

 24 Best solution obtained for R 0m = 85% in Experiment #3

	Break		Repairperson	SM plan	R * m (%)	C *
	m	T	S	P
	1		(×1)	E21(2),E32(3)	85.21	5020.22
	2		(× 1)	E21(2), E31(1)	86.11
				(× 1) E12(2), E33(3), E42(2)
	3	(× 1)		E32(3)	85.67
			(× 2)	S1: E11(2), E21(2), E52(2)
				S2: E42(3), E51(3)
	4		(× 2)	S1: E22(4)	85.18
				S2:E33(2), E53(4)
				(× 1) E12(4), E31(3)
	5		(× 2)	S1:E12(2)	85.19
				S2:E21(2), E33(3), E42(3)
	6	(× 1)		E21(4)	85.85
			(× 1)	E42(3), E51(3)
				(× 1) E13(4),E33(4)
	7		(× 2)	S1: E32(4)	85.20
				S2: E33(1)
				(× 1) E12(3), E41(4), E54(2)
	8		(× 2)	S1: E21(3)	86.55
				S2: E31(4)
				(× 1) E42(3), E53(4)
	9		(× 1)	E21(2), E31(1)	85.02
				(× 1) E12(2), E32(4)
	10		(× 2)	S1: E21(2), E42(3)	85.16
				S2: E33(4)
				(× 1) E12(4)

Table 25 :

 25 Best solution obtained for R 0m = 80% in Experiment #3

	Break		Repairperson	SM plan	R * m (%)	C *
	m	T	S	P
	1		(×1)	E21(2)	81.21	4393.72
	2		(× 2)	S1: E32(3), E51(2)	80.62
				S2: E21(3)
	3			(× 1) E12(3), E31(4), E42(3)	81.33
	4	(× 1)		E53(2)	80.32
			(× 2)	S1: E13(3)
				S2: E21(2), E33(3)
	5		(× 1)	E41(4)	80.02
				(× 1) E12(2), E21(1), E32(3)
	6		(× 1)	E21(3), E53(4)	81.63
				(× 1) E31(4), E42(1)
	7	(× 1)		E12(3), E54(2)	81.34
			(× 1)	E32(1)
				(× 1) E21(3), E51(4)
	8		(× 1)	E21(2), E31(3), E42(3)	80.08
	9	(× 1)		E12(2) E21(4)	80.35
				(× 1) E33(4)
	10		(× 1)	E21(2), E54(2)	80.25
				(× 1) E12(3), E32(4)

Table 26 :

 26 Best solution obtained for R 0m = 75% in Experiment #3

	Break		Repairperson	SM plan	R * m (%)	C *
	m	T	S	P
	1			DN	79.36	3741.1
	2		(× 1)	E21(3), E33(3)	75.61
	3			(× 1) E12(2), E31(2), E42(3), E51(2)	78.57
	4		(× 1)	E21(2), E32(4)	76.00
	5	(× 1)		E31(4)	75.19
			(× 2)	S1: E33(2), E53(4)
				S2: E12(1), E31(1), E54(2)
	6		(× 1)	E21(3), E33(2), E42(1)	76.31
	7		(× 2)	S1: E32(4)	75.24
				S2: E12(3), E51(4)
	8		(× 1)	E21(2), E31(2), E42(2)	77.09
	9		(× 1)	E12(1) E21(1), E42(2)	77.36
				(× 1) E33(3), E53(2)
	10			(× 1) E12(3), E21(2), E32(2)	75.01

Table 28 :

 28 Performance of the GA: case of mixed cohort in Experiment #3

	R0m (%) Best Cost Mean cost Worst cost	σ	Cv(%) CP Ut
	85	5020.22	5242.73	5475.28	125.39	2.30	284
	80	4393.72	4528.79	4712.11	103.46	2.20	240
	75	3741.10	3866.90	3968.70	78.80	2.04	123
	70	3438.33	3604.36	3693.20	64.16	1.78	341