Alice Berthier
email: alice.berthier@utt.fr

Alice Yalaoui

Hicham Chehade

Farouk Yalaoui

Lionel Amodeo

Christian Bouillot

An unrelated parallel machines rescheduling problem: an industrial case study

Keywords: Rescheduling, Unrelated parallel machines, Genetic algorithm

This study tackles an unrelated parallel machines rescheduling problem. Sequence and machine dependent setup times and limited resources are taken into consideration. The study focuses on the objective of proposing an efficient and stable rescheduling solution. The resolution approach is explained and illustrated. Different indicators to optimize the rescheduling planning are tested and results are analyzed. The problem is inspired from a concrete case of textile industry.

Introduction

The purpose of this paper is to propose a method to adapt the production scheduling when disruption occurs. The stake is to have a limited impact on the workshop organization and the productivity performances. The workshop considered in this study is composed of unrelated parallel machines with machine and sequence dependent setup times. Two types of limited resources are considered: number of operators available that limits the number of parallel machines able to run at the same time and number of adjusters that limits the number of setup. Each adjuster can do only one setup at a time. The objective is to reschedule a known number of jobs after a disruption. An initial production planning is provided by a scheduling algorithm already implemented with c max minimization. However, perfect production conditions are very unrealistic, disruptions can occur and the initial planning may no longer be up to date. The different disruptions that can occur in this problem are:

-Arrival of a new job -Deletion of a job -Machine breakdown -Lack of human resources (operator or adjuster)

The rescheduling objective is to find the best possible planning to finish all the jobs as soon as possible by keeping stability in the planning initially provided and integrating work in progress information. This is why this study is focused on the objective of maximizing performance (min c max ) while maintaining stability. This problem stems from a real case encountered in textile industry, facing the industry 4.0 revolution. The development of online-business requires more and more flexibility and reactivity, specially with the COVID 19 crisis context as e-business have greatly increased (+100% in 2020). The adaptation needed is reflected in the entire process of the clothing manufacturing industry from the knitting of the fabric to the assembling stage. The rest of the study is organized as follows. Section 2 gives a literature review on this kind of problem. Section 3 provides contribution of this study with the exposition of resolution method. The next section gives results obtained. A conclusion end up the study in section 5.

State of the art

In the literature, only two works tackle problems on the same type of system studied in this paper. Work of Berthier et al. [START_REF] Berthier | Machines group and load balancing: an industrial case[END_REF] deals with a dynamic layout problem in the same industrial environment. The importance of flexibility in such workshop may be encountered at a tactical level. In [START_REF] Berthier | Unrelated parallel machines scheduling with dependent setup times in textile industry[END_REF], the authors propose a complete study of the scheduling problem (MILP and AG) to deal with c max minimization in such systems. In fact, often, real-world scheduling problems are dynamic systems and they need to respond to exogenous events [START_REF] Framinan | Architecture of manufacturing scheduling systems: Literature review and an integrated proposal[END_REF]. Different rescheduling approaches are proposed in the literature. The first one is to use a standard scheduling method with the new data after disruption. This can rich high quality solution on the performance objective. However, solution stability is not guarantee [START_REF] Mohan | A review of dynamic job shop scheduling techniques[END_REF]. On real life production, getting a totally different schedule is very unfavorable to a good workshop organization and management. The second one is to use a proactive scheduling. This is generated by inserting idle time between the pre-scheduling activities, enabling the disruptions to be smoothed out through the system in order to maintain the schedule quality [START_REF] Arnaout | Rescheduling of parallel machines with stochastic processing and setup times[END_REF]. Stochastic approaches are an other way to do it [START_REF] Ruszczyński | Stochastic programming models[END_REF]. The last one is reactive scheduling, commonly referred to as rescheduling. It is a procedure to modify the existing schedule during processing to adapt to changes in a production or operational environment. Kim [START_REF] Kim | Rescheduling of unrelated parallel machines with jobdependent setup times under forecasted machine breakdown[END_REF] recently studies a rescheduling problem of unrelated parallel machines with job-dependent setup times under forecasted machine breakdown. To briefly review some authors that tackle similar problems in literature: [START_REF] Yin | Dynamic rescheduling expert system for hybrid flow shop with random disturbance[END_REF] consider an hybrid flowshop with random disturbance and develop and implement a heuristic on an expert system software. In [START_REF] Olteanu | A dynamic rescheduling algorithm for resource management in large scale dependable distributed systems[END_REF] incoming workflows to be executed on a large-scale distributed system are modeled as directed graphs, where tasks may fail their computations. Heuristics for the problem have been implemented in a specific application simulator. In [START_REF] Sun | An effective approach for the scheduling of refining process with uncertain iterations in steel-making and continuous casting process[END_REF] a steel making continuous caster process is considered with uncertain tasks and a Lagrangian Decomposi-tion method is developed to solve it. [START_REF] Zhang | Flexible job-shop scheduling/rescheduling in dynamic environment: a hybrid mas/aco approach[END_REF] consider a flexible job shop with partial and total rescheduling to deal with rush orders, job cancellations and machine breakdowns. Finally, [START_REF] Baykasoglu | Solving comprehensive dynamic job shop scheduling problem by using a grasp-based approach[END_REF] tackles a dynamic job shop with new orders, rush orders, order cancellations, due date changes and machine breakdowns. Rescheduling is event driven and it is carried out considering different criteria in lexicographic order. On a majority of studies, contrary to scheduling problems, the complexity comes from the combination of two conflicting objectives. Rescheduling problems are taken into consideration: performance and stability measurements. Multiple indicators in the literature are proposed. The definition and use of appropriate performance metrics or quality indicators is crucial. Currently, there are many proposed metrics [START_REF] Jiang | Consistencies and contradictions of performance metrics in multiobjective optimization[END_REF] that can be classified into unary, which assign each nondominated set a number that reflects a certain quality aspect, and binary, which assign a number to a pair of Pareto approximations. However, each industry has its own characteristics that involves specific indicators. For example, stability can be evaluate by the number of jobs processed on different machines in the original and new schedules ( [START_REF] Azizoglu | Parallel-machine rescheduling with machine disruptions[END_REF]). Other approaches defined stability in terms of deviation of job starting times between the original and revised schedules and the difference of job sequences ( [START_REF] Pfeiffer | Stability-oriented evaluation of hybrid rescheduling methods in a job-shop with machine breakdowns[END_REF], [START_REF] Liu | Cost allocation in rescheduling with machine unavailable period[END_REF]).

Resolution method

The contribution of this study is to explore multiple evaluation metrics of the rescheduling solution. First, instances are randomly generated and schedule with the genetic algorithm developed by Berthier et al. [START_REF] Berthier | Unrelated parallel machines scheduling with dependent setup times in textile industry[END_REF]. This algorithm is based on makespan minimization. The entire production has to be finished as soon as possible without any priority under the time horizon. Then, disturbances are generated. The new problem is to reschedule with two conflicting objectives : keep a good performance but also guarantee stability. The flowchart of the resolution approach is given in figure 1. The stability in rescheduling is more complex to evaluate than c max evaluation.

Fig. 1: Flowchart of the resolution approach

This study focuses on the measurement of the number of machine assignment differences between the initial scheduling and the rescheduling planning for each job. To illustrate this, an example is given in the following. This example does not take human resources limitation into consideration. Table 1 gives the processing time p mi of each machine m = (1...M ) for each job i = (1...N ) per unite and the quantity of each job is given in the last row of the table. Table 2 gives the setup times between jobs i = (0...N ) and j = (1...N ) on machine 1, 2 and 3. Figure 2 shows the result of the example instance scheduling. The c max value reached is 29. Now, supposing that M2 becomes unavailable from time 7 to time 37. If jobs 4 and 5 are shifted after the disruption, the c max value is increasing up to 54 but stability measurement is equal to 0: no job is machine changed (figure 3). But if machine job assignment is change as in figure 4, the stability indicator is degraded to 2. However, the performance measure c max is improve to 42. This stability indicator is explored in different combinations and analyzed in order to get the most pertinent and efficient rescheduling planning to the company. These indicator can be compared to a limit parameter. If the limit is crossed, the objective function is penalized. This allows a tolerance and plays up on the performance objective. -Generation of initial instances that will be scheduled using the scheduling genetic algorithm of the previous study with this industrial partner -Generation of disturbances on this solution Table 3 details the different step to generate an instance inspired from [START_REF] Yalaoui | An efficient heuristic approach for parallel machine scheduling with job splitting and sequence-dependent setup times[END_REF]. Each instance is generated with the probability law indicated in table 3. The data generated are the number of jobs to schedule, machine eligibility for each job, processing times and setup times. This instance is scheduled with these initial data. After that, one disturbance for this instance is generated randomly among the four disturbance types. The rescheduling process used in this study is event-driven. Each time a disturbance occurs, rescheduling is triggered. Only one disturbance is generated for each instance. After disturbance, the new availability of each resource has to be calculated. In the example of figure 3, when disturbance occurs at time 7, M1 (resp. M3) is unavailable until time 16 (resp. time 16) to finish job 6 (resp. 1). Taking into consideration M1 and M3 availability and the unavailability of M2, the instance is rescheduled.

Objective functions

The resolution method is based on the same genetic algorithm (GA) used in [START_REF] Berthier | Unrelated parallel machines scheduling with dependent setup times in textile industry[END_REF]. The solution representation uses in this GA is given in figure 5. The representa- In the first row, each job is assigned once and the order will be the sequencing decoding order. The machine assigned for each job is indicated in the second row. To initialize the population of solution, as in numerous papers, a randomized initialization is used.

Job 6 4 3 5 1 2 Machine 1 2 3 2 2 3

Fig. 5: Solution chromosome example

Only the objective function is changed. Different objective functions have been studied in this paper. The first goal of planning rescheduling is to keep the efficiency of the solution, which corresponds to the optimization of the makespan value (c max ). This is still the main objective, as performance is more important than stability for the industrial partner. Thus, the first objective function studied is only to minimize the value of c max . The other objective functions are composed of two elements to optimize: the c max and a metric to guarantee as much as possible the stability between the initial planning and the rescheduling one. The indicator of stability chosen is the number of assignment job/machine differences. A penalty cost is used for stability, to allow a tolerance threshold. A fixed number of disturbance over the total number of jobs can be tolerated. When changes occur right after the rescheduling date, it can disturb the organization already in place: setups are made, workers already have information about what they have to do next. However, when changes occur at the end of the horizon, the schedule can change. So it is not really disturbing. Finally, seven objective functions are compared, each one is linear combination:

1. Minimization of c max 2. Minimization of c max and assignment differences 3. Minimization of c max and penalty if assignment differences are up to 10% of jobs 4. Minimization of c max and penalty if assignment differences are up to 30% of jobs 5. Minimization of c max and penalty if assignment differences are up to 50% of jobs 6. Minimization of c max and penalty if assignment differences occur less than 24 hours after rescheduling date 7. Minimization of c max and penalty if assignment differences occur less than 48 hours after rescheduling date

Results

60 instances with different disturbances have been generated. The average size of instances is 189 jobs to schedule initially. For each instance, all the objective functions have been applied and compared. After disturbance, 111 jobs in average has to be reschedule. The disturbance date is generated in average at 26% of the c max value of the initial schedule optimization. Instances have been grouped, related to the type of disturbance. For each group, results are the average of the solution evaluation. Table 4 shows for each group of instances, the performance objective c max reached when it is the single objective function. Stability value of the solution is the reference to evaluate the other objective functions (Section 3.2). This value is calculated as follow : for each job i, if the initial machine assignment is different than after rescheduling, a i = 1, else a i = 0. The value in table 4 is : N i=1 a i . For performance and stability, standard deviation are given. Depending on the disruption and instance data, performance and stability may be affected in different ways. Distribution of performance (c max ) are given by box plots (Figure 6). The performance value distribution change from one group to another. Averages and minimum are still very closed but maximum and quartiles diverge.

In table 5, the results with the application of the six other objective functions are given as deviation compared to the references. It can be observed that the efficiency objective is not very degraded compared to when it is the only one optimize. However, the stability is considerably increased by up to 97% when job/machine assignment differences are taken into consideration. As previously, for each job i, if the initial machine assignment is different than after rescheduling with the specific objective function, b i = 1, else b i = 0. The stability value in table 5 is calculated by : (

N i=1 b i - N i=1 a i )/ N i=1 a i .
If the stability value is 0%, it means that the number of job assignment changes is the same than Fig. 6: Performance value distribution for each type of disturbance rescheduling optimization when stability are not considered. For instance, for group Arrival of a new job, if in table 5, stability value reaches -100%, it means that 40 jobs has no assignment changes compared to initial scheduling planning. When the value is superior to 0%, it means that more machine assignment changes are generated than in the reference rescheduling solution. It happens when assignment changes penalized only beyond a large percentage of jobs (

Conclusion

The originality of the problem studied in this paper is the specific application to the textile industry. New evaluation methods of stability in rescheduling problem have been tested in order to offer the industrial partner an efficient solution. The continuity of this study is to allow to the company to reschedule the workshop production every time an unpredictable disruption occurs. The method has to be tested on real instances. It is the next step of this study with the industrial partner. Generated several disruptions on same instances is an other perspective. It is also a first step on the road to the 4.0 factory transformation. Different evaluation functions have been tested and evaluated thanks to a random instance generator. With random generated data similar to real material, the company will be able to choose the evaluation scenario most appropriate to rescheduling, knowing the impact on both efficiency and stability. To future perspective, the Pareto front can be determine in order to let the industrial choose the solution between a set of solutions that is the most pertinent.

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig. 2: Initial scheduling of the example

Table 1 :

 1 Processing times p mi

Table 2 :

 2 Setup times s mij

Table 3 :

 3 Instances random generator tion chosen is an array table with two rows and as many columns as scheduled jobs.

Table 5 ,

 5 Arrival of a new job, objective function 4 and 5).

	Objective function	Instance size	1
	Disturbance	Initial number of jobs Performance Stability
	Arrival of a new job	183±101	23 129±22 068 40±27
	Deletion of a job	204±57	21 492±11 212 23±26
	Machine breakdown	181±87	21 607±17 964 43±40
	Lack of human resources	188±59	18 966±11 188 27±27
	Total	189±72	20 786±14 626 32±31

Table 4 :

 4 Results reference on optimization of c max for the different disturbances

Table 5 :

 5 Efficiency and stability results with the different objective function for the different disturbancesIt is a very important prerequisite to have an agile and reactive production plan.

Acknowledgment

This research was supported by ANRT (Association National Recherche Technologie). These acknowledgements also go to our industrial partner. Their participation allows us to work on research topics applied to the current industrial context.


David Lemoine
email: david.lemoine@imt-atlantique.fr

Bruno Castanier
email: bruno.castanier@univ-angers.fr

Keywords: Tactical planning, Predictive maintenance, Faisability criterion, Stochastic Simulation-based approach

Elaborating some Master Scheduling Programs to maximize the customer product demands in respect to the different logistics costs and optimizing maintenance policies are classically addressed independently by two scientific communities whose interests may in some cases diverge. The objective of this paper is to build, based on recent contributions from the field of maintenance optimization, an integrated optimization approach for tactical production plans and maintenance decisions. This goal of integrating as much information as possible into more holistic approaches and more relevant decisions is clearly one of the challenges that can be found in the precepts of the "Industry of the Future". In this paper, we propose, through a modeling of the effects of the degradation of the production system on its production efficiency, to show a benefit of the simultaneous consideration of both concerns. Feasibility criteria are also proposed to ensure the robustness of given tactical plans against the hazards of degradation and failure of the production system.

Tactical planning and predictive maintenance:

towards an integrated model based on ε-reliability

1 Introduction
Tactical Production Planing problems are often modelled by so-called "Lot-Sizing" models which are based on an estimated capacity of the production system. This leads to plans that are at best sub-optimised, at worst unfeasible, and many studies have attempted to refine this estimate by taking into account operational constraints, in particular by integrating scheduling constraints [START_REF] Dauzère-Péres | Integration of lotsizing and scheduling decisions in a job-shop[END_REF]. Nevertheless, few researches take into account the impact of the ageing of this system and the loss of capacity caused by maintenance operations, whether preventive or curative. Moreover, when maintenance is considered, it is only seen as a capacity-consuming activity (when maintenance models are based exclusively on lifetime laws, block or age-based policies can be used to define periodicity constraints that facilitate capacity reservation [START_REF] Pandey | A methodology for joint optimization for maintenance planning, process quality and production scheduling[END_REF]). Planning maintenance therefore amounts to determining the best compromise between minimising disruption to production and guaranteeing reliable performance at the lowest cost. But, if we consider that ageing performance can vary according to the use and the operations carried out, it is possible to differentiate degradation behaviours according to the items whose production is planned. Thus, we see an interdependence between the two types of planning: maintenance decisions are made according to the planned production as planned production is a function of the maintenance decisions. Thus, just like the production planning process, maintenance planning is also a performance lever for the production system and it makes sense to consider them into an integrated process.

In the search for performance and the associated instrumentation of production tools, conditional type approaches for which the maintenance decision is defined according to a health state estimated with the information collected on the system [START_REF] Jafari | Joint optimal lot sizing and preventive maintenance policy for a production facility subject to condition monitoring[END_REF], are based on state indicators (typically the performance level of the system) or prognosis indicators such as the Residual Lifetime (RUL). These indicators allow to define optimality structures for maintenance decision policies. The use of the Residual Lifetime clearly serves as a basis for predictive maintenance whose importance is growing in the context of Industry 4.0. That is why, from [START_REF] Castanier | A Preliminary Integrated Model for Optimizing Tactical Production Planning and Condition-Based Maintenance[END_REF], we propose a new methodology based on this indicator (more precisely on a reliability threshold estimated over a production period instead of a limit degradation level).

Problem modeling

In order to provide feasible production plans, tactical production planning requires a good estimate of production capacities. However, these estimates may vary throughout the planning horizon, especially due to degradation of production resources that may lead to decreases in machine efficiency but also to a range of breakdowns. Thus, in a first part we give a general lot sizing model with finite capacities which allows to obtain different production plans (end items as well as components). It is known to be the classical model which extends material requirements planning (MRP) concept by taking into account production capacities [START_REF] Comelli | A review of tactical planning models[END_REF]. Then we will detail in a second part the way we model the loss of performance related to the degradation of machines.

The tactical planning model

The dynamic multi-level capacitated lot-sizing problem (MLCLSP) was introduced in [START_REF] Billington | Mathematical Programming Approaches to Capacity-Constrained MRP Systems: Review, Formulation and Problem Reduction[END_REF]. Here, we detail the formulation given by [START_REF] Buschkühl | Dynamic capacitated lotsizing problems: a classification and review of solution approaches[END_REF]. Parameters and decision variables of the model are summarized in Table 1.

(s i X i,t + h i I i,t ) (1) s.t.
I i,t = I i,t-1 + Q i,t-li - j∈Si a ij Q j,t -D i,t ∀(i, t) ∈ N × T (2) i∈N (p i Q i,t + b i X i,t ) ≤ C t ∀(k, t) ∈ K × T (3)
Q i,t ≤ C t X i,t ∀(i, t) ∈ N × T (4)
Q i,t , I i,t ≥ 0 ∀(i, t) ∈ N × T (5)
X i,t ∈ {0, 1} ∀(i, t) ∈ N × T (6)
(1) is the logistic costs we seek to minimize. ( 2) is the inventory balance constraint and (3) is the capacity constraint concerning production and setup time.

(4) ensures that production of item i takes place at period t, only if the resource is setup for this item. ( 5) and ( 6) are positivity and integrity constraints.

In such a model, it can be seen that p i , the capacity consumed to produce one item i, is independent of the performance level of production system. The aim of the next section is to explain the way we use to tackle fill this gap.

Modeling of the performance degradation for production system

Here, we consider that the production system degrades as it is used: this degradation results in a loss of real capacity by increasing operating times. We denote

x τ the effective performance rate at workload level τ since the last preventive maintenance, which we assume to be measurable for all τ . We also assume that the degradation of this rate is a continuous, workload-dependent, increasing stochastic process and that its expectation is a function of the load induced by the manufactured product (its unit processing time). More formally, we note {X τ , τ > 0} the stochastic process modeling the evolution of the degradation of the performance rate of the production system where τ represents the workload already processed by the production system. We assume that at the new state (X 0 ) the rate is equal to 1 and that the process is strictly decreasing until 0 (state in which the machine is so degraded that it is not able to produce anymore). We suppose that on an interval [τ, τ + ∆τ ], the loss of performance is modeled by a random variable which follows a Gamma distribution Γ (α∆τ, β) where α and β were previously estimated. This modelling in the form of a homogeneous gamma process is classical in maintenance [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF]. However, as the operating times are particularly low with respect to the planning horizon, we assume in our modeling that the performance rate is constant during the manufacturing of a product unit and that its evolution only occurs at the end of the production of this unit.

As previously mentioned, we also take into account the increase in the duration of the operating times as a function of the level of the performance rate of the system. More precisely, if x τ represents the performance rate of the production system at the load level τ , we note p i (x τ ) the unit production capacity of an item i associated with this rate and we will thus have, for a non-maintained system, p i (x τ ) the production capacity of an item i associated with this rate :

-x τ +pi(xτ ) < x τ , -p i x τ +pi(xτ ) ≥ p i (x τ ).
We also take into account the failures (breakdowns) of the production system, failures that we always assume to be cataleptic and requiring corrective maintenance (that takes C Corr units of capacity) to get the system back on line. We assume that these failures are distributed according to a failure rate λ (x) which depends on the performance rate of the system. Moreover, we assume that this rate decreases in x. Thus, the lower the performance rate of the system, the higher the probability of failures. Since we have made the assumption that the performance rate is constant during the manufacturing of an item, it will be the same for the failure rate and if the manufacturing of an item i starts at workload level τ , then the expected number of failures during its manufacturing will be λ (x τ ) p i (x τ ).

Fig. 1 illustrates the modeling of the performance rate for a given production period, considering the system as new at the beginning of the period. It can be seen from this model that the integration of the system degradation implies taking into account the sequencing of the production lots.

Workload τ

Performance rate 1 0 p 1 (x 0 )p 1 (x 1 ) p 1 (x 2 ) p 2 (x 3 ) + C Corr p 2 (x 4 ) p 3 (x 5 ) p 3 (x 6 )

CCorr

A tactical planning period

• x0 x1 • x2 • x3 • x4 • x5 • x6 • x7 • Item 1 Item 2 Item 3
Real performance rate modeled performance rate machine breakdown Fig. 1. Example of a scenario of the realization of a production plan according to the evolution of the performance degradation

The performance indicator: the ε-reliability

In this section, we define a feasibility indicator that ensures the robustness of the production plan with a certain level of probability ε. Based on the ε-feasible indicator provided by [START_REF] Castanier | A Preliminary Integrated Model for Optimizing Tactical Production Planning and Condition-Based Maintenance[END_REF], we recall the concept of ε-feasibility for one period and we define the ε-reliability indicator on which we base our proposal.

Definition 1 (ε-feasible Period). A period t ∈ [1, T ] is said to be ε-feasible if and only if the probability of not exceeding the production capacity (taking into account all the capacity consumption, including maintenance) is greater than or equal to ε.

Using this indicator, [START_REF] Castanier | A Preliminary Integrated Model for Optimizing Tactical Production Planning and Condition-Based Maintenance[END_REF] qualify a production plan as ε-feasible if all its periods are T √ ε-feasible. This definition allows them to develop an algorithmic solution based on the simulation of the entire production plan. However, the obligation to have an identical probability of success at each period induces a rigidity that we propose to remove thanks to the ε-reliability Definition 2 (ε-reliable production plan). A production plan is called εreliable if and only if each production period t ∈ [1, T ] is ε t -feasible and

T t=1 ε t ≥ ε.
Indeed, by using such a definition, we allow more flexibility on the distribution of the robustness over the periods of the planning horizon while ensuring the same overall level of robustness of the production plan.

Our approach

Conditional maintenance approaches allow maintenance to be organised as best as possible, essentially on the basis of the degradation state of the system itself. A predictive strategy can be seen as conditional maintenance for which the decision variable is usually a residual life. This residual lifetime is characterised by a conditional reliability that is a function of this same state. In [START_REF] Castanier | A Preliminary Integrated Model for Optimizing Tactical Production Planning and Condition-Based Maintenance[END_REF], authors studied a conditional scheme for which the decision threshold was optimised to guarantee the best compromise between performance rate and available capacity. The maintenance decision can then be made at any time during production, regardless of organisational constraints. Here, we propose to develop a predictive maintenance policy allowing a restoration of the production system according to its state of performance while taking into account the customer's requests translated in the tactical plan. We will consider that this maintenance can be done at most once per period and can be carried out either at the beginning or at the end of the production period, thus joining a number of operational practices.

Our maintenance policy is based on the ε-reliability indicator. Therefore, at the beginning of each period t, this allows us to adjust its ε t -feasibility according to the previous periods according to the allocation principle defined below.

Definition 3 (Feasibility allocation).

For each period t ∈ [1, T ], we define

ε t = T -(t-1) √ ε t j=1 F R j (x j-1 ) (7)
where F R j (x j ) is the probability of achieving the production plan at period j knowing the performance level x j-1 of the production system at the end of manufacturing in precedent period.

Thus, we propose to define a predictive maintenance strategy based on the estimation of the ability of the system to achieve its production plan at each period. The maintenance decision is no longer defined on the basis of lifetime or calendar parameters but is a function of the current state of the system estimated after the production of each item. Even if its implementation in an operational context is more difficult than traditional approaches, these predictive approaches, which are similar to conditional maintenance under certain assumption, offer real potential for improving performance, particularly in economic point of view [START_REF] Castanier | A Sequential Condition-Based Repair/Replacement Policy with Non Periodic Inspections for a System subject to Continuous Wear[END_REF].

Then, our methodology can be summarized into five iterative steps :

Step 1 : The datas are readen and the lot-sizing model is initialized with the estimated production capacities C t for each period t. The effective performance rate for the production system at the beginning is x 0 .

Step 2 : Thanks to the lot-sizing model, an optimal production plan is elaborated (by using, for instance, a mixed integer linear program or an optimization method), t ← 1.

Step 3 : 1. ε t is computed thanks to the feasibility-allocation formula [START_REF] Buschkühl | Dynamic capacitated lotsizing problems: a classification and review of solution approaches[END_REF] 2. The production plan is simulated at period t according to the performance rate x t-1 (the performance rate at the beginning of period t).

The probability of achieving the production plan at period t F R t (x t-1 ) is then computed. 3. if F R t (x t-1 ) ≥ ε t then go to Step 4 else * if there is no preventive maintenance at the beginning of period t, schedule one: x t-1 = 1. Go back to 2. * if there is a preventive maintenance at the beginning of period t, C t is the decreased and the lot-sizing model is updated. Got to Step 2.

Step 4 : x t is computed as the the median of all observed performance rates at the end of period t thank to the simulation model. t ← t + 1. Go to Step 5 Step 5 : If t = T + 1 then Stop (a feasible plan has been elaborated) else go to Step 3.

At each period t, in the simulation process two kinds of maintenance are considered:

-If a failure occurs, minimal corrective maintenance (with no effect on the performance rate of the production system) is performed (in this case a capacity consumption corresponding to the maintenance time) and a corrective maintenance cost is added. -If a preventive maintenance occurs, then a capacity consumption corresponding to the maintenance preventive is taken into account and a preventive maintenance cost is added.

A software tool

To improve the applicability of our approach in an industrial context or at least to get their feeling, the algorithm is directly implemented in a computer program with an ergonomic and user-friendly interface. The final results are numerically and graphically presented. This interface is divided into four modules:

1. a data management module for loading, saving or importing data from an external database; 4. a graphical interface for the specification of the problem with all the parameters (capacities, costs, degradation, etc.), the presentation of the intermedi-ate results obtained during the optimisation procedure and the final solution through different graphs presenting the associated indicators (Fig. 3).

Fig. 3. The GUI of the planning tool

Conclusion and further research

In this paper, a new methodology for integrating tactical planning and maintenance has been presented. Using a model of the performance level of the production system thanks to degradation or ageing of the production system on the capacity consumed for manufacturing, and based on a new indicator called ε-reliability, an original maintenance strategy, based on considerations developed in a predictive maintenance context, has been proposed. A Feasibility-Allocation process has been designed in order to ensure a certain robustness level for elaborated tactical plans. Thus, our model extends classical approaches combining production and maintenance essentially focused on the unavailability times generated by preventive maintenance, with a stronger contribution on the interaction between the system degradation and its productivity. This has been integrated into a demonstration software platform for industrial purposes.

The model presented opens up many areas of development for future work in both the modelling and optimisation fields. One of the first points that would be interesting to address is the possibility of modulating the maintenance decision rules in order to offer the possibility of an optimised positioning of preventive maintenance according to opportunities linked, for example, to changes in series time or others. As we pointed out that scheduling has an impact on the performance level of the production system, integrating sequencing decision on the manufacturing of each production lot could be an interesting area of improvement. Another perspective of research could be to develop a model for which the preventive maintenance decision is not restricted to a total recovery of performance but to define the effectiveness of the operation to be implemented according to the capacities not consumed by the production plans, the effectiveness being able to be correlated with an effective maintenance time. On the field of optimisation and more precisely on the criteria to be taken into account, even if the feasibility partially captures the volatility of the global planning problem, it could be interesting to extend the average maintenance criteria.

Fig. 2

 2 Fig. 2 sketches our methodology.

Fig. 2 .

 2 Fig. 2. Optimization scheme for the considered predictive maintenance policy

Table 1 .

 1 Notations of the MLCLSP

	Index sets
	N	: Set of items, N = {1, • • • , N }
	T	: Set of periods, T = {1, • • • , T }
	Si	: Set of direct successors of item i in the Bill Of Material (BOM)
	Parameters
	aij	: Quantity of item i directly required to produce one unit of item j
	Ct	: Available capacity of production system at period t
	Di,t	: External demand of item i at period t
	hi	: Unitary holding cost of item i per unit and period
	si	: Setup cost of item i
	pi	: Production time per unit of item i
	bi	: Setup time of item i
	li	: planned lead time for item i
	Variables
	Xi,t	: Binary setup variable of item i at period t
	Qi,t	: Production quantity of item i at period t
	Ii,t	: Inventory of item i at the end of period t
	Thus, we can write the model as follows:
	Minimize
		t∈T i∈N