Condition based Maintenance applied to Reduce Unavailability of Machines in Flexible Job Shop Scheduling Problem

7th IFAC Conference on Manufacturing Modelling, Management, and Control International Federation of Automatic Control June 19-21, 2013. Saint Petersburg, Russia

Condition based Maintenance applied to Reduce Unavailability of Machines in Flexible Job Shop Scheduling Problem Yahong ZHENG. Khaled MESGHOUNI. Simon COLLART DUTILLEUL LAGIS, UMR CNRS 8219, Ecole Centrale de Lille, BP 48, 9LOOHQHXYH G¶Ascq, France. (Tel:+33 3 20 33 54 85, e-mail: [email protected], [email protected], simon.collart [email protected]) Abstract: Maintaining the reliability of a system is one of the most critical and challenging tasks for manufacturing system during production. In this paper, we suggest to use condition based maintenance (CBM) in order to reduce unavailability of machines. CBM is a kind of preventive maintenance (PM), based on condition monitoring. This paper focuses on solving the flexible job shop scheduling problem with preventive maintenance (FJSPPM), adding PM into FJSP. Application of CBM makes the scheduling problem dynamic, as the moment of carrying on maintenance is not deterministic. In order to solve this dynamic FJSPPM (DFJSPPM), firstly, we apply an integrated genetic algorithm (IGA) for FJSP, and then, we use an inserting algorithm (IA) to add PM into solutions of FJSP found by (IGA). To demonstrate efficiency of IA for treating maintenance scheduling, we compare it with our simultaneous scheduling algorithm (SSA) and with neighborhood search called hGA in numerical examples of literature. A new better solution for an instance in benchmark of FJSP is obtained with our approach. Keywords: Diagnosis and maintenance of manufacturing systems, production planning and scheduling, Integrated Genetic Algorithm, Inserting Algorithm

1. INTRODUCTION Job shop is a common and basic type of manufacturing, especially to the production with multi type of processing in contemporary era. Job shop scheduling problem (JSSP) has become important research topic since 1960s. It was demonstrated NP-complete when P • (m is the number of machines) (Garey et al., 1976). Flexible job shop scheduling problem (FJSP) is an extension and generalization of JSSP, it is more complicated. The FJSP extends job shop scheduling problem (JSSP) with alternative machines routings, by assuming that, each operation can be performed by several machines (Nasr et al., 1990, Najid et al. 1991, Hussain et al., 1998, Thomalla., 2001). This flexibility can be total or partial. Both were studied in (Kacem et al., 2002) which developed two new approaches: the approach by localization and an evolutionary approach controlled by the assignment model. Most literature working on job shop scheduling problem (JSSP) do not consider unavailability of machines. However, in most practical manufacturing environments, there are always some uncertainties inducing unavailability of machines, such as absence of staff, operational errors of staff, machine breakdown, and so on. The reasons of staff and other subjective aspects are usually not considered as optimization section. Meanwhile, the objective aspects, like PDFKLQHV¶ SHUIRUPDQFHV DUH IRFXVHG PXFK LQ UHVHDUFK 0DFKLQH¶V SHUIRUPDQFH GHSHQGV RQ LWV SDUDPHWHUV ZKLFK DUH fixed at birth and cannot be improved. In contrary, it decreases with time in its lifetime. Besides the final scrap, breakdown cannot be avoided, especially in term of modern 978-3-902823-35-9/2013 © IFAC

machine, which is always with a sophisticated structure. In most cases, breakdown is caused by very small errors, imprecision between the two joint parts, or attrition of some parts, etc. Therefore, ensuring machines performing in good condition attracts many researchers and practitioners. Among them, preventive maintenance (PM) is widely studied. PM is an effective method to increase availability of machines. However, there has not been much research on JSSP considering maintenance activities. We have referred several researchers working on JSSP integrated with preventive maintenance (PM) (Wang et al., 2010, Gao et al., 2006). Other similar works study influence of maintenance activity in some other production modes, like parallel machines (Liao et al., 2006, Berrichi et al., 2010). For FJSP with machine maintenance in consideration, a hybrid genetic algorithm was proposed by (Gao et al., 2006) to solve FJSP with non-fixed availability constraints, which used partial representation method to represent only a part of a candidate solution and leaves the lasting part decided by the heuristic method for strengthen the inheritability of this candidate solution. (Moradi et al., 2010) investigated integrated FJSP with preventive maintenance activities under multi-objective optimization approaches, in this study four multi-objective optimization methods are compared to find the Paretooptimal solution. In our work, we suggest to apply condition based maintenance (CBM). CBM is the maintenance policy in which preventive maintenance is triggered after identifying a symptom of impending failure with the aid of conditionmonitoring techniques. Its attribute of real-time makes FJSP dynamic. For dynamic FJSPPM (DFJSPPM), we propose an

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inserting algorithm (IA) to add PM to FJSP. We will compare IA with our Integrated GA with SSA and also to compare it with hGA used in literature (Gao et al., 2006). This methods integrate maintenance tasks in the list of jobs to be scheduled (3 jobs and one maintenance task, we consider it like 4 jobs) The remaining contents of this paper are organized as follows: in section II, CBM is introduced and the concept of DFJSPPM is interpreted; mathematical modeling of problem is described in section III; solution approach is detailed in section IV; in section V we present a numerical example to demonstrate effects of proposed approach; a conclusion and future directions are given at the end of this paper. 2. CONDITION BASED MAINTENANCE Although PM has been studied in FJSP to increase availability of machines and reliability of the entire production system, the maintenance activity interval is mainly given as a constant or the maintenance must be executed in a predicted interval. Moreover, the theoretical basis of the maintenance interval is not specified, i.e., PM is only an abstract activity. The type of PM we use in FJSP is Condition Based Maintenance (CBM). In the groundbreaking report of (Neal et al. 1979), differences in maintenance strategies (e.g., breakdown, planned, etc.) were illustrated and CBM was suggested. Since 1990s condition monitoring techniques has been increasingly used to aid planning replacement (Aven., 1996) and preventive maintenance (Christer et al., 1997). CBM has become widely accepted as one of the drivers to reduce maintenance costs and increase machines availability (Al-Najjar., 1991, Bengtsson., 2002). The main idea of CBM is executing maintenance activity in the period of potential machine failure, in order to avoid malfunction of machines. CBM is based on using real-time data to prioritize and optimize maintenance resources. Observing the state of the system is known as condition monitoring. Such a system will determine the equipment's health, and act only when maintenance is actually necessary. So, how to define the health of equipment? If some types of them can easily be observed by measuring simple values as vibration, temperature or pressure, it is not simple to convert it into usable knowledge. Many failure modes are not correlated to age, but most of them give a warning sign that they are occurring or about to happen. If proof can be found that process is in the final stage of failure, it may be possible to take action to prevent it from failing completely or to avoid the consequences. For more details about CBM, can refer to (Sethiya., 2006). The PíF curve, which describes the deterioration process of machine, showed in (Fig. 1), where A: the moment when failure began; P: the moment when hidden failure is detected; F: the moment when malfunction occurs; 7 LQWHUYDO RI 3 í ) WKH SURFHVV RI WKH KLGden failure developing to malfunction. state

A P

Tt F 0 T

Fig. 1. P-F curve

t

The P-F interval is the interval between occurrence of a potential failure and its degeneration into a functional failure. It is also known as the warning period. If we want to detect the potential failure before it becomes a functional failure, the interval between monitoring checks must be less than the P-F interval. To find the right moment to execute maintenance is the essence of CBM Advantage of CBM is that it transforms state of maintenance from passiveness to activeness. It gets rid of the disadvantage of the cyclic maintenance (low or too maintenance). CBM is based on the real-time monitoring of the state of the production system, the maintenance is dynamic, without static periods or intervals. Therefore, different to traditional FJSPPM, our problem is a dynamic FJSPPM (DFJSPPM). Before we detected some hidden failures (point P), we do not consider maintenance tasks in scheduling problem. We execute manufacturing according to a scheduling solution of normal FJSP, which is called the preschedule. When arriving point P, we have to consider adding maintenance task. We need now to find a solution for a FJSPPM. 3. PROBLEM DESCRIPTION The first step of solving DFJSPPM is to specify the time windows of PM tasks. This step transforms the problem into FJSPPM. We use the mathematical description proposed by (Gao et al 2006). This description present the problem as follows: n jobs are to be scheduled on m machines. Each job i represents a number of ni non preemptable ordered operations. Execution of each operation k of job i (noted as Oik) requires one machine selected from a set of available machines, called Aik, and will occupy that machine tikj time units until the operation is completed. There are Lj maintenance tasks which have to be processed on machine j during the planning horizon. The maintenance task corresponds to a predefined time window T, within which the starting time of the maintenance task can be moved. We assume that: (1) All the ni ordered operations of the ith job have to be processed; (2) Activity of each operation Oik requires one machine selected from a set of available machines, called Aik; (3) Every machine processes only one activity (job operation or maintenance task) at a time; (4) Set-up and unloading times for operations are machine independent and are included in the processing time of operation; (5) Maintenance task is considered as a special job, the duration of maintenance is included in makespan of jobs; (6) Machine is restored to good working condition after maintenance. (7) Jobs are processed before their due dates. Therefore, there is no penalty of tardiness and storage cost. Production cost concerns only with processing cost. The different notations and parameters used in this paper are: i, h: index of jobs, i, h = 1, 2, · · · ,n; j: index of machines, j = 1, 2, · · · ,m; k, g: index of operation sequence, k, g =1, 2,· · · , ni; l: index of maintenance tasks, l = 1, 2, · · · ,Lj . n: total number of jobs; m: total number of machines;

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ni: total number of operations of Job i; Lj: total number of maintenance tasks on Machine j; Oik: the kth operation of Job i; PMjl: lth potential maintenance on Machine j; p: maximum number of maintenances needed on one machine; Aik: set of available machines for Operation Oik; tikj: processing time of Oik on Machine j; CMj: unit processing cost on machine j; pjl: duration of Maintenance task PMjl; [tEjl , tLjl ]: time window associated with maintenance, where tEjl is the earliest starting time, and tLjl is the latest starting time of Maintenance task PMjl -1, if Machine j is selected for operation oik ,½ xikj ® ¾ ¿ ¯0, otherwise; cik: completion time of the Operation Oik; yjl: completion time of Maintenance task PMjl. The objective function is to optimize the makespan of jobs with PM tasks. the model is given as follows: min F1 max§¨ max cini , max y j L ·¸ 1d j d m j ¹ © 1di dn

>c s.t. ›>c >y c ›> c hg

ik

jl

@

cik t hgj xhgj xikj t 0 ,

ik

@

Table 1. Operating processing time

J1

(1)

i, j , h, g , j

@

p jl xijk t 0

y jl tikj xijk t 0 ,

In our integrated GA, the two sub problems, machine assignment and sequence scheduling, are integrated in one GA procedure. The overall structure of integrated GA is described as follows: (1) Coding: the genes of chromosome represents machine assignment of operations. The chromosome is represented by a matrix where each row is an ordered series of the operating sequence of this job, each element of this row contain the machine which performs this operation. We illustrate this representation by the following example. We consider three jobs and five machines. The operating sequences of these jobs and their operating processing time are presented on Table 1.

J2

chg t ikj xhgj xikj t 0 ,

ik

4.1 Integrated Genetic Algorithm for FJSP

i, j , ( j, l

@

(2) J3 (3)

¦ xikj 1, j • Aik , i, k

(4)

t Ejl d y jl d t Ljk , l , j

(5)

xikj • >0,1@, i, k

(6)

M1

M2

M3

M4

M5

O1,1

1

9

3

7

5

O1,2 O1,3

3 6

5 7

2 1

6 4

4 3

O2,1 O2,2

1 2

4 8

5 4

3 9

8 3

O2,3

9

5

1

2

4

O3,1 O3,2

1 5

5 9

9 2

3 4

2 3

One possible chromosome is indicated as follows:

Table 2. Machine assignment to operations Jobs/Operation

cik t 0, i, k

(7)

y jl t 0, j, l

(8)

Objective function (1) is minimization of makespan, which is the max value between latest completion time of job and that of maintenance. Inequality (2) ensures no-overlapping constraints between operations on the same machine. Inequality (3) ensures no-overlapping constraints between preventive maintenance tasks and operations on the same machine. Equation (4) states that only one machine should be selected from the set of available machines for each operation. Inequality (5) states that the preventive maintenance tasks have to be executed within their time windows. (7) and (8) ensure the feasibility of the two variables. 4. SOLUTION APPROACH To solve the FJSPPM, we proceed in two stages; the first one is to find a schedule (preschedule) for the FJSP using integrated genetic algorithms. The second step is to add the PM tasks in this preschedule using an insertion algorithm (IA).

Oi,1

Oi,2

Oi,3

J1

3

3

4

J2

1

5

2

J3

1

5

(2) Initial population: It is initialized that machine for each operation is randomly selected from the set of available machines, Aij, to guarantee its feasibility. (3) Offspring generation: Crossover and mutation are to realize variety of population and to avoid local optimal solution. In crossover procedure, we use entire row or column crossover, to keep feasibility of individuals. Firstly, randomly select two individuals as one pair of parents. Secondly, randomly select the manner of crossover, either row crossover or column crossover. Finally, choose randomly the cut position of row or column and carry on crossover to generate two new individuals. In mutation procedure, individual and its mutation position is selected randomly and within the mutation probability. In order to ensure feasibility of solutions, the mutated value is also in the set of available machines. (4) Fitness evaluation: The evaluation function is the link between the genetic algorithms and the problem to be solved. In our case, the makespan is used as fitness value, which will be obtained in the decoding procedure presented on (Fig.2).

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Operations on the same machine are sequenced according to the position of occurrence. Procedure: priority-based Decoding Input: Chromosome X Output: an active schedule Begin for i=1 to m Q18X(:,1) //machines used for the first operation of each job Schedule the first operation of each job, in the case where more than one operation using the same machines, according to operations priority, assign operations on the machine. for i=1 to m for k=1 to n Q18X(:,k) //machines used for the kth operation of each job Schedule the rest 2 to n operations of each job, initialize the stating time of each operation as the end time of its preceding operation; if the initialized starting time is earlier than the end time of other operations proceeded on the same machine, set its starting as this end time. Out the active schedule end

(1). All PM tasks are initialized to start at the latest moment in time windows. (2). When one operation meets a PM, compact PM task to the left as possible, and then schedule the operation after PM. 5. NUMERICAL EXPERIMENTS

Fig. 2. Decoding procedure (5) Sequence scheduling: Firstly, according to processing sequence to schedule all operations of all jobs, e.g., the first operation of each job will be firstly scheduled, and then the second operation, and so on, until all operations are scheduled. For scheduling operations on the same machine, there are two traditional methods. One is scheduling operations according to the sequence of number of jobs (Mesghouni et al., 2004, Gen et al., 2000), e.g., in the example in Table 2, operations O21 and O31 are all assigned to machine 1, O21 will be scheduled before O31. Decoding procedure is described as in (Fig. 3). The other one is giving a vector of priority of operations, as a part of chromosome, and then sequencing operations on the basis of the priority (Gao et al., 2006). In our work, we try to find a promising sequencing method for operations on the same machine. In section 5.1, we present a comparison between four scheduling sequence methods: (a) order of number of jobs; (b) reverse order of number of jobs; (c) random order; (d) order of available time of jobs. (6) Selection: In our work, to keep good individuals in population pool, they will join in competition with offspring. 2N individuals are generated until this step. N (Population size) individuals will be selected from 2N ones by selection method. We will use tournament, which was proposed by (Goldberg et al., 1992). (7) Stop criteria: the program stops when fixed number of generation is reached. The best chromosome, together with corresponding schedule, is outputted as results. Otherwise, the program iterates steps (3)-(6). 4.2 Solution for FJSPPM Based on the approach for FJSP, we use IA to add PM in the program. In IA, PM tasks will be inserted in the idle intervals of jobs after all jobs have been scheduled. We aim to make full use of the idle intervals, which always exist and are unavoidable in scheduling. IA is described as follows: (1) Find effective idle time intervals from left to right on the scheduling sequence of each machine, in the range of time windows of PM task. (2) If the maximal effective idle time interval cannot satisfy the duration time of the PM task, perform PM task at the beginning of the maximal idle time interval, and then delay all its posterior sequence of operations. We will compare our approach with SSA proposed in (Gao et al., 2006), which is as follows:

To verify effectiveness of our approach for FJSPPM, we use the benchmark in (Gao et al., 2006), in which the example of FJSP is cited from (xia et al., 2005). There are three instances: J8M8 is an example for FJSP with partial flexibility, 8 jobs with 27 operations processed on 8 machines; J10M10 is an instance of total flexibility, 10 jobs with 30 operations processed on 10 machines; J15M10, instance of total flexibility, 15 jobs with 56 operations processed on 10 machines. All our experiments are tested on a computer with Intel (R) Core (TM) 2 CPU (2.66 GHz 2.67GHz), GA program is realized in Matlab R2008a. The results obtained in subsequent experiments through executing the program 10 times to get an average value. The parameters used in GA in our simulation are listed in Table 3. Experiments of GA in subsequent sections will also use the same environment and parameters. Table 3. Parameters of GA Parameters Population size (N) Generation (M) Crossover probability (Pc) Mutation probability (Pm)

Value 2000 200 0.9 0.1

5.1 Test of comparing sequence scheduling methods We use the examples J8M8 and J15M10 presented in (Xia et al., 2005) to compare the efficiency of the four scheduling sequence methods. The results are in Table 4 and Table 5. Table 4. Result of comparison scheduling sequence methods in integrated GA on example J8M8

(a) Order (b) Reverse (c) Random (d)Order availability

Best value 15 14 14 15

Mean value 15.4 14 15.3 15.2

Mean CPU time (second) 229.63 260.61 578.55 567.32

Table 5. Result of comparison scheduling sequence methods in integrated GA on example J15M10

(a) Order (b) Reverse (c) Random (d) Order availability

Best value 12 13 13 12

Mean value 13.5 13.9 14.1 12.9

Mean CPU time (second) 386.15 433.29 833.85 866.27

From results in Table 4, we can see that scheduling sequence method (b) is superior to the others. But in Table 5 sequence method (d) is the best. That may be corresponding to size of problem. We cannot arbitrarily define the best sequence scheduling method, but it is obvious (b) and (d) performs

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relatively better among the four. Therefore, we will use order of available time of jobs as scheduling sequence method for large scale problem, while for small scale problem inverse order of jobs is preferred. 5.2 Test of integrated GA applied for FJSP We compare performance of our proposed integrated GA with solutions obtained in (Xia et al., 2005), and (Gao et al., 2006). Results from our approach and best solutions in literature are illustrated in Table 6. PSO + SA in the work (Xia et al., 2005), a hybrid approach of Simulated Annealing and Particle Swarm Optimization was used for FJSP. GA with neighborhood search called hGA in (Gao et al., 2006). We can see that, our proposed approach integrated GA performs better than the other two approaches in small scale instance. For instances of large scale, it is also competitive. Our best result for J8M8 is shown in (Fig. 3).

Table 9. PM tasks of FJSPPM problem J15M10 PM11 PM21 [tEjl , tLjl ] [2,5] [3,7] Duration 1 1

J8M8 14 15 15

J10M10 7 7 7

Table 10. Result of approach applied on FJSPPM Integrated GA with SSA Integrated GA with IA hGA with SSA

J15M10 12 12 11

Makespan Execution time Makespan Execution time Makespan

O72

6

O42 O11

5

O23

O42

O81

O84

2

O61

3

4

O21

5

6

7

8

9

10

11

O33 12

13

14

Fig. 3. Gantt chart of best solution of J8M8

O24

O84

O73

O22

O61

PM41

O43 O82

O51

O81

1

2

10

O31

9

O41

PM31

PM21

O33

O63

PM11

3

4

5

6

7

8

9

10

11

12

13

O42

14

15

16

17

O32

Table 8. PM tasks of FJSPPM problem J10M10 PM41 PM51 PM61 PM71 PM81 PM91 PM101 [2,5] [3,7] [2,6] [2,5] [3,7] [2,5] [3,6] 2 1 2 2 3 2 3

O92

O11

1 0

1

2

O73 O23

O83

PM11

O71

O53

PM31

O82

2

PM21

O13

3

4

5

6

7

8

Fig. 5. Gantt chart of best solution of J10M10PM 10

O91

9

O51

O131 O32 O121

8

O21

7

O41

O42

O31

2

PM31

0

1

O112

PM61

2

O92 3

O44

4

O124 O43

PM51 O83

O34

O13

O53

PM21 O62

5

6

O54

O144

O154

PM41 O113

PM32 O104

O134 O102

7

O114

O153

O24

O152

PM11

O94 O14

O93

O33

O141

O103

PM82

O143

O22

O72

PM101

O123

O122

O12

O11

1

O133

PM81

O82

O71

3

PM91

O52

O61

4

O23

O142

PM71

O101

5

O132

O151 O111

O81

6

8

O84 9

10

11

12

Fig. 6. Gantt chart of best solution of J15M10PM

Table 7. PM tasks of FJSPPM problem J8M8 PM11 PM21 PM31 PM41 PM51 PM61 PM71 PM81 [tEjl , tLjl ] [5, 10] [6, 9] [10, 15] [9, 17] [3, 10] [8, 16] [4, 14] [7, 13] Duration 4 3 5 3 3 5 3 4

O43

PM41

O12

O101

PM61

PM51

O103 O102

O33

O93

O52

O81

3

PM71

O22 O91

4

O63

PM81

O21 O61

5

We compare the two approaches, IA and SSA on the three examples of (Xia et al., 2005). In SSA methods; our integrated GA and hGA of (Gao et al., 2006), jobs and PM tasks are scheduled simultaneously. In our IA approach, we use Inserting Algorithm (IA) to insert PM after all jobs have been scheduled. Both the two methods are based on the same initialization of starting time of PM which is set at the latest moment of time window. The three examples are J8M8PM and J10M10PM, with one maintenance task on each machine, and J15M10PM with one maintenance task on each machine except for machine 3 and 8 with 2 maintenances tasks. Data of PM tasks from (Gao et al., 2006) are detailed in table 7, table 8 and table 9.

PM91

O72

7

5.3 Test of IA applied for FJSPPM

PM101 O62

O51

6

PM31 [1,6] 1

O54 PM61

O53

O52

O71

8

PM11 PM21 [tEjl , tLjl ] [2,4] [2,7] Duration 2 1

PM81

PM71

O63

O43 2

O83

Fig. 4. Gantt chart of best solution of J8M8PM

O32

O82

1

12

O24

O51 0

8

O73

O22

O21

O41

1

17

PM51

O12

O41

0 O52

O71

3

13 387.15

O13

O53

O12

4

9 252.56

O54

1 O11

18 251.91

O13

O32

2

5

J15M10 12 915.72

O62

O31

7

3 O62

6

J10M10 8 556.31

O83

O31

7

J8M8 17 541.10

O72

8

4 8

PM32 PM41 PM51 PM61 PM71 PM81 PM82 PM91 PM101 [5,11] [3,10] [2,8] [1,6] [3,7] [1,5] [7,11] [2,5] [3,8] 2 3 2 1 1 1 1 1 1

The results obtained are presented in Table 10. The value obtained by hGA with SSA is in (Gao et al., 2006). Gantt chart of our best solution for the three examples of FJSPPM is in (Fig. 4), (Fig. 5) and (Fig. 6) respectively.

Table 6. Result of integrated GA for FJSP Integrated GA SA+PSO hGA

PM31 [1,3] 1

The results are promising, the SSA methods give better results, but the results of IA are enough good, our proposed IA has much more advantage in executing speed than SSA. Moreover, the approach for solving FJSP and IA to add PM into FJSP are two independent modules. Within the independent modules, we can add or reduce elements conveniently. 1409

2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia

6. CONCLUSIONS In order to ensure availability of machines in FJSP, we add condition based maintenance (CBM), a kind of preventive maintenance (PM), into a preschedule obtained through integrated GA. The real-time attribute of CBM makes FJSPPM dynamic. We propose an insertion algorithm (IA) for solving this dynamic FJSPPM. Moreover, we propose four different sequence scheduling algorithms in decoding procedure of integrated GA. Results of experiments on examples of FJSP have shown its good performance. We obtain a new better solution for an instance of FJSP, J8M8. Integrated GA with SSA can get competitive results for FJSPPM. Although integrated with IA obtain inferior results, it performs much better in executing speed. With IA, we used the empty internal between jobs. It is a module independent from the module of solving FJSP. We can easily change either of them to improve the solution. This advantage will be useful when the scheduling problem needed to connect to other management system, like ERP. Even though results of using IA for treating emerging maintenance task are not good as that of SSA, IA is obviously more suitable for DFJSPPM than SSA. For future research, we should try to develop more effective algorithm of computation for FJSPPM with IA, which is suitable for DFJSPPM. 7. ACKNOWLEDGEMENTS This research is partially supported by the project LIA 2MCSI. We sincerely thank the kindly assistance. REFERENCES Al-Najjar. B. (1991), On the selection of condition based maintenance for mechanical systems, Operational Reliability and System Maintenance, pp. 153±73. Aven. T (1996). Condition based replacement policies-a counting process approach, Reliability Engineering & System Safety, vol. 51, no. 3, pp.275±281. Bengtsson, M.(2002). Condition based maintenance on rail vehicles IDPMTR, vol. 2, p. 06. Berrichi, A., Yalaoui, F., Amodeo, L and Mezghiche, M. (2010). Bi-objective ant colony optimization approach to optimize production and maintenance scheduling, Computers & Operations Research, vol. 37, no. 9, pp. 1584±1596, Christer, A., Wang, W. and Sharp, J.(1997). A state space condition monitoring model for furnace erosion rediction DQG UHSODFHPHQW ´ European Journal of Operational Research, vol. 101, no. 1, pp. 1±14. Gao, J., Gen, M., and Sun, L.(2006). Scheduling jobs and maintenances in flexible job shop with a hybrid genetic DOJRULWKP ´ Journal of Intelligent Manufacturing, vol. 17, no. 4, pp. 493±507. Garey, M., Johnson, D., and Sethi, R. (1976). The complexity of flowshop and jobshop scheduling, Mathematics of operations research, pp. 117±129. Gen, M. and Cheng, R. (2000). Genetic algorithms and engineering optimization. Wiley-interscience, vol. 7.

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