Abstract - Reuse of facilities can bring manufacturers not only less investment but also green images, whereas it requires much maintenance to guarantee the reusability of facilities, which could lead to higher maintenance cost and more production lost. In this paper, a preventive maintenance scheduling method for complex series-parallel system is proposed under group maintenance policy utilizing intelligent algorithms. Hybrid Genetic Algorithm (HGA) and Tabu Search (TS) are employed and compared in terms of time complexity and effectiveness. A case study is then presented. It is verified that group maintenance policy can enhance the reuse of facilities as well as reduce maintenance cost and production lost in the long run. In addition, it can be concluded that HGA is more effective but more time consuming compared with TS.

Keywords - Genetic Algorithm, group maintenance

policy, maintenance scheduling, reuse, Tabu Search

I. INTRODUCTION Reuse of facilities is considered as one of the most economical and effective ways to reduce resource exploitation and environmental impact of manufacturing. However, reuse of facilities calls for much effort on maintenance, which results in higher maintenance cost and more production lost caused by idling of facilities. Therefore, it is crucial for manufacturers to implement maintenance scheduling beforehand to minimize maintenance cost and production lost. At present more and more attention has been paid on preventive maintenance (PM) scheduling [1]. However, methods for PM scheduling so far lack versatility since they have focused more on simple production systems such as serial or parallel production systems instead of complex series-parallel systems. The research on maintenance policies is mainly focused on opportunistic maintenance policy and group maintenance policy. Opportunistic maintenance policy, which means the failure of one subsystem results in possible opportunity to undertake maintenance on other subsystems, has been widely studied and applied. Meanwhile, group maintenance policy is also of great importance in complex production systems, and draws much attention. Assaf et al. [2] proposed optimal group maintenance policies under different circumstances of facility inspections. Sheu et al. [3] and Wildeman et al. [4] divided group maintenance activity into separated phases. However, these researches lack practical value since neither structure dependency of facilities in series-

parallel production systems nor production lost caused by facility idling was taken into consideration. The approaches of solving maintenance scheduling problems include traditional optimization algorithms such as dynamic programming, and intelligent algorithms such as Genetic Algorithm (GA) [5]. The effectiveness of traditional optimization algorithms gets unsatisfactory as maintenance scheduling problems become more and more complex. GA is widely used in solving maintenance scheduling problems due to its strong global searching ability, whereas its disadvantage is also obvious, which is fast initial convergence followed by progressive slower improvements. It is suggested that GA should be combined with other algorithms such as Simulated Annealing (SA), which forms Hybrid Genetic Algorithms (HGAs), to overcome the disadvantage. Esbensen et al. [6] combined GA with SA, and this algorithm is verified to be more effective than simple GA. Initially proposed by Glover, Tabu Search (TS) is another promising algorithm with high performance in solving Maintenance Scheduling problems. Gopalakrishnan et al. [7] proposed a TS based method for PM scheduling problem to maximize the total priority of the scheduled tasks subject to resource availability constraints. In this paper, a maintenance scheduling method for complex series-parallel system is proposed under group maintenance policy combined with opportunistic maintenance policy utilizing both HGA, which combines GA with SA, and TS. The performance degradation of facilities is modeled with Weibull distribution. The effect of maintenance is modeled by employing and improving Malik’s Proportional Age Reduction model. The production system topology is built by using process sequence information. A case study is presented to validate the effectiveness of the proposed method.

II. PRODUCTION SYSTEM MODELING A. Performance Degradation Modeling The performance degradation of a facility is modeled by reliability. Weibull distribution, which is characterized by two parameters, a scale (η ) and a slope (β ), is utilized to produce reliability function since it is widely used in reliability analysis. If t stands for life time of a facility, its reliability, R(t), can be calculated by (1).

( ) exp[ ( ) ]tR t β

η= − (1)

Reuse Oriented Group Maintenance Scheduling Based on Hybrid Genetic Algorithm and Tabu Search

Jihong Yan, Dingguo Hua, and Zimo Wang

Department of Industrial Engineering, Harbin Institute of Technology, Harbin, China (jyan@hit.edu.cn; hwadingo@yahoo.com.cn; princeink@126.com)

978-1-4577-0739-1/11/$26.00 ©2011 IEEE 1524

Fig. 1. Illustration of Proportional Age Reduction model

Fig. 2. Topology of a simple series-parallel system

B. Maintenance Effect Modeling In this paper, four types of maintenance actions are considered, which are minor maintenance, medium maintenance, overhaul, and replacement. The definition and effect of each type are as follows. Minor maintenance is to clean, adjust, and lubricate a facility. It has no improvement effect on reliability of the facility, whereas it can slow down reliability degradation. Medium maintenance is to repair or replace a few components that have been most seriously damaged. It has minor improvement effect on reliability of a facility. Overhaul is to inspect all the components of a facility and replace those which are likely to fail in the next operation period. It can improve the reliability of the facility significantly but not to “as good as new”. Replacement is to replace a facility with a new one. The four types of maintenance are triggered by referring to three thresholds of reliability, T 1

h , T 2 h , and T 3

h , as shown in Fig. 1. When the reliability of a facility falls into minor or medium maintenance zones, minor maintenance or medium maintenance should be carried out correspondingly. When reliability falls between T3

h and zero, namely falls into overhaul zone, overhaul should be carried out. When improvement value of reliability is below a certain level, denoted as LR, then the reuse of facility as a whole is considered no longer economic, and replacement should be carried out. To model the reliability improvement after preventive maintenance, Malik’s Proportional Age Reduction (PAR) model [8] is employed. According to Malik’s model, the kth effective maintenance is presumed to reduce the last operation time (tk − tk-1) to (1 − I ) · (tk − tk-1). In this model, the improvement factor I denotes the effect of the kth maintenance and is set between 0 and 1. It is also assumed that the improvement by the kth maintenance has no effect on that by the (k − 1)th maintenance. Therefore, after the kth maintenance, the reliability of a facility between tk and tk+1 can be calculated by (2).

( ) exp{ [( ) / ] }k k kR t t t T βη= − − + (2) where

11(1 ) ( )k

k j jjT I t t −=

= − ⋅ −∑ (3)

However, limitation exists when PAR model is utilized to model the effect of minor maintenance and overhaul, and PAR model is adjusted correspondingly. To model the effect of minor maintenance, the reliability improvement after utilizing PAR model is subtracted. For instance, according to PAR model, if a minor maintenance is carried out at time t1, the reliability of facility will be improved as the blue dashed line shows, as presented in Fig. 1. According to the adjusted PAR model, the reliability improvement, shown as the green dashed line in Fig. 1, is subtracted. Then the reliability of facility after minor maintenance is as the non-dashed blue line shows. According to PAR model, after several maintenances are carried out on a facility, its reliability can not be improved to a level higher than that reached by the last maintenance, which is not reasonable. Therefore, we adjust (3) into (4) to model the effect of overhaul.

1 1(1 ) ( )a b ak k k kT F t I t F t− −= ⋅ + − ⋅ − ⋅

(4)

where, F , with a range of (0, 1), is compensation factor which is used to magnify the effect of overhaul;

1a

kt − and bkt are calculated by inverse function of (1) with

reliability values after the (k − 1)th maintenance and before the kth maintenance. C. Production System Topology To build the topology of a production system, each facility in the system is assigned a facility number and a data unit, which records the facility numbers of the facility and its upstream and downstream facilities. It must be satisfied that the facility number of a facility should be bigger than those of its upstream facilities. The input and output of a production system are also considered as facilities and assigned facility numbers. The topology of one simple series-parallel production system is shown in Fig. 2 where facility 5 has facilities 2, 3, and 4 as its upstream facilities, and facility 6, the output, as its downstream facility. If multiple production lines exist in one production system, each facility has to be assigned with a production line number. For example, in the production system shown in Fig. 2, facilities 1, 2, and 3 compose production line 1 to manufacture part A; facility 4 itself composes production line 2 to manufacture part B; and facility 5 composes production line 3 to assemble parts A and B.

III. MAINTENANCE POLICY A. Overall Maintenance Policy In this paper, a novel group maintenance policy combined with opportunistic maintenance policy is proposed. To introduce this policy, definition of maintenance activity is presented here first. Maintenance activity, which includes a number of maintenance actions, is defined as an activity composed of set-up of

0

0 1 23

0 4 5

1 2 5

1 3 5 234

5

6

Input Output

6

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maintenance, implementation of maintenance actions, and production recovery. One maintenance activity is triggered when at least one facility need overhaul or replacement, after which opportunistic maintenance is carried out conditionally on the facilities that need minor or medium maintenance. According to group maintenance policy, the facilities should be maintained in units of groups formed following certain rules. Two types of groups are considered in this paper, namely static groups and dynamic groups. In addition, when all the facilities in one static group need only minor maintenance, they are not maintained in the present maintenance activity. In this paper, it is assumed that group maintenance can reduce the time of each maintenance action by 40 percent. It is also assumed that the production of a system can not be recovered, with entire production capacity or a portion of it, until the whole production procedure can be implemented. The rules presented in the next section are intended to minimize the time needed to recover production in a maintenance activity. B. Static Group Static groups, which will not change once formed, are formed according to the topology of a production system. Five rules should be followed to form static groups. Rule I - According to the layout of a workshop, only facilities that are closely located should be included into one static group. Rule II - Facilities that carry out similar production processes should be included into one static group preferably under Rule I. Rule III - According to the logistics of a workshop, only facilities that are in the same flow of work-in-process should be included into one static group. Rule IV - The production capacity in one static group should be balanced. Rule V - The number of facilities in one static group should be neither less than 2 nor more than 5. B. Dynamic Group After all the possible static groups are formed, there may be some facilities that are left not included in any statics groups. In addition, in a maintenance activity, it is possible that not all the facilities in one static group need maintenance and hence they can not be maintained as a group. Therefore, dynamic groups are formed to maintain as many as possible these facilities as groups. Three rules should be followed to form dynamic groups. Rule I - Only facilities of the same type should be included into one dynamic group. Rule II - Only facilities in the same production line should be included into one dynamic group. Rule III - The number of facilities in one dynamic group should be neither less than 2 nor more than 5.

Fig. 3. Flow chart of HGA

1

(1 ) ( )j

j

tk k jp p ctL L P r t dt

−

= + − ⋅ ⋅∫

Fig. 4. Flow chart for production lost calculation

IV. HYBRID GENETIC ALGORITHM IMPLEMENTATION A flow chart for the HGA, which combines GA with SA, is shown in Fig. 3. Initial population is created after initialization of parameters. Then the fitness of the population is evaluated, after which the optimal individual, i.e. the optimal solution, of this generation is recorded. Subsequently, selection, crossover, and mutation operations are carried out. The optimization process by HGA is stopped when the population has evolved for Gm generations. A. Solution Encoding In this paper, either a single facility that doesn’t belong to any groups or a maintenance group is considered as a maintenance unit, which is assigned with an integer unit number. Hence integer encoding is utilized. In addition, the string that contains an individual’s encoding is referred to as a chromosome, and each integer is referred to as a gene. If there are M units in one maintenance activity, feasible chromosomes are obtained by putting integers from 1 to M in random orders. B. Fitness Function To calculate the fitness value of one individual, cost of the kth maintenance activity under the corresponding schedule should be obtained first. The cost of a maintenance activity consists of maintenance cost C k

m and production lost Lk

p. Maintenance cost, which includes cost of maintenance actions for maintenance units C k

u and set-up cost C k

s , can be calculated by (5) and (6). k k km s uC C C= + (5)

, , , ,1( )Mk k i k i k i k i

u p t h eiC C C C C

== + + +∑ (6)

where C k,i p , C k,i

t , C k,i h , and C k,i

e are costs of spare parts, tools, maintainers, and energy for maintenance unit i. In this paper it is assumed that production lost should be calculated from the perspective of the entire system.

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Fig. 5. Flowchart of mutation operation with SA

Fig. 6. Flow chart of TS

Hence the production recovery situation should be checked after maintenance for a unit is finished at tj. The portion of production recovery at time tj is denoted as P j

c , and t0 is set as zero. The production lost L k

p can be calculated following the flow chart in Fig. 4. In Fig. 4, r(t) stands for lost rate, the production lost per unit time, with entire system shut down, which can be calculated by (7).

/0.50( ) 1.1 tr t r ⎢ ⎥⎣ ⎦= ⋅ (7)

The lost rate increases from its initial value r0 with maintenance time since the longer the production stays shut down partly or entirely, the bigger the opportunity cost is. For a number of maintenance schedules, once their maintenance cost and production lost are obtained, their fitness values can be calculated. Suppose there are Q maintenance schedules, for the qth schedule, its fitness value fq can be calculated by (8).

, , , ,11 / [( ) / ( )]Qk k k k

q m q p q m r p rrf C L C L

== + +∑ (8)

C. Crossover and Mutation Operations In this paper the idea of SA is integrated into the crossover and mutation operations by accepting or discarding the degenerate chromosomes, which become worse after crossed or mutated, according to SA methods. For instance, the mutation operations with SA are presented in Fig. 5. The population has N individuals. Additionally, in this paper, order crossover (two-point crossover) and inverse mutation are used.

V. TABU SEARCH ALGORITHM IMPLEMENTATION

A flow chart for Tabu Search (TS) is shown in Fig. 6 where S is the maximum iteration times. An initial solution is created by the integer encoding method described in Section IV. A neighbor of x is created by swaping two randomly selected integers in its solution string. Tabulist keeps a record of the swap moves by which x is created. In terms of aspiration criterion, a solution created by a swap move in the Tabulist is accepted if the solution is better than the best solution x* obtained so far. The size of the neighbor is set as V, and the length of the Tabulist is set as V/2.

VI. CASE STUDY A. Case Description In this section, a case study on maintenance scheduling of a bearing production workshop is presented. The production system has 45 facilities, as shown in Fig. 7. Facilities 0, 46 and 17 are input, output, and heat treatment respectively, and it is assumed that their performances do not degrade with time. Blue arrows and red arrows connect production lines for outer race and inner race respectively. Black arrows connect assembling line of bearings. In Fig. 7 different colors stand for different types of facilities, and the layout of the workshop is also as presented in Fig. 7. Facilities 1 to 16 carry out lathing processes, 18 to 35 milling, 36 and 37 grinding, 38 and 39 demagnetizing, 40 and 41 cleaning, 42 and 43 sizing, 44 assembling, and 45 riveting. Based on the above information, topology of the system is built, and accordingly 15 static groups are formed, which are marked with dashed ellipses in Fig. 7. B. Parameter Settings For each facility, the two parameters for reliability modeling, η and β, are derived from randomly generated simulation data; so it is with the facility’s price and the resources and time needed for its maintenance actions. In this paper, the three thresholds, T 1

h , T 2 h , and T 3

h , are set as 0.9, 0.75, and 0.6 respectively; and LR is set as 0.2; improvement factors for minor maintenance, medium maintenance, and overhaul are set as 0.2, 0.6, and 0.7; compensation factor F is calculated by (9) where On is number of overhauls for a single facility.

(1/(1 ))0.1 nOF += (9) The set-up cost Cs of each maintenance activity is set as 1000; the unit prices for four types of maintenance resources, i.e. spare parts, tools, maintainers, and energy, are set as 100, 50, 50, and 10 respectively; the initial value of production lost rate r0 is set as 1000. For the Hybrid Genetic Algorithm, population size N is set as (M − 1) ⋅ M / 2, where M is the number of maintenance units in a maintenance activity; maximum evolution generation Gm is set as 100; crossover probability Pc and mutation probability Pm are set as 0.7 and 0.05 respectively; the initial value of Simulated

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Annealing Temperature T is set as 104, and cool-down factor α is set as 0.9. For Tabu Search, the maximum iteration times S is set as 100, the size of neighbor V is set as (M − 1) ⋅ M / 2. C. Simulation and Result Analysis To verify the effectiveness of the proposed method, maintenance scheduling of the production system from the 1st to the 3000th day of operation is carried out with maintenance resource constraints. During the 3000 days, 84 maintenance activities are carried out if group maintenance combined with opportunistic maintenance (GM&OM) policy is followed, and 85 maintenance activities are carried out if only opportunistic maintenance (OM) policy is followed. In addition, about 74% of the facilities have longer reuse periods under GM&OM policy. For instance, the reuse period of facility 2 was extended from 1932 days to 2730 days if GM&OM policy is followed, as shown in Fig. 8. In each maintenance activity, the maintenance tasks are scheduled using HGA and TS respectively. Fig. 9 is an example of Gantt chart under GM&OM policy, where yellow bars stand for group maintenance.

Fig. 7. Production system for bearings

Fig. 8. Reuse situation of facility 2 under two maintenance policies

Fig. 9. Gantt chart of maintenance scheduling under GM&OM policy

TABLE I OPTIMIZATION RESULT OF MAINTENANCE COST RATES

GM&OM policy OM policy HGA 835 953 TS 841 958

To evaluate the effectiveness of the two algorithms employed in this paper, maintenance cost rate Rc, which can be calculated by (10), is used.

1( ) /K k k

c m p skR C L T

=⎡ ⎤= +⎣ ⎦∑ (10)

where K is total number of maintenance activities and Ts is simulation time. The maintenance cost rates optimized by HGA and TS under two types of maintenance policies are shown in Table I, from which it can be concluded that HGA is more effective in optimization than TS on maintenance scheduling of series-parallel production system. However, statistics shows that the time consumed in optimization with TS is only 40% of that with HGA, which shows that TS is more time efficient. In addition, it can also be concluded that maintenance cost rate under GM&OM policy is approximately 12% smaller than that under OM policy.

VII. CONCLUSION In this paper, a novel methodology for maintenance scheduling is proposed. HGA and TS are employed and it is verified that HGA is more effective but less time efficient than TS. In addition, a group maintenance policy combined with opportunistic maintenance policy is proposed, which can not only extend the reuse period of facilities but also reduce the maintenance cost rate significantly. In the future, more research should be done on methods of grouping facilities.

ACKNOWLEDGMENT This research is funded by the National Natural Science Foundation of China (#70971030).

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