Multi-objective modeling for preventive

J Intell Manuf DOI 10.1007/s10845-013-0766-6

Multi-objective modeling for preventive maintenance scheduling in a multiple production line V. Ebrahimipour · A. Najjarbashi · M. Sheikhalishahi

Received: 12 November 2012 / Accepted: 21 March 2013 © Springer Science+Business Media New York 2013

Abstract The change of market and the explosion of product variety have led to increased automation and the need for complex equipment. Therefore, the reliability and the maintainability of the equipment play an important role in controlling both the quantity and quality of the products. In order to ensure specified availability and reliability of production process, preventive maintenance (PM) should be taken during production process. However, taking unscheduled PM can impose high costs to the firm, and adversely decrease the reliability of production line. In this paper, a multi-objective PM scheduling problem in a multiple production line is considered. Reliability of production lines, costs of maintaining, failure and downtime of system are measured as multiple objectives, and different thresholds for available manpower, spare part inventory and periods under maintenance is applied. Production system in this paper consists of serial and parallel machines. Finally, a test problem has been solved to show the effectiveness of the proposed model. Keywords Reliability · Maintenance cost · Series-parallel system · Effective age · Multi-objective optimization · UGF Introduction Preventive maintenance is a broad term that encompasses a set of activities aimed at improving the overall reliability V. Ebrahimipour (B)· A. Najjarbashi · M.Sheikhalishahi Department of Industrial Engineering, University college of Engineering, University of Tehran, Tehran, Iran e-mail: [email protected]; [email protected] V. Ebrahimipour Mathematics and Industrial Engineering Department, Ecole Polytechnique de Montreal, Montreal, QC, Canada

and availability of a system. All types of systems, from conveyors to cars to overhead cranes, have prescribed maintenance schedules set forth by the manufacturer that aim to reduce the risk of system failure. Preventive maintenance activities generally consist of inspection, cleaning, lubrication, adjustment, alignment, and/or replacement of subcomponents that wear-out. Preventive maintenance activities can be categorized in one of two ways, component maintenance or component replacement. An example of component maintenance would be maintaining proper air pressure in the tires of an automobile. Note that this activity changes the aging characteristics of the tires and, if done correctly, ultimately decreases their rate of occurrence of failure. An example of component replacement would be simply replacing one or more of the tires with new ones. Due to the importance of reliability applications in industries, especially in complex systems including nuclear powers, powerhouses, etc., it is subject of great number of research in recent years (Luxhoj and Shyur 1997; Oh et al. 2012). Obviously, preventive maintenance involves a basic tradeoff between the costs of conducting maintenance/replacement activities and the cost savings achieved by reducing the overall rate of occurrence of system failures. Designers of preventive maintenance schedules must weigh these individual costs in an attempt to minimize the overall cost of system operation. They may also be interested in maximizing the system reliability, subject to some sort of budget constraint. Other criteria such as availability and demand satisfaction might be considered as the objective functions. The general problem of preventive maintenance scheduling is to find the best sequence of maintenance actions for each component in the system in each period over a planning horizon such that overall costs are minimized subject to a constraint on reliability or the reliability of the system is maximized subject to a constraint on budget (Moghaddam 2008).

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Goel et al. (1973) presented a simulation model and developed a statistical analysis that considers three different types of preventive maintenance activities for components by defining stochastic and deterministic decision variables as well as unavailability and cost as the objectives. Malik (1979) introduced improvement factor in preventive maintenance scheduling problem for the first time. This factor allows for a variable effect of maintenance on the aging of a system. When α=0, the effect of maintenance is to return the system to a state of “good-as-new”. When α=1, maintenance has no effect, and the system remains in a state of “badas-old” (Moghaddam 2008). Liao et al. (2010) developed a reliability-centred sequential preventive maintenance model for monitored repairable deteriorating system. It is supposed that system’s reliability could be monitored continuously and perfectly, whenever it reaches the threshold R, the imperfect repair must be performed to restore the system. Ascher and Feingold (1984) considered components failures as a non-homogenous poisson process (NHPP) where failure rate is a function of time. Canfield (1986) studied preventive maintenance optimization models via focusing on different aspects of failure function on systems reliability. He mentioned that preventive maintenance actions do not change or affect deterioration behavior of failure rate, so the developed failure function is constant with maintenance actions. He considered increasing failure rate based on the Weibull distribution for his study and determines the optimal cost of maintenance policies. McClymonds and Winge (1987) presented methods to achieve optimal preventive maintenance scheduling for nuclear power plants, though they have not been applied successfully. They considered the plant availability and reliability as the objective functions and develop models based on assigning resources to preventive and corrective maintenance activities. Nakagawa (1988) presented a basic and notable approach for models that utilize improvement factor. He developed two analytical models in order to find the optimal preventive maintenance schedule based on an assumption of increasing failure rate over time. The first model, called a preventive maintenance hazard rate model, calculates the average failure cost of minimal repairs along with costs of preventive maintenance and replacement under the assumption that preventive maintenance actions reduce the next effective age to zero, the failure rate is assumed to increase with the increasing the frequency of preventive maintenance actions. The second model, called an age reduction preventive maintenance model, considers the average failure cost of minimal repairs as well as costs of preventive maintenance and replacement by assuming the age reduction after each minimal repair. Jayabalan and Chaudhuri (1992) proposed another referenced work on age reduction and improvement factors models. They develop an optimization model and a branching algorithm that minimizes the total cost of preventive maintenance and replacement activities. They

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assume a constant improvement factor and define a required failure rate. In addition, they assume a zero failure cost. Kralj and Petrovic (1995) developed a large-scale multi-objective combinatorial optimization model. They considered minimization of total fuel costs, maximization of reliability in term of expected unserved energy, and minimization of technological concerns as the objective functions. In addition, they define maintenance duration, maintenance continuity, maintenance season, maintenance sequence of thermal units of the same class, limitation on simultaneous maintenance of thermal units, and limitation on total capacity on maintenance due to labor and resources as the constraints. Dedopoulos and Smeers (1998) developed a nonlinear optimization model to find the best preventive maintenance schedule by considering the degree of age reduction as the variable in the model. They defined improvement factor, time and duration of preventive maintenance activities as decision variables, and considered fixed cost and variable cost for maintenance actions. Levitin and Lisnianski (2000) presented an optimization model in order to determine the optimal replacement scheduling in multi-state series-parallel systems. They considered an increasing failure rate based on the expected number of failures during time intervals and defined summation of maintenance activities cost along with cost of unsupplied demand due to failures of components as the objective function. They utilized universal generating function (UGF) approach and applied genetic algorithm to find the optimal maintenance policy. Cavory et al. (2001) presented an optimization model to schedule preventive maintenance tasks of machines in a single product manufacturing production line. They considered the total throughput of the line as the objective function. At the first step, they formulate the optimization model and analyze it via analytical approach. Then, the researchers applied genetic algorithm in order to find the best solution of the problem. Bris et al. (2003) considered cost and availability as the systems criteria in their research. They optimize a model including cost in the objective function and availability as the constraint by using a genetic algorithm to find the best preventive maintenance schedule. They utilized MATLAB to calculate the system availability via a Monte Carlo simulation approach. Leou (2003) presented an optimization model to find the optimal preventive maintenance schedule for a multicomponent system. He considered total cost of operations and maintenance activities along with reliability as the criteria of the system. In addition, he defines maintenance crew and duration of maintenance as the system’s constraints. According to the literature, a large number of researchers have worked on preventive maintenance scheduling problems and utilized evolutionary algorithms and simulation approaches to solve them. several other studies related to

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this paper have been performed by Shalaby et al. (2004), Samrout et al. (2005), Wang and Tsai (2012), BartholomewBiggs et al. (2006), El-Ferik and Ben-Daya (2006), Jin et al. (2006), Lapa et al. (2006), Suresh and Kumarappan (2006), Tam et al. (2006), Alardhi et al. (2007), Che-Hua (2007), Schutz et al. (2011), Hagmark and Virtanen (2007), Lim and Park (2007), Quan et al. (2007), Shirmohammadi et al. (2007), Taboada et al. (2008), Lin and Wang (2012). In the following of this section we discuss some recent papers in this field. Pereira et al. (2009) proposed a PSO approach for preventive maintenance scheduling optimization. They focused on reliability and cost, and flexible intervals between maintenance were allowed. They applied the approach for a system containing seven main components: three pumps and four valves. Tian et al. (2009) developed a physical programming based approach to deal with the multi-objective conditionbased maintenance (CBM) optimization problem for a single unit using proportional hazards model. They solved an example of CBM of shear pump bearings and showed that with the proposed approach, the decision maker can make a good tradeoff between the cost and reliability objective. In an additional research, Tian and Liao (2011) performed a similar work on multi-component systems. The study performed by Harrou et al. (2009) is an excellent work in the field. They formulated the problem of imperfect maintenance optimization for series-parallel transmission system structure. Their work focused on selecting the optimal sequence of intervals to perform preventive maintenance (PM) actions to improve the availability. They utilized Harmony Search and Genetic algorithms to solve the problem. Lin and Wang (2010) presented a method of non-periodic PM policy based on failure limit policy for series-parallel systems and a new importance measure of unit-cost extended life (UCEL) to evaluate extent to which maintaining components benefit the system. Reviewing literature shows that although few authors have worked on preventive maintenance scheduling for seriesparallel systems, none of them have considered a multiple production line. In addition, almost all of the researchers paid no attention to time required for maintenance and replacement actions and assumed that it’s negligible. We cover these research gaps in our paper and incorporate some other new issues that are aspects of real world problem of preventive maintenance scheduling, to have a comprehensive model for the problem. The following of the paper is organized as follows: in section “Problem definition”, problem description and formulation is presented. In section “Illustrative examples”, we propose the optimization method to the problem. Efficiency of optimization method is examined thorough solving a test problem in section “Illustrative examples”. Eventually, Concluding remarks and suggestions for future research are presented in section “Summary an conclusions”.

Problem definition The system under study is a multiple production line consists of N series-parallel components, each subject to degradation. Some components are common among all production lines and some of them belong to a specific line. Each component is going to fail with a failure rate that is a function of time. We assumed that components have a failure rate follows weibull distribution (λ(t) = βλ(λt)β−1 ). Preventive maintenance activities in this paper are adjustment and replacement. We seek to establish a schedule of future adjustment and replacement activities for each component over the period [0, T ]. The interval [0, T ] is segmented into J discrete intervals, each of length T /J . At the end of period j, the system is either, adjusted, replaced, or no action is taken. It is assumed that adjustment activity reduces the “effective age” of component with a coefficient α that is between 0 and 1, and thus affects its failure rate. But replacement activities bring component to its initial state i.e. the component is “as good as new” - the failure rate of component after replacement is equal to the failure rate at time zero. And if we do nothing at the end of period, the effective age of component has no change. Additionally, each adjustment and replacement activity takes an amount of time that has discrete uniform distribution between a lower and upper bound. Mathematical modeling Notation i: j, x : Pl : f i (t) : pi, j (z) :

Component; Time period; Set of components in production line l; Failure rate of component i; Probability mass function of working state for component i in period j; Utility function for component i in period u i, j (z) = Utility function for component i in period ( pi, j (z))z 1 + (1 − pi, j (z))z 0 : j, where z 1 is the state that the component is normal and z 0 is the state when the component is failed; Effective age of component i at the start ti, j : of period j; Effective age of component i at the end of ti, j : period j;  t E[Ni, j ] = ti,i,jj Expected number of failure for component i in period j; f i (t) : α: Age reduction factor of adjustment activities; M: Human resource required for maintenance activities; R: Human resource required for replacement;

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H: B:

Total available human resource; Maximum budget for replacement activities; Number of spare parts required for each adjustment; Number of required spare parts for each replacement; Total available spare parts; Duration of each adjustment; Duration of each replacement; 1, if component i at period j is adjusted, 0, otherwise; 1, if component i at period j is replaced, 0, otherwise; Unit cost of adjustment on component i; Unit cost of adjustment on component i; Fixed cost of downtime of system.

S1 : S2 : S: dm : dr : m i, j : ri, j : Mi : Ri : Z:

Mathematical model Minimi zeCost :

N  T 

Fi E[Ni, j ]

i=1 j=1

+

N  T  (Mi .m i, j + Ri .ri, j ) i=1 j=1

+

T 

Z (1 −

j=1

N 

(1 − (m i, j + ri, j ))) (1)

i=1

Maximi zer eliabilit y =

T  i

[⊗ ∅

j=1

(u 1, j (1), u 2, j (1), ..., u n, j (1))] (2) s.t. ti,1 = 0

i = 1, ..., N

ti, j = (1 − m i. j−dm )(1 − ri. j−dr )ti, j−1 +m i, j−dm (α.ti, j−dm ) i = 1, ..., N ; j = 1, ..., T  i = 1, ..., N ; j = 1, ..., T ti, j = ti, j + T /J N  T 

(3) (4) (5)

(M.m i, j + R.ri, j ) ≤ H

(6)

(S1 .m i, j + S2 .ri, j ) ≤ S

(7)

(dm .m i, j + dr .ri, j ) ≤ 0.3T

(8)

i=1 j=1 N  T 

N  T 

Ri .ri, j ≤ B

(10)

i=1 j=1

Equation (1) represents the cost function to be minimized. The first term on the right side of equation is failure cost during time period, the second term is adjustment and replacement costs, and the third term represents downtime cost of production line. Equation (2) defines reliability function to be maximized. Reliability of whole system composed of reliability of production lines. Equation (3) assures that initial age of each component at the start of period 1 equals to zero. Equation (4) determines effective age of component i at the start of period j with respect to previous adjustment or replacement activities performed on it. Assuming that maintenance activities are performed on components at the end of period, Eq. (5) expresses the relation between effective ages at the start and end of the period. Equations (6) and (7) express limitation of maximum available manpower and spare parts for maintenance activities, respectively. Equation (8) determines a threshold for downtime of system due to maintenance activities. Equation (9) assures that either adjustment or replacement can be performed on a component in each period and until it has not finished, the next activity may not start. Finally, Eq. (10) defines that performing replacement activities on components is limited by maximum available budget. Integrated approach In the previous section, a mixed integer non-linear programming (MINLP) model has been proposed for multi-objective preventive maintenance (PM) scheduling problem. One of the objectives represents total cost of the system and the other represents reliability of the whole system. These two objectives don’t have similar scales and have big differences in value and the meaning of that value. In addition, calculating reliability of a series-parallel system is much more complicated than reliability of a pure series or parallel system. Thus in the following section universal generation function (UGF) method for reliability calculation is described. Furthermore multi objective optimization algorithm is presented. UGF method to calculate reliability of a system

i=1 j=1 N  T  i=1 j=1 x+max{d m ,dr }

m i, j + ri, j ≤ 1

j=x

i = 1, ..., N ; x = 1, ..., (T − max{dm , dr })

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(9)

There are various methods of quantitative estimation for reliability of systems consisting of devices that have a range of working levels. In this paper, UGF method is used to estimate reliability of the system. This method, convenient for numerical implementation, proved to be very effective for high dimension combinatorial problems (Harrou and Zeblah 2009). In this method, the z-transform

J Intell Manuf

Fig. 1 A binary-state series-parallel system

of each random variable X i that implies component i in the system, represents its probability mass function (p.m.f.) (x , ..., xik ), ( pi0 , ..., pik )in the polynomial form u i (z) = i0ki xi j in which x represents working level of comij j=0 pi j z ponent i and pi j is the probability that the component works at this level. Reliability of component i is equal to derivative of u i (z) = 1at z=1, that represents its average working level actually. In a similar way one can obtain the z-transform representing the p.m.f. of the arbitrary function f on a system of components by a more general composition operator ⊗ f over z-transform representations of p.m.f. of n independent variables (springer UGF): ⎛ ⊗f ⎝

ki 

⎞

=

k2 

j1 =0 j2 =0

...

∅ par (X N 1 , . . . , X N n )) Rsystem = U  (1)

(12)

Multi-objective optimization Engineering optimization problems typically involve multiple conflicting objectives. A general multi-objective optimization problem tries to find the design variables that optimize a set of different objectives over the feasible design space. A mathematical formulation of the multi-objective optimization problem is given as follows (Huang et al. 2005): Subjecttox ∈ X

n kn   jn =0

U (z) = ∅ser (∅ par (X 11 , . . . , X 1n ), . . . ,

Minimi ze f (x) = { f 1 (x), f 2 (x), ..., f m (x)}

pi ji z xi ji ⎠

ji =0 k1 

u i (z) = p1 z 1 + (1 − p1 )z 0

 pi ji z f (xi j1 ,...,xn jn )

(11)

i=0

In the context of this technique, the z-transform of a random variable for which the operator ⊗ f is defined is referred to as u-function. The u-function of variable X i is u i (z) and the u-function of the function f (X 1 , . . . , X n )is U (z) = ⊗ f (u 1 (z), u 2 (z), . . . , u n (z)). Reliability of this system of components is equal to derivative of Uz at z=1. While the most effective applications of the UGF method lie in the field of the MSS reliability, it can also be used for evaluating the reliability of binary-state systems. The system of components in this paper is a binary-state series-parallel system. The general form of a series-parallel system is illustrated in Fig. 1 As discussed earlier, in a binary system, each component has two states: working or failure. Thus, for the u-function of component i and the system we have:

(13)

where xis an n-dimensional vector of design variables, X is the feasible design space, mis the number of design objectives, f i (x)is the objective function for the i th design objective, and f (x) is the design objective vector. Because of the conflicting nature among different design objectives, it is typically impossible to achieve the best values for all the objectives simultaneously. One of the most widely used methods for multi-objective optimization is the Weightedsum Method. It converts a multi-objective optimization problem into a single-objective optimization problem by using a weighted sum of all the objective functions as the single objective. The mathematical model of the Weighted-sum Method takes the following form: Minimi ze f =

m 

wi f i (x)

i=1

Subjecttox ∈ X

(14) m

Where wi is the weight of objective i, and i=1 wi = 1, wi ≥ 0, i = 1, ..., m (Tian et al. 2009). However, due to different scales of cost objective and reliability objective, we

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J Intell Manuf Table 2 Maximum human resources, spare parts and replacement budget in different planning horizons

Fig. 2 Series-parallel manufacturing system of test problem number 1

cannot use this method directly. The objectives have to be scaled first. Our approach is to relax one of the two objectives, and solve the optimization problem with one objective. After finding the optimal amount of each objective function, we divide the function by its optimal value and then apply the Weighted-sum method to solve the problem with integrated objective function as follows: Minimi ze f = w1 (Z 1 /(Z 1∗ )) + w2 (Z 2 /(Z 2∗ ))

T

H

S

B

30

80

350

2,000

90

240

1,100

3,700

300

800

3,500

5,000

Illustrative examples In this section, three test problems of manufacturing systems consist of components in series-parallel structure have been solved to evaluate the proposed model and performance of the proposed algorithm. Three different amounts for planning horizon is considered to solve test problems in; 30, 90, and 300 time periods. Each time period can represent a day, a week, or any amount of time regarding to our planning levels.

(15)

Finally, the proposed preventive maintenance scheduling problem can be summarized as follows • Obtaining or generating problem data such as system structure, costs of adjustment, replacement, failure, etc., failure rates of components, age reduction factor and so on; • Calculating reliability function of system using UGF method; • Solving the problem once by considering cost as single objective and once by considering reliability to find the optimum value for each of the two objectives (Z i∗ ); • Assign proper weights to objective functions and solving the problem with

integrated objective function f = w1 zz∗1 + w2 zz∗2 using an exact solver; 1 2 • Propose the optimal schedule of maintenance activities which is consist of adjustment and replacement times for system under study.

Test problem number 1 Consider a series-parallel manufacturing system with two production lines consist of seven components. Figure 2 illustrates the structure of this system. Components 1 and 2 are the same. Production line 1 is composed of components 1, 2, 3, 4, 5 and 6, while production line 2 is composed of components 1, 2, 3, 4, 5 and 7. All components can be adjusted or replaced. Table 1 illustrates problem parameters for each component. Maximum available human resource “H”, spare parts “S” and replacement budget “B” for different amounts of planning horizon are represented in Table 2. In addition, downtime cost “Z” is 500. Numerical results have been reported in Tables 3, 4, 5, 6. Table 3 shows the results for test problem number 1. Tables 4, 5 and 6 show the results of preventive maintenance scheduling for different time periods. In these tables, repair and adjustment are shown by rep and adj, respectively.

Table 1 Test problem number 1’ parameters Components

Lambda

Beta

Alpha

F

M

R

hm

hr

dm

dr

Sone

Stwo

1

0.062

2.2

0.32

250

35

200

1

1

1

2

4

6

2

0.062

2.2

0.32

250

35

200

1

1

1

2

4

6

3

0.085

2

0.28

240

32

210

1

1

2

2

5

8

4

0.088

2.05

0.25

270

65

245

1

1

1

1

6

8

5

0.084

1.9

0.2

210

42

180

1

2

2

2

5

6

6

0.082

1.75

0.18

220

50

205

1

2

1

2

3

5

7

0.068

2.1

0.35

280

38

235

1

2

1

2

3

5

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J Intell Manuf Table 6 PM schedule for T = 300

Table 3 Obtained results for test problem number 1 T

Cost

R1

30

7,619

90

33,741

67.00

300

148,200

202.20

R2

25.46

Components

R1

R2

25.71

0.848667

0.857000

68.93

0.744444 0.674000

204.7

Periods

1

0.765889

33

adj

0.682333

38

2

9

67 3

4

5

rep

19

rep

adj

rep

6

7

adj

rep

adj adj

1

126

2

3

4

5

6

7

adj

adj

166

adj

adj

175

rep

176

141 11

rep

33

rep

rep

rep

41 47 64

rep rep

rep adj

69

rep

adj rep

178 rep

adj

adj

adj rep adj

adj adj

adj

Here we have a series-parallel manufacturing system with two production lines consist of 14 components. Figure 3 illustrates this system. Components 2, 6, 8; 3, 7, 9, 11, 12 are the same. Production line 1 is composed of components 1, 2, 3, 4, 5 and 13, while production line 2 is composed of components 6, 7, 8, 9, 10, 11, 12 and 14. All components can be adjusted or replaced. Table 7 illustrates problem parameters for each component. Maximum available human resource “H”, spare parts “S” and replacement budget “B” for different amounts of planning horizon are represented in Table 8 . In addition, downtime cost “Z” is 500. Numerical results have been reported in Tables 9, 10, 11, 12. Table 9 shows the results for test problem number 2. Tables 10, 11 and 12 show the results of preventive maintenance scheduling for different time periods. In these tables, repair and adjustment are shown by rep and adj, respectively.

adj

adj

adj

adj rep rep

rep adj adj

rep

adj

adj

adj

rep

rep adj adj

adj

246 252

rep adj

221 234

adj

adj

215

Test problem number 2

rep adj

adj

adj

adj

adj

adj adj

adj

adj rep

259 273

adj

rep

adj

193 204

adj

adj adj

103 106

adj

adj adj

76

Components

7

adj

rep

98 Table 5 PM schedule for T = 90

6

adj

71

93

Periods

5

75

adj rep

4

rep

56

Components 1

3

39

Table 4 PM schedule for T = 30

Periods

2

adj rep

rep

adj

adj

Test problem number 3 In this test problem, there is a series-parallel manufacturing system with three production lines consist of 20 components. Figure 4 illustrates this system. Components 1–5, 6, 7, 9, 10, 11, 12, 13, 14, 18, 19 are the same. Production line 1 is composed of components 1–5, 6, 7, 11, 12, 15, 16, 17, and production line 2 is composed of components 1–5, 8, 13, 14,

Fig. 3 Series-parallel manufacturing system of test problem number 2

15, 16, 18, 19, and finally production line 3 is composed of components 1–5, 9, 10, 13, 14, 15, 16, 20. All components can be adjusted or replaced.

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J Intell Manuf Table 7 Test problem number 2’ parameters Component

Lambda

Beta

Alpha

F

M

R

hm

Hr

dm

dr

sm

sr

1

0.038

2.1

0.35

280

38

235

3

4

2

2

7

12

2

0.054

1.9

0.2

210

42

180

1

1

1

1

4

7

3

0.015

2.25

0.45

200

45

175

1

1

1

2

3

6

4

0.052

1.75

0.18

220

50

205

1

1

2

3

6

8

5

0.035

1.85

0.22

215

48

210

1

1

1

1

2

5

6

0.054

1.9

0.2

210

42

180

2

2

1

1

4

7

7

0.055

2

0.28

240

32

210

1

2

1

1

2

5

8

0.054

1.9

0.2

210

42

180

2

2

1

1

4

7

9

0.055

2

0.28

240

32

210

1

2

1

1

2

5

10

0.012

1.8

0.38

225

30

215

2

1

1

1

4

8

11

0.032

2.2

0.32

250

35

200

1

1

1

1

1

3

12

0.032

2.2

0.32

250

35

200

1

1

1

1

1

3

13

0.058

2.05

0.25

270

65

245

1

2

2

3

8

15

14

0.03

2.15

0.37

255

55

250

1

3

2

3

6

11

Table 13 illustrates problem parameters for each component. Maximum available human resource “H”, spare parts “S” and replacement budget “B” for different amounts of planning horizon are represented in Table 14. In addition, downtime cost “Z” is 500. Numerical results have been reported in Tables 15, 16, 17. Table 15 shows the results for test problem number 3. Tables 16 and 17 show the results of preventive maintenance scheduling for different time periods. In these tables, repair and adjustment are shown by rep and adj, respectively.

Table 8 Maximum human resources, spare parts and replacement budget in different planning horizons T

H

S

B

30

210

850

3,000

90

450

1,600

5,000

300

1,500

5,500

9,000

Table 9 Obtained results for test problem number 2 T

Cost

30

3,785

90

29,361

300

137,000

R1

R1

R2

26.99

0.817

0.899666667

58.51

74.2

0.650111111

0.824444444

159.41

222.17

0.531366667

0.740566667

24.51

R2

Table 10 PM schedule for T = 30

According to Tables 3, 9 and 15 which represent optimum values of cost and reliability for each test problem, its

Components Periods

1

2

14

rep

rep

Table 11 PM schedule for T = 90

3

4

5

Adj

adj

6

7

8

9

10

11

12

13

14

rep

Components Periods

1

2

15

rep

adj

41

rep

rep

44 54

123

Computational results

rep

rep

3

4

5

6

rep

rep

rep

7

8

9

10

11

12

rep

rep

rep

13

14

adj rep

Rep

rep rep

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Table 12 PM schedule for T = 300

Components Periods

1

2

38

rep

rep

3

4

58 63

rep

5

6

rep

rep

7

8

9

rep

adj

rep

adj

adj

rep

rep

rep

adj

rep rep rep

adj

157

rep

adj

180

rep

adj

148

rep

rep

rep

adj rep

rep

rep rep

rep

rep

rep rep

rep

adj

185

rep

197

adj

rep

rep

rep

rep

203

adj

205

rep

208

rep

225

adj rep

adj

rep

239 258

14

adj

rep

109

229

13

rep

103

201

12

adj

99

135

11

adj

83

104

10

rep rep

rep

adj

adj rep

rep adj

Fig. 4 Series-parallel manufacturing system of test problem number 3

concluded that the proposed integrated approach is efficient both in solutions quality and time consumption. Reasonable costs, semi-high and high average reliabilities of provided PM schedules validate the performance of solution approach.

The minimum average value obtained for reliability functions in test problems is 0.54 and the average of all reliability function values is about 0.79 which is a high reliability for production lines. Furthermore, the maximum time consumed

123

J Intell Manuf Table 13 Test problem number 3’ parameters Component

Lambda

Beta

Alpha

F

M

R

hm

hr

dm

dr

sm

sr

1

0.045

1.85

0.22

215

48

210

1

1

2

2

3

6

2

0.045

1.85

0.22

215

48

210

1

1

2

2

3

6

3

0.045

1.85

0.22

215

48

210

1

1

2

2

3

6

4

0.045

1.85

0.22

215

48

210

1

1

2

2

3

6

5

0.045

1.85

0.22

215

48

210

1

1

2

2

3

6

6

0.065

2

0.28

240

32

210

1

2

1

1

2

5

7

0.065

2

0.28

240

32

210

1

2

1

1

2

5

8

0.032

1.8

0.38

225

30

215

1

2

2

2

6

11

9

0.048

2.1

0.35

280

38

235

2

2

1

2

1

3

10

0.048

2.1

0.35

280

38

235

2

2

1

2

1

3

11

0.035

2.25

0.45

200

45

175

2

1

2

3

3

7

12

0.035

2.25

0.45

200

45

175

2

1

2

3

3

7

13

0.042

2.2

0.32

250

35

200

1

1

1

1

2

5

14

0.042

2.2

0.32

250

35

200

1

1

1

1

2

5

15

0.042

2.2

0.32

250

35

200

1

1

1

1

2

5

16

0.04

2.15

0.37

255

55

250

2

3

2

2

5

9

17

0.064

1.9

0.2

210

42

180

3

3

1

2

3

5

18

0.062

1.75

0.18

220

50

205

2

4

2

3

1

5

19

0.068

2.05

0.25

270

65

245

1

1

1

1

4

7

20

0.068

2.05

0.25

270

65

245

1

1

1

1

4

7

to solve test problems was less than 2.5 min (172 s). In the proposed case studies, a highly reliable system was more preferred to have a low-cost PM schedule, thus we assigned weight of 0.6 to reliability function and 0.4 to cost. We used BARON, BONMIN, and DICOPT solvers in GAMS 23.5 to solve above test problems on a PC with Intel core i5 CPU, 2.67 GHz processor, 4 GB memory, and Windows 7 Professional operating system.

Table 14 Maximum human resources, spare parts and replacement budget in different planning horizons T

H

S

B

30

225

675

4,500

90

675

2,000

7,000

300

2,250

6,750

18,000

and β are reasonable and very close to the values obtained before. Sensitivity analysis In this section sensitivity analysis has been done to verify the robustness of the proposed approach in scheduling preventive maintenance. Two factors which seems to have a significant effect on the solution are, age reduction factor, and, the maximum available budget for replacement. Reducing makes adjustment more desirable for scheduler and vice versa. However, increasing allows having more replacements which can renew the component and reduce its age to zero. We have tested sensitivity of solutions to a reduction in and an increase in on two different periods of test problem number 1 and one period for test problems number 2 and 3. Obtained solutions are shown in Table 18. Table 18 ensures that variations in age reduction factor and maximum available budget for replacement do not affect the efficiency of the proposed solution approach. In comparison with Tables 3, 9 and 15, obtained solutions after changing α

123

Summary and conclusions In this paper, scheduling of preventive maintenance activities in a series-parallel manufacturing system composed of multiple production lines considering reliability of the production lines and costs of maintaining, failure and downtime of system as multiple objectives and thresholds for available manpower, spare part inventory, replacement budget and off periods of the system due to maintenance activities is studied. We consider two types of maintenance activities: adjustment and replacement. Adjustment reduces the effective age of component with a coefficient, and replacement bring component back to its initial working state and makes the effective age zero. Finally, a test problem has been solved to examine effects of proposed schedule on its performance factors. A mixed-integer non-linear multi-objective model

J Intell Manuf Table 15 Obtained results for test problem number 3

T

Cost

R1

R2

R3

R1

R2

R3

30

9,413

26.5

27.5

26.5

0.883333333

0.916666667

0.883333333

90

62,158

72

75.74

75.57

0.8

0.841555556

0.839666667

300

–

–

–

–

–

–

–

6

7

adj

adj

6

7

Table 16 PM schedule for T = 30 Components Periods

1

2

3

4

5

12

8

9

10

adj

adj

9

10

11

12

13

14

adj

15

16

17

rep

rep

adj

15

16

17

rep

rep

rep

rep

rep

rep

18

19

20

adj

rep

19

20

Table 17 PM schedule for T = 90 Components Periods

1

2

3

4

5

8

11

12

13

14

18

18

adj

19 40

rep

adj

rep

44

rep

59

rep

rep rep

rep

rep

rep

rep

rep

adj

66

rep

74

rep

75

Table 18 Obtained results after sensitivity analysis

rep

Test Problem

T

Cost

R1

R2

R3

R1

R2

R3

1

30

5,624

24.2

24.89

–

0.8066

0.8296

–

1

90

30,533

68.52

70.55

–

0.7613

0.7838

–

2

30

3,120

21.06

26.09

–

0.702

0.8696

–

3

30

8,455

26.60

27.396

26.356

0.8867

0.9132

0.8785

have been proposed for the problem and three test problems solved by an integrated approach based on weighted sum method using GAMS software. In addition, we applied UGF method to estimate the reliability of the system. The proposed approach is applied for three different case studies, and the results are shown for different time periods. Moreover periodic maintenance schedule consisting repair and adjustment decisions are shown. The proposed approach of this study would help managers to identify the preferred strategy considering and investigating various parameters and policies. Acknowledgments The authors are grateful for the valuable comments and suggestion from the respected reviewers. Their valuable comments and suggestions have enhanced the strength and significance of our paper. The authors would like to acknowledge the financial support of University of Tehran for this research under Grant Number 27775/01/06.

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