Metaheuristic-based inspection policy for a one-shot ... - Springer Link

Journal of Mechanical Science and Technology 28 (10) (2014) 3947~3955 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-014-0905-9

Metaheuristic-based inspection policy for a one-shot system with two types of units† Won Young Yun*, Li Liu and Young Jin Han Department of Industrial Engineering, Pusan National University, Busan, 609-735, Korea (Manuscript Received January 20, 2014; Revised June 2, 2014; Accepted June 2, 2014) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In this paper, we address an inspection policy problem for a one-shot system with two types of units, namely, Type 1 units that fail at random times and Type 2 units that degrade with time. Interval availability and life cycle cost are used as optimization criteria and estimated by simulation. We determine inspection intervals, preventive replacement ages of Type 1 units, and preventive maintenance thresholds of Type 2 units that have minimal life cycle cost and satisfy the target interval availability during inspection periods. A simulation-based optimization procedure using a hybrid genetic algorithm is proposed to find near-optimal solutions. Numerical examples are studied to investigate the effects of model parameters on optimal solutions and compare the hybrid genetic algorithm with the general genetic algorithm. Keywords: One-shot system; Inspection interval; Gamma process; Simulation; Hybrid genetic algorithm ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction One-shot systems, such as missile systems, fire extinguishers, and airbags, are usually stored for a long period of time and are operational at most once during their life cycle. The maintenance of high reliability of these one-shot systems in storage is important to successfully perform their objectives. However, the reliability of one-shot systems in storage deteriorates with time, and failure is only revealed when they are required. Thus, inspection must be carried out periodically to immediately detect system failure. However, determining the most suitable inspection intervals that will balance inspection frequency and maintenance costs is difficult. Thus, determining inspection intervals is an optimization problem in the formulation of inspection and maintenance policies for complex repairable systems. Several researchers had proposed various inspection policies for a one-shot system. Hariga [5] considered two possible scenarios: the determination of a good state of repair upon inspection and the detection of a system failure upon inspection. A procedure that will determine optimal inspection interval T* was proposed to maximize the expected profit per unit of time. Ito and Nakagawa [6] addressed the periodic inspection policy problem for a storage system with two types of units, namely, Type 1 unit that is maintained upon inspection and Type 2 unit that is degraded over time. An inspection policy *

Corresponding author. Tel.: +82 51 510 2421, Fax.: +82 51 512 7603 E-mail address: [email protected] This paper was presented at the ICMR2013, Emeishan, Sichuan, China, July15-18, 2013. Recommended by Guest Editor Dong Ho Bae © KSME & Springer 2014 †

was also proposed in which the system is replaced if its reliability becomes lower than a pre-determined level. Optimal inspection time T*, which minimizes the average cost until overhaul, was determined. Ito and Nakagawa [7] proposed another inspection policy in which the storage system is replaced either upon the detection of failure or at a predetermined time (N + 1)T, depending on which occurs first. Optimal inspection time T*, which minimizes the total expected cost until replacement, was determined. Under the previously mentioned inspection policy, Ito and Nakagawa [8] determined optimal inspection times, with which the total expected cost, including the testing and lost costs, is minimized until system failure is detected. Ito and Nakagawa [9] then considered the testing and overhaul costs and determined optimal inspection times with which the mean time to overhaul is maximized and the average cost until overhaul is minimized. Ito et al. [10] likewise proposed an extended inspection model for a storage system consisting of three types of units, namely, Type 1 that is maintained at time interval T, Type 2 that is replaced at time interval NT, and Type 3 that is neither maintained nor replaced. The optimal inspection time T* and optimal inspection number N*, which minimize the average cost until overhaul, were analytically determined. Kaio et al. [11] considered an inspection policy problem for a single-unit system and assumed that the system will fail in the inspection process. Optimal inspection intervals were determined to minimize the expected cost until system failure is detected. Hariga [5] studied a periodic inspection policy for a single-unit system and proposed a heuristic method to determine optimal inspection intervals that will maximize the ex-

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W. Y. Yun et al. / Journal of Mechanical Science and Technology 28 (10) (2014) 3947~3955

pected profit per unit of time for cases of exponential distribution and Weibull distribution of the time to failure. Nakagawa and Mizutani [13] considered three inspection policies, namely, periodic, sequential, and asymptotic inspection policies for a single-unit system over a finite time span. Nakagawa et al. [14] proposed periodic, random, and sequential inspection policies for a deteriorating system with random working times and determined optimal inspection intervals that will minimize the total expected cost. Taghipour and Banjevic [16] proposed an inspection optimization model for a multi-unit system over both finite and infinite time horizons. They proposed an inspection policy in which the failed units are either minimally repaired or replaced at pre-determined times as a function of their age. The optimal inspection interval T* was indicated as a means of minimizing the expected cost, with a penalty cost being incurred for the time interval between failure and detection. To date, some researchers have considered the periodic inspection policy for a storage system but regular inspection intervals may not always be profitable. Inspection intervals often need to be reduced as the age of the system increases. Thus, the non-periodic inspection policy for a storage system has been proposed for a storage system. Golmakani and Moakedi [4] addressed a non-periodic inspection policy problem for a multi-unit system over a finite horizon. A search algorithm was suggested to determine optimal inspection intervals with which the expected total cost is minimized. The non-periodic inspection policy was also compared with the periodic inspection policy. Yun et al. [17] considered a oneshot system consisting of two types of units, namely, Type 1 units that fail at random times and Type 2 units that are replaced by new ones at pre-determined times. Interval availability was used as an optimization criterion to maintain high system availability between inspection times. Given the replacement times of Type 2 units, optimal inspection points were determined to minimize the life cycle cost and satisfy the target interval availability during inspection periods using a genetic algorithm. Meanwhile, the units in a one-shot system can be classified into two types, namely, Type 1 units that fail at random times and Type 2 units that degrade with time. For example, in the case of a missile system, the guidance and control units are Type 1 units, whereas the ignition unit of the rocket motor is a Type 2 unit. In general, the inspection process only involves checking whether the system normally operates or not, thus the formulation of a preventive maintenance policy for the units in the one-shot system is necessary to improve the reliability of the units. Li and Pham [12] proposed a conditionbased maintenance model for a storage system and two randomized degradation functions to describe the degradation of the units in the system. Inspection intervals and preventive maintenance thresholds of the units were both determined to minimize the expected total maintenance cost. Markov processes, such as the Brownian motion with drift, compound Poisson process, and gamma process, have also been used to

describe the degradation of Type 2 units. Grall et al. [2] attempted to solve a condition-based maintenance policy problem for a single-deteriorating system and considered a gamma process for the degradation of system. Optimal inspection times and preventive maintenance thresholds were determined, and a periodic inspection scheme was compared with a dynamic inspection scheme. Noortwijk [15] reviewed the applications of gamma processes in time-based and conditionbased maintenance optimization problems. Yun et al. [18] studied an inspection policy for a one-shot system and used the compound Poisson process to describe the degradation of Type 2 units. Inspection intervals and preventive maintenance thresholds of Type 2 units, which minimize the life cycle cost and satisfy the target interval availability, were determined. In the present paper, we aim to address a non-periodic inspection policy problem for a one-shot system with two types of units: Type 1 units that fail at random times and Type 2 units that degrade with time. The preventive maintenance policies for Type 1 and 2 units are considered to improve reliability. We aim to optimally determine inspection intervals, preventive replacement ages of Type 1 units, and preventive maintenance thresholds of Type 2 units. This paper is organized as follows. The inspection interval model for a one-shot system with two types of units is explained in Sec. 2. A simulation-based optimization procedure using a hybrid genetic algorithm is proposed in Sec. 3. Numerical examples are studied to investigate the effects of model parameters and compare the hybrid genetic algorithm with the general genetic algorithm in Sec. 4. Finally, the conclusions of the study are presented in Sec. 5.

2. Inspection interval model for a one-shot system with two types of units In this section, we introduce the inspection interval model for a one-shot system consisting of two types of units. The preventive maintenance policies for Type 1 and 2 units are explained to improve the storage system reliability. The appropriate performance measures in the evaluation of the oneshot system are also detailed in terms of interval availability and life cycle cost. Table 1 shows the notation for the model. A one-shot system usually consists of several units, and the units can be classified into two types: Type 1 units that fail at random times and Type 2 units that degrade with time and require stringent storage environments to prevent their degradation in storage. Therefore, system failure is only revealed when the system is operational, but the one-shot system performs its objective at most once during its life cycle. Keeping high system reliability in storage is important, and inspection must be carried out periodically to immediately detect system failure. However, finding appropriate inspection intervals is difficult because inspection frequency and maintenance costs must be balanced. While frequent inspection can keep high system reliability in storage, the high frequency of inspection leads to high maintenance costs and in some cases,


Table 1. Notation for the inspection interval model of a one-shot system Notation

Definition

i

Index of units ( i = 1, 2,3,..., N )

p

Index of periods ( p = 1, 2,3,..., P )

AI p

Interval availability of pth period

AIT

Target interval availability

CI

Inspection operation cost

i CCM

C

i PM

Corrective maintenance cost of unit i Preventive maintenance cost of unit i

LC

Life cycle cost

TI

Total number of inspections

E[CM i ] E[ PM i ]

Expected number of corrective maintenances of unit i Expected number of preventive maintenances of unit i

e

Allowable gap

q

Uniform random variable ( 0 < q < 1 )

Mu

Mutation rate

Asys

System availability

ASTD

Standard deviation of interval availability

ASTD _ T

Target standard deviation of interval availability

DASTD

Increment in system availability

DCost

Increment in life cycle cost

the process of testing can degrade Type 2 units [6]. By contrast, when the one-shot system is inspected less frequently, less maintenance costs are incurred but these are in exchange of the decrease in system reliability and the increase in system failure risk. Hence, inspection intervals, which represent an appropriate balance between inspection frequency and maintenance costs for the one-shot system, should be optimally determined. Each Type 1 unit in the one-shot system may have different failure distributions, including an exponential distribution or a Weibull distribution. If the failure of a Type 1 unit follows an exponential distribution, preventively maintaining the unit upon inspection is unnecessary because the failure rate function is always constant. Whereas, if the failure time of the Type 1 unit has a Weibull distribution with increasing failure rate function, the unit deteriorates with time. In addition, the Type 2 unit whose degradation level exceeds the failure threshold cannot normally operate; hence, the unit is said to be in a “failure” state. This phenomenon may increase the time interval between system failure and its detection. Therefore, a preventive maintenance policy for Type 1 and 2 units must be considered to improve reliability of these units. However, determining the appropriate preventive maintenance frequency of the units in the one-shot system is also difficult because of the corresponding maintenance costs. Hence, preventive maintenance frequency of the units in the system must be determined, with which we aim to optimally determine the preventive replacement ages of Type 1 units and preventive maintenance thresholds of Type 2 units. In general, the steady state (average) availability is inter-

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preted as the average proportion of a long period of time when the system is able to function and is usually used as an optimization criterion for optimizing inspection and maintenance policies of complex repairable systems, such as railway systems, aircrafts, and automobiles. However, presenting the performance of the one-shot system is not appropriate because the one-shot system suddenly performs a mission between inspection times and at most once during the life cycle. Therefore, maintaining high system availability over inspection periods is important, allowing the system to correctively operate when required. Hence, we consider interval availability as the probability that the one-shot system is in a functioning state in the inspection times as an optimization criterion. According to the state of Type 1 and 2 units in the one-shot system upon inspection, two possible maintenance actions can be performed, namely, corrective maintenance (CM) and preventive maintenance (PM). Therefore, inspections in the one-shot system may incur maintenance costs such as inspection operation cost, CM and PM costs of Type 1 and 2 units. Life cycle cost is also used as an optimization criterion to assess system performance, along with the inspection operation cost and CM and PM costs of units in the one-shot system. The assumptions are as follows: • A one-shot system consists of two types of units. • Failure of all units in the system is independent. • Life cycle time of the one-shot system is finite and given. • System failure is only detected by inspection. • Inspection to detect system failure is always perfect. • Failure thresholds of Type 2 units are given. • A perfect repair model is used for CM and PM. Given these assumptions, we determine inspection intervals, preventive replacement ages of Type 1 units, and PM thresholds of Type 2 units in the one-shot system, which result in the reduction of life cycle cost and satisfy the target interval availability over inspection periods. The inspection interval optimization problem of a one-shot system with two types of units is given as N

{

}

i i Min E [ LC ] = ( CI ´ TI ) + å ( CCM ´ E[CM i ]) + ( CPM ´ E[ PM i ]) . i =1

Subject to AI p ³ AIT , "p .

(1) (2)

3. A simulation-based optimization method using a hybrid genetic algorithm In this section, we propose a method to optimally determine inspection intervals of a one-shot system. First, a simulationbased optimization procedure using a hybrid genetic algorithm is proposed. Second, we explain a hybrid genetic algorithm with a heuristic method to generate alternatives. 3.1 A simulation-based optimization procedure using a hybrid genetic algorithm To quantitatively evaluate the performance of the one-shot

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Fig. 2. Example of a solution representation.

Fig. 1. Simulation-based optimization procedure using a hybrid genetic algorithm.

Fig. 3. Example of a one-cut point crossover.

3.2 General genetic algorithm system, we consider interval availability and life cycle as optimization criteria and conduct a simulation method to estimate them. Given that each Type 1 unit in the system may have different failure distributions and the failure and repair of the units at the next inspection time depend on the ones at the previous inspection and thus, analytically obtaining system performance measures is difficult. The gamma process describes the degradation process of Type 2 units in the one-shot system, and a gamma-bridge sampling method is applied to simulate the gamma process [1]. Simulation methods basically provide limited statistics related to system performance measures, and obtaining analytic solutions by simulation alone is impossible. Therefore, appropriate optimization methods are required to find near-optimal alternatives; thus, we propose a hybrid genetic algorithm with a heuristic method. Fig. 1 shows a simulation-based optimization procedure using a hybrid genetic algorithm, and the detailed procedure is as follows: Step 1: Input simulation data, such as failure and repair distributions with units and maintenance costs. Step 2: Set the target interval availability. Step 3: Generate alternatives using the hybrid genetic algorithm of a heuristic method. 3.1: Generate inspection intervals, preventive replacement ages of Type 1 units, and PM thresholds of Type 2 units using the genetic algorithm. 3.2: Improve the interval availability of alternatives by the heuristic method. Step 4: Estimate the interval availability and life cycle cost by simulation. 4.1: If AI p ³ AIT (for all p), set the current best solution as the global best solution and terminate the procedure. 4.2: If AI p < AIT (for at least one period), go to Step 3 to generate new alternatives.

A general genetic algorithm is used to generate alternative inspection intervals, preventive replacement ages of Type 1 units, and PM thresholds of Type 2 units in the one-shot system. One year is considered as the unit time of the inspection interval and a gene in a chromosome represents an inspection time of the one-shot system. Thus, the total number of genes in a chromosome represents the system life cycle. The value of each gene is either “0’”or “1”. If the value of the hth gene in the chromosome is “1”, the inspection is performed at h year. The integer strings are used to represent the preventive replacement ages of Type 1 units and the PM thresholds of Type 2 units in the one-shot system. Fig. 2 shows an example of the solution representation and illustrates the inspection of a one-shot system at 4, 12, 14, 26, and 29 years. Two Type 1 units are replaced when their age reaches the preventive replacement ages of 9 and 13 years and two Type 2 units are replaced when their degradation level reaches the PM thresholds of 69 and 84. At the beginning of the genetic algorithm, an initial population of chromosomes is generated as the “initial generation”, and the initial solutions are randomly generated. To balance exploration and exploitation in the search space, three genetic operators are used: crossover, mutation, and selection. In the crossover operation, two new chromosomes (offspring) are produced by combining two chromosomes (parents) that have survived from the previous generation. A single-point crossover is used to generate new chromosomes (Fig. 3). In the mutation operation, the deterministic adaptation is considered, and the mutation rate of the current generation decreases from 0.5 to 0.2, as the number of generations increases [3]. The values of the genes in the chromosome are replaced relative to whether the target interval availability is satisfied or not. The detailed procedure of the mutation operation is as follows:

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Step 1: Select a chromosome that has survived from the previous generation. Step 2: Select a period that is not yet sequentially checked, and let the period be the pth period. 2.1: Generate a uniform random value, q . 2.1.1: If q £ Mu , go to Step 2.2. Otherwise, go to Step 2.5. 2.2: Check whether the target interval availability is satisfied or not. 2.2.1: If AIT - e £ AI p £ AIT + e , go to Step 2.5. 2.2.2: If AI p < AIT - e , select a gene that represents the pth inspection point and go to Step 2.3. 2.2.3: If AI p > AIT + e , select a gene that represents the pth inspection point and go to Step 2.4. 2.3: Increase the interval availability of the pth period. 2.3.1: Randomly select a gene that is contained beth tween ( p - 1) and the pth period. 2.3.2: Exchange the value of each gene. 2.3.3: Estimate the interval availability by simulation and go to Step 2.5. 2.4: Decrease the interval availability of the pth period. 2.4.1: Randomly select a gene that is contained beth tween pth and ( p + 1) period. 2.4.2: Exchange the value of each gene. 2.4.3: Estimate the interval availability by simulation and go to Step 2.5. 2.5: If all periods are searched, go to Step 3. Otherwise, go to Step 2. Step 3: Randomly select a gene that represents the PM of a unit in the one-shot system. 3.1: Generate a uniform random value, q . If q £ Mu , go to Step 3.2. Otherwise, go to Step 3.4. 3.2: If AIT - e £ Asys £ AIT + e , go to Step 3.4. 3.3: If Asys < AIT - e , decrease the value of the gene as much as q . 3.4: If Asys > AIT + e , increase the value of the gene as much as (1 - q ) and go to Step 3.5. 3.5: Estimate the interval availability and the life cycle cost by simulation. 3.6: If all the genes are selected, terminate the procedure. Otherwise, estimate the interval availability and go to Step 3. 3.3 Heuristic method After performing the mutation operation, a heuristic method is performed to improve the alternatives. In the heuristic method, the target standard deviation of the interval availability is considered and the heuristic procedure is terminated, either if the target standard deviation is satisfied or the number of replications for improving the alternative is reached. The detailed procedure for this method is as follows: Step 1: Select a chromosome that is not yet chosen. 1.1: If ASTD £ ASTD _ T , terminate the procedure. 1.2: If ASTD > ASTD _ T , generate the number of replications

and go to Step 2. Step 2: Select the first period that does not satisfy the target interval availability and let the period be the pth period. 2.1: If AI p < AIT - e , go to Step 2.2. Otherwise, go to Step 2.3. 2.2: Increase the interval availability of the pth period. 2.2.1: Obtain DASTD / DCost by simulation when the inspection time of the pth period is shortened as much as the unit time. 2.2.2: Obtain DASTD / DCost by simulation when the value of each gene that represents the PM of the unit is decreased as much as the improvement unit. 2.2.3: Select the alternative with the highest score among the alternatives generated in Steps 2.2.1 and 2.2.2. Go to Step 3. 2.3: Decrease the interval availability of the pth period. 2.3.1: Obtain DASTD / DCost by simulation when the inspection time of the pth period is lengthened as much as the unit time. 2.3.2: Obtain DASTD / DCost by simulation when the value of each gene that represents the PM of the unit is increased as much as the improvement unit. 2.3.3: Select the alternative with the lowest score among the alternatives generated in Steps 2.3.1 and 2.3.2. Go to Step 3. Step 3: Check whether the target standard deviation is satisfied. 3.1: If ASTD £ ASTD _ T , terminate the procedure. Otherwise, go to Step 3.2. 3.2: If the current replication number is less than the number of replications, increase the current replication number by one and go to Step 2. Otherwise, terminate the procedure. Fitness evaluation in the genetic algorithm is used to check the value of the objective function that is subject to constraint as the target interval availability. Considering the mathematical difficulty, simulation is conducted to estimate the life cycle cost, and the fitness value of a chromosome is the expected life cycle cost of a one-shot system during its life cycle. Simulation is performed repeatedly to estimate the expected number of CM and PM of each unit in the system. Penalty cost is also added to the chromosome whose interval availability does not satisfy the target interval availability. P

Penalty cost =

å ( AI p =1

T

)

- AI p ´ LC ´ w

(3)

where w is the current generation number in the genetic algorithm. In the selection process, the better chromosomes in each generation are identified, and in the elite selection method, the

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best chromosome from parents and offspring is retained. The termination condition is the number of generations in the genetic algorithm.

Table 2. Failure and repair distributions of units in the system. Name

Failure distribution (yr)

Repair distribution (hr)

A

Exp (27.5)

Exp (12)

B

Exp (26.0)

Exp (12)

4. Numerical examples

C

Exp (28.0)

Exp (10)

To evaluate the performance of the hybrid genetic algorithm with the heuristic method, coding is done using C++ programming language and the numerical examples are executed on an IBM-PC compatible with an Intel Core 3.3 GHz. For numerical examples, we consider a one-shot system consisting of 10 units (A to J) with a series structure and having a life time of 30 years (262,800 hours). Table 2 shows the failure and repair distributions of units in the system. We assume that the time to failure of Type 1 units follows either an exponential distribution or a Weibull distribution. For the degradation of Type 2 units, the shape parameter of the gamma distribution is a = c × t b , where c = 1.1 and b = 0.05 for Unit I and c = 1.2 and b = 0.01 for Unit J. The scale parameters of the gamma distribution β are 0.001 and 0.002 for Units I and J, respectively. The failure thresholds of Units I and J are 110 and 130, respectively. Table 3 shows the CM and PM costs of units in the one-shot system, and the inspection operation cost is 1,000,000. The simulation length is 30 years (262,800 hours), and the number of replications is 100. The parameters for the genetic algorithm are as follows: crossover rate is 0.7, mutation rate is 0.5, population size is 50, and generation size is 50. For the heuristic method, the improvement unit is 10%, the allowable gap ε is 0.01, and the target standard deviation of interval availability is 0.02. In numerical examples, we initially consider different values of target interval availability, which are 0.75, 0.80, and 0.85 (CASE 1), and then decrease the scale parameters of the failure distribution of Type 1 units by 15%, 30%, and 45% (CASE 2). Lastly, the failure thresholds of Type 2 units are increased by 15%, 30%, and 45% (CASE 3). Based on the numerical results, we analyze the effect of model parameters on the near-optimal solutions and compare the performance of the hybrid genetic algorithm with the general genetic algorithm.

D

Exp (25.5)

Exp (10)

E

Exp (29.0)

Exp (4)

F

Exp (25.0)

Exp (6)

G

Weibull (27.0, 1.8)

Exp (6)

H

Weibull (26.5, 2.0)

Exp (4)

I

-

Exp (10)

J

-

Exp (12)

Table 3. Maintenance costs of units in the system. Name

CM cost

A

350,000

PM cost

B

320,000

-

C

270,000

-

D

300,000

-

E

280,000

-

F

340,000

-

G

280,000

190,000

H

330,000

170,000

I

360,000

200,000

J

290,000

210,000

4.1 CASE 1: Different values of target interval availability We initially consider different values of target interval availability: 0.75, 0.80, and 0.85. To satisfy high target interval availability, inspection must be carried out more frequently, incurring high maintenance costs. As a result, the life cycle cost and total number of inspections increase as the target interval availability increases (Fig. 4 and Table 4). In addition, improving the reliability of the units by frequent PM is necessary to satisfy high interval availability. Thus, the preventive replacement ages of Units G and H and PM thresholds of Units I and J should be decreased as the interval availability increases (Table 5). From the numerical results the hybrid genetic algorithm results in 7.13% lower life cycle cost on the

Fig. 4. Life cycle cost for different values of target interval availability.

average than the general genetic algorithm. However, the general genetic algorithm requires 42.37% smaller CPU time on the average than the hybrid genetic algorithm (Table 6). 4.2 CASE 2: Decreasing the scale parameters of the failure distribution of Type 1 units In this case, target interval availability is 0.75, and the scale

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Table 4. Optimal inspection points of a one-shot system. AIT 0.75 0.80

0.85

Table 7. Optimal inspection points of a one-shot system.

Method

Inspection points (yr)

General GA

3, 7, 8, 14, 20, 21, 23, 28

Decrement

Hybrid GA

4, 7, 11, 23, 24

Hybrid GA

4, 7, 11, 23, 24

General GA

3, 6, 10, 15, 16, 18, 22, 25, 28

General GA

3, 6, 10, 15, 16, 18, 22, 25, 28

Hybrid GA

4, 7, 10, 19, 25, 27

General GA

3, 4, 8, 9, 11, 12, 14, 16, 19, 21, 22, 26, 27, 28, 29

Hybrid GA

4, 5, 13, 19, 20, 22, 23, 28, 29

0% 15%

Table 5. Optimal preventive replacement ages of Units G and H and preventive maintenance thresholds of units of Units I and J. AIT 0.75 0.80 0.85

Method

G (yr)

H (yr)

I

J

General GA

25.02

22.11

78

68

Hybrid GA

25.96

23.33

92

73

General GA

22.99

20.12

62

56

Hybrid GA

23.54

21.83

89

63

General GA

19.73

18.62

49

43

Hybrid GA

21.94

20.07

72

57

30%

45%

Decrement 0%

Table 6. CPU time to obtain optimal solutions.

0.75 0.80 0.85

Method

CPU time (sec)

General GA

3,790

Hybrid GA

6,493

General GA

3,950

Hybrid GA

6,894

General GA

4,209

Hybrid GA

7,354

30%

45%

Hybrid GA

4, 7, 10, 19, 25, 27

General GA

3, 4, 8, 9, 11, 12, 14, 16, 19, 21, 22, 26, 27, 28, 29

Hybrid GA

4, 5, 13, 19, 20, 22, 23, 28, 29

General GA

2, 5, 6, 8, 9, 12, 13, 15, 16, 19, 20, 21, 24, 25, 27, 28

Hybrid GA

3, 6, 13, 17, 19, 20, 22, 23, 24, 26, 27

Method

G (yr)

H (yr)

I

J

General GA

25.02

22.11

78

68

Hybrid GA

25.96

23.33

92

73

General GA

22.62

19.65

75

70

Hybrid GA

23.54

21.83

90

71

General GA

17.91

17.42

80

71

Hybrid GA

18.11

18.09

95

74

General GA

17.07

16.88

79

67

Hybrid GA

17.39

17.52

93

69

Table 9. CPU time to obtain optimal solutions. Decrement 0%

parameters of the failure distribution of Units G and H are decreased by 15%, 30%, and 45%. As the reliability of Type 1 units decreases, the time interval between system failure and its detection may increase and can be reduced by frequent inspection. As a result, the life cycle cost and total number of inspections increase as the scale parameters of the failure distribution of Type 1 units decrease (Fig. 5 and Table 7). In addition, the PM of Units G and H must be performed frequently to improve their reliability. The PM frequency of Units I and J is not affected because the failure of each unit occurs independently. Thus, the preventive replacement ages of Units G and H only decrease as the scale parameters of the failure distribution of Type 1 units decrease (Table 8). From the numerical results, the hybrid genetic algorithm provides 5.70% lower life cycle cost on the average, but requires 27.46% larger CPU time on the average than the general genetic algorithm (Table 9).

Inspection points (yr) 3, 7, 8, 14, 20, 21, 23, 28

Table 8. Optimal preventive replacement ages of Units G and H and preventive maintenance thresholds of units of Units I and J.

15%

AIT

Method General GA

15% 30% 45%

Method

CPU time (sec)

General GA

3,790

Hybrid GA

6,493

General GA

4,809

Hybrid GA

6,709

General GA

5,019

Hybrid GA

6,935

General GA

5,368

Hybrid GA

7,298

4.3 CASE 3: Increasing the failure thresholds of Type 2 units In this last case, the target interval availability is 0.75, and the failure thresholds of Units I and J are increased by 15%,

Fig. 5. Life cycle costs for different scale parameters of the failure distribution.

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Table 10. Optimal inspection points of a one-shot system. Increment 0% 15%

Method

Inspection points (yr)

General GA

3, 7, 8, 14, 20, 21, 23, 28

45%

Increment

Method

G (yr)

H (yr)

I

J

General GA

25.02

22.11

78

68 73

Hybrid GA

4, 7, 11, 23, 24

General GA

4, 9, 14, 20, 25, 27

Hybrid GA

25.96

23.33

92

4, 10, 16, 25, 27

General GA

25.13

21.93

85

73

Hybrid GA

25.89

23.21

103

91

Hybrid GA 30%

Table 11. Optimal preventive replacement ages of Units G and H and preventive maintenance thresholds of units of Units I and J.

General GA

4, 10, 18, 24, 26

Hybrid GA

4, 14, 16, 20

General GA

4, 11, 20, 26

Hybrid GA

4, 13, 20, 27

0% 15% 30% 45%

General GA

24.92

21.72

89

80

Hybrid GA

26.06

23.31

111

116

General GA

24.82

22.06

95

91

Hybrid GA

25.91

23.18

124

126

Table 12. CPU time to obtain optimal solutions. Increment 0% 15% 30% 45%

Fig. 6. Life cycle costs for different failure thresholds of Type 2 units.

30%, and 45%. By increasing the time to failure of Units I and J, frequent inspection is not necessary and thus the life cycle cost and total number of inspections decrease (Fig. 6 and Table 10). Increasing the failure threshold of Units I and J does not affect the reliability improvement of Units G and H. Therefore, the preventive replacement ages of Units G and H do not change considerably. In addition, frequent PM of Units I and J to improve their reliability when their failure threshold increases is not necessary to be carried out. Table 11 shows the optimal preventive replacement ages of Units G and H and the PM thresholds of Units I and J that minimize the life cycle cost and satisfy the target interval availability of 0.75. The hybrid genetic algorithm also provides 5.57% lower life cycle cost on the average, but requires larger CPU time at 36.15% on the average than the general genetic algorithm (Table 12).

5. Conclusions A one-shot system is usually stored for long time and performs a mission at most once during its life cycle. The units in the one-shot system can be classified into two types of units: Type 1 units that fail at random times and Type 2 units that degrade with time. System reliability in storage deteriorates with time, thus requiring inspection to be performed periodically to immediately detect system failure. System reliability can also be improved by PM of units in the system. However,

Method

CPU time (sec)

General GA

3,790

Hybrid GA

6,493

General GA

4,136

Hybrid GA

6,873

General GA

4,509

Hybrid GA

7,096

General GA

4,963

Hybrid GA

7,318

maintenance frequency and maintenance costs must be balanced, forming the optimization problem in inspection and PM policies for a one-shot system. In this paper, we addressed an inspection interval problem for a one-shot system with two types of units and considered the degradation process of Type 2 units. To quantitatively evaluate the performance of the oneshot system, the interval availability over inspections and the life cycle cost, including the inspection operation, CM, and PM costs, were used as optimization criteria and estimated by simulation. A hybrid genetic algorithm with a heuristic method was proposed to find inspection intervals, preventive replacement ages of Type 1 units, and PM thresholds of Type 2 units in the one-shot system, which minimize the life cycle cost and satisfy the target interval availability. In the numerical examples, we first considered different values of target interval availability, and the results showed that the life cycle cost and total number of inspections increase as the target interval availability increases. In addition, the preventive replacement ages of Type 1 units and PM thresholds of Type 2 units decrease to satisfy high target interval availability. Subsequently, we decreased the time to failure of Type 1 units, and the results showed that the life cycle cost and total number of inspections increase whereas preventive replacement ages of Type 1 units decrease. However, the PM thresholds of Type 2 units do not considerably change because the failure of each unit is independent. Finally, as the failure thresholds of Type 2 units increase, the life cycle cost and the total number of inspections decrease, and the PM thresholds of Type 2 units increase. From the results of the numerical


experiments, the hybrid genetic algorithm has 6.13% lower life cycle cost on the average, but requires 35.33% larger CPU time on the average than the general genetic algorithm. In practical application, many one-shot systems are deployed at operational sites and maintenance sites have a certain amount of maintenance resources, such as spare parts, engineers, and maintenance equipment. Therefore, we need to consider the maintenance delay time contributed by the shortage of maintenance resources at the maintenance sites when the inspection interval of one-shot systems are determined. For further study, an inspection scheduling problem for one-shot systems under the constraint of maintenance resources may be explored.

Acknowledgment This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2060066).

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Won Young Yun is a Professor at the Department of Industrial Engineering, Pusan National University, Busan, Korea. He received his B.S. degree in Industrial Engineering from Seoul National University, Korea, and master’s and doctorate degrees from the Korea Advanced Institute of Science and Technology (KAIST). His current research interests focus on system reliability, simulation based optimization in reliability and maintenance, and logistics. Li Liu is a master’s candidate at the Department of Industrial Engineering, Pusan National University, Busan, Korea. Her research interests focus on inspection policy of one-shot systems and simulation-based optimization in inspection and maintenance.

Young Jin Han is a Ph.D. candidate at the Department of Industrial Engineering, Pusan National University, Busan, Korea. His research interests focus on system reliability and simulation-based optimization in reliability and maintenance.