A novel integrated model is proposed to optimize the redundancy allocation problem (RAP) and the reliabilitycentered maintenance (RCM) simultaneously. A system of both repairable and nonrepairable components has been considered. In this system, electronic components are nonrepairable while mechanical components are mostly repairable. For nonrepairable components, a redundancy allocation problem is dealt with to determine optimal redundancy strategy and number of redundant components to be implemented in each subsystem. In addition, a maintenance scheduling problem is considered for repairable components in order to identify the best maintenance policy and optimize system reliability. Both active and cold standby redundancy strategies have been taken into account for electronic components. Also, net present value of the secondary cost including operational and maintenance costs has been calculated. The problem is formulated as a biobjective mathematical programming model aiming to reach a tradeoff between system reliability and cost. Three metaheuristic algorithms are employed to solve the proposed model: Nondominated Sorting Genetic Algorithm (NSGAII), Multiobjective Particle Swarm Optimization (MOPSO), and Multiobjective Firefly Algorithm (MOFA). Several test problems are solved using the mentioned algorithms to test efficiency and effectiveness of the solution approaches and obtained results are analyzed.
In general, reliability is defined as ability of a system to meet required performance standards under specified conditions during a determined time horizon. It has a significant effect on manufacturing cost, company’s fame, production efficiency and environment, and so forth. There are two main approaches to enhance system reliability: implementing a proper maintenance policy and using effective redundancy strategies. Applying these approaches leads to increases in system reliability along with increasing costs of other resources. Thus, reaching a tradeoff between system reliability, cost, volume, weight, and so forth is significantly important [
There are two types of maintenance policies: corrective maintenance (CM) and preventive maintenance (PM). In corrective maintenance, the system is repaired or replaced after failure. However, prescheduled periodic maintenance actions are taken in preventive maintenance. It is obvious that preventive maintenance prevents major failures that impose high costs on the system. In recent years, many authors conducted variety of research on preventive maintenance scheduling problems. A mathematical model has been proposed by Goel and Gupta [
Martorell et al. [
However, selecting a proper maintenance policy is not all we can do to maximize system reliability. Identifying and implementing the best redundancy strategy is another way to optimize system reliability. One of the famous problems in field of reliability optimization is redundancy allocation problem (RAP). Redundant components are incorporated into the system to back up different parts of the system and prevent system breakdown under different redundancy strategies. There are two main redundancy strategies: (1) active redundancy, in which all redundant components are implemented in a parallel structure together from time zero and only one component is required to work at any given time, and (2) standby redundancy, in which a sequential order is determined for using the redundant components at component failure time. Three variants of the standby redundancy strategy are called cold, warm, and hot. Each strategy can be implemented in a different part of a system. RAPs are proved to be NPhard by Chern [
Coit [
A large number of studies on redundancy allocation problem have been conducted after 2010. Among those who studied multiobjective RAP (MORAP), Zio and Bazzo [
Firefly Algorithm as a new metaheuristic optimization method was introduced in 2008 by Yang [
In this paper, a novel mathematical model of a system of repairable and nonrepairable components is formulated. The model contains two objectives: firstly, it aims to select a proper redundancy strategy for nonrepairable part of the system and secondly, it offers a maintenance policy for repairable part of the system. Minimizing net present value of total cost and maximizing system reliability are objectives of the problem. In addition, different types of redundancy strategies, repair, and replacement actions are considered in order to model the problem as realistic as possible. Other practical constraints such as available budget for purchasing redundant components, volume, weight, and maximum allowed failure rate in each inspection period are taken into account. Due to NPhardness of the problem, the authors tried to employ metaheuristic methods to solve proposed model. Three common solution approaches called NSGAII, MOPSO, and MOFA were selected based on the Vanoye and Parra classification. RuizVanoye and DíazParra [
Remainder of the paper is organized as follows. In Section
In this section, a new integrated mathematical model is proposed for redundancy allocation and reliabilitycentered maintenance problems. Objective of the reliability problems could be one or a set of the following objectives: maximizing system reliability and minimizing cost, weight, and volume of the system. In this paper, system reliability and costs including maintenance and operational costs are considered as objectives.
In most articles, the system under study includes either repairable or nonrepairable components. However, systems usually consist of repairable and nonrepairable components simultaneously in real world [
In this paper, a system of electronic and mechanical components has been considered. Figure
A system of mechanical and electronic subsystems.
The system is comprised of two subsystems in series: mechanical components and electronic components (Figure
Selecting optimal maintenance policy for mechanical components is considered in order to maximize system reliability.
Selecting a proper redundancy strategy, active or cold standby, and determining the number of redundant components in the electronic section is taken into account aimed at improving system reliability. Selecting active redundancy strategy adds operational costs to the system cost while selecting cold standby redundancy strategy threatens system performance by imperfect switching.
It is possible to use different type of components with different initial and operational costs and failure rates for the electronic subsystem.
Since unstable market and economic conditions may have serious effects on results, inflation rate and time value of money are considered in computations.
Required resources such as financial resources, human resources, volume, and weight are known deterministically.
Timetofailure distributions of components are independent.
A fixed amount of budget is available at time zero to purchase electronic components (initial cost).
Secondary cost of the system is calculated by taking operational costs of the electronic subsystem and maintenance costs of the mechanical subsystem into account during the system running period (mission time).
The system mission time is finite.
Repair and replacement times and restoration times are calculated as the system downtime cost.
In cold standby strategy, redundant components do not fail before their activation. In addition, failure rates in active redundancy strategy are larger than cold standby strategy because active redundant components are exposed to the operational stresses. (i.e.,
Failure detection mechanism and switching are imperfect.
The constraints of the problem can be formulated as follows:
The proposed model contains two objectives:
Maximizing system reliability in each period by selecting optimal redundancy strategy and maintenance policy.
Minimizing secondary cost of the system including operational costs of the electronic subsystem and maintenance costs of the mechanical subsystem during the system mission.
The system under study consists of two electronic and mechanical subsystems connected in series according to Figure
Let
Total cost of the system (
It should be noted that, in active redundancy, expected value for failure of each component in each period (
As mentioned in Introduction and Literature Review, the solution methods were selected based on the RuizVanoye and DíazParra [
The proposed model for the problem contains two conflicting objectives. We try to make a tradeoff between these objectives to achieve a desired level of optimality for each objective. One of the common approaches to solve multiobjective problems is the weightedsum method that converts the problem into a single objective problem by making a weighted linear combination (WLC) of objectives. Although it is a very popular method due to its simplicity and ease of implementation, it has some major disadvantages such as determining weight of each objective and lack of information about it. Thus, another method called Pareto set has been developed. Pareto set method produces set of solutions within the feasible region of the problem that dominate other feasible solutions. The nondominated solution sets are called Pareto optimal solutions and other inferior solutions are called dominated solutions. The decision maker selects the final set of Pareto optimal solutions according to his/her preferences and considered criteria. In this paper, three metaheuristic algorithms have been employed to produce Pareto optimal solutions.
The NSGAII is developed for solving multiobjective problems by adding two operators to the classic Genetic Algorithm (GA) to find Pareto optimal sets instead of finding the unique optimal solution [
In our study, the proposed chromosome has two parts. The first part represents the electronic section and the second part represents the mechanical section. The electronic section contains
Electronic section matrix.

 

1  2  ⋯ 

Redundancy strategy  
1  2  0  ⋯  1  1 
2  1  1  ⋯  2  2 
⋮  ⋮  ⋮  ⋯  ⋮  ⋮ 

0  2  ⋯  1  1 
The proposed chromosome for the mechanical section is indicated by two matrixes with
Crossover in Genetic Algorithm.
The Multiobjective Particle Swarm Optimization (MOPSO) is a metaheuristic algorithm capable of producing high quality nondominated Pareto optimal solutions with high diversities for multiobjective problems. The MOPSO is widely used by researchers due to its simplicity and successful performance in continuous optimization problems. The idea of this algorithm is inspired by a swarm of birds looking for food [
In MOPSO, all objective functions are calculated and evaluated for each particle and the nondominated solutions (based on Pareto set concept) found by the particles are stored in a repository. The size of repository is limited and is set by decision maker. In addition, the search space is divided into hypercubes with a fitness value. Fitness value is inversely proportional to the number of particles it contains [
Firefly Algorithm (FA) is the last method applied in this study. It was introduced by Yang in 2010 [
All fireflies are unisex and attracting a firefly by another one is regardless of its sex.
Attractiveness is proportional to the brightness and both of these features will decrease with increasing distance. Less bright fireflies are always attracted to the brighter one and if there is no brighter one, the move will be randomly.
The brightness of fireflies is defined according to the objective function (like fitness function in Genetic Algorithm).
Single objective form of Firefly Algorithm (FA) was developed to Multiobjective Firefly Algorithm (MOFA) in 2013 by Yang [
Define objective functions:
Generate initial population of fireflies
Formulate light intensity
Define absorption coefficient
While
for
Evaluate approximations
if
Move firefly
if pervious position doesn’t dominate new one
New position replaced with old one
end if
end if
end for
Update and pass non dominated solution to next generation
Update
End while
Since that, variables of our problem are binary and integer and MOPSO algorithm and MOFA find solutions in continuous space; round function is used to convert real number to the integer, and to change detected solutions to the binary style, sigmoid function is applied.
In this section, assigning proper values to the parameters of algorithms and using comparison metrics for evaluating solution methods are discussed first. Then, three different sets of test problems (small, medium, and large size) are tackled and solved using the chosen solution methods. Finally, obtained results by each algorithm have been compared and the obtained results for an example are explained.
Setting proper values for the control parameters of metaheuristic algorithms has a significant effect on their desirable performance. Welltuned parameters empower the algorithms in producing better solutions within shorter computation times. Thus, setting proper values for control parameters is a critical task [
Range of the main parameters.
NSGAII  MOPSO  MOFA  

Parameter  Range  Parameter  Range  Parameter  Range 
Pop. size ( 
20–100  Pop. size ( 
20–100  Pop. size ( 
10–100 
Max iteration number (Maxit)  50–200  Max iteration number (Maxit)  50–200  Max iteration number (Maxit)  50–200 
Cross rate (Cr)  0.5–0.9  Inertia weight ( 
0.4–0.9  Randomization parameter 
0.1–0.9 
Mutation rate (Mr)  0.01–0.3  Cognitive factor 
12  Fixed light absorption coefficient 
1–3 
Social factor 
12 
In the next step, the response variables should be determined. Three performance metrics are chosen as response variables which are CPU time, number of nondominated solutions (NNS), and Diversification Metric (DM). These metrics were selected based on the two features, the convergence speed and diversity of the detected solutions. More details on the performance metrics can be found in Section
Central composite design (CCD) with 6 center points is applied for the experiments. Experiments are run by Design Expert 9. According to the number of input variables and type of the design, different number of experiments should be run. For instance, in case of the four parameters and 6 center points’ design, 46 experiments are required. After performing the experiments, analysis of variance (ANOVA) is applied to fit an adequate model to the experimental data. Last step is setting goals for responses to generate optimal condition (optimal level of the parameters). Here, we aim to minimize CPU time and maximize NNS and DM.
Figure
Optimum values of the algorithms parameters.
NSGAII  MOPSO  MOFA  

Pop. size  100  Pop. size  100  Pop. size  25 
Max iteration number  50  Max iteration number  50  Max iteration number  200 
Cross rate  0.9 

0.9 

0.1 
Mutation rate  0.3 

1 

1 

2 
Counter plot for MOFA: desirability versus Maxit and
Other parameters are set according to the literatures as follows:
MOPSO:
According to Deb et al. [
In order to examine performance of the algorithms, three sets of test problems with different sizes (small, medium, and large) are simulated. The dimensions of these sets are shown in Table
Test problems dimensions.
Case  Number of electronic components; 
Number of mechanical components; 
Number of component types in the electronic subsystem; 
Number of component types in the mechanical subsystem; 

Small size 




Medium size 




Large size 




Mean and standard deviation of the metrics for the different sizes of the test problems.
Problem size  Time (seconds)  NNS  DM  MS  

NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  
Small  1.91 

3.50 

12.3 
3.3 
55.35 

22.60 
0.01493 

0.04230 
Medium  2.59 

6.99 

11.90 
3.85 
39.02 

28.65 
0.0238 

0.0482 
Large  31.49 

91.25 

12.84 
4.11 
51.35 

34.35 
0.00419 

0.07408 
Four metrics are considered to evaluate algorithms performance (Table
It also can be concluded from Table
Obtained values for the DM and MS indicate higher diversity solutions of the produced nondominated solutions by MOPSO in all three cases. This means that the MOPSO method has a wider spread.
Our model has two conflicting objectives, reliability and cost. As the reliability of the system goes higher its cost increases too. We try to find an optimal tradeoff between these objectives. To compare quality of solutions generated by three approaches, mean and lower and upper bounds of objectives are reported in Tables
Boundary values for reliability.
Method  Small size problems  Medium size problems  Large size problems  

NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  
Mean  0.991548  0.945431  0.96181  0.985369  0.914826  0.938127  0.99101  0.86809  0.90353 
Lower bound  0.981583  0.908162  0.939269  0.970327  0.860268  0.909683  0.98771  0.80074  0.86255 
Upper bound  0.996518  0.988152  0.980975  0.994157  0.974125  0.957834  0.99190  0.94698  0.93663 
Boundary values for cost.
Method  Small size problems  Medium size problems  Large size problems  

NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  
Mean  1713.687  1734.326  1870.389  2944.94  3035.20  3189.41  4725.122  4900.882  5166.778 
Lower bound  1652.382  1632.584  1790.569  2891.92  2883.30  3070.47  4662.533  4586.47  5015.414 
Upper bound  1848.888  2014.234  1987.349  3065.41  3356.64  3346.01  4889.001  5529.174  5369.501 
As it can be concluded from Tables
In remainder of this section, an example is considered based on the proposed model and the results are explained. Tables
This example is solved using three approaches for ten times and the mean and standard deviation values for the performance metrics and objective functions boundary are reported in Tables
Mean and standard deviation values for performance metrics.
Algorithm  NSGAII  MOPSO  MOFA  

Mean  Std Dev  Mean  Std Dev  Mean  Std Dev  
Time  2.74  0.12 

0.06  6.84  0.49 
NNS 

5.19  13.20  3.65  3.70  1.06 
DM  44.96  10.79 

15.11  30.17  12.21 
MS  0.0068  0.0165 

0.0132  0.0508  0.0218 
Objectives boundary values for the sample.
Method  Reliability  Cost  

NSGAII  MOPSO  MOFA  NSGAII  MOPSO  MOFA  
Mean  0.999421  0.87153  0.90719  3848.62  3900.48  4285.62 
Lower bound  0.999825  0.802065  0.841807  3760.333  3485.712  4044.209 
Upper bound  0.999833  0.977786  0.951062  4027.349  4488.861  4585.62 
As concluded before, MOPSO method provides better values for the metrics (time, NNS, DM, and MS) and wider boundary values for the objectives but as Table
The proposed model in this paper aims at finding proper maintenance policies and effective redundancy strategies. Table
Selected maintenance actions and redundancy strategies by the NSGAII.
Electronic section  Mechanical section  




 
1  2  3  4  Strategy  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  
1  0  0  1  0  A 
















2  1  1  0  0  C 
















3  1  0  0  0  A 
















4  0  1  0  0  C 
















5  0  1  0  0  A 
















6  0  1  0  1  A 
















7  0  0  1  0  A 
















8  0  1  0  0  A 
















9  1  0  1  0  A 
















10  1  0  0  0  C 
















11  0  0  1  0  C 
















12  2  0  0  1  A  
13  0  1  0  0  C  
14  1  0  0  0  A 
Component data for electronic subsystem.

Choice 1 ( 
Choice 2 ( 
Choice 3 ( 
Choice 4 ( 













 
1  0.00532  0.004  2  0.000726  0.0012  1  0.00499  0.004  2  0.00818  0.007  3 
2  0.00818  0.007  3  0.000619  0.0001  1  0.00431  0.003  2  —  —  — 
3  0.0133  0.012  3  0.0110  0.011  3  0.0124  0.011  3  0.00466  0.004  2 
4  0.00741  0.006  2  0.0124  0.012  3  0.00683  0.006  2  —  —  — 
5  0.00619  0.005  1  0.00413  0.004  2  0.00818  0.007  3  —  —  — 
6  0.00436  0.003  3  0.00567  0.005  3  0.00268  0.002  2  0.000408  0.0004  1 
7  0.0105  0.01  3  0.00466  0.004  2  0.00394  0.003  2  —  —  — 
8  0.0105  0.01  3  0.00105  0.0006  1  0.0105  0.01  3  —  —  — 
9  0.00268  0.002  2  0.000101  0.00005  1  0.00408  0.003  1  0.000943  0.00005  1 
10  0.0141  0.013  3  0.00683  0.006  2  0.00105  0.0005  1  —  —  — 
11  0.00394  0.003  2  0.00355  0.003  2  0.00314  0.002  2  —  —  — 
12  0.00236  0.001  1  0.00769  0.007  2  0.0133  0.012  3  0.0110  0.01  3 
13  0.00215  0.001  2  0.00536  0.005  3  0.00665  0.006  3  —  —  — 
14  0.0110  0.001  3  0.00834  0.003  1  0.00355  0.003  2  0.00436  0.004  3 

Choice 1 ( 
Choice 2 ( 
Choice 3 ( 
Choice 4 ( 

















 
1  10  1  3  3  10  1  4  4  20  2  2  2  20  2  5  5 
2  20  2  8  8  10  1  10  10  10  1  9  9  —  —  —  — 
3  20  2  7  7  30  3  5  5  10  1  6  6  40  4  4  4 
4  30  3  5  5  40  4  6  6  50  5  4  4  —  —  —  — 
5  20  2  4  4  20  2  3  3  30  3  5  5  —  —  —  — 
6  30  3  5  5  30  3  4  4  20  2  5  5  20  2  4  4 
7  40  4  7  7  40  4  8  8  50  5  9  9  —  —  —  — 
8  30  3  4  4  50  5  7  7  60  6  6  6  —  —  —  — 
9  20  2  8  8  30  3  9  9  40  4  7  7  30  3  8  8 
10  40  4  6  6  40  4  5  5  50  5  6  6  —  —  —  — 
11  30  3  5  5  40  4  6  6  50  5  6  6  —  —  —  — 
12  20  2  4  4  30  3  5  5  40  4  6  6  50  5  7  7 
13  20  2  5  5  30  3  5  5  20  2  6  6  —  —  —  — 
14  40  4  4  6  40  4  6  7  50  5  6  6  60  6  5  9 
Component data for mechanical subsystem.
Parameters 
 

1  2  3  4  5  6  7  8  9  10  11  

—  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004167  0.0004167  0.0004167 




0.00056  0.00056  0.00056  0.00056  0.00056  0.00056  0.00056  0.00056  0.0005834  0.0005834  0.0005834 

0.00085  0.00085  0.00085  0.00085  0.00085  0.00085  0.00085  0.00085  0.0008784  0.0008784  0.0008784  




0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004167  0.0004167  0.0004167 

0.0005  0.0005  0.0005  0.0005  0.0005  0.0005  0.0005  0.0005  0.0005167  0.0005167  0.0005167  



—  2.5  2.5  2.5  2.5  2.5  2.5  2.5  2.6  2.6  2.6  2.4 




2  1.5  2.5  3  1.8  3.5  3  2  2.5  1.5  2 

1  1  1.5  1.5  1  2  1.5  1  1.5  0.5  1  




4  3  5  6  4  7  6  4  3.5  3  4.5 

3  1.5  3  2  2  4  3  1.5  2  1.5  2.5  




1  1  2  1  2  1  1  2  1  1  2 

2  1  1  1  1  2  1  1  2  1  1  




1  2  1  2  2  1  2  2  1  2  1 

1  2  2  1  2  2  1  2  2  1  2  




2  2  2  2  2  2  2  2  2  2  2 

1.5  1.5  1.5  1  1  1.5  1.5  1  1  1.5  1.5  




8  7  6  8  9  7  6  5  7  6  9 

3  2  5  4  5  5  2  3  3  2  3 
Upper bound of parameters.
Parameter 













Value  5  3  4  0.0009  1000  20  200  200  0.001  0.03  0.99 
Reliabilitycost Pareto solutions obtained by (a) NSGAII and (b) MOPSO.
In this paper, a biobjective reliability model by integrating redundancy allocation problem (RAP) and reliabilitycentered maintenance (RCM) problem for a system of nonrepairable electronic components and repairable mechanical components has been proposed. Objectives of the problem are maximizing the system reliability and minimizing the system operational and maintenance costs. In order to improve system reliability, active and cold standby redundancy strategies and periodic maintenance actions are considered for electronic section and mechanical section, respectively. Total system cost includes initial costs for purchasing the required equipment and secondary costs such as operational costs of the electronic section and maintenance costs of the mechanical section. Initial costs are taken into account by setting a budget constraint and operational costs are considered as the second objective of the problem. Three metaheuristic algorithms, NSGAII, MOPSO, and MOFA, are used to solve the proposed model. Different sets of test problems (small, medium, and large size) have been generated for evaluating solution methods. Obtained results indicate that MOPSO algorithm requires less time to produce Pareto optimal sets with high diversities according to the DM and MS. NSGAII outperforms MOPSO and MOFA in terms of generating more nondominated solutions (NNS) with better values for system reliability and cost. Finally an example was solved and the obtained results were explained.
Electronic components,
Mechanical components,
Redundancy strategy; 1 for active and 2 for cold standby,
Component type in the electronic subsystem,
Component type in the mechanical subsystem,
Number of available types for component
Number of available types for component
Type of maintenance activities performed on component
Time,
Maximum allowed weight and volume for system
Maximum allowed number of components in subsystem
Initial available budget to purchase electronic components
Available operators to perform maintenance activities in each period
Weight of type
Volume of type
Purchasing cost for type
Operational cost for type
Cost of repair type
Replacement cost for type
Cost of system downtime due to performing repair type
Cost of system downtime due to replacing for type
Number of operators required to perform repair type
Number of operators required to replace type
Failure rate of type
Failure rate of component
Failure rate of type
Maximum allowed failure rate for each mechanical component in each period
Rate of increase in failure rate for mechanical component
Compound interest rate based on time periods
System mission time
Number of inspections during each time unit.
Number of components
1 if redundancy strategy
If repair type
If type
Failure rate of component
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewer for the precious and constructive comments which led to significant improvement in the paper.