# Reliability Optimization and Importance Analysis of Circular-Consecutivek-out-of-n System

Authors: Shuai Zhang; Jiang-bin Zhao; Hong-guang Li; Ning Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The circular-consecutivek-out-of-n:F(G) system (Cir/Con/k/n:F(G) system) usually consists of n components arranged in a circle where the system fails (works) if consecutive k components fail (work). The optimization of the Cir/Con/k/n system is a typical case in the component assignment problem. In this paper, the Birnbaum importance-based genetic algorithm (BIGA), which takes the advantages of genetic algorithm and Birnbaum importance, is introduced to deal with the reliability optimization for Cir/Con/k/n system. The detailed process and property of BIGA are put forward at first. Then, some numerical experiments are implemented, whose results are compared with two classic Birnbaum importance-based search algorithms, to evaluate the effectiveness and efficiency of BIGA in Cir/Con/k/n system. Finally, three typical cases of Cir/Con/k/n systems are introduced to demonstrate the relationships among the component reliability, optimal permutation, and component importance.


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## Body

## 1. Introduction

The component assignment problem (CAP) [1] is a kind of classic problem in the optimization of system reliability. The system is composed of n positions and n components with different reliabilities, and then the n components should be assigned into n different positions to find the optimal assignment with the maximum system reliability. So, the CAP is usually a kind of combinatorial optimization and generally NP-hard problem [2]. The optimization of consecutive k-out-of-n system (Con/k/n system) is a typical case of CAP, and its purpose is to ensure that the system reliability remains largest. Con/k/n system contains linear-consecutive k-out-of-n:F(G) system and circular-consecutive k-out-of-n:F(G) system.A linear-consecutivek-out-of-n:F(G) system (Lin/Con/k/n:F(G) system) is an ordered sequence of n components arranged in a line such that the system fails (works) if and only if at least k consecutive components fail (work). Lin/Con/k/n systems are used in most system designs, such as telecommunication or pipeline networks, the allocation of microwave towers and street lamps, and arrangement of spacecraft relay stations [3]. For a pipeline network, the petroleum or gas is sent to other places from the origin through the pipeline, which assumes that the pumps are arranged in the equidistant spacing. Each pump has the ability to send the gas to the following k pumps. If a pump is failed, the pipeline system also can work normally unless the k consecutive pumps all failed. Actually, the pipeline system is a Lin/Con/k/n:F system which is widely used in the practical engineering.A circular-consecutivek-out-of-n:F(G) system (Cir/Con/k/n:F(G) system) is an ordered sequence of n components arranged in a circle such that the system fails (works) if and only if at least k consecutive components fail (work). Cir/Con/k/n systems are used in the camera system of nuclear accelerator and computer ring network [3]. For a nuclear accelerator, k high-speed cameras are arranged around the accelerator to record the motion state of various particles. If and only if at least k cameras work properly, the complete motion of particles can be recorded successfully. Actually, the camera system of nuclear accelerator is a Cir/Con/k/n:G system.In recent years, many researchers have tried to optimize the Con/k/n system by combining the heuristic algorithm with importance measure.The concept of importance measure was first proposed by Birnbaum for binary systems in 1969 [4]. Then, it made substantial progress and was applied to broad engineering practice. Vesely [5] and Fussell [6] put forward the Fussell-Vesely importance based on the fault tree analysis. Nakashima and Yamato [7] raised the uncertainty importance for the component to discriminate the component which affected the system reliability significantly. Hong and Lie [8] proposed the joint importance to evaluate the effect on system reliability which is caused by the interaction of the components. Borgonovo and Apostolakis [9] designed the differential importance measure for the probabilistic safety assessment. Si et al. [10–12] introduced the integrated importance measure and do some research of integrated importance measure in various situations. Based on the integrated importance measure, Dui et al. [13] extended the integrated importance measure from unit time to system lifetime and to different life stages; then Dui et al. [14] also studied how the transition of component states affects system performance under the semi-Markov process. Si et al. [15] studied the component reliability importance with the changes of the optimal component sequence for Lin/Con/k/n systems. Liu et al. [16] proposed the generalized Griffith importance measure to evaluate the accurate contribution of the components for continuous state systems.For the optimal assignment, Kontoleon [17] gave an iterative algorithm to calculate the optimal permutation. This algorithm could assign the component with the lowest reliability into all the positions in the system and order the component based on the Birnbaum importance and finally place the component into the system through iterative allocation. Zuo and Kuo [18] proposed two similar heuristic algorithms, called ZKA and ZKB, which could be applied into the Lin/Con/k/n systems. Then, Zhu et al. [19] put forward two new algorithms, referred to as ZKC and ZKD, to improve the ZK algorithms. Lin and Kuo [20] established a greedy algorithm (which is called LKA) based on the Birnbaum importance by assigning components one by one into the system. On the basis of LKA, Yao et al. [21] built the LK type algorithms, which included LKA, LKB, LKC, and LKD, and designed a Birnbaum importance-based two-stage approach (BITA) according to the numerical experimentation of ZK and LK algorithms. BI is one of the most widely investigated importance measures and has been applied to the CAP especially in [17–21]; therefore, this article focuses on the BI applied to Cir/Con/k/n systems for the CAP.Evolutionary algorithm is a kind of advanced heuristic algorithm, which combines the random algorithm with local search. Problem-independent technique is used to solve various complex problems, such as genetic algorithm [22], simulated annealing algorithm [23], particle swarm optimization [24], tabu search algorithm [25], and neural net algorithm [26]. The research of combining metaheuristics with importance measure is also developed to solve CAP. Cai et al. [27] proposed an improved genetic algorithm based on heuristic method (BGA) to deal with the CAP, and the research illustrates that BGA is more effective for the systems with arbitrary reliable components. Yao et al. [28] constructed a Birnbaum importance-based genetic local search algorithm (BIGLS), which is a comprehensive genetic algorithm to reduce the solution space of the optimal solution based on the local search. When the components of CAP are less, local search could improve the accuracy and convergence speed of the algorithm, but it will take longer time. Cai et al. [29] proposed a Birnbaum importance-based genetic algorithm (BIGA) to analyze the performance of the algorithm in the Lin/Con/k/n systems, which is stable and feasible to solve the general CAP with stronger robustness.The rest of this paper is organized as follows. In Section2, BIGA is introduced to optimize the reliability of Cir/Con/k/n systems which can break through the limitation of local optimal solution. By comparing with BITA and BIGLS, the numerical examples are implemented to discuss the optimization results of BIGA in the small and large systems. In Section 3, three typical cases with different n, k, and p in Cir/Con/k/n system are implemented, and the relationships among the Birnbaum importance and the optimal assignment are discussed. Finally, conclusions of the research are summarized in Section 4.


## 2. Reliability Optimization Method for Cir/Con/k/n System

### 2.1. BIGA for Cir/Con/k/n System

In order to optimize the reliability of Cir/Con/k/n systems efficiently, the BIGA [29] is introduced in this section. The detailed process of BIGA is as follows, and the flowchart is shown as in Figure 1.(1)
For the optimization problem, the objective function is system reliability of Cir/Con/k/n system, and the solution is the permutation of the components with maximum reliability. The system reliability can be calculated based on the literature [30].(2)
Choose the real-number encoding method and determine the encoding space for individuals.(3)
Generate an initial parent populationp(t) which contains M individuals. Perform Birnbaum importance-based local search on all the M individuals, and update the initial parent population p(t).(4)
For the populationp(t), calculate the fitness of each individual.(5)
When the generation meets the termination condition 1, which is the limit of the generation scale, the algorithm will be terminated and the optimal solution will be output. If not, go to Step (6).(6)
Selection is performed on the current populationp(t), and the best chromosome will be selected and saved.(7)
Measure the fitness scaling of each individual.(8)
For the current population, when the termination condition 2 is satisfied, the process will be terminated and the optimal solution will be output. The termination condition refers to the convergence degree of populations; that is,(favg-fmin)·Sf+(fmax-favg)<α, where fmin, favg, and fmax represent the minimum, average, and maximum fitness of chromosomes, respectively. Sf>1 is conversion factor, and α is a very small positive number. If not, go to Step (9).(9)
For the current populationp(t), select M new individuals according to the fitness of each individual.(10)
Perform the crossover on the previous selectedM individuals to generate offspring population which contains M individuals.(11)
Perform mutation on the previous selected offspring population to generate the new offspring populationp(t+1).(12)
Measure the fitness of every individual in the offspring population.(13)
Perform the BITA on the offspring chromosomes. If the fitness of individual is larger than that of the optimal chromosome, the individual will not perform the BITA.(14)
Perform the elitist strategy, replace the lowest fitness chromosome of offspring populationp(t+1) by the optimal chromosome, and then update the offspring population p(t+1).(15)
Replace the initial populationp(t) by the new offspring population p(t+1), and turn back to Step (5).Figure 1
The flow chart of BIGA.


### 2.2. Numerical Experiments

System reliability depends not only on its inherent structure, but also on the reliabilities of components and the assigned positions of components in the system. The low, high, and arbitrary reliable components are introduced into the experiments to test the performance of BIGA. Their reliabilities are randomly distributed on[0.01,0.2], [0.8,0.99], and [0.01,0.99], respectively.This paper defines that the number of components for the small system is below 8 (including 8), and the number is more than 15 for the large system. Cir/Con/k/n systems include F systems and G systems. In the experiments, the small system cases include 6 systems (g1,g2,g3,g4,f1,f2), and the large system cases include 12 F systems and 16 G systems. The symbols of small systems and large systems are shown in Tables 1 and 2, respectively.Table 1
The symbolic representation of small systems.
g
1: Cir/con/2/5:G
g
2: Cir/con/2/6:G
g
3: Cir/con/3/5:G
g
4: Cir/con/3/6:G
f
1: Cir/con/3/5:F
f
1: Cir/con/3/6:FTable 2
The symbolic representation of large systems.
System
Mark
Cir/Con/3/20:F
F
1
Cir/Con/4/20:F
F
2
Cir/Con/5/20:F
F
3
Cir/Con/3/30:F
F
4
Cir/Con/4/30:F
F
5
Cir/Con/5/30:F
F
6
Cir/Con/3/50:F
F
7
Cir/Con/4/50:F
F
8
Cir/Con/5/50:F
F
9
Cir/Con/3/100:F
F
10
Cir/Con/4/100:F
F
11
Cir/Con/5/100:F
F
12
Cir/Con/2/20:G
G
1
Cir/Con/3/20:G
G
2
Cir/Con/4/20:G
G
3
Cir/Con/5/20:G
G
4
Cir/Con/2/30:G
G
5
Cir/Con/3/30:G
G
6
Cir/Con/4/30:G
G
7
Cir/Con/5/30:G
G
8
Cir/Con/2/50:G
G
9
Cir/Con/3/50:G
G
10
Cir/Con/4/50:G
G
11
Cir/Con/5/50:G
G
12
Cir/Con/2/100:G
G
13
Cir/Con/3/100:G
G
14
Cir/Con/4/100:G
G
15
Cir/Con/5/100:G
G
16Considering the fact that a single instance of algorithm may have error, thus 100 instances of one case as a test have been regarded. For each test, the reliabilities of components are randomly generated based on the type of components, which is the same in a test. Two typical experiments are introduced to compare the performance of the three algorithms, BIGA, BIGLS, and BITA, and all the three types of components should be considered in the experiments. Experiment 1 is used to compare the performance of three algorithms in small systems, and Experiment 2 is used for large systems. In all experiments, the BIGLS and BIGA are implemented with initial population sizeM=20, the conversion factor Sf=3, maximum generations T=200, crossover probability pc=0.8, mutation probability pm=0.05, and α=0.0001.Experiment 1.
In order to verify the performance of BIGA for the CAP in small systems, the experiment is implemented to compare with three algorithms, respectively, for six small systems. In this experiment, the times of achieving the optimal solution are recorded, and the optimal solution is received by enumeration method. When the solution of algorithm is better than that of enumeration method, the value of “achieve” will be increased by 1.Experiment 2.
In order to verify the performance of BIGA for the CAP in large systems, all analysis methods are similar to Experiment 1. In the large system, the optimal solution cannot get in a reasonable time by enumeration method. In order to compare the performance of BIGA and BIGLS, the optimal solution is gained by BITA. When the solution of BIGA and BIGLS is better than that of BITA, the improved number (IN) will be increased by 1.


### 2.3. Experimental Results Analysis

In this paper, all the results are implemented on MATLAB 7.11, and the computer configuration is the Intel (R) Core (TM) 2.3 GHZ processor and 2 GB of memory.Experiment 1.
The results of small systems are shown as Table3. According to the result, we can construct Figure 2 to tell which algorithm is better. From Figure 2, it can be inferred that three algorithms can obtain the optimal solution, and the “achieve” of BIGA and BIGLS is better than that of BITA. For further analysis of Figure 2, the curve volatility of BIGLS and BIGA is stable in addition to the high reliability systems. In order to illustrate the performance of BIGA and BIGLS for different types of components, the mean achievement values of the two algorithms are calculated, and the results are shown in Figure 3. It can be clearly seen that BIGLS is slightly more efficient than BIGA in the low reliable systems, but less effective and inferior to BIGA in the high and arbitrary systems.Table 3
Experimental results of three algorithms for small systems.
System
Algorithm
Achieve
Low
High
Arbitrary
g
1
BITA
73
79
67
BIGLS
100
95
99
BIGA
100
96
100
g
2
BITA
57
76
52
BIGLS
99
87
99
BIGA
98
89
100
g
3
BITA
72
95
66
BIGLS
100
98
100
BIGA
100
100
100
g
4
BITA
81
79
60
BIGLS
100
100
99
BIGA
99
100
99
f
1
BITA
86
85
89
BIGLS
100
98
100
BIGA
100
100
100
f
2
BITA
68
66
64
BIGLS
99
100
98
BIGA
100
100
100Figure 2
The achievement of three algorithms for small systems.
(a)
Low reliable components
(b)
High reliable components
(c)
Arbitrary reliable componentsFigure 3
Average of achievement for BIGLS and BIGA.Experiment 2.
The experimental results of the large systems are shown in Table4. According to the IN values in Table 4, Figure 4 is obtained to illustrate the IN values of BIGLS and BIGA in the large systems. In each subgraph, the abscissa expresses the large systems, and the ordinate shows the IN values of BIGLS and BIGA. From the subgraphs (a) and (d), the IN values of two algorithms are close, and the IN value of BIGLS is worse than that of BIGA except 4 low reliable systems (F3, F7, F8, and F11) and 3 high reliable systems (G11, G12, and G15). Observing other subgraphs in Figure 4, the IN value of BIGA is better than that of BIGLS, which shows that BIGA can improve the performance of reliability optimization. In subgraph (d), it is not hard to find that the IN values of the two algorithms are close and small when n is more than 50. In these systems, the solution generated by the BITA is the optimal solution or approximate optimal solution, and the optimization space of BIGA and BIGLS is relatively small. Therefore, the three algorithms are all good for optimization of Cir/Con/k/n systems.Table 4
Experimental results of the large systems by BIGA and BIGLS.
System
Algorithm
IN
Low
High
Arbitrary
F
1
BIGLS
44
74
65
BIGA
45
81
73
F
2
BIGLS
49
63
97
BIGA
49
79
98
F
3
BIGLS
56
52
99
BIGA
53
57
99
F
4
BIGLS
63
50
79
BIGA
64
86
92
F
5
BIGLS
59
48
75
BIGA
59
82
88
F
6
BIGLS
55
28
99
BIGA
56
29
100
F
7
BIGLS
41
20
23
BIGA
32
98
97
F
8
BIGLS
96
16
44
BIGA
95
84
98
F
9
BIGLS
69
9
97
BIGA
77
11
100
F
10
BIGLS
19
18
21
BIGA
20
99
97
F
11
BIGLS
97
14
41
BIGA
96
87
98
F
12
BIGLS
75
3
90
BIGA
79
4
100
G
1
BIGLS
32
3
38
BIGA
53
3
62
G
2
BIGLS
23
46
78
BIGA
32
53
84
G
3
BIGLS
23
90
65
BIGA
30
92
71
G
4
BIGLS
24
91
63
BIGA
25
91
66
G
5
BIGLS
15
0
8
BIGA
46
0
19
G
6
BIGLS
7
11
53
BIGA
22
21
87
G
7
BIGLS
4
43
62
BIGA
21
45
86
G
8
BIGLS
4
69
75
BIGA
8
66
83
G
9
BIGLS
12
0
3
BIGA
41
0
5
G
10
BIGLS
1
3
13
BIGA
9
7
42
G
11
BIGLS
1
7
60
BIGA
2
6
88
G
12
BIGLS
1
9
77
BIGA
1
8
90
G
13
BIGLS
0
0
3
BIGA
31
0
5
G
14
BIGLS
0
1
10
BIGA
7
1
39
G
15
BIGLS
0
2
55
BIGA
1
1
90
G
16
BIGLS
1
4
82
BIGA
1
4
93Figure 4
Experimental results of BIGA and BIGLS.
(a)
F system with low reliable components
(b)
G system with low reliable components
(c)
F system with high reliable components
(d)
G system with high reliable components
(e)
F system with arbitrary reliable components
(f)
G system with arbitrary reliable componentsThrough the typical experiments, it is clear that the BIGA can get better performance of reliability optimization than BIGLS and BITA. In the low reliable systems, sometimes BIGLS is better than BIGA, but the results of two algorithms are nearly identical. In the high or arbitrary reliable systems, the BIGA is better than BIGLS or BITA. The BIGA can improve the effectiveness and efficiency of solving the CAP.


## 2.1. BIGA for Cir/Con/k/n System

In order to optimize the reliability of Cir/Con/k/n systems efficiently, the BIGA [29] is introduced in this section. The detailed process of BIGA is as follows, and the flowchart is shown as in Figure 1.(1)
For the optimization problem, the objective function is system reliability of Cir/Con/k/n system, and the solution is the permutation of the components with maximum reliability. The system reliability can be calculated based on the literature [30].(2)
Choose the real-number encoding method and determine the encoding space for individuals.(3)
Generate an initial parent populationp(t) which contains M individuals. Perform Birnbaum importance-based local search on all the M individuals, and update the initial parent population p(t).(4)
For the populationp(t), calculate the fitness of each individual.(5)
When the generation meets the termination condition 1, which is the limit of the generation scale, the algorithm will be terminated and the optimal solution will be output. If not, go to Step (6).(6)
Selection is performed on the current populationp(t), and the best chromosome will be selected and saved.(7)
Measure the fitness scaling of each individual.(8)
For the current population, when the termination condition 2 is satisfied, the process will be terminated and the optimal solution will be output. The termination condition refers to the convergence degree of populations; that is,(favg-fmin)·Sf+(fmax-favg)<α, where fmin, favg, and fmax represent the minimum, average, and maximum fitness of chromosomes, respectively. Sf>1 is conversion factor, and α is a very small positive number. If not, go to Step (9).(9)
For the current populationp(t), select M new individuals according to the fitness of each individual.(10)
Perform the crossover on the previous selectedM individuals to generate offspring population which contains M individuals.(11)
Perform mutation on the previous selected offspring population to generate the new offspring populationp(t+1).(12)
Measure the fitness of every individual in the offspring population.(13)
Perform the BITA on the offspring chromosomes. If the fitness of individual is larger than that of the optimal chromosome, the individual will not perform the BITA.(14)
Perform the elitist strategy, replace the lowest fitness chromosome of offspring populationp(t+1) by the optimal chromosome, and then update the offspring population p(t+1).(15)
Replace the initial populationp(t) by the new offspring population p(t+1), and turn back to Step (5).Figure 1
The flow chart of BIGA.


## 2.2. Numerical Experiments

System reliability depends not only on its inherent structure, but also on the reliabilities of components and the assigned positions of components in the system. The low, high, and arbitrary reliable components are introduced into the experiments to test the performance of BIGA. Their reliabilities are randomly distributed on[0.01,0.2], [0.8,0.99], and [0.01,0.99], respectively.This paper defines that the number of components for the small system is below 8 (including 8), and the number is more than 15 for the large system. Cir/Con/k/n systems include F systems and G systems. In the experiments, the small system cases include 6 systems (g1,g2,g3,g4,f1,f2), and the large system cases include 12 F systems and 16 G systems. The symbols of small systems and large systems are shown in Tables 1 and 2, respectively.Table 1
The symbolic representation of small systems.
g
1: Cir/con/2/5:G
g
2: Cir/con/2/6:G
g
3: Cir/con/3/5:G
g
4: Cir/con/3/6:G
f
1: Cir/con/3/5:F
f
1: Cir/con/3/6:FTable 2
The symbolic representation of large systems.
System
Mark
Cir/Con/3/20:F
F
1
Cir/Con/4/20:F
F
2
Cir/Con/5/20:F
F
3
Cir/Con/3/30:F
F
4
Cir/Con/4/30:F
F
5
Cir/Con/5/30:F
F
6
Cir/Con/3/50:F
F
7
Cir/Con/4/50:F
F
8
Cir/Con/5/50:F
F
9
Cir/Con/3/100:F
F
10
Cir/Con/4/100:F
F
11
Cir/Con/5/100:F
F
12
Cir/Con/2/20:G
G
1
Cir/Con/3/20:G
G
2
Cir/Con/4/20:G
G
3
Cir/Con/5/20:G
G
4
Cir/Con/2/30:G
G
5
Cir/Con/3/30:G
G
6
Cir/Con/4/30:G
G
7
Cir/Con/5/30:G
G
8
Cir/Con/2/50:G
G
9
Cir/Con/3/50:G
G
10
Cir/Con/4/50:G
G
11
Cir/Con/5/50:G
G
12
Cir/Con/2/100:G
G
13
Cir/Con/3/100:G
G
14
Cir/Con/4/100:G
G
15
Cir/Con/5/100:G
G
16Considering the fact that a single instance of algorithm may have error, thus 100 instances of one case as a test have been regarded. For each test, the reliabilities of components are randomly generated based on the type of components, which is the same in a test. Two typical experiments are introduced to compare the performance of the three algorithms, BIGA, BIGLS, and BITA, and all the three types of components should be considered in the experiments. Experiment 1 is used to compare the performance of three algorithms in small systems, and Experiment 2 is used for large systems. In all experiments, the BIGLS and BIGA are implemented with initial population sizeM=20, the conversion factor Sf=3, maximum generations T=200, crossover probability pc=0.8, mutation probability pm=0.05, and α=0.0001.Experiment 1.
In order to verify the performance of BIGA for the CAP in small systems, the experiment is implemented to compare with three algorithms, respectively, for six small systems. In this experiment, the times of achieving the optimal solution are recorded, and the optimal solution is received by enumeration method. When the solution of algorithm is better than that of enumeration method, the value of “achieve” will be increased by 1.Experiment 2.
In order to verify the performance of BIGA for the CAP in large systems, all analysis methods are similar to Experiment 1. In the large system, the optimal solution cannot get in a reasonable time by enumeration method. In order to compare the performance of BIGA and BIGLS, the optimal solution is gained by BITA. When the solution of BIGA and BIGLS is better than that of BITA, the improved number (IN) will be increased by 1.


## 2.3. Experimental Results Analysis

In this paper, all the results are implemented on MATLAB 7.11, and the computer configuration is the Intel (R) Core (TM) 2.3 GHZ processor and 2 GB of memory.Experiment 1.
The results of small systems are shown as Table3. According to the result, we can construct Figure 2 to tell which algorithm is better. From Figure 2, it can be inferred that three algorithms can obtain the optimal solution, and the “achieve” of BIGA and BIGLS is better than that of BITA. For further analysis of Figure 2, the curve volatility of BIGLS and BIGA is stable in addition to the high reliability systems. In order to illustrate the performance of BIGA and BIGLS for different types of components, the mean achievement values of the two algorithms are calculated, and the results are shown in Figure 3. It can be clearly seen that BIGLS is slightly more efficient than BIGA in the low reliable systems, but less effective and inferior to BIGA in the high and arbitrary systems.Table 3
Experimental results of three algorithms for small systems.
System
Algorithm
Achieve
Low
High
Arbitrary
g
1
BITA
73
79
67
BIGLS
100
95
99
BIGA
100
96
100
g
2
BITA
57
76
52
BIGLS
99
87
99
BIGA
98
89
100
g
3
BITA
72
95
66
BIGLS
100
98
100
BIGA
100
100
100
g
4
BITA
81
79
60
BIGLS
100
100
99
BIGA
99
100
99
f
1
BITA
86
85
89
BIGLS
100
98
100
BIGA
100
100
100
f
2
BITA
68
66
64
BIGLS
99
100
98
BIGA
100
100
100Figure 2
The achievement of three algorithms for small systems.
(a)
Low reliable components
(b)
High reliable components
(c)
Arbitrary reliable componentsFigure 3
Average of achievement for BIGLS and BIGA.Experiment 2.
The experimental results of the large systems are shown in Table4. According to the IN values in Table 4, Figure 4 is obtained to illustrate the IN values of BIGLS and BIGA in the large systems. In each subgraph, the abscissa expresses the large systems, and the ordinate shows the IN values of BIGLS and BIGA. From the subgraphs (a) and (d), the IN values of two algorithms are close, and the IN value of BIGLS is worse than that of BIGA except 4 low reliable systems (F3, F7, F8, and F11) and 3 high reliable systems (G11, G12, and G15). Observing other subgraphs in Figure 4, the IN value of BIGA is better than that of BIGLS, which shows that BIGA can improve the performance of reliability optimization. In subgraph (d), it is not hard to find that the IN values of the two algorithms are close and small when n is more than 50. In these systems, the solution generated by the BITA is the optimal solution or approximate optimal solution, and the optimization space of BIGA and BIGLS is relatively small. Therefore, the three algorithms are all good for optimization of Cir/Con/k/n systems.Table 4
Experimental results of the large systems by BIGA and BIGLS.
System
Algorithm
IN
Low
High
Arbitrary
F
1
BIGLS
44
74
65
BIGA
45
81
73
F
2
BIGLS
49
63
97
BIGA
49
79
98
F
3
BIGLS
56
52
99
BIGA
53
57
99
F
4
BIGLS
63
50
79
BIGA
64
86
92
F
5
BIGLS
59
48
75
BIGA
59
82
88
F
6
BIGLS
55
28
99
BIGA
56
29
100
F
7
BIGLS
41
20
23
BIGA
32
98
97
F
8
BIGLS
96
16
44
BIGA
95
84
98
F
9
BIGLS
69
9
97
BIGA
77
11
100
F
10
BIGLS
19
18
21
BIGA
20
99
97
F
11
BIGLS
97
14
41
BIGA
96
87
98
F
12
BIGLS
75
3
90
BIGA
79
4
100
G
1
BIGLS
32
3
38
BIGA
53
3
62
G
2
BIGLS
23
46
78
BIGA
32
53
84
G
3
BIGLS
23
90
65
BIGA
30
92
71
G
4
BIGLS
24
91
63
BIGA
25
91
66
G
5
BIGLS
15
0
8
BIGA
46
0
19
G
6
BIGLS
7
11
53
BIGA
22
21
87
G
7
BIGLS
4
43
62
BIGA
21
45
86
G
8
BIGLS
4
69
75
BIGA
8
66
83
G
9
BIGLS
12
0
3
BIGA
41
0
5
G
10
BIGLS
1
3
13
BIGA
9
7
42
G
11
BIGLS
1
7
60
BIGA
2
6
88
G
12
BIGLS
1
9
77
BIGA
1
8
90
G
13
BIGLS
0
0
3
BIGA
31
0
5
G
14
BIGLS
0
1
10
BIGA
7
1
39
G
15
BIGLS
0
2
55
BIGA
1
1
90
G
16
BIGLS
1
4
82
BIGA
1
4
93Figure 4
Experimental results of BIGA and BIGLS.
(a)
F system with low reliable components
(b)
G system with low reliable components
(c)
F system with high reliable components
(d)
G system with high reliable components
(e)
F system with arbitrary reliable components
(f)
G system with arbitrary reliable componentsThrough the typical experiments, it is clear that the BIGA can get better performance of reliability optimization than BIGLS and BITA. In the low reliable systems, sometimes BIGLS is better than BIGA, but the results of two algorithms are nearly identical. In the high or arbitrary reliable systems, the BIGA is better than BIGLS or BITA. The BIGA can improve the effectiveness and efficiency of solving the CAP.


## 3. Discussion of Optimal Permutation and Importance Measure

### 3.1. Optimal Assignment for Cir/Con/k/n System

The structure optimization problem of the system is through reassigning the positions of then components to get the optimal system reliability. The optimal assignment also needs to reassign the n components with different reliabilities into the n different positions.Definition 1.
If the sequenceτ∗∈S is the optimal system structure, τ∗ should satisfy the following conditions:(1)φτ∗,p=maxτ∈S⁡φτ,p.
The reliability of componentτ(i) changes from pτ(i) to pτ(i)∗, and the system reliability after changing is presented as follows:(2)φτi∗,p=pτi∗pτjR1i,1j,p+pτi∗1-pτjR1i,0j,p+1-pτi∗1-pτjR0i,0j,p+1-pτi∗pτjR0i,1j,p.
However, when the reliability ofpτ(i) is changed, sequence τ∗∈S may not be the optimal structure. The system reliability, after exchanging the positions i and j in the sequence τ, is as follows:(3)φτij∗,p=pτi∗pτjR1i,1j,p+pτj1-pτi∗R1i,0j,p+1-pτi∗1-pτjR0i,0j,p+1-pτjpτi∗R0i,1j,p.
In order to compare the improvement of system reliability in these two methods, the difference of system can be calculated by (4), which is as follows:(4)φτi∗,p-φτij∗,p=pτi∗1-pτjR1i,0j,p+1-pτi∗pτjR0i,1j,p-pτj1-pτi∗R1i,0j,p-1-pτjpτi∗R0i,1j,p=pτi∗-pτjR1i,0j,p-pτi∗-pτjR0i,1j,p=pτi∗-pτjR1i,0j,p-R0i,1j,p.
φ
(
τ
i
∗
,
p
)
-
φ
(
τ
i
j
∗
,
p
)
<
0 means that the method exchanging the positions of components is better than that of improving the component reliability. Thus, the components in the positions i and j in the sequence τ should be exchanged on this occasion.
The importance of component is used to measure the degree of impact that the change of component reliability has on the system reliability. Birnbaum [4] first proposed the calculation methods of the component importance(5)Ii=∂Rp∂pi=R1i,p-R0i,p.
When the reliability of componentichanges from pi to pi+Δ, the system reliability after changing the component reliability is shown as follows:(6)Rpi+Δi,p=pi+ΔR1i,p+1-pi-ΔR0i,p=piR1i,p+1-piR0i,p+Δ·R1i,p-ΔR0i,p=Rp+Δ·Ii.
From (6), when the difference of component reliability is Δ, the difference of system reliability will become Δ·Ii. When the reliability of specific component changes, the initial optimal assignment also may change.
Kuo et al. [30] presented the calculation method of the BI of component i in the Cir/Con/k/n systems. It was shown as follows:(7)IiG=Rn;k-Ri-1;k-R′n-i;k+Ri-1;kR′n-i;kpi.
Papastavridis [31] presented the calculation of the BI of component i in the Cir/Con/k/n systems. It was shown as follows:(8)IiF=Ri-1;kR′n-i;k-Rn;kqi.
Zuo and Kuo [18] gave the completed invariant permutations of Cir/Con/k/n system, which is shown in Table 5. If the system reliability only depends on the order of the component reliability, and when the reliability meets the condition p1<p2<⋯<pn, the optimal assignment will be invariant. Whatever the value of component reliability is, the permutation of component reliability is invariant when the order of component reliability is determined, and the design of the system according to Table 5 will be the optimal assignment.
If the invariant permutation exits, we can find the optimal assignment. However, if it does not exist, we can use the BIGA to find the approximate optimal assignment.Table 5
The invariant permutation of Cir/Con/k/n systems.
k
F system
G system
k
=
1
Any permutation
Any permutation
k
=
2
(1,n-1,3,n-3,…,n-4,4,n-2,2,n,1)
Inexistence
2
<
k
<
(
n
/
2
)
Inexistence
Inexistence
(
n
/
2
)
≤
k
<
n
-
2
Inexistence
Inexistence
k
=
n
-
2
(1,n-1,3,n-3,…,n-4,4,n-2,2,n,1)
(1,3,5,…,n,…,6,4,2,1)
k
=
n
-
1
Any permutation
Any permutation
k
=
n
Any permutation
Any permutation


### 3.2. Analysis of Component Importance with Varied Reliability

In practical engineering, the components reliabilities usually degrade, respectively. So, the change of the optimal assignment and the component importance will be discussed in this condition. According to the values of component reliability, the order of then components in the system is p1<p2<⋯<pn. The reliability of component n is 0.99, and the reliability of component i is pi=0.99-0.025(n-i) when n≤40. In order to verify the changes of importance measure in the optimal assignment by the instances of Cir/Con/k/n systems, we construct three typical cases and set n=5 (small system), n=9 (medium system), and n=15 (large system). According to Table 5, we set k=2,2<k<n/2,n/2≤k≤n-2,k=n-2 and k=n-1 to analyze the changes of importance measure and the reasons of the changes.Case 1 (Cir/Con/k/5 system).
Considering all kinds of Cir/Con/k/5 systems, the value of k is 2, 3, and 4. Figure 5 represents the relationship between the importance measure of component 5 and its reliability in the optimal permutation for Cir/Con/k/5 systems, and p1=0.89, p2=0.915, p3=0.94, and p4=0.965. With decreasing of p5, I5G will drop sharply at the node where the permutation of components is reassigned. Such as, for Cir/Con/3/5 G system, when p5=0.91575, I5G=p2p4p3+q2p4p3p1=0.8528, and the optimal permutation is 5-4-3-2-1-5; but when p5=0.9141, I5G=q1p4p3p2=0.0865, and the optimal permutation is 5-3-4-2-1-5.
For Cir/Con/k/5 systems, Figure 6 demonstrates the changes of all components’ importance measure with different p5 in the optimal permutations. When the reliability of component 5 is highest, the BI of component 5 is the largest. With decreasing of p5, the BI of component 5 will decrease. In order to get the maximum system reliability, the position with highest BI should be assigned the component with largest reliability. For example, in the Cir/Con/2/5 F system, when p5∈[0.962,0.99), component 5 should be given greater priority to be assigned to position 5; when p5∈[0.913,0.962), component 4 should be given greater priority to be assigned to position 5; when p5∈[0.889,0.913), component 3 should be given greater priority to be assigned to position 5. Similarly, we can find the phenomenon in other Cir/Con/k/5 systems. Therefore, in the Cir/Con/k/5 systems, the BI of components will drop sharply at the node where the components are reassigned, which demonstrates that the changes of optimal permutation will have an effect on the BI of components.Figure 5
Changes ofI5 with different p5 in the Cir/Con/k/5 systems.Figure 6
Changes of BI for all components with differentp5 in the Cir/Con/k/5 systems.Case 2 (Cir/Con/k/9 system).
Considering all kinds of Cir/Con/k/9 systems, the value of k is 2, 3, 4, 5, 6, 7, and 8. From Figure 7, the BI of component 9 has volatility with decreasing of p9 in the optimal permutation. In the Cir/Con/2/9 system, the volatility of I9 in G system is more obvious than that of F system. We can learn from Table 5 that the Cir/Con/2/9:G system does not have the invariant permutation because of 2≤k<(n/2); we can only find the approximate optimal assignment based on the BIGA. In order to analyze the volatility of I9, we only study the fact that p9 is in the interval [0.9302,0.9500] for the Cir/Con/2/9 G system. The I9 and the optimal permutation with different p9 are shown in Table 6, and the changes of I9 are shown in Figure 8. As shown in the table, when p9=0.9401, the optimal permutation is 2-8-7-5-9-3-4-6-1-2, and R(2875934612)-R(2874395612)=-1.8652×10-5, which illustrates that 2-8-7-5-9-3-4-6-1-2 is not the optimal permutation; when p9=0.9385, the system optimal permutation is 2-8-7-4-3-9-5-6-1-2, R(2875934612)-R(2874395612)=1.8964×10-5, which illustrates that permutation 2-8-7-5-9-3-4-6-1-2 is superior to permutation 2-8-7-4-3-9-5-6-1-2.
In the Cir/Con/2/9F(G) system, the change of system reliability with the decrease of p9 is shown in Figures 9 and 10, respectively. With the p9 decreasing, the system reliability of F system and G system both decrease, but the system reliability of G system is more fluctuant. In the Cir/Con/2/9 G system, the system reliability with p9=0.9401 is lower than that of p9=0.9385. As shown in Table 6, when p9=0.9401, the permutation 2-8-7-5-9-3-4-6-1-2 is not the optimal assignment. Therefore, the permutation at the abnormal floating point is not necessarily the optimal assignment.Table 6
Optimal permutation andI9 with different p9 in the Cir/Con/2/9 G system.
p
9
I
9
Optimal permutation
0.9500
0.0008900
2-8-7-5-9-3-4-6-1-2
0.9484
0.0008571
2-8-7-5-9-4-3-6-1-2
0.9467
0.0008623
2-8-7-5-9-4-3-6-1-2
0.9451
0.0008661
2-8-7-5-9-4-3-6-1-2
0.9434
0.0008709
2-8-7-4-3-9-5-6-1-2
0.9418
0.0008762
2-8-7-5-9-3-4-6-1-2
0.9401
0.0008802
2-8-7-5-9-3-4-6-1-2
0.9385
0.0008509
2-8-7-4-3-9-5-6-1-2
0.9368
0.0008693
2-8-7-5-9-3-4-6-1-2
0.9352
0.0008889
1-6-5-4-3-9-7-8-2-1
0.9335
0.0009091
1-6-5-4-3-9-7-8-2-1
0.9319
0.0008952
1-6-5-4-3-9-7-8-2-1
0.9302
0.0009128
1-6-5-4-3-9-7-8-2-1Figure 7
Changes ofI9 with different p9 in the Cir/Con/k/9 systems.Figure 8
Changes ofI9 with p9 in the interval [0.9302, 0.9500] for Cir/Con/2/9 G systems.Figure 9
Changes of system reliability with differentp9 for Cir/Con/2/9 F systems.Figure 10
Changes of system reliability with differentp9 for Cir/Con/2/9 G systems.Case 3 (Cir/Con/k/15 system).
Considering all kinds of Cir/Con/k/15 systems, the value of k is 2, 6, 10, 13, and 14. From Figure 11, I15 has some obvious fluctuation, when the value of p15 is in the interval [0.75,0.8] or [0.85,0.99]. In order to analyze the change of I15 in the Cir/Con/2/15 G system clearly, we choose that p15 is in the intervals [0.75,0.8], [0.85,0.9], and [0.9,0.99]. From Figure 12, the change of I15 does not show the monotonicity. The value of p15 is especially in the interval [0.75,0.8], the floating of I15 is more significant, which reflects that the invariance of the optimal permutation is not fixed.
Whenp15 is decreasing from 0.7804 to 0.7639 in the Cir/Con/2/15:G system, the I15 and the optimal permutation are shown in Table 7. When the system has the same optimal permutation, the change of I15 is very small. However, when the optimal permutation changes, the change of I15 will be larger. Therefore, the BI of component is influenced by the optimal permutation.
Considering Figures11 and 12, in the Cir/Con/2/15 system, the volatility of I15 for G system is clearer than that of F system. We can learn from Table 5 that the Cir/Con/2/15:G system does not have the invariant permutation because of 2≤k<(n/2), and we can find the approximate optimal assignment based on the BIGA. The change of system reliability for Cir/Con/2/15 F(G) system with decreasing of p15 is shown in Figures 13 and 14, respectively. With the p15 decreasing, the system reliability of F system and G system will decrease, but changes of the system reliability for G system are more fluctuant. The system reliability with p15=0.7689 is lower than p15=0.7672 in the G system. Therefore, the permutation at the abnormal floating point is not the optimal permutation.Table 7
Optimal permutation of component 15 with different reliabilities ofp15.
p
15
Birnbaum importance
Optimal arrangement
0.7804
8.348 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7788
8.354 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7772
7.948 × 10−5
1-10-13-14-12-8-15-5-3-4-6-7-9-11-2-1
0.7755
8.366 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7738
8.598 × 10−5
2-10-11-13-14-12-15-7-9-8-5-3-4-6-1-2
0.7722
8.378 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7705
8.384 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7689
8.060 × 10−5
1-12-14-13-11-10-9-15-3-4-7-6-5-8-2-1
0.7672
8.342 × 10−5
1-8-6-15-7-5-3-4-9-11-10-12-14-13-2-1
0.7656
8.358 × 10−5
1-8-6-4-15-10-11-13-14-12-7-3-5-9-2-1
0.7639
8.728 × 10−5
2-11-10-9-13-14-12-15-3-4-5-6-7-8-1-2Figure 11
Changes ofI15 with different p15 in the Cir/Con/k/15 system.Figure 12
Changes ofI15 with different p15 in the Cir/Con/2/15:G system.Figure 13
System reliability of Cir/Con/2/15:F system with different p15.Figure 14
System reliability of Cir/Con/2/15:G system with different p15.From Table5, there is no invariant permutation for Cir/Con/6/15:F system. So the system optimal permutation depends on the component reliability, and the I15 is connected with the optimal permutation. We can find that I15 has the larger fluctuation in Figure 11, when p15 is in the intervals [0.8,0.99], [0.7,0.75], and [0.6,0.65]. In order to analyze the change of I15 obviously, we choose I15 in the intervals [0.95, 0.9], [0.9, 0.95], [0.85, 0.9], [0.8, 0.85], [0.7, 0.75], and [0.6, 0.65]. From Figure 15, the change of I15 does not show monotonicity, and I15 is influenced by the change of the optimal permutation.Figure 15
Changes ofI15 with different p15 for the Cir/Con/6/15:F system.Figure16 demonstrates the reliability of Cir/Con/6/15 system with changing of p15. The system reliability with p15=0.7474 is lower than that of p15=0.7458, which illustrates that the assignment at the floating point p15=0.7474 is not the optimal assignment.Figure 16
System reliability of Cir/Con/6/15:F system with different p15.With regard to the Cir/Con/k/n systems, from the above, we can get the relationship of component reliability, importance measure, and the system optimal assignment as follows:(1)
If the component reliability changes, the initial optimal permutation of system may not be the optimal.(2)
With the change of the component reliability, the optimal assignment will be changed, and the Birnbaum importance measure of component and the optimal permutation have the stronger relevance.(3)
The volatility of component Birnbaum importance measure does not show the monotonicity, and the fluctuation of importance measure illustrates the change of the optimal assignment.


## 3.1. Optimal Assignment for Cir/Con/k/n System

The structure optimization problem of the system is through reassigning the positions of then components to get the optimal system reliability. The optimal assignment also needs to reassign the n components with different reliabilities into the n different positions.Definition 1.
If the sequenceτ∗∈S is the optimal system structure, τ∗ should satisfy the following conditions:(1)φτ∗,p=maxτ∈S⁡φτ,p.
The reliability of componentτ(i) changes from pτ(i) to pτ(i)∗, and the system reliability after changing is presented as follows:(2)φτi∗,p=pτi∗pτjR1i,1j,p+pτi∗1-pτjR1i,0j,p+1-pτi∗1-pτjR0i,0j,p+1-pτi∗pτjR0i,1j,p.
However, when the reliability ofpτ(i) is changed, sequence τ∗∈S may not be the optimal structure. The system reliability, after exchanging the positions i and j in the sequence τ, is as follows:(3)φτij∗,p=pτi∗pτjR1i,1j,p+pτj1-pτi∗R1i,0j,p+1-pτi∗1-pτjR0i,0j,p+1-pτjpτi∗R0i,1j,p.
In order to compare the improvement of system reliability in these two methods, the difference of system can be calculated by (4), which is as follows:(4)φτi∗,p-φτij∗,p=pτi∗1-pτjR1i,0j,p+1-pτi∗pτjR0i,1j,p-pτj1-pτi∗R1i,0j,p-1-pτjpτi∗R0i,1j,p=pτi∗-pτjR1i,0j,p-pτi∗-pτjR0i,1j,p=pτi∗-pτjR1i,0j,p-R0i,1j,p.
φ
(
τ
i
∗
,
p
)
-
φ
(
τ
i
j
∗
,
p
)
<
0 means that the method exchanging the positions of components is better than that of improving the component reliability. Thus, the components in the positions i and j in the sequence τ should be exchanged on this occasion.
The importance of component is used to measure the degree of impact that the change of component reliability has on the system reliability. Birnbaum [4] first proposed the calculation methods of the component importance(5)Ii=∂Rp∂pi=R1i,p-R0i,p.
When the reliability of componentichanges from pi to pi+Δ, the system reliability after changing the component reliability is shown as follows:(6)Rpi+Δi,p=pi+ΔR1i,p+1-pi-ΔR0i,p=piR1i,p+1-piR0i,p+Δ·R1i,p-ΔR0i,p=Rp+Δ·Ii.
From (6), when the difference of component reliability is Δ, the difference of system reliability will become Δ·Ii. When the reliability of specific component changes, the initial optimal assignment also may change.
Kuo et al. [30] presented the calculation method of the BI of component i in the Cir/Con/k/n systems. It was shown as follows:(7)IiG=Rn;k-Ri-1;k-R′n-i;k+Ri-1;kR′n-i;kpi.
Papastavridis [31] presented the calculation of the BI of component i in the Cir/Con/k/n systems. It was shown as follows:(8)IiF=Ri-1;kR′n-i;k-Rn;kqi.
Zuo and Kuo [18] gave the completed invariant permutations of Cir/Con/k/n system, which is shown in Table 5. If the system reliability only depends on the order of the component reliability, and when the reliability meets the condition p1<p2<⋯<pn, the optimal assignment will be invariant. Whatever the value of component reliability is, the permutation of component reliability is invariant when the order of component reliability is determined, and the design of the system according to Table 5 will be the optimal assignment.
If the invariant permutation exits, we can find the optimal assignment. However, if it does not exist, we can use the BIGA to find the approximate optimal assignment.Table 5
The invariant permutation of Cir/Con/k/n systems.
k
F system
G system
k
=
1
Any permutation
Any permutation
k
=
2
(1,n-1,3,n-3,…,n-4,4,n-2,2,n,1)
Inexistence
2
<
k
<
(
n
/
2
)
Inexistence
Inexistence
(
n
/
2
)
≤
k
<
n
-
2
Inexistence
Inexistence
k
=
n
-
2
(1,n-1,3,n-3,…,n-4,4,n-2,2,n,1)
(1,3,5,…,n,…,6,4,2,1)
k
=
n
-
1
Any permutation
Any permutation
k
=
n
Any permutation
Any permutation


## 3.2. Analysis of Component Importance with Varied Reliability

In practical engineering, the components reliabilities usually degrade, respectively. So, the change of the optimal assignment and the component importance will be discussed in this condition. According to the values of component reliability, the order of then components in the system is p1<p2<⋯<pn. The reliability of component n is 0.99, and the reliability of component i is pi=0.99-0.025(n-i) when n≤40. In order to verify the changes of importance measure in the optimal assignment by the instances of Cir/Con/k/n systems, we construct three typical cases and set n=5 (small system), n=9 (medium system), and n=15 (large system). According to Table 5, we set k=2,2<k<n/2,n/2≤k≤n-2,k=n-2 and k=n-1 to analyze the changes of importance measure and the reasons of the changes.Case 1 (Cir/Con/k/5 system).
Considering all kinds of Cir/Con/k/5 systems, the value of k is 2, 3, and 4. Figure 5 represents the relationship between the importance measure of component 5 and its reliability in the optimal permutation for Cir/Con/k/5 systems, and p1=0.89, p2=0.915, p3=0.94, and p4=0.965. With decreasing of p5, I5G will drop sharply at the node where the permutation of components is reassigned. Such as, for Cir/Con/3/5 G system, when p5=0.91575, I5G=p2p4p3+q2p4p3p1=0.8528, and the optimal permutation is 5-4-3-2-1-5; but when p5=0.9141, I5G=q1p4p3p2=0.0865, and the optimal permutation is 5-3-4-2-1-5.
For Cir/Con/k/5 systems, Figure 6 demonstrates the changes of all components’ importance measure with different p5 in the optimal permutations. When the reliability of component 5 is highest, the BI of component 5 is the largest. With decreasing of p5, the BI of component 5 will decrease. In order to get the maximum system reliability, the position with highest BI should be assigned the component with largest reliability. For example, in the Cir/Con/2/5 F system, when p5∈[0.962,0.99), component 5 should be given greater priority to be assigned to position 5; when p5∈[0.913,0.962), component 4 should be given greater priority to be assigned to position 5; when p5∈[0.889,0.913), component 3 should be given greater priority to be assigned to position 5. Similarly, we can find the phenomenon in other Cir/Con/k/5 systems. Therefore, in the Cir/Con/k/5 systems, the BI of components will drop sharply at the node where the components are reassigned, which demonstrates that the changes of optimal permutation will have an effect on the BI of components.Figure 5
Changes ofI5 with different p5 in the Cir/Con/k/5 systems.Figure 6
Changes of BI for all components with differentp5 in the Cir/Con/k/5 systems.Case 2 (Cir/Con/k/9 system).
Considering all kinds of Cir/Con/k/9 systems, the value of k is 2, 3, 4, 5, 6, 7, and 8. From Figure 7, the BI of component 9 has volatility with decreasing of p9 in the optimal permutation. In the Cir/Con/2/9 system, the volatility of I9 in G system is more obvious than that of F system. We can learn from Table 5 that the Cir/Con/2/9:G system does not have the invariant permutation because of 2≤k<(n/2); we can only find the approximate optimal assignment based on the BIGA. In order to analyze the volatility of I9, we only study the fact that p9 is in the interval [0.9302,0.9500] for the Cir/Con/2/9 G system. The I9 and the optimal permutation with different p9 are shown in Table 6, and the changes of I9 are shown in Figure 8. As shown in the table, when p9=0.9401, the optimal permutation is 2-8-7-5-9-3-4-6-1-2, and R(2875934612)-R(2874395612)=-1.8652×10-5, which illustrates that 2-8-7-5-9-3-4-6-1-2 is not the optimal permutation; when p9=0.9385, the system optimal permutation is 2-8-7-4-3-9-5-6-1-2, R(2875934612)-R(2874395612)=1.8964×10-5, which illustrates that permutation 2-8-7-5-9-3-4-6-1-2 is superior to permutation 2-8-7-4-3-9-5-6-1-2.
In the Cir/Con/2/9F(G) system, the change of system reliability with the decrease of p9 is shown in Figures 9 and 10, respectively. With the p9 decreasing, the system reliability of F system and G system both decrease, but the system reliability of G system is more fluctuant. In the Cir/Con/2/9 G system, the system reliability with p9=0.9401 is lower than that of p9=0.9385. As shown in Table 6, when p9=0.9401, the permutation 2-8-7-5-9-3-4-6-1-2 is not the optimal assignment. Therefore, the permutation at the abnormal floating point is not necessarily the optimal assignment.Table 6
Optimal permutation andI9 with different p9 in the Cir/Con/2/9 G system.
p
9
I
9
Optimal permutation
0.9500
0.0008900
2-8-7-5-9-3-4-6-1-2
0.9484
0.0008571
2-8-7-5-9-4-3-6-1-2
0.9467
0.0008623
2-8-7-5-9-4-3-6-1-2
0.9451
0.0008661
2-8-7-5-9-4-3-6-1-2
0.9434
0.0008709
2-8-7-4-3-9-5-6-1-2
0.9418
0.0008762
2-8-7-5-9-3-4-6-1-2
0.9401
0.0008802
2-8-7-5-9-3-4-6-1-2
0.9385
0.0008509
2-8-7-4-3-9-5-6-1-2
0.9368
0.0008693
2-8-7-5-9-3-4-6-1-2
0.9352
0.0008889
1-6-5-4-3-9-7-8-2-1
0.9335
0.0009091
1-6-5-4-3-9-7-8-2-1
0.9319
0.0008952
1-6-5-4-3-9-7-8-2-1
0.9302
0.0009128
1-6-5-4-3-9-7-8-2-1Figure 7
Changes ofI9 with different p9 in the Cir/Con/k/9 systems.Figure 8
Changes ofI9 with p9 in the interval [0.9302, 0.9500] for Cir/Con/2/9 G systems.Figure 9
Changes of system reliability with differentp9 for Cir/Con/2/9 F systems.Figure 10
Changes of system reliability with differentp9 for Cir/Con/2/9 G systems.Case 3 (Cir/Con/k/15 system).
Considering all kinds of Cir/Con/k/15 systems, the value of k is 2, 6, 10, 13, and 14. From Figure 11, I15 has some obvious fluctuation, when the value of p15 is in the interval [0.75,0.8] or [0.85,0.99]. In order to analyze the change of I15 in the Cir/Con/2/15 G system clearly, we choose that p15 is in the intervals [0.75,0.8], [0.85,0.9], and [0.9,0.99]. From Figure 12, the change of I15 does not show the monotonicity. The value of p15 is especially in the interval [0.75,0.8], the floating of I15 is more significant, which reflects that the invariance of the optimal permutation is not fixed.
Whenp15 is decreasing from 0.7804 to 0.7639 in the Cir/Con/2/15:G system, the I15 and the optimal permutation are shown in Table 7. When the system has the same optimal permutation, the change of I15 is very small. However, when the optimal permutation changes, the change of I15 will be larger. Therefore, the BI of component is influenced by the optimal permutation.
Considering Figures11 and 12, in the Cir/Con/2/15 system, the volatility of I15 for G system is clearer than that of F system. We can learn from Table 5 that the Cir/Con/2/15:G system does not have the invariant permutation because of 2≤k<(n/2), and we can find the approximate optimal assignment based on the BIGA. The change of system reliability for Cir/Con/2/15 F(G) system with decreasing of p15 is shown in Figures 13 and 14, respectively. With the p15 decreasing, the system reliability of F system and G system will decrease, but changes of the system reliability for G system are more fluctuant. The system reliability with p15=0.7689 is lower than p15=0.7672 in the G system. Therefore, the permutation at the abnormal floating point is not the optimal permutation.Table 7
Optimal permutation of component 15 with different reliabilities ofp15.
p
15
Birnbaum importance
Optimal arrangement
0.7804
8.348 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7788
8.354 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7772
7.948 × 10−5
1-10-13-14-12-8-15-5-3-4-6-7-9-11-2-1
0.7755
8.366 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7738
8.598 × 10−5
2-10-11-13-14-12-15-7-9-8-5-3-4-6-1-2
0.7722
8.378 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7705
8.384 × 10−5
1-5-3-4-6-15-7-8-9-10-12-14-13-11-2-1
0.7689
8.060 × 10−5
1-12-14-13-11-10-9-15-3-4-7-6-5-8-2-1
0.7672
8.342 × 10−5
1-8-6-15-7-5-3-4-9-11-10-12-14-13-2-1
0.7656
8.358 × 10−5
1-8-6-4-15-10-11-13-14-12-7-3-5-9-2-1
0.7639
8.728 × 10−5
2-11-10-9-13-14-12-15-3-4-5-6-7-8-1-2Figure 11
Changes ofI15 with different p15 in the Cir/Con/k/15 system.Figure 12
Changes ofI15 with different p15 in the Cir/Con/2/15:G system.Figure 13
System reliability of Cir/Con/2/15:F system with different p15.Figure 14
System reliability of Cir/Con/2/15:G system with different p15.From Table5, there is no invariant permutation for Cir/Con/6/15:F system. So the system optimal permutation depends on the component reliability, and the I15 is connected with the optimal permutation. We can find that I15 has the larger fluctuation in Figure 11, when p15 is in the intervals [0.8,0.99], [0.7,0.75], and [0.6,0.65]. In order to analyze the change of I15 obviously, we choose I15 in the intervals [0.95, 0.9], [0.9, 0.95], [0.85, 0.9], [0.8, 0.85], [0.7, 0.75], and [0.6, 0.65]. From Figure 15, the change of I15 does not show monotonicity, and I15 is influenced by the change of the optimal permutation.Figure 15
Changes ofI15 with different p15 for the Cir/Con/6/15:F system.Figure16 demonstrates the reliability of Cir/Con/6/15 system with changing of p15. The system reliability with p15=0.7474 is lower than that of p15=0.7458, which illustrates that the assignment at the floating point p15=0.7474 is not the optimal assignment.Figure 16
System reliability of Cir/Con/6/15:F system with different p15.With regard to the Cir/Con/k/n systems, from the above, we can get the relationship of component reliability, importance measure, and the system optimal assignment as follows:(1)
If the component reliability changes, the initial optimal permutation of system may not be the optimal.(2)
With the change of the component reliability, the optimal assignment will be changed, and the Birnbaum importance measure of component and the optimal permutation have the stronger relevance.(3)
The volatility of component Birnbaum importance measure does not show the monotonicity, and the fluctuation of importance measure illustrates the change of the optimal assignment.


## 4. Conclusions

This paper solves the reliability optimization problem for Cir/Con/k/n system by introducing the BIGA method, which combines the genetic algorithm with Birnbaum importance. The numerical experiments are implemented and comparison results with BIGLS verify the superiority of BIGA. By analyzing three typical cases with different n, k, and p in Cir/Con/k/n systems, the relationships among the Birnbaum importance and the optimal assignment are discussed. With the change of the component reliability, the optimal assignment will be changed, while the importance measure of component has strong relevance with the optimal permutation. The volatility of importance measure does not show the monotonicity with the decrease of component reliability, and the fluctuation of importance measure illustrates the change of the optimal assignment.



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