Reliability Optimization and Importance Analysis of Circular-Consecutive 𝑘 -out-of- 𝑛 System

The circular-consecutive 𝑘 -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.


Introduction
The component assignment problem (CAP) [1] is a kind of classic problem in the optimization of system reliability.The system is composed of  positions and  components with different reliabilities, and then the  components should be assigned into  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 -outof- system (Con// system) is a typical case of CAP, and its purpose is to ensure that the system reliability remains largest.Con// system contains linear-consecutive -out-of-:() system and circular-consecutive -out-of-:() system.
A linear-consecutive -out-of-:() system (Lin/Con/ /:() system) is an ordered sequence of  components arranged in a line such that the system fails (works) if and only if at least  consecutive components fail (work).Lin/Con// 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  pumps.If a pump is failed, the pipeline system also can work normally unless the  consecutive pumps all failed.Actually, the pipeline system is a Lin/Con//: system which is widely used in the practical engineering.
A circular-consecutive -out-of-:() system (Cir/Con/ /:() system) is an ordered sequence of  components arranged in a circle such that the system fails (works) if and only if at least  consecutive components fail (work).Cir/Con// systems are used in the camera system of nuclear accelerator and computer ring network [3].For a nuclear accelerator,  high-speed cameras are arranged around the accelerator to record the motion state of various particles.If and only if at least  cameras work properly, the complete motion of particles can be recorded successfully.Actually, the camera system of nuclear accelerator is a Cir/Con//: system.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// systems, which is stable and feasible to solve the general CAP with stronger robustness.
The rest of this paper is organized as follows.In Section 2, BIGA is introduced to optimize the reliability of Cir/Con// 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 , , and  in Cir/Con// 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.

Reliability Optimization Method for
Cir/Con// System 2.1.BIGA for Cir/Con// System.In order to optimize the reliability of Cir/Con// 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// 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 population () which contains  individuals.Perform Birnbaum importancebased local search on all the  individuals, and update the initial parent population ().
(4) For the population (), 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 population (), 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, ( avg −  min ) ⋅   + ( max −  avg ) < , where  min ,  avg , and  max represent the minimum, average, and maximum fitness of chromosomes, respectively.  > 1 is conversion factor, and  is a very small positive number.If not, go to Step (9).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 population ( + 1) by the 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// systems include  systems and  systems.In the experiments, the small system cases include 6 systems (1, 2, 3, 4, 1, 2), and the large system cases include 12  systems and 16  systems.The symbols of small systems and large systems are shown in Tables 1 and 2, respectively.
Considering 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 size  = 20, the conversion factor   = 3, maximum generations  = 200, crossover probability   = 0.8, mutation probability   = 0.05, and  = 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.
Experiment 1.The results of small systems are shown as Table 3.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.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  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// systems.

Experiment 2. The experimental results of the large systems are shown in
Through 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.

Discussion of Optimal Permutation and Importance Measure
3.1.Optimal Assignment for Cir/Con// System.The structure optimization problem of the system is through reassigning the positions of the  components to get the optimal system reliability.The optimal assignment also needs to reassign the  components with different reliabilities into the  different positions.

High reliable components
Arbitrary reliable components BIGLS BIGA 99.67 99.5 96.67 97 99.83 99.17 Definition 1.If the sequence  * ∈  is the optimal system structure,  * should satisfy the following conditions: The reliability of component () changes from  () to  * () , and the system reliability after changing is presented as follows: However, when the reliability of  () is changed, sequence  * ∈  may not be the optimal structure.The system reliability, after exchanging the positions  and  in the sequence , is as follows: 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: (  * , ) − ( *  , ) < 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  and  in the sequence  should be exchanged on this occasion.
Large systems Large systems (e)  system with arbitrary reliable components 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 When the reliability of component  changes from   to   + Δ, the system reliability after changing the component reliability is shown as follows: From (6), when the difference of component reliability is Δ, the difference of system reliability will become Δ ⋅   .
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  in the Cir/Con// systems.It was shown as follows: Papastavridis [31] presented the calculation of the BI of component  in the Cir/Con// systems.It was shown as follows: Zuo and Kuo [18] gave the completed invariant permutations of Cir/Con// 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  1 <  2 < ⋅ ⋅ ⋅ <   , 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.

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 the  components in the system is  1 <  2 < ⋅ ⋅ ⋅ <   .The reliability of component  is 0.99, and the reliability of component  is   = 0.99 − 0.025( − ) when  ≤ 40.In order to verify the changes of importance measure in the optimal assignment by the instances of Cir/Con// systems, we construct three typical cases and set  = 5 (small system),  = 9 (medium system), and  = 15 (large system).According to Table 5, we set  = 2, 2 <  < /2, /2 ≤  ≤ −2,  = −2 and  =  − 1 to analyze the changes of importance measure and the reasons of the changes.
For Cir/Con//5 systems, Figure 6 demonstrates the changes of all components' importance measure with different  5 in the optimal permutations.When the reliability of component 5 is highest, the BI of component 5 is the largest.With decreasing of  5 , 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  system, when  5 ∈ [0.962, 0.99), component 5 should be given greater priority to be assigned to position 5; when  5 ∈ [0.913, 0.962), component 4 should be given greater priority to be assigned to position 5; when  5 ∈ [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//5 systems.Therefore, in the Cir/Con//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.
Case 2 (Cir/Con//9 system).Considering all kinds of Cir/Con//9 systems, the value of  is 2, 3, 4, 5, 6, 7, and 8. From Figure 7, the BI of component 9 has volatility with decreasing of  9 in the optimal permutation.In the Cir/Con/2/9 system, the volatility of  9 in  system is more obvious than that of  system.We can learn from Table 5 that the Cir/Con/2/9: system does not have the invariant permutation because of 2 ≤  < (/2); we can only find the approximate optimal assignment based on the BIGA.In order to analyze the volatility of  9 , we only study the fact that  9 is in the interval [0.9302, 0.9500] for the Cir/Con/2/9  system.The  9 and the optimal permutation with different  9 are shown in Table 6, and the changes of  9 are shown in Figure 8.
In the Cir/Con/2/9 () system, the change of system reliability with the decrease of  9 is shown in Figures 9  and 10, respectively.With the  9 decreasing, the system reliability of  system and  system both decrease, but the system reliability of  system is more fluctuant.In the    Cir/Con/2/9  system, the system reliability with  9 = 0.9401 is lower than that of  9 = 0.9385.As shown in Table 6, when  9 = 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.
From Figure 11,  15 has some obvious fluctuation, when the value of  15 is in the interval [0.75, 0.8] or [0.85, 0.99].In order to analyze the change of  15 in the Cir/Con/2/15  system clearly, we choose that  15 is in the intervals [0.75, 0.8], [0.85, 0.9], and [0.9, 0.99].From Figure 12, the change of  15 does not show the monotonicity.The value of p 15 is especially in the interval [0.75, 0.8], the floating of  15 is more significant, which reflects that the invariance of the optimal permutation is not fixed.When  15 is decreasing from 0.7804 to 0.7639 in the Cir/Con/2/15: system, the  15 and the optimal permutation are shown in Table 7.When the system has the same optimal permutation, the change of  15 is very small.However, when the optimal permutation changes, the change of  15 will be larger.Therefore, the BI of component is influenced by the optimal permutation.Considering Figures 11 and 12, in the Cir/Con/2/15 system, the volatility of  15 for  system is clearer than that of  system.We can learn from Table 5 that the Cir/Con/2/15: system does not have the invariant permutation because of 2 ≤  < (/2), and we can find the approximate optimal assignment based on the BIGA.The change of system reliability for Cir/Con/2/15 () system with decreasing of  15 is shown in Figures 13 and 14, respectively.With the  15 decreasing, the system reliability of  system and  system will decrease, but changes of the system reliability for  system are more fluctuant.The system reliability with  15 = 0.7689 is lower than  15 = 0.7672 in the  system.Therefore, the permutation at the abnormal floating point is not the optimal permutation.From Table 5, there is no invariant permutation for Cir/Con/6/15: system.So the system optimal permutation depends on the component reliability, and the  15 is connected with the optimal permutation.We can find that  15 has the larger fluctuation in Figure 11, when  15 is in the intervals [0.8, 0.99], [0.7, 0.75], and [0.6, 0.65].In order to analyze the change of  15 obviously, we choose  15 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  15 does not show monotonicity, and  15 is influenced by the change of the optimal permutation.
Figure 16 demonstrates the reliability of Cir/Con/6/15 system with changing of  15 .The system reliability with  15 = 0.7474 is lower than that of  15 = 0.7458, which illustrates that the assignment at the floating point  15 = 0.7474 is not the optimal assignment.
With regard to the Cir/Con// 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.

Conclusions
This paper solves the reliability optimization problem for Cir/Con// 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 , , and  in Cir/Con// 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.

Figure 1 :
Figure 1: The flow chart of BIGA.

( 9 )( 12 )
For the current population (), select  new individuals according to the fitness of each individual.(10) Perform the crossover on the previous selected  individuals to generate offspring population which contains  individuals.(11) Perform mutation on the previous selected offspring population to generate the new offspring population ( + 1).Measure the fitness of every individual in the offspring population.(13) Perform the BITA on the offspring chromosomes.

Figure 2 :
Figure 2: The achievement of three algorithms for small systems.

Figure 3 :
Figure 3: Average of achievement for BIGLS and BIGA.

( 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.

Table 1 :
The symbolic representation of small systems.

Table 4 .
According to the IN values inTable 4, Figure 4 is obtained to illustrate the IN values of BIGLS and

Table 3 :
Experimental results of three algorithms for small systems.
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 (3, 7, 8, and 11) and 3 high reliable systems (11, 12, and 15).Observing other subgraphs in Figure

Table 4 :
Experimental results of the large systems by BIGA and BIGLS.