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Abstract

The reliability optimization problem arises along with the increasing demands for products’ performance in practical engineering. Importance measures are capable of selecting critical components to gain the greatest improvement in system reliability with the constraint of maintenance cost. The characteristics of importance measure for four kinds of typical systems are discussed to illustrate the usage of importance measure in the practical reliability optimization. A reliability optimization model is eventually established and importance measure–based genetic algorithms are developed to solve the reliability optimization problem efficiently. Finally, two numerical experiments are implemented based on the smoke alarm systems. Experiment I is to illustrate the effectiveness of the importance measure–based genetic algorithms compared with standard genetic algorithm. Finally, the relationship between the order of component reliability improvement and the component parameters is analyzed in Experiment II.

Keywords

Reliability optimization importance measure genetic algorithm cost efficiency

Introduction

In reliability engineering, importance measures are generally used to identify the weakest components, and the system reliability can be improved through increasing the component reliability. The concept of importance measures is originally proposed by Birnbaum,¹ which stands for the contribution of component reliability changes to the system reliability. Some representative and latest importance measures include the uncertainty importance measure,^2,3 the generalized Griffith importance measure,⁴ and the integrated importance measure.^5–8 With the importance values of all components, proper actions can be applied to the weakest component to improve system reliability. By now, importance measures have been widely used in practice to identify the weaknesses of different kinds of systems. Many importance measures have been proposed with respect to the diverse considerations of system performance, reflecting different probabilistic interpretations and potential applications.⁹ Barker et al.¹⁰ proposed the component importance measure for the network based on resilience. Baroud et al.¹¹ demonstrated a time-dependent paradigm for resilience and the associated stochastic metrics in a waterway transportation context, and deployed two stochastic resilience-based component importance measures. A novel probability density function estimation–based method is proposed to efficiently evaluate the moment-independent index.¹² Zhai et al.¹³ presented a general formulation of the moment-independent importance measures in reliability and safety engineering. Lisnianski et al.¹⁴ considered a reliability importance evaluation for components in an aging multi-state system to reduce the computational burden of solving the real-world problem. Shen and Cui¹⁵ considered an importance measure calculation for the circular consecutive-k-out-of-n system with sparse d. Wu et al.¹⁶ introduced an importance measure, called component maintenance priority, which is used to select components for preventive maintenance. Dui et al.¹⁷ proposed a new importance measure for analyzing the impact of external factors on system performance. In view of the negative impact of functional dependency, Wang et al.¹⁸ provided an alternative importance measure for measuring the importance of components in mechatronic systems. The variance-based global importance measure is used to evaluate the importance of a component to the mission reliability of phased mission systems while the reliabilities of all components vary randomly.¹⁹

The reliability optimization problem involves minimizing the system cost subject to reliability requirement or maximizing the system reliability under resource constraints (such as cost, weight, or volume), which has been proved to be NP-hard.²⁰ Peng et al.²¹ investigated the reliability analysis and optimal structure of series–parallel phased mission systems that maximizes the system reliability. Meng et al.²² proposed an evidence-based collaborative reliability optimization method to handle epistemic uncertainties of complex engineering systems. Li et al.²³ put forward a hybrid reliability optimization method for various pallets with random parameters, which aimed to provide an effective computational tool for reliability design of the pallet system. Liu et al.²⁴ presented an approach of joint redundancy and imperfect maintenance strategy optimization for multi-state systems. Li et al.²⁵ proposed a two-stage approach by applying a multiple objective evolutionary algorithm for solving multi-objective system reliability optimization problems.

Heuristics methods, such as genetic algorithm (GA), have been widely applied for this type of problem owing to their effectiveness and robustness. Based on genetic evolutionary mechanism, GA has been applied to solve the multi-objective discrete reliability optimization problem in a k-dissimilar-unit non-repairable cold-standby redundancy system.²⁶ A modified non-dominated sorting GA is used to search the Pareto optimal solutions of the multi-objective multi-stage formulation for reliability growth planning.²⁷ The application of importance measures can direct the evolution and limit the randomness in heuristics algorithms by prioritizing the components with higher importance value for reliability improvement. Based on importance measures, heuristics have been developed to solve the component assignment problem in Lin/Con/k/n systems, where system reliability is maximized by optimally assigning interchangeable components to different positions.^28–31

This article concentrates on estimating the reliability improvement of each component to maximize the system reliability under cost constraint. The remainder of this article is structured as follows. Section “Typical importance measures” introduces two traditional importance measures and the relationship between them. In section “Characteristics of importance measures for typical systems,” changes of component importance in typical systems are demonstrated to guide practical reliability optimization. Model of reliability optimization and the importance measure–based GAs are developed in section “Reliability optimization formulations.” In section “Case study,” some smoke alarm systems are introduced to illustrate the effectiveness of importance measure–based GAs and the relationship between the order of component reliability improvement and component parameters is also analyzed. Section “Conclusion” summarizes this work.

Typical importance measures

The typical importance measures such as Birnbaum importance measure (BM) and the Δ-importance measure (DM) are introduced in this section.

BM

The BM is used to evaluate the significance of a component to system reliability¹ and is the effect of the changes of component reliability on the system reliability. The BM of component i in a system is defined in equation (1), where $R_{s}$ is the system reliability and $r_{i}$ is the reliability of component i

I_{i}^{B} = \frac{\partial R_{s}}{\partial r_{i}}

(1)

DM

The DM³² is developed to evaluate the increment of the system reliability due to the improvement of the component reliability. And $r'_{i}$ is the reliability of component i after the improvement

I_{i}^{Δ} = ({r'}_{i} - r_{i}) \cdot \frac{\partial R_{s}}{\partial r_{i}} = ({r'}_{i} - r_{i}) \cdot I_{i}^{B}

(2)

From equation (2), it is clear that DM is the BM times the component reliability improvement. The BM actually is taking the partial derivative of the system structural function with respect to component reliability, but the DM also needs to consider the component reliability improvement.

Characteristics of importance measures for typical systems

Importance measures are used to evaluate the importance of various components in a system. The importance measure value may not be as useful as their relative ranking. Characteristics of importance measure for components in typical systems, which include series, parallel, series–parallel, and parallel–series systems, are of great significance to guide the practical optimization procedure. The changes of BM and DM with the increase in the objective component reliability are discussed as follows.

Series system

According to the calculation of BM in the series system, the lower a component’s reliability is, the larger the BM of the component is. But DM should consider the component reliability improvement and BM. The component with the largest importance is regarded as the objective component, and assume component 1 as the objective component to analyze the changes of BM and DM. As the reliability of the objective component increases, BM and DM of the remaining components also increase (Figure 1). In Figure 1, the bold up arrow represents the changes of objective component reliability, and the other arrows represent the changes of BM for the corresponding components. DM of the objective component is increasing with the increase in objective component reliability, but BM of the objective component is unchanging. If the importance of another component is larger than that of the original objective component, then it is defined as the new objective component. This iteration is a better way to improve the system reliability in the actual optimization problems.

Figure 1.

Changes of BM and DM with the increasing reliability of objective component 1 in the series system.

Parallel system

In a parallel system, the higher a component’s reliability is, the larger the BM of the component is. But DM should consider the component reliability improvement and BM. As the reliability of the objective component increases, BM and DM of the remaining components decrease instead (Figure 2). DM of the objective component is increasing with the increase in objective component reliability, but BM of the objective component is unchanging. In order to obtain the maximum system reliability effectively, the reliability of component 1 needs to improve its reliability as much as possible. If the system reliability cannot satisfy the reliability constraint, the component with the largest importance measure after the improvement should be considered as the new objective component to improve its reliability.

Figure 2.

Changes of BM and DM with the increasing reliability of objective component 1 in the parallel system.

Series–parallel system

A series–parallel system is a series of parallel systems. The lower a subsystem’s reliability is, the larger the BM of the subsystem is. The subsystem with the largest importance measure is defined as the objective subsystem, and the component with the highest importance measure in the subsystem is regarded as the objective component. For the series–parallel system, assume that component 1 in the left first subsystem is the objective component. As the reliability of the objective component increases, BM and DM of components in series with the objective subsystem increase, but those of components in parallel with the objective component decrease instead, as shown in Figure 3. DM of the objective component is increasing with the increase in objective component reliability, but BM of the objective component is unchanging. In Figure 3, the objective subsystem is the left first subsystem and the objective component is component 1 in this subsystem.

Figure 3.

Changes of BM and DM with the increasing reliability of objective component 1 in the series–parallel system.

Parallel–series system

A parallel–series system is the parallel of series systems. The higher a subsystem’s reliability is, the larger the BM of the subsystem is. The subsystem with the highest importance measure is defined as the objective subsystem, and the component with the highest importance measure in the subsystem is the objective component. In Figure 4, as the reliability of component 1 in the top subsystem increases, BM and DM of components in the other parallel subsystems decrease, but those of components, which are in series with component 1 in the top subsystem, increase. However, DM of the objective component is increasing with the increase in objective component reliability, but BM of the objective component is unchanging.

Figure 4.

Changes of BM and DM with the increasing reliability of objective component 1 in the parallel–series system.

Reliability optimization formulations

Assumptions of the reliability optimization problem

The reliability optimization problem under study belongs to the reliability allocation category, which is proved to be NP-hard and noted as Problem P1.³³ With the limitation of total cost, the objective of Problem P1 is to maximize system reliability by increasing the reliability of individual components. The assumptions of the reliability optimization problem studied in this article are shown as follows:

The system is coherent and monotonous;

Components are statistically independent;

System and components only have two disjoint states, namely, failed and operational;

The overall cost is the summation of individual component costs.

Cost function

Mettas³⁴ established the relationship among component reliability, feasibility of reliability improvement, and cost. If the component reliability is known, the cost can be calculated based on the maximum achievable reliability and the minimum acceptable reliability of the component. The cost of component i with reliability $r_{i}$ can be calculated based on equation (3)

c_{i} (r_{i}) = e^{[(1 - f_{i}) \frac{r_{i} - r_{i, \min}}{r_{i, \max} - r_{i}}]}

(3)

Note that this penalty function acts as a weighting factor that captures the difficulty of increasing the component reliability from $r_{i, \min}$ to $r_{i}$ . $r_{i, \max}$ is the maximum achievable reliability of component i and $r_{i, \min}$ is the minimum acceptable reliability of component i. $f_{i}$ describes the feasibility of reliability improvement of component i, and a smaller $f_{i}$ implies that it is more difficult to improve the reliability of component i.

Therefore, the cost of component from $r_{i}$ to $r'_{i}$ can be calculated as follows

\begin{matrix} Δ c_{i} = c_{i} (r_{i} + Δ r_{i}) - c_{i} (r_{i}) \\ = e^{[(1 - f_{i}) \frac{r_{i} + Δ r_{i} - r_{i, \min}}{r_{i, \max} - (r_{i} + Δ r_{i})}]} - e^{[(1 - f_{i}) \frac{r_{i} - r_{i, \min}}{r_{i, \max} - r_{i}}]} \end{matrix}

where $c_{i} (r_{i} + Δ r_{i})$ is the cost of increasing reliability of component i from $r_{i, \min}$ to $r_{i} + Δ r$ and $Δ c_{i}$ is the cost of increasing reliability of component i from $r_{i}$ to $r_{i} + Δ r$ .

Reliability optimization model

Subject to overall cost constraint $C_{o}$ , Problem P1 is a kind of reliability optimization problem, which is to obtain the maximum system reliability with the constraints of maintenance cost, and it can be represented as a non-linear programming model as follows

\begin{matrix} obj . \max R_{s} (Δ r; r) = R_{s} (Δ r_{1} + r_{1}, \dots, Δ r_{i} + r_{i}, \dots, Δ r_{n} + r_{n}) \\ s . t . {\begin{matrix} \sum_{i = 1}^{n} Δ c_{i} ⩽ C_{o} \\ 0 ⩽ Δ r_{i} ⩽ r_{i, \max} - r_{i} (i = 1, 2, \dots, n) \end{matrix} \end{matrix}

(4)

For the reliability optimization model, the objective function is to maximize the system reliability, and the decision variable is the improvement of each component reliability, which is noted as $(Δ r_{1}, \dots, Δ r_{i}, \dots, Δ r_{n})$ . $R_{s} (Δ r; r)$ is the objective function, which is the system reliability when the component reliability $r$ is increased by $Δ r$ . $\sum_{i = 1}^{n} Δ c_{i}$ is the total cost of increasing reliability of all components from $r_{i}$ to $r_{i} + Δ r$ , which should not exceed $C_{o}$ , and $C_{o}$ is the maximum overall cost. The improvement of component i should be less than $r_{i, \max} - r_{i}$ . In this reliability optimization model, we did not consider adding the importance measure into the model, and the importance measure is just the tool for solving the reliability optimization problem.

Mechanism of applying importance measure

When the reliabilities of n components can be improved, the question is how to appropriately determine the reliability improvement of each component to maximize the system reliability with the constraints of overall cost. BM can be used to express the improvement of system reliability if the component reliability is increasing.

Suppose that a system consists of n components. $R_{s} = R_{s} (r_{i})$ actually is the function of $r_{i}$ . According to the multivariate composite function derivation rule, taking derivatives of t on both sides of $R_{s} = R_{s} (r_{i})$ we obtain

\frac{d R_{s}}{dt} = \frac{\partial R_{s}}{\partial r_{i}} \cdot \frac{d r_{i}}{dt} = I_{i}^{B} \cdot \frac{d r_{i}}{dt}

(5)

From equation (5), if the reliability increment of component i is determined, the larger the $I_{i}^{B}$ , the more the contribution of component i. The changes of system reliability can be represented by the BM times the component reliability increment, which is shown as follows

Δ R_{s} \approx \sum_{i = 1}^{n} I_{i}^{B} \cdot Δ r_{i} = \sum_{i = 1}^{n} I_{i}^{D}

(6)

As shown in equation (6), with the same reliability increment, the component with the highest BM can bring the highest improvement of system reliability. If the increment of all components is not the same, the component with the highest DM can bring the highest improvement of system reliability. Therefore, the component with larger BM or DM should be given more priority for reliability improvement.

Algorithms for the reliability optimization model

Since Problem P1 is a non-linear, non-convex optimization model, heuristic algorithms, such as GA, have been widely applied to solve this type of problem. Based on the advantages of importance measure, the importance measure–based GA is developed to solve Problem P1 by taking into account the variability of reliability limits and overall cost constraint. In these algorithms, local search is executed based on BM and DM. The flowchart of the importance measure–based GA is shown in Figure 5, and its detailed procedure is as follows:

Step 1. Confirm the objective function and solution space

The objective function is the system reliability after improvement, and the solution space is a set of the reliability increment of n components, which can be recorded as $Δ r = (Δ r_{1}, Δ r_{2}, \dots, Δ r_{n})$ , and the reliability range of component i is $Δ r_{i} \in [0, r_{i, \max} - r_{i}]$ .

Step 2. Confirm the encoding and decoding method

We use the real number encoding method to represent the reliability increment, where a gene is the real value of the component reliability increment, and all the n genes are treated as a chromosome. Because the gene is the real value of reliability increment, the decoding method need not be used in this algorithm.

Step 3. Initialize the population

According to the population size, s chromosomes are generated based on the solution space, and each chromosome is the potential solution of the problem, which has n positions recording the increment of n components’ reliability.

Step 4. Calculate the fitness of each chromosome and identify the best chromosome

Considering the improvement cost, if the cost is higher than the overall cost constraint, the fitness will be modified with the penalty. Therefore, the fitness function is viewed as $F_{j} = 1 / R_{s} + k \times \max {0, 1 - C_{o} / C}$ . The penalty coefficient k is chosen as an enormous positive number. The value of the fitness $F_{j}$ depends on the combination of reliability increment of all components and their corresponding improvement cost. The less the $F_{j}$ is, the better the corresponding chromosome j is. The chromosome with the smallest fitness in the population is identified as the optimal solution of the current generation, and the solution is saved for the subsequent execution.

Step 5. Inspect whether the termination condition is satisfied

If the algorithm reaches the maximum iteration M, the algorithm will be terminated and output the solution. Generally, the value of M is suggested to vary in the range of [100, 1000].

Step 6. Perform selection on current population

According to the roulette strategy, the offspring population is generated through selecting s chromosomes from the current population. For selecting appropriate chromosomes with the higher system reliability and the constraints of overall cost, the normalization is conducted to modify the fitness of each chromosome by

F'_{j} = \frac{F_{j, \max} - F_{j} + α}{F_{j, \max} - F_{j, \min} + α}

where $F_{j, \min}$ and $F_{j, \max}$ represent the best and worst fitness in the current population, respectively; $α$ is a very small positive number in the case the equation is zero. As a result, the chromosome with the greater value of $F'_{j}$ has more opportunity to be selected as the offspring population.

Step 7. Perform crossover on population

Generate a random number between 0 and 1; if the number is less than $p_{c}$ , the single-point crossover is conducted on a pair of chromosomes. The genes after the crossover point will be exchanged in these two chromosomes. Otherwise, the chromosome remains unchanged.

Step 8. Perform mutation on population

Mutation is performed on all chromosomes to maintain the diversity of the population with a probability of $p_{m}$ . One gene on the mutation point in a chromosome will be replaced with a random number a, $a \in [0, r_{i, \max} - r_{i}]$ .

Step 9. Perform local search based on importance measure

According to section “Mechanism of applying importance measure,” the components with larger importance measure should be given priority for the reliability improvement during the optimization process. Local search based on importance measure is introduced to give the optimization direction. The importance measure of each gene is calculated; the gene with the maximum importance measure will be replaced with a random number $b, b \in [0, r_{i, \max} - r_{i}]$ . Then the fitness values before and after the local optimization are calculated and compared. If the fitness after local optimization is better, the chromosome will be updated; otherwise, the chromosome remains unchanged.

Step 10. Perform the elitism strategy and generate the new offspring population

The chromosome with the worst fitness should be replaced with the best chromosome saved in Step 4. This process makes the offspring in the current population not worse than that in the previous population.

Figure 5.

The flowchart of the importance measure–based genetic algorithm.

In this article, the Birnbaum importance measure–based genetic algorithm (BGA) and Δ-importance measure–based genetic algorithm (DGA) are introduced to solve the optimization problem. The procedures of BGA and DGA are the same as the importance measure–based GAs except for Step 9. The differences of BGA and DGA focus on the local search process, and the other processes of these two algorithms are all the same. For DGA, the importance measure in Step 7 should be replaced by the DM, which is the local search based on DM to find the optimal solution. For the BGA, the importance measure in Step 9 should be replaced by BM. The difference of BGA and DGA is in whether the local search considers the improvement of component reliability in the importance measure or not.

Case study

The smoke alarm systems are the typical mixed system, which includes monitors, sensors, logical processors, and controlling elements. The monitor is used to monitor the states of components in the system; the sensors are used to detect the smoke density and collect the smoke data; the logical processers are used to deal with the collected data; the controlling elements are used to trigger the alarms in the system.

In order to improve the system reliability, a company selects a kind of alarm system which is an integration of two sets of smoke alarm systems from different distributors. Therefore, the smoke alarm system from the first distributor is noted as Case I, the smoke alarm system from the second distributor is noted as Case II, and the smoke alarm system that the company is using is noted as Case III. The structures of these three cases are shown in Figure 6. Case I is a typical series–parallel system, which has two sensors and two controlling elements. Case II is a simple mixed system because sensors 4 and 5 are connected in series. Case III actually is the integration of Cases I and II, which is a complex mixed system.

Figure 6.

The smoke alarm system structure of the three cases.

The improvement cost varies depending on the component type, including feasibility, minimum reliability, maximum reliability, and current reliability. Data of the components are shown in Table 1.

Table 1.

Parameters of each component.

Component	$f_{i}$	$r_{i, \min}$	$r_{i, \max}$	$r_{i}$
1	0.8	0.90	0.95	0.91
2	0.6	0.88	0.96	0.92
3	0.6	0.90	0.99	0.94
4	0.75	0.80	0.90	0.85
5	0.7	0.85	0.94	0.88
6	0.8	0.88	0.98	0.92
7	0.7	0.88	0.98	0.92
8	0.6	0.85	0.95	0.90
9	0.5	0.90	0.99	0.90
10	0.6	0.90	0.97	0.92
11	0.5	0.90	0.98	0.91
12	0.6	0.93	0.99	0.94

Design of experiments

In order to demonstrate that BGA and DGA are better than GA, the first experiment is designed through comparing the solutions of GA, BGA, and DGA. The second experiment is presented to determine the order of component reliability improvement to obtain the optimal solution.

Experiment I. The parameters in the three algorithms are the same, with $s = 100$ , $M = 300$ , $p_{c} = 0.9$ , and $p_{m} = 0.08$ . The overall cost constraint is set at $C 1 = 200, C 2 = 400, \dots, C 20 = 4000$ which is an arithmetic progression that totally includes 20 instances. For each instance of the three cases, 100 trials will be performed to obtain the results based on GA, BGA, and DGA.

Experiment II. For each case, select five instances C4, C8, C12, C16, and C20 and choose the optimal solution of DGA from the 100 trials randomly. The relationship of improvement order of components with component parameters is analyzed.

Results of Experiment I

GA is a kind of random algorithm, and the mean optimal results and the robustness of the results are two important factors of the effectiveness of algorithm. Therefore, the results of the experiment mainly consider the mean system reliability and the robustness of the three algorithms for different cases.

Comparisons of the mean optimal solutions

Mean system reliability of Case I

For 20 instances in Case I, the mean system reliability after improvement and the mean number of generations to obtain the solutions obtained by GA, BGA, and DGA are shown in Table 2.

Table 2.

The mean system reliability and the mean number of generations for Case I.

Case I	Mean system reliability			Mean convergence generation
Case I	GA	BGA	DGA	GA	BGA	DGA
C1	0.92021	0.92047	0.92047	221	205	206
C2	0.92154	0.92175	0.92173	198	199	202
C3	0.92218	0.92238	0.92236	218	192	187
C4	0.92256	0.92274	0.92277	207	183	189
C5	0.92279	0.92299	0.92299	204	188	208
C6	0.92300	0.92321	0.92325	209	191	195
C7	0.92314	0.92340	0.92343	198	179	194
C8	0.92328	0.92348	0.92354	210	190	196
C9	0.92339	0.92362	0.92362	213	186	197
C10	0.92350	0.92372	0.92376	212	194	197
C11	0.92368	0.92379	0.92382	210	188	197
C12	0.92375	0.92385	0.92390	200	201	193
C13	0.92373	0.92393	0.92397	211	199	197
C14	0.92385	0.92399	0.92404	208	196	192
C15	0.92396	0.92406	0.92405	205	192	194
C16	0.92395	0.92413	0.92414	223	200	201
C17	0.92395	0.92419	0.92418	222	192	192
C18	0.92407	0.92423	0.92425	210	195	203
C19	0.92410	0.92425	0.92429	219	198	195
C20	0.92412	0.92430	0.92431	221	198	195

GA: genetic algorithm; BGA: Birnbaum importance measure–based genetic algorithm; DGA: Δ-importance measure–based genetic algorithm.Note: The bold values signifies that the best mean value of system reliability and convergence generation obtained by three algorithms for Case 1.

From Table 2, we can find that the mean system reliability values of BGA and DGA are both larger than that of GA for 20 instances. And the mean convergence generation values of BGA and DGA are always less than that of GA except for instance C2. In order to compare the effectiveness of BGA and DGA, we can find that it is more possible to obtain better solution with DGA because we can find that there are more instances that the mean system reliability of DGA is larger than that of BGA, which is shown in Figure 7. In terms of mean system reliability, BGA obtained the highest reliability in 5 of 20 cases and DGA obtained the highest reliability in 15 of 20 cases. In terms of the mean number of generations to obtain the near-optimal solution, BGA obtained the fewest number of mean convergence generations in 11 of 20 cases and DGA obtained the fewest number of generations in 8 of 20 cases. However, the mean convergence generation of DGA is near to that of BGA, with the difference being less than 10.

Figure 7.

Comparison of the mean system reliability for Case I.

Mean system reliability of Case II

For 20 instances in Case II, the mean system reliability after improvement and the mean number of generations to obtain the solutions runs by GA, BGA, and DGA are shown in Table 3.

Table 3.

The mean system reliability and the mean number of generations for Case II.

Case II	Mean system reliability			Mean convergence generation
Case II	GA	BGA	DGA	GA	BGA	DGA
C1	0.92731	0.92751	0.92761	216	214	224
C2	0.92863	0.92886	0.92894	219	218	219
C3	0.92926	0.92951	0.92956	210	213	216
C4	0.92962	0.92983	0.92993	226	209	221
C5	0.92994	0.93013	0.93018	228	206	201
C6	0.93013	0.93035	0.93038	221	205	214
C7	0.93036	0.93046	0.93060	215	196	208
C8	0.93046	0.93065	0.93069	214	212	211
C9	0.93058	0.93073	0.93078	219	213	217
C10	0.93070	0.93082	0.93090	229	213	200
C11	0.93082	0.93090	0.93100	219	209	215
C12	0.93087	0.93100	0.93107	215	224	218
C13	0.93096	0.93104	0.93112	221	213	203
C14	0.93102	0.93114	0.93119	228	219	217
C15	0.93108	0.93118	0.93129	219	209	208
C16	0.93106	0.93123	0.93129	225	225	205
C17	0.93120	0.93130	0.93138	229	214	202
C18	0.93123	0.93133	0.93141	232	214	215
C19	0.93125	0.93141	0.93147	226	219	201
C20	0.93132	0.93146	0.93147	222	223	207

From Table 3, we can find that the mean system reliability values of BGA and DGA are both larger than that of GA for 20 instances. And the mean convergence generation values of BGA and DGA are almost less than that of GA except for instances C2 and C12. In order to compare the effectiveness of BGA and DGA, we can find that DGA can obtain the better solution than BGA because we can find that the mean system reliability of DGA is larger than that of BGA for all 20 instances, which is shown in Figure 8. In terms of mean system reliability, BGA obtained the highest reliability in none of 20 cases and DGA obtained the highest reliability in all the 20 cases. In terms of the mean number of generations to obtain the near-optimal solution, BGA obtained the fewest number of mean convergence generations in 7 of 20 cases and DGA obtained the fewest number of generations in 11 of 20 cases.

Figure 8.

Comparison of the mean system reliability for Case II.

Mean system reliability of Case III

For 20 instances in Case III, the mean system reliability after improvement and the mean number of generations to obtain the solutions runs by GA, BGA, and DGA are shown in Table 4.

Table 4.

The mean system reliability and the mean number of generations for Case III.

Case III	Mean system reliability			Mean convergence generation
Case III	GA	BGA	DGA	GA	BGA	DGA
C1	0.94701	0.94723	0.94722	209	200	196
C2	0.94737	0.94757	0.94758	210	184	185
C3	0.94749	0.94774	0.94774	203	192	194
C4	0.94760	0.94784	0.94784	206	182	185
C5	0.94769	0.94791	0.94791	199	187	193
C6	0.94779	0.94796	0.94796	209	185	193
C7	0.94778	0.94801	0.94801	204	188	172
C8	0.94786	0.94804	0.94804	210	185	188
C9	0.94790	0.94807	0.94807	205	191	189
C10	0.94790	0.94810	0.94809	204	183	189
C11	0.94791	0.94812	0.94812	207	195	186
C12	0.94795	0.94814	0.94814	206	195	183
C13	0.94795	0.94816	0.94816	207	181	181
C14	0.94800	0.94817	0.94818	204	179	191
C15	0.94801	0.94819	0.94819	209	188	186
C16	0.94799	0.94821	0.94820	212	190	191
C17	0.94802	0.94822	0.94822	203	176	187
C18	0.94804	0.94823	0.94823	217	187	180
C19	0.94802	0.94824	0.94824	194	189	195
C20	0.94806	0.94825	0.94825	206	178	187

From Table 4, we can find that the mean system reliability values of BGA and DGA are both larger than that of GA for 20 instances. And the mean convergence generation values of BGA and DGA are less than that of GA. In order to compare the effectiveness of BGA and DGA, we can find that the mean system reliability of DGA is almost the same as that of BGA, which is shown in Figure 9. In terms of mean system reliability, BGA obtained the highest reliability in 11 of 20 cases and DGA obtained the highest reliability in 9 of 20 cases. In terms of the mean convergence generations to obtain the near-optimal solution, BGA obtained the fewest number of mean convergence generations in 12 of 20 cases and DGA obtained the fewest number of generations in 8 of 20 cases. We can also find that BGA is better to solve the instances with the lower cost constraints (instances 1–10) and DGA is better to solve the instances with higher cost constraints (instances 11–20).

Figure 9.

Comparison of the mean system reliability for Case III.

Comparisons of the robustness

The coefficient of variation (CV) is applied to evaluate the robustness of the three algorithms, which is presented in Table 5. The calculation formula of CV is as follows

CV = \frac{Standard deviation}{Mean}

(7)

Table 5.

Comparisons of the three algorithms in terms of coefficient of variation.

Cases	Case I			Case II			Case III
Cases	GA	BGA	DGA	GA	BGA	DGA	GA	BGA	DGA
C1	0.0409	0.0179	0.0178	0.0458	0.0260	0.0197	0.0213	0.0033	0.0043
C2	0.0375	0.0194	0.0200	0.0412	0.0241	0.0204	0.0218	0.0039	0.0027
C3	0.0300	0.0189	0.0189	0.0365	0.0244	0.0184	0.0255	0.0032	0.0034
C4	0.0228	0.0182	0.0151	0.0482	0.0346	0.0202	0.0267	0.0037	0.0032
C5	0.0343	0.0234	0.0177	0.0467	0.0254	0.0210	0.0231	0.0048	0.0032
C6	0.0329	0.0198	0.0149	0.0429	0.0295	0.0216	0.0167	0.0026	0.0030
C7	0.0345	0.0199	0.0121	0.0367	0.0321	0.0181	0.0233	0.0032	0.0029
C8	0.0394	0.0209	0.0152	0.0415	0.0259	0.0228	0.0177	0.0028	0.0034
C9	0.0351	0.0171	0.0215	0.0361	0.0282	0.0215	0.0178	0.0032	0.0039
C10	0.0357	0.0171	0.0156	0.0371	0.0295	0.0226	0.0213	0.0032	0.0033
C11	0.0250	0.0223	0.0175	0.0312	0.0335	0.0197	0.0217	0.0035	0.0036
C12	0.0291	0.0240	0.0211	0.0386	0.0297	0.0201	0.0242	0.0041	0.0024
C13	0.0365	0.0235	0.0209	0.0393	0.0274	0.0222	0.0232	0.0028	0.0030
C14	0.0317	0.0201	0.0179	0.0351	0.0282	0.0197	0.0210	0.0035	0.0028
C15	0.0259	0.0210	0.0215	0.0392	0.0347	0.0193	0.0208	0.0037	0.0037
C16	0.0273	0.0170	0.0215	0.0425	0.0311	0.0246	0.0235	0.0023	0.0033
C17	0.0390	0.0197	0.0168	0.0373	0.0298	0.0216	0.0208	0.0029	0.0031
C18	0.0284	0.0195	0.0153	0.0319	0.0290	0.0229	0.0190	0.0025	0.0026
C19	0.0277	0.0179	0.0187	0.0400	0.0282	0.0206	0.0220	0.0030	0.0031
C20	0.0344	0.0173	0.0220	0.0335	0.0226	0.0261	0.0164	0.0026	0.0032

As shown in Table 5, the CV is very low for the three methods, which indicates that the robustness of GA, BGA, and DGA are all pretty good. For Case I and Case II, the CV of DGA is very likely less than that of BGA and GA. For Case I, BGA obtained the lowest CV in 7 of 20 cases and DGA obtained the lowest CV in 13 of 20 cases. For Case II, BGA obtained the lowest CV in 1 of 20 cases and DGA obtained the lowest CV in 19 of 20 cases. For Case III, the CV for BGA seems better than that of DGA, but the value is almost the same as that of DGA.

For the results of Experiment I, the effectiveness of BGA and DGA is better than that of GA, that is to say the local search based on the importance measure can obtain the better solution. DGA can solve Cases I and II more effectively, especially for Case II. For Case III, although the BGA performs better than DGA, the solution is almost the same. Therefore, the DGA is better to solve this optimization problem P1 than BGA.

Results of Experiment II

According to the results of Experiment I, the solution of DGA is selected to discuss the relationship of the improvement order of components with component parameters. The solution of instances C4, C8, C12, C16, and C20 for the three cases is selected randomly from the experiments, which is generated by DGA. The optimal solution of the cases based on DGA is shown in Table 6.

Table 6.

Optimal solution of cases based on DGA.

Instances		Solution	Cost	System reliability
Case I	C4	[0.0384, 0.0311, 0.0406, 0.0546, 0.0410, 0.0595]	795.55	0.92242
	C8	[0.0385, 0.0327, 0.0420, 0.0557, 0.0417, 0.0620]	1594.30	0.92364
	C12	[0.0384, 0.0337, 0.0422, 0.0560, 0.0447, 0.0640]	2395.20	0.92396
	C16	[0.0385, 0.0341, 0.0441, 0.0561, 0.0453, 0.0624]	3188.30	0.92430
	C20	[0.0378, 0.0350, 0.0445, 0.0562, 0.0456, 0.0644]	3990.76	0.92378
Case II	C4	[0.0383, 0.0401, 0.0513, 0.0550, 0.0416, 0.0776, 0.0461]	799.91	0.92994
	C8	[0.0382, 0.0432, 0.0541, 0.0556, 0.0402, 0.0811, 0.0468]	1599.08	0.93079
	C12	[0.0387, 0.0417, 0.0521, 0.0555, 0.0406, 0.0816, 0.0465]	2396.96	0.93096
	C16	[0.0383, 0.0458, 0.0549, 0.0556, 0.0425, 0.0827, 0.0470]	3152.46	0.93128
	C20	[0.0384, 0.0442, 0.0529, 0.0568, 0.0415, 0.0834, 0.0471]	3998.21	0.93163
Case III	C4	[0.0384, 0.0142, 0.0361, 0.0320, 0.0463, 0.0535, 0.0536, 0.0260, 0.0737, 0.0350, 0.0452, 0.0458]	799.97	0.94784
	C8	[0.0386, 0.0284, 0.0365, 0.0417, 0.0546, 0.0541, 0.0542, 0.0350, 0.0761, 0.0418, 0.0597, 0.0446]	1596.34	0.94806
	C12	[0.0387, 0.0282, 0.0374, 0.0417, 0.0543, 0.0549, 0.0532, 0.0413, 0.0812, 0.0443, 0.0492, 0.0446]	2397.37	0.94813
	C16	[0.0387, 0.0250, 0.0426, 0.0410, 0.0491, 0.0554, 0.0538, 0.0364, 0.0818, 0.0393, 0.0614, 0.0462]	3193.66	0.94823
	C20	[0.0388, 0.0300, 0.0361, 0.0361, 0.0502, 0.0559, 0.0558, 0.0366, 0.0774, 0.0394, 0.0592, 0.0455]	3993.49	0.94826

DGA: Δ-importance measure–based genetic algorithm.

In order to analyze which component should be selected first to improve its reliability to maximize the system reliability, the improvement percentage of each component i is introduced to evaluate the utilization of component reliability improvement. The calculation of improvement percentage for component i can be obtained by equation (8). The improvement percentage of each component is shown in Table 7

I P_{i} = \frac{Δ r_{i}}{r_{i, m a x} - r_{i}} 100 %

(8)

Table 7.

Improvement percentage of each component for the selected instances.

		Improvement percentage
Case I	C4	[96.11%, 77.85%, 81.29%, 91.07%, 81.92%, 85.05%]
	C8	[96.11%, 77.85%, 81.29%, 91.07%, 81.92%, 85.05%]
	C12	[96.22%, 81.70%, 84.01%, 92.80%, 83.33%, 88.58%]
	C16	[95.94%, 84.20%, 84.46%, 93.25%, 89.39%, 91.46%]
	C20	[96.26%, 85.19%, 88.11%, 93.55%, 90.59%, 89.16%]
Case II	C4	[95.91%, 86.02%, 88.42%, 92.84%, 82.61%, 90.31%, 93.37%]
	C8	[95.78%, 80.23%, 85.46%, 91.73%, 83.30%, 86.19%, 92.24%]
	C12	[95.46%, 86.33%, 90.17%, 92.64%, 80.42%, 90.15%, 93.51%]
	C16	[96.67%, 83.45%, 86.86%, 92.43%, 81.24%, 90.66%, 92.98%]
	C20	[95.63%, 91.70%, 91.52%, 92.70%, 85.10%, 91.85%, 93.98%]
Case III	C4	[96.64%, 62.92%, 75.50%, 77.00%, 84.82%, 91.28%, 90.21%, 70.13%, 86.71%, 79.93%, 78.47%, 90.68%]
	C8	[96.09%, 35.61%, 72.29%, 63.91%, 77.21%, 89.25%, 89.30%, 52.03%,81.90%, 69.92%, 64.57%, 91.51%]
	C12	[96.56%, 71.04%, 73.03%, 83.38%, 90.94%, 90.16%, 90.36%, 70.06%, 84.61%, 83.58%, 85.25%, 89.26%]
	C16	[96.77%, 70.43%, 74.76%, 83.38%, 90.55%, 91.52%, 88.72%, 82.52%, 90.18%, 88.58%, 70.27%, 89.20%]
	C20	[96.85%, 62.42%, 85.15%, 82.09%, 81.78%, 92.33%, 89.67%, 72.77%, 90.88%, 78.70%, 87.75%, 92.50%]

When the reliability of component i is near to the $r_{i} + Δ r_{i}$ , the cost will increase sharply, thus the improvement of component reliability is hard to reach $r_{i, \max}$ . If the improvement percentage of the component reaches 90%, it means that the component should be considered first. Based on the phenomenon, we can find the order of the component to improve the component reliability from Table 7. For Case I, we need to consider components 1 and 7 in instances C4 and C8; components 1 and 11 should be considered in instance C12; components 11 and 10 should be considered in instance C16; components 10 and 3 should be considered in instance C20. For Case II, we need to consider components 1, 12, 6, and 9 in instances C4 and C8; components 9, 5, and 6 should be considered in instances C12 and C16; components 6, 4, and 8 should be considered in instance C20. For Case III, we need to consider components 1, 6, 7, and 12 in instances C4 and C8; components 4, 5, 9, 10, and 11 should be considered in instance C12; components 4, 8, and 10 should be considered in instance C16; component 3 should be considered in instance C20.

From the improvement percentage, we can obtain the general order of component reliability improvement. In order to build the relationship of the improvement order of components with component parameters, the greedy strategy is easy to determine and we need to improve the component reliability with the lower unit cost. The improvement of system reliability with unit cost $Δ R_{s} / Δ r_{i}$ for component i can be evaluated as follows

\begin{matrix} \frac{Δ R_{s}}{Δ r_{i}} \approx \frac{(r_{i, \max} - r_{i}) \cdot \frac{\partial R_{s}}{\partial r_{i}}}{(r_{i, \max} - r_{i}) \cdot \frac{\partial c_{i}}{\partial r_{i}}} = \frac{{(r_{i, \max} - r_{i})}^{2} \cdot I_{i}^{B}}{c_{i} \cdot (r_{i, \max} - r_{i, min})} \\ = \frac{(r_{i, \max} - r_{i}) \cdot I_{i}^{D}}{c_{i} \cdot (r_{i, \max} - r_{i, min})} \end{matrix}

Based on the component parameters, we can obtain the improvement of system reliability with unit cost for each component as shown in Table 8.

Table 8.

Improvement of system reliability with unit cost of each component for the three cases.

Cases	$Δ R_{s} / Δ r_{i}$	Ranking
Case I	[0.090706, 0.003110, 0.004608, 0.072831, 0.005953, 0.010939]	[1-7-11-10-3-2]
Case II	[0.086934, 0.007116, 0.009286, 0.008188, 0.003994, 0.011982, 0.089572]	[12-1-9-5-6-4-8]
Case III	[0.754878, 0.001474, 0.002365, 0.002113, 0.003007, 0.010198, 0.038939, 0.002557, 0.007628, 0.004024, 0.004787, 0.061800]	[1-12-7-6-9-11-10-5-8-3-4-2]

From Table 8, we can find the order of component reliability improvement for the three cases, and the order is consistent with the analysis of the solution. For Case I, the improvement order is 1-7-11-10-3-2; for Case II, the improvement order is 12-1-9-5-6-4-8; for Case III, the improvement order is 1-12-7-6-9-11-10-5-8-3-4-2. Therefore, the order of component reliability improvement can be determined based on $Δ R_{s} / Δ r_{i}$ , which is related with the component parameters and importance measure.

Conclusion

In this article, BM and DM are introduced into the GA for solving the reliability optimization problem with component improvement cost effectively. Numerical experiments are implemented to evaluate and justify the effectiveness of the importance measure–based GAs based on the smoke alarm systems. The results of Experiment I show that BGA and DGA perform better than GA, which can reach the near global solution in a more effective way, and DGA is better than BGA in general. The results of Experiment II show that the order of component reliability improvement can be determined by the system reliability improvement with unit cost, which is related with the component parameters and importance measure. The future work will focus on the applications of importance measure for more practical complex systems such as the reconfigurable systems or the phased mission systems.

Footnotes

Handling Editor: Xihui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 71401016) and the Fundamental Research Funds for Central Universities of Chang’an University (Nos 300102228110 and 300102228402).

ORCID iD

Jiang-bin Zhao

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Reliability optimization of systems with component improvement cost based on importance measure

Abstract

Keywords

Introduction

Typical importance measures

BM

DM

Characteristics of importance measures for typical systems

Series system

Parallel system

Series–parallel system

Parallel–series system

Reliability optimization formulations

Assumptions of the reliability optimization problem

Cost function

Reliability optimization model

Mechanism of applying importance measure

Algorithms for the reliability optimization model

Case study

Design of experiments

Results of Experiment I

Comparisons of the mean optimal solutions

Mean system reliability of Case I

Mean system reliability of Case II

Mean system reliability of Case III

Comparisons of the robustness

Results of Experiment II

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References