Preventive Maintenance Optimization and Comparison of Genetic Algorithm Models in a Series–Parallel Multi-State System

1.1 Context

The preventive maintenance (PM) consists of actions that improve the condition of system elements before they fail. The optimization of the policy of preliminary planned PM actions is the subject of much research activities. The aim of this paper is to optimize, for each component of a multi-state series–parallel system, the periodic maintenance policy minimizing the system cost function, with respect to the system availability constraint A₀, and for a given mission time T_M. The specificity of the first time inspection is considered. The first time inspection is determined based on Birnbaum important factor. Each component state becomes “as good as new” after each inspection.

Although genetic algorithms (GAs) can rapidly locate the region in which the global optimum exists, they take a relatively long time to locate the exact local optimum in the region of convergence. A combination of a GA and a local search (LS) method can speed up the search to locate the exact global optimum. In such a hybrid, applying a LS to the solutions that are guided by a GA to the most promising region can accelerate convergence to the global optimum.

The improper choice of control parameters is another source of the limitation of GAs in solving real-world problems due to its detrimental influence on the trade-off between exploitation and exploration. Depending on these parameters, the algorithm can either succeed in finding an optimum solution in an efficient way or fail. Choosing the correct parameter values is a time-consuming task. In addition, the use of rigid, constant control parameters is in contradiction to the evolutionary spirit of GAs. For this reason, other search techniques [like fuzzy logic (FL)] can be utilized to set the values of these parameters, while the search is progressing.

Applications of GA for optimization problems are widely known. FL may be used to dynamically control the parameters of the GA as in Refs. [5, 7, 10, 11, 17], or [6]. Dahiya et al. [3] showed that combining GA with LS making the use of domain-specific knowledge enhances the speed of convergence of GA and can be made competitive with others when the search space is too large to explore (see for some applications and comparisons: Refs. [2, 4, 15, 16, 18], etc.). In this paper, we propose to use a GA (without the guide of FL), a hybridization of GA and LS GA-LS, an evolutionary algorithm called FL controlled GA (FLC-GA), and hybridization (FLC-GA) and LS (FLC-GA-LS) to solve the optimization problem so as to compare the performance of these algorithms. The universal generating function (UGF) is used to assess the availability of the studied system. The solution comprises both the availability and the cost evaluation. Levitin and Lisnianski [8] studied a similar optimization problem applied on a series–parallel multi-state system taking into account imperfect component PM actions. This model does not consider the first time of inspection, and the age of component is reduced by a factor after each imperfect maintenance action. The GA was used to solve the optimization problem [1] have developed a method to determine an optimal periodic maintenance policy in a series–parallel system but not for multi-states system. The Monte Carlo simulation was used to assess the system availability.

1.2 System Description

We adopt a series–parallel multi-state system with non-identical binary independent components. The system maintenance cost is a function of the inspection cost of each component. In our study, the cost of inspection for each component is the same for all the mission periods. All components are immediately and perfectly repairable after failure. The failure times of each component for a fixed load occur according to an exponential distribution (constant rate). Once the maintenance period of a system component is fixed, one can evaluate the component performance distribution and its maintenance cost. Hence, the system availability, the system performance, and its total maintenance cost are evaluated. The objective function (the system cost) is known. The optimization procedure seeks for the components’ maintenance periods that maximize the objective function under system availability constraint. A numerical example is considered, and the process is applied for the different required availability values. Section 2 describes the PM model for general the series–parallel system. Section 3 formulates the optimization problem. Section 4 provides the details of GA, FL, and LS. In Sections 5 and 6, simulation results and conclusions are addressed.

2.1 Maintenance Model for Basic Components

We assume that the PM actions improve the reliability of the basic component to as good as new; thus, the component’s age is restored to zero. The problem to find the optimal vector T_p is closely connected with another problem, i.e., to find the optimal first inspection time vector T₀: because it makes no sense to carry out inspections in the beginning of the life of a system, when both the system and its basic components are very reliable according to Bris et al. [1]. Thus, the optimal vector T₀ must be found for each of the basic components. The optimal vector T₀ must be constructed so that it takes into account both cost and reliability point of views.

2.2 General Series–Parallel Structure

In this paper, we adopt a series–parallel system structure that is shown in Figure 1. K is the number of series sub-systems, and E_K is the number of parallel components in the k-th series sub-system.

Figure 1:

General Series-Parallel Structure; K is the Number of Series Sub-Systems, and E_K is the Number of Parallel Components in the k-th Series Sub-System. T_M, T₀, and T_p are, Respectively, the Mission Time, the Time of the First Inspection, and the Maintenance Period of the Component (the Bracket Notation ⌊x⌋ is the Integer Value of x).

2.3 Cost Model

The cost of the presented PM policy of a series–parallel system can be calculated as a function of the inspections costs of the system components. One can write:

η_(e(i,k)) is the number of total inspections of the i-th component in the k-th series sub-system (parallel block) during the mission time. c_l(e(i, k)) is the cost of the l-th inspection of the i-th component in the k-th series sub-system. K is the number of series sub-systems, and E_K is the number of parallel components in the k-th series sub-system. N=∑k=1K∑i=1Eke(i, k) is the number of system components. We assume that the inspection cost of the component is constant during the mission time. Hence, one can write:

c(e(i, k) is the cost inspection of the i-th component in the k-th series sub-system with:

4.1 Genetic Algorithm

Our problem is characterized by a large space of solutions. As the objective function (the quality of the solution) is the only available information, the resolution should be done through meta-heuristic methods. The GA is one of the most powerful meta-heuristic methods. It has been adopted to solve many reliability optimization problems [1, 8, 9, 12] etc. It is inspired from the genetic biology. It is based on the principle of evolutionary search.

The solutions are represented by chromosomes in the form of strings. Any maintenance policy can be represented as an N-length integer string x in which any element determines the maintenance period for system component j.

The number of a population at each iteration is constant, and it is N_s. After crossover and mutation, two solutions are obtained. Each solution is a PM policy. Then, the cost of PM and availability of the entire system corresponding to each policy are calculated. Thus, the objective function of each solution is evaluated. The solutions are sorted in ascending order according to the value of their objective function (maintenance cost). The solutions satisfying the constraint of availability and having minimum values of objective function are the best solutions among the population and consequently are ranked first. The solutions that do not satisfy the required constraint are all classified after those satisfying the required constraint and in ascendant order according to their objective function. The first solutions (among two solutions obtained) are saved, and the others are discarded.

The adopted GA can be described by the following steps:

1. Build an initial population of N_s solutions generated randomly.

2. Evaluate each chromosome in the population.

3. Obtain new solutions by using crossover and mutations with probability p_m and p_c, respectively.

4. Decode the string and evaluate the solution corresponding to the obtained maintenance policy (determine the system availability and the PM cost).

5. The objective value is used to compare different solutions. The solutions are ordered from the best to the worst. The best solutions join the population, and the other ones are discarded.

6. Repeat N_r times steps 2 to 5.

The reproduction of the parents leads to two newborns. Parents are selected two by two. Two adjacent parents are prepared for the crossover procedure to get two newborns. The reproduction of parents provides new solutions. These solutions are put into a new list. All parents are subject to the crossover procedure. A crossing point is fixed to combine different parts of parental chromosomes and build a new born. Figure 2 illustrates this procedure.

Figure 2:

Crossover Procedure.

4.2 Fuzzy Logic

In order to improve the search performance of evolutionary algorithm, it was proposed to use FL for adjusting the crossover rate and the mutation rate in 10 consecutive generations as used by the authors [7, 10, 11]. The average fitness of the population and the population diversity were taken as inputs to fuzzy controllers for controlling crossover and mutation rates. It is because appropriate crossover and mutation rates can improve the search performance reflected by the average fitness values of the population and control the population diversity to prevent premature convergence from happening in the population as cited by Lau et al. [7]. The output of the controller is the changes of the two probabilities p_c and p_m. Lau et al. [7] introduced a FL controller (FLC), which aimed to set proper parameter values of the GA, in order to solve a transportation problem. Li et al. [11] used a similar FLC to solve the parallel machine-scheduling problem.

In this work, to solve the PM optimization problem, we used a similar FLC than that adopted by Lau et al. [7]. This FLC is adopted in GA to change its crossover and mutation probabilities in order to improve the search ability of the algorithm. Hence, our modified algorithm is called FLC-GA.

The implementation of FLC-GA is shown as follows. Let f_a(t−9), f_a(t), f_a(t+1), and f_a(t+10) be the average fitness values of the solutions of the population obtained at generations (t−9), t, (t+1), and (t+10), respectively, d(t) and d(t+10) be the degree of population diversity at generations t and (t+10), respectively, p_c(t) and p_m(t) be the crossover rate and mutation rate adopted at generation t, Δp_c(t+1) and Δp_m(t+1) be the change of crossover rate and mutation rate at generation (t+1), respectively.

When the generation t is reached, the values of f_a(t−9), f_a(t), and d(t) are included in the two fuzzy controllers. d is the average of bit difference of all pairs of chromosomes in the population. This can be calculated using the following formula:

N_s is the total number of chromosomes in the population, N is the chromosome length, g_ik is the value of the k-th gene of the i-th chromosome, and δ(g_ik, g_jk)=1 if g_ik=g_jk, 0 otherwise. Then, the two fuzzy controllers will determine the values of Δp_c(t+1) and Δp_m(t+1) as the outputs.

The FL, which was suggested by Zadeh [19], is a powerful useful tool for performing reasoning in decision-making problems involving uncertainty and vagueness. The FLC is composed of three steps: fuzzification, decision making, and defuzzification. Triangular membership functions, which are commonly used, have been adopted in this study. There are nine fuzzy sets forming the decision table. The IF-THEN rule of Lau et al. [7] is used. The center of gravity (COG) defuzzification method is adopted here, as it is the best-known defuzzification according to Li et al. [11]. The details of these steps are explained in the work of Maatouk et al. [13].

4.3 Local Research

The main errors that a traditional GA suffers from are slow convergence and premature convergence. To improve upon these defects, a new GA known as hybrid GA was developed. It is a combination of simple GA and LS algorithm. The technique of hybridization of knowledge and global GA is called memetic algorithm (MA). The GA first performs the search, and then, the refinement is done by the LS algorithm. After the n iteration of LS, the local optimal solution is injected into the current generation. A LS method locates the local minima, which complements the GA to capture global minima.

In our case, the crossover operator is associated with a small diversification through LS. The procedure consists of a small change in the value of the genes of the resulting chromosomes after the crossover procedure. A small value (compared to the mission time of the system) is added or reduced from the components of the resulting chromosome (which represents periods of maintenance of the system components).

This change has no great effect in terms of the components’ maintenance plan. Often, a small change in the period of maintenance of a component keeps constant the number of PM actions of this component and, thus, the cost of its maintenance. However, it can sometimes have a great effect on the cost of maintenance of a component by adding or removing one PM action, without making major changes to the system availability. This allows a jump of the optimal maintenance cost of the system.

The value to add or to reduce (ϵ) must be a compromise between the local operating processes (LS) and the overall operating process (GA). Several tests (about 20 tests) were conducted to find the variation interval I of (ϵ), which improves the result of our case studied.

After the creation of two new born N₁ and N₂ by the crossover procedure of the two parents P₁ and P₂, a random value (ϵ) selected from the interval I is reduced from each value of the gene of the first born N₁. If the new value of the maintenance period is negative, then it is replaced by a randomly chosen value between 1 and 3. The new solution is evaluated (by calculating the cost of maintenance and the availability of the system subject to the corresponding maintenance plan). The values obtained are compared with those of the solution N₁. The best solution is then retained (which corresponds to the minimum cost). The operation is repeated by adding to the genes of the second born N₂ an (ϵ) value chosen randomly from the interval I. If the new value of the maintenance period exceeds the mission time (in our case T_M=50), then, it is replaced by a value randomly chosen between 47 and 49. We will always be close to the initial solutions, so that diversifying the solutions (obtained by crossing procedures) keeps the parent properties. This procedure allows the search for a local optimum in the space of the best solutions. The following algorithm shows the steps of the LS:

1. After crossover of selected parents P₁ and P₂, new borns N₁ and N₂ are created.

2. Add small variation to the gene of chromosomes of N₁ (N₁⇒N₁+ϵ). Many tests were run to choose the best value of ϵ.

3. Evaluation (cost and availability), comparison (with and without mutation), best solution is retained, N₂⇒N₂+ϵ.

4. Evaluation (cost and availability), comparison (with and without mutation), best solution is retained.

5. Repeat the whole process after each crossover operation.

5.1 Using GA

The optimization problem is treated under the availability constraint A₀=0.9 in the first time using GA. Many tests have been produced for the simulation with different values of parameters (N_S, N_r, p_c, and p_m: population size, generation number, probability of crossover, and probability of mutation, respectively). p_m must be low (supposed to be less than 0.4), and p_c must be high (supposed to be more than 0.8). The following Table 2 shows the minimal maintenance cost obtained for different values of mutation and crossover probabilities and for N_S=100, N_r=2000.

The best solution was obtained for N_S=100, N_r=2000, p_c=0.8, and p_m=0.05.

The number of chromosomes N=9. The corresponding optimal vector of maintenance period is:

T_P=[14.9738 1.7376 42.9512 13.4853 14.3283 20.7269 2.7772 3.9822 9.5986].

The best solution cost is C_PM=561.64.

Many runs of GAs with different required system availability have been done: A₀=0.9, A₀=0.8, A₀=0.7, and A₀=0.6. The effect of this factor on the optimal PM cost is studied. Results are summarized in Table 3.

• A₀=0.9:

  T_P=[14.97 1.74 42.95 13.49 14.33 20.73 2.78 3.98 9.60]

  C_PM=561.64

• A₀=0.8:

  T_P=[15.64 2.69 7.06 5.42 8.75 10.37 2.74 18.48 5.57]

  C_PM=555.94

• A₀=0.7:

  T_P=[4.21 10.48 18.66 3.70 13.36 12.26 2.80 19.69 12.49]

  C_PM=453.41

• A₀=0.6: T_P=[6.34 7.33 35.90 30.85 4.86 14.58 3.89 10.52 20.06]

  C_PM=376.14

The overall process is repeated using hybridization FLC-GA, GA-LS, and FLC-GA-LS. In order to produce fair performance comparisons between these algorithms, in the parameter setting of the GA, we used the same values of population size, generation number, probability of crossover, and probability of mutation (N_S=100, N_r=2000, p_c=0.8, and p_m=0.05).

5.2 Using FCL-GA

The solutions are presented in Table 4.

• A₀=0.9:

  T_P=[8.05 31.89 2.92 22.95 5.42 19.34 2.70 22.64 3.49]

  C_PM=544.47

• A₀=0.8:

  T_P=[4.01 4.50 41.45 5.93 5.42 7.30 46.29 5.36 4.96]

  C_PM=491.32

• A₀=0.7:

  T_P=[4.81 4.85 38.27 5.40 21.63 7.41 5.35 42.25 5.52]

  C_PM=410.87

• A₀=0.6:

  T_P=[3.55 19.08 19.68 14.11 12.31 3.52 25.05 10.22 12.56]

  C_PM=359.09

5.3 Using Hybridization GA and LS

The solutions are presented in Table 5.

• A₀=0.9:

  T_P=[43.55 3.03 47.7 40.9 1.74 3.52 32.98 43.33 10.16]

  C_PM=491.74

• A₀=0.8:

  T_P=[3.83 4.41 21.19 4.63 8.69 20.40 4.79 8.05 6.04]

  C_PM=487.41

• A₀=0.7:

  T_P=[3.5 15.90 13.66 10.30 6.97 7.43 4.60 6.05 41.63]

  C_PM=409.57

• A₀=0.6:

  T_P=[7.74 9.38 6.40 11.10 11.74 7.33 9.93 20.95 9.04]

  C_PM=340.59

5.4 Using Hybridization FLC-GA and LS

The solutions are presented in Table 6. The results show that hybridization FLC-GA and LS outperforms the GA and the FLC-GA.

• A₀=0.9:

  T_P=[47.17 1.61 9.77 17.22 3.72 34.98 42.07 32.26 7.72]

  C_PM=453.00

• A₀=0.8:

  T_P=[46.82 12.45 2.57 23.74 16.67 11.87 30.67 6.57 7.13]

  C_PM=398.74

• A₀=0.7:

  T_P=[23.12 20.20 5.24 47.00 4.64 46.23 8.55 31.74 5.89]

  C_PM=343.29

• A₀=0.6:

  T_P=[9.51 16.20 10.22 4.79 20.91 29.32 30.67 45.81 7.70]

  C_PM=227.18

5.5 Interpretations of Result

The results for the different used algorithms are summarized in Figure 4.

It can be seen that the cost of the optimal maintenance policy decreases quickly when the required availability decreases in all cases (using GA, FLC-GA, GA-LS, or hybrid FLC-GA and LS). This is due to the fact that the number of PM actions of components required to answer the availability constraints decreases (components can be less available); hence, the cost of PM of the system components, and thus, the entire system maintenance cost decreases.

The hybridization of genetic algorithm with another algorithm (LS or FL) shows an improvement of the minimal cost, which is reduced with respect to that obtained using GA, and for all required availability constraints. Using this hybridization, the performance of the genetic algorithm was enhanced.

This is because the GAs’ problem solution power can be increased by local searching and/or FL. A simple GA may suffer from slow convergence and instability of results.

Figure 4 shows that both GA-FL and GA-LS gives approximately the same results and for different values of required availability.

It can be seen also that the optimal maintenance cost obtained using hybrid FLC-GA and LS is less than that obtained using GA, using GA-LS, and less than that obtained using FLC-GA, and for all different values of required availability. Thus, hybrid FLC-GA and LS outperforms GA GA-LS, and FLC-GA in all of these cases. The reason is that LS maintains the current solution and search its neighborhoods for a better one so that combined with GA, LS is suited for fine-tuning solutions, which are very close to optimal ones. FL can balance the degree of convergence and diversity of the population no matter what the current state of the population is. This can provide a good convergence and prevent trend to local optimization.

The interest on LS comes from the fact that they may effectively and quickly explore the basin of attraction of optimal solutions, finding an optimum with a high degree of accuracy and within a small number of iterations. GAs are optimization techniques that use a population of candidate solutions. They explore the search space by evolving the population through four steps: parent selection, crossover, mutation, and replacement. GAs have been seen as search procedures that can locate high-performance regions of vast and complex search spaces, but they are not well suited for fine-tuning solutions. However, the components of the GAs may be specifically designed, and their parameters tuned, in order to provide effective LS behavior.

In order to improve the search performance of evolutionary algorithm, it was proposed to use FL for adjusting the crossover rate and the mutation rate. Appropriate crossover and mutation rates can improve the search performance reflected by the average fitness values of the population and control the population diversity to prevent premature convergence from happening in the population. Using the beneficial advantages of the three algorithms (GA, FL, and LS) simultaneously, hybridization FLC-GA with LS outperforms all other tested algorithms.

We have also tested the execution time of the different instances (with and without hybridization (FLC-GA and/or LS)). When using hybridization, the execution time does not have a significant increase. The best solution cost has a significant reduction. Thus, it is interesting to apply the hybridization ion in this optimization problem.