**A Hybrid Parallel Genetic Algorithm for Reliability Optimization**

Ki Tae Kim and Geonwook Jeon *Korea National Defense University, Republic of Korea* 

### **1. Introduction**

126 Real-World Applications of Genetic Algorithms

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989-998.

Reliability engineering is known to have been first applied to communication and transportation systems in the late 1940's and early 1950's. Reliability is the probability that an item will perform a required function without failure under stated conditions for a stated period of time. Therefore a system with high reliability can be likened to a system which has a superior quality. Reliability is one of the most important design factors in the successful and effective operation of complex technological systems. As explained by Tzafestas (1980), one of the essential steps in the design of multiple component systems is the problem of using the available resources in the most effective way so as to maximize the system reliability, or so as to minimize the consumption of resources while achieving specific reliability goals. The improvement of system reliability can be accomplished using the following methods: reduction of the system complexity, the allocation highly reliable components, and the allocation of component redundancy alone or combined with high component reliability, and the practice of a planned maintenance and repair schedule. This study deals with reliability optimization that maximizes the system reliability subject to resource constraints.

This study suggests mathematical programming models and a hybrid parallel genetic algorithm (HPGA). The suggested algorithm includes different heuristics such as swap, 2 opt, and interchange (except for reliability allocation problem with component choices (RAPCC)) for an improvement solution. The component structure, reliability, cost, and weight were computed by using HPGA and the experimental results of HPGA were compared with the results of existing meta-heuristics and CPLEX.

### **2. Literature review**

The goal of reliability optimization is to maximize the reliability of a system considering some constraints such as cost, weight, and so on. In general, reliability optimization divides into two categories: the reliability-redundancy allocation problem (RRAP) and the reliability allocation problem with component choices (RAPCC).

### **2.1 The reliability-redundancy allocation problem (RRAP)**

The RRAP is the determination of both optimal component reliability and the number of component redundancy allowing mixed components to maximize the system reliability

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 129

This study deals with the reliability-redundancy allocation problem in a series-parallel

The relationship between the system reliability ( *RS* ) and the reliability of subsystem *i* ( *Ri* ),

1

The relationship between the reliability of subsystem *i* ( *Ri* ) and the reliability of

1 1 1 *ij m x*

*i ij j R r* =

component *j* available for subsystem *i* ( *ij r* ), in a parallel system, is shown in Eq. (2).

<sup>=</sup> ∏ (1)

=− − ∏ (2)

*n S i i R R* =

*RS* : system reliability

0, otherwise

system as shown in Fig. 1.

Fig. 1. Series-parallel system

in a series system, is shown in Eq. (1).

 = *ij*

*Ri* : reliability of subsystem *i*

*CS* : system-level constraint limits for cost *WS* : system-level constraint limits for weight

*j i <sup>x</sup>*

*ij r* : reliability of component *j* available for subsystem *i*

( ) 1, if component usedinsubsystem for RAPCC

**3.1 Reliability-redundancy allocation problem (RRAP)** 

*ij c* : cost of component *j* available for subsystem *i wij* : weight of component *j* available for subsystem *i ui* : maximum number of components used in subsystem *i ij x* : quantity of component *j* used in subsystem *i* (for RRAP)

under cost and weight constraints. It is known as the NP-hard problem suggested by Chern (1992).

A variety of algorithms, as summarized in Tillman et al. (1977), and more recently by Kuo & Prasad (2000), Kuo & Wan (2007), including exact methods, heuristics and meta-heuristics have already been proposed for the RRAP. An exact optimal solution is obtained by exact methods such as cutting plane method (Tillman, 1969), branch-and-bound algorithm (Chern & Jan, 1986; Ghare & Taylor, 1969), dynamic programming (Bellman & Dreyfus, 1958; Fyffe et al., 1968; Nakagawa & Miyazaki, 1981; Yalaoui et al., 2005), and goal programming (Gen et al., 1989). However, as the size of problem gets larger, such methods are difficult to apply to get a solution and require more computational effort. Therefore, heuristics and metaheuristics are used to find a near-optimal solution in recent research.

The research using heuristics is as follows. Kuo et al. (1987) present a heuristic method based on a branch-and-bound strategy and lagrangian multipliers. Jianping (1996) has developed a method called a bounded heuristic method. You & Chen (2005) proposed an efficient heuristic method. Meta-heuristics such as genetic algorithm (Coit & Smith, 1996; Ida et al., 1994; Painton & Campbell, 1995), tabu search (Kulturel-Konak et al., 2003), ant colony optimization (Liang & Smith, 2004), and immune algorithm (Chen & You, 2005) have been introduced to solve the RRAP.

### **2.2 The reliability allocation problem with component choices (RAPCC)**

The RAPCC is the determination of optimal component reliability to maximize the system reliability under cost constraint. A problem is formulated as a binary integer programming model with a nonlinear objective function (Ait-Kadi & Nourelfath, 2001), which is equivalent to a knapsack problem with multiple-choice constraint, so that it is the NP-hard problem (Garey & Johnson, 1979). Some algorithms for such knapsack problems with multiple-choice constraint have been suggested in the literature (Nauss, 1978; Sinha & Zoltners, 1979; Sung & Lee, 1994).

A variety of algorithms including exact methods, heuristics, and meta-heuristics have already been proposed for the RAPCC. An exact optimal solution is obtained by branchand-bound algorithm (Djerdjour & Rekab, 2001; Sung & Cho, 1999). Meta-heuristics such as neural network (Nourelfath & Nahas, 2003), simulated annealing (Kim et al., 2004; Kim et al., 2008), tabu search (Kim et al., 2008), and ant colony optimization (Nahas & Nourelfath, 2005) have been introduced to solve the RAPCC. Also, Kim et al. (2008) solved the largescale examples by using a reoptimization procedure with tabu search and simulated annealing.

### **3. Mathematical programming models**

Notations and decision variables in the mathematical programming model are as follows.

*n* : the number of subsystems *m* : the number of components


under cost and weight constraints. It is known as the NP-hard problem suggested by Chern

A variety of algorithms, as summarized in Tillman et al. (1977), and more recently by Kuo & Prasad (2000), Kuo & Wan (2007), including exact methods, heuristics and meta-heuristics have already been proposed for the RRAP. An exact optimal solution is obtained by exact methods such as cutting plane method (Tillman, 1969), branch-and-bound algorithm (Chern & Jan, 1986; Ghare & Taylor, 1969), dynamic programming (Bellman & Dreyfus, 1958; Fyffe et al., 1968; Nakagawa & Miyazaki, 1981; Yalaoui et al., 2005), and goal programming (Gen et al., 1989). However, as the size of problem gets larger, such methods are difficult to apply to get a solution and require more computational effort. Therefore, heuristics and meta-

The research using heuristics is as follows. Kuo et al. (1987) present a heuristic method based on a branch-and-bound strategy and lagrangian multipliers. Jianping (1996) has developed a method called a bounded heuristic method. You & Chen (2005) proposed an efficient heuristic method. Meta-heuristics such as genetic algorithm (Coit & Smith, 1996; Ida et al., 1994; Painton & Campbell, 1995), tabu search (Kulturel-Konak et al., 2003), ant colony optimization (Liang & Smith, 2004), and immune algorithm (Chen & You, 2005) have

The RAPCC is the determination of optimal component reliability to maximize the system reliability under cost constraint. A problem is formulated as a binary integer programming model with a nonlinear objective function (Ait-Kadi & Nourelfath, 2001), which is equivalent to a knapsack problem with multiple-choice constraint, so that it is the NP-hard problem (Garey & Johnson, 1979). Some algorithms for such knapsack problems with multiple-choice constraint have been suggested in the literature (Nauss, 1978; Sinha &

A variety of algorithms including exact methods, heuristics, and meta-heuristics have already been proposed for the RAPCC. An exact optimal solution is obtained by branchand-bound algorithm (Djerdjour & Rekab, 2001; Sung & Cho, 1999). Meta-heuristics such as neural network (Nourelfath & Nahas, 2003), simulated annealing (Kim et al., 2004; Kim et al., 2008), tabu search (Kim et al., 2008), and ant colony optimization (Nahas & Nourelfath, 2005) have been introduced to solve the RAPCC. Also, Kim et al. (2008) solved the largescale examples by using a reoptimization procedure with tabu search and simulated

Notations and decision variables in the mathematical programming model are as follows.

heuristics are used to find a near-optimal solution in recent research.

**2.2 The reliability allocation problem with component choices (RAPCC)** 

been introduced to solve the RRAP.

Zoltners, 1979; Sung & Lee, 1994).

**3. Mathematical programming models** 

*n* : the number of subsystems *m* : the number of components

*i* : index for subsystems ( *i n* = 1,2, , ) *j* : index for components ( *j m* = 1,2, , )

annealing.

(1992).

*RS* : system reliability *Ri* : reliability of subsystem *i CS* : system-level constraint limits for cost *WS* : system-level constraint limits for weight *ij r* : reliability of component *j* available for subsystem *i ij c* : cost of component *j* available for subsystem *i wij* : weight of component *j* available for subsystem *i ui* : maximum number of components used in subsystem *i ij x* : quantity of component *j* used in subsystem *i* (for RRAP) ( ) 1, if component usedinsubsystem for RAPCC 0, otherwise = *ij j i <sup>x</sup>*

### **3.1 Reliability-redundancy allocation problem (RRAP)**

This study deals with the reliability-redundancy allocation problem in a series-parallel system as shown in Fig. 1.

Fig. 1. Series-parallel system

The relationship between the system reliability ( *RS* ) and the reliability of subsystem *i* ( *Ri* ), in a series system, is shown in Eq. (1).

$$R\_S = \prod\_{i=1}^{n} R\_i \tag{1}$$

The relationship between the reliability of subsystem *i* ( *Ri* ) and the reliability of component *j* available for subsystem *i* ( *ij r* ), in a parallel system, is shown in Eq. (2).

$$R\_i = 1 - \prod\_{j=1}^{m} \left[1 - r\_{ij}\right]^{\chi\_{\bar{y}}} \tag{2}$$

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 131

The objective function is to maximize the system reliability in a series system. Eq. (9) shows the cost constraint, Eq. (10) represents the multiple-choice constraint which is that the problem prohibits component redundancy, and Eq. (11) defines the decision variables.

The genetic algorithm is a stochastic search method based on the natural selection, reproduction, and evolution theory proposed by Holland (1975). The parallel genetic algorithm paratactically evolves by operating several sub-populations. This study suggests a hybrid parallel genetic algorithm for reliability optimization with resource constraints. The suggested algorithm includes different heuristics such as swap, 2-opt, and interchange (except for RAPCC) for an improvement solution. The suggested process of a hybrid parallel

The gene representation has to reflect the properties of the system structure. The suggested

The subsystem in Table 1 indicate the nominal number of subsystem. However, it is not necessary for this number to be one for the composition of a substantial objective function. The "Redundancy & Component" row represents the number of components available for each subsystem. For example, as shown Table 1, subsystem 1 consists of two components of C1, one component of C2, one component of C3. Table 1 can be expressed as shown in Fig. 4. The suggested algorithm for the RAPCC represents a gene by one string as shown in Table 2. The subsystem in Table 2 indicates the nominal number of subsystems. However, it is not necessary for the composition of a substantial objective function. The "Component" row represents the available component number for each subsystem. For example, as shown Table 2, a series system uses component No.3 in subsystem 1, component No.2 in subsystem

The population of a parallel genetic algorithm consists of an initial population and several sub-populations. The initial population is usually generated by the random and the heuristic generation method. The heuristic generation method tends to interrupt global search.

Redundancy & Component 2 1 1 0 1 0 2 0 1 3 0

Subsystem(Component Alternatives) 1(4) 2(3) 3(4)

algorithm for the RRAP represents a gene by one string as shown in Table 1.

<sup>=</sup> , 1,2, , *i n* <sup>=</sup> (10)

*xij* = {0,1} , 1,2, , *i n* = , *j m* = 1,2, , (11)

1

 1

**4. Hybrid parallel genetic algorithm** 

genetic algorithm is shown in Fig. 3.

Table 1. Gene representation (RRAP)

2, …, and component No.5 in subsystem 6.

**4.2 Population** 

**4.1 Gene representation** 

*m ij j x* =

Using Eqs. (1) and (2), the mathematical programming model of the RRAP in a seriesparallel system is as follows.

$$\text{Maximize}\quad R\_S = \prod\_{i=1}^{n} R\_i = \prod\_{i=1}^{n} \left\{ 1 - \prod\_{j=1}^{m} \left[ 1 - r\_{ij} \right]^{x\_{\bar{\eta}}} \right\} \tag{3}$$

$$\text{Subject to } \sum\_{i=1}^{n} \sum\_{j=1}^{m} c\_{ij} \cdot x\_{ij} \le C\_S \tag{4}$$

$$\sum\_{i=1}^{n} \sum\_{j=1}^{m} w\_{ij} \cdot x\_{ij} \le \mathcal{W}\_S \tag{5}$$

$$1 \le \sum\_{j=1}^{m} \mathbf{x}\_{ij} \le u\_i \quad i = 1, 2, \cdots, n \tag{6}$$

$$\infty\_{ij} \ge 0 \text{ , } i = 1, 2, \cdots, n \text{ , } \ j = 1, 2, \cdots, m \text{ , integer} \tag{7}$$

The objective function is to maximize the system reliability in a series-parallel system. Eqs. (4) and (5) show the resource constraints with cost and weight. Eq. (6) shows the maximum and minimum number of components that can be used for each subsystem. Eq. (7) shows the integer decision variables.

#### **3.2 Reliability allocation problem with component choices (RAPCC)**

As shown in Fig. 2, a series system consisting of n subsystems where each subsystem has several component alternatives which can perform same functions with different characteristics is considered in this study. The problem is proposed to select the optimal combination of component alternatives to maximize the system reliability given the cost. Only one component will be adopted for each subsystem.

Fig. 2. Series system

Using Eq. (1), the mathematical programming model of the RAPCC in a series system is as follows.

$$\text{Maximize}\quad R\_S = \prod\_{i=1}^{n} \left( \sum\_{j=1}^{m} r\_{ij} \cdot \mathbf{x}\_{ij} \right) \tag{8}$$

$$\text{Subject to } \sum\_{i=1}^{n} \sum\_{j=1}^{m} c\_{ij} \cdot \mathbf{x}\_{ij} \le \mathbf{C}\_{\mathbf{S}} \tag{9}$$

$$\sum\_{j=1}^{m} \mathbf{x}\_{ij} = \mathbf{1}, \; i = \mathbf{1}, \; 2, \; \cdots, m \tag{10}$$

$$\mathbf{x}\_{ij} = \{0, 1\} \text{ , } \ i = 1, 2, \cdots, m \text{ , } \ j = 1, 2, \cdots, m \tag{11}$$

The objective function is to maximize the system reliability in a series system. Eq. (9) shows the cost constraint, Eq. (10) represents the multiple-choice constraint which is that the problem prohibits component redundancy, and Eq. (11) defines the decision variables.

### **4. Hybrid parallel genetic algorithm**

The genetic algorithm is a stochastic search method based on the natural selection, reproduction, and evolution theory proposed by Holland (1975). The parallel genetic algorithm paratactically evolves by operating several sub-populations. This study suggests a hybrid parallel genetic algorithm for reliability optimization with resource constraints. The suggested algorithm includes different heuristics such as swap, 2-opt, and interchange (except for RAPCC) for an improvement solution. The suggested process of a hybrid parallel genetic algorithm is shown in Fig. 3.

### **4.1 Gene representation**

130 Real-World Applications of Genetic Algorithms

Using Eqs. (1) and (2), the mathematical programming model of the RRAP in a series-

*Subject to*

1 1

*ij i*

The objective function is to maximize the system reliability in a series-parallel system. Eqs. (4) and (5) show the resource constraints with cost and weight. Eq. (6) shows the maximum and minimum number of components that can be used for each subsystem. Eq. (7) shows

As shown in Fig. 2, a series system consisting of n subsystems where each subsystem has several component alternatives which can perform same functions with different characteristics is considered in this study. The problem is proposed to select the optimal combination of component alternatives to maximize the system reliability given the cost.

Using Eq. (1), the mathematical programming model of the RAPCC in a series system is as

1 1

*i j*

= =

*n m*

1 1

*n m S ij ij i j R rx* = =

 = ⋅ 

*ij ij S*

*cx C*

∏ (8)

⋅ ≤ (9)

*x u*

*i j*

1

*j*

**3.2 Reliability allocation problem with component choices (RAPCC)** 

*Subject to*

Only one component will be adopted for each subsystem.

 *Maximize*

=

1 *m*

= =

*n m*

111

*ij ij S*

*wx W*

1 1

*i j*

= =

*n m*

*S i ij iij RR r* ===

1 1 *ij x n nm*

*ij ij S*

*cx C*

∏∏∏ (3)

⋅ ≤ (4)

⋅ ≤ (5)

≤ ≤ , 1,2, , *i n* <sup>=</sup> (6)

0 *ij x* ≥ , 1,2, , *i n* = , *j m* = 1,2, , , Integer (7)

= = −−

parallel system is as follows.

the integer decision variables.

Fig. 2. Series system

follows.

 *Maximize*

The gene representation has to reflect the properties of the system structure. The suggested algorithm for the RRAP represents a gene by one string as shown in Table 1.


Table 1. Gene representation (RRAP)

The subsystem in Table 1 indicate the nominal number of subsystem. However, it is not necessary for this number to be one for the composition of a substantial objective function. The "Redundancy & Component" row represents the number of components available for each subsystem. For example, as shown Table 1, subsystem 1 consists of two components of C1, one component of C2, one component of C3. Table 1 can be expressed as shown in Fig. 4.

The suggested algorithm for the RAPCC represents a gene by one string as shown in Table 2.

The subsystem in Table 2 indicates the nominal number of subsystems. However, it is not necessary for the composition of a substantial objective function. The "Component" row represents the available component number for each subsystem. For example, as shown Table 2, a series system uses component No.3 in subsystem 1, component No.2 in subsystem 2, …, and component No.5 in subsystem 6.

### **4.2 Population**

The population of a parallel genetic algorithm consists of an initial population and several sub-populations. The initial population is usually generated by the random and the heuristic generation method. The heuristic generation method tends to interrupt global search.

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 133

The fitness function to evaluate the solutions is commonly obtained from the objective function. Penalty functions were used for infeasible solutions by the random generation method in this study. Eqs. (12) and (13) show cost and weight penalty functions, respectively.

1, total cost

<sup>≤</sup> <sup>=</sup>

1, total weight <sup>1</sup> , otherwise <sup>≤</sup> <sup>=</sup>

The multiplication of system reliability and penalty functions related to its cost and weight (except for RAPCC) were used to calculate the fitness of the solutions in the suggested

The selection method to choose the pairs of parents is applied by the roulette wheel method in the suggested algorithm. The roulette wheel method is one of the most common proportionate selection schemes. In this scheme, the probability to select an individual is proportional to its fitness. It is also stochastically possible for infeasible solutions to survive. The suggested algorithm applies the elitism strategy for the survival of an optimum solution

The crossover is the main genetic operator. It operates on two individuals at a time and generates offspring by combining both individuals' features. The crossover operator applies a uniform crossover in the suggested algorithm as shown in Fig. 5. The steps of the uniform

**Step 1.** Random numbers were generated for individuals and the individual for crossover

**Step 3.** For each bit of the mated individuals was generated a random number of either 0 or 1. **Step 4.** The two offspring bits were generated through a crossover of the two parents' bits

was selected by comparing the crossover rate for each individual.

when the random number associated with those bits was 1.

**Step 2.** The selected individuals were mated between themselves.

cos

*total t*

*total weight*

by generation in order to avoid the disappearance of an excellent solution.

 

*C*

 

*W*

*P*

*P*

<sup>1</sup> , otherwise

*s*

*s*

*fitness R P P* = ⋅⋅ *SCW* (14)

*W*

(12)

(13)

*C*

Table 2. Gene representation (RAPCC)

algorithm as shown in Eq. (14).

**4.4 Selection** 

**4.5 Crossover** 

crossover are as follows.

**4.3 Fitness** 

Subsystem 1 2 3 4 5 6 Component 3 2 4 1 3 5

Therefore, the initial population is generated by the random generation method in this study. The initial population is composed 500 individuals with 100 individuals allocated for each sub-population.

Fig. 4. System structure of Table 1


Table 2. Gene representation (RAPCC)

#### **4.3 Fitness**

132 Real-World Applications of Genetic Algorithms

Therefore, the initial population is generated by the random generation method in this study. The initial population is composed 500 individuals with 100 individuals allocated for

each sub-population.

Fig. 3. Hybrid parallel genetic algorithm

Fig. 4. System structure of Table 1

The fitness function to evaluate the solutions is commonly obtained from the objective function. Penalty functions were used for infeasible solutions by the random generation method in this study. Eqs. (12) and (13) show cost and weight penalty functions, respectively.

$$P\_c = \begin{cases} 1, & \text{total cost} \le C\_s\\ \frac{1}{total\\_cost}, & \text{otherwise} \end{cases} \tag{12}$$

$$P\_w = \begin{cases} 1, & \text{total weight} \le W\_s\\ \frac{1}{\text{total weight}}, & \text{otherwise} \end{cases} \tag{13}$$

The multiplication of system reliability and penalty functions related to its cost and weight (except for RAPCC) were used to calculate the fitness of the solutions in the suggested algorithm as shown in Eq. (14).

$$\text{fitness} = \mathbf{R}\_S \cdot \mathbf{P}\_C \cdot \mathbf{P}\_W \tag{14}$$

#### **4.4 Selection**

The selection method to choose the pairs of parents is applied by the roulette wheel method in the suggested algorithm. The roulette wheel method is one of the most common proportionate selection schemes. In this scheme, the probability to select an individual is proportional to its fitness. It is also stochastically possible for infeasible solutions to survive. The suggested algorithm applies the elitism strategy for the survival of an optimum solution by generation in order to avoid the disappearance of an excellent solution.

#### **4.5 Crossover**

The crossover is the main genetic operator. It operates on two individuals at a time and generates offspring by combining both individuals' features. The crossover operator applies a uniform crossover in the suggested algorithm as shown in Fig. 5. The steps of the uniform crossover are as follows.


A Hybrid Parallel Genetic Algorithm for Reliability Optimization 135

Migration parameter Sub-population size Migration period Migration rate Value 100 50 0.2

The genetic parameters include the population size, crossover rate (Pc), mutation rate (Pm), and the number of generations. It is hard to find the best parametric values, so the following parameters were obtained by repeated experiments. The genetic parameters are shown in

> Crossover rate(Pc)

Value 500 0.8 0.02 1,000~3,000

The suggested algorithm includes different heuristics such as swap, 2-opt, and interchange (except for RAPCC) for improvement of the solution. The swap heuristic was used to exchange each bit which selected two solutions among the five solutions generated by the parallel genetic algorithm. After applying the swap heuristic, a solution of the parallel genetic algorithm was selected by using best fitness. In a selected solution, the 2-opt heuristic performed the exchanging of two bits to enable improvement. The interchange heuristic was applied to each subsystem to exchanging sequences of bits. Finally, a solution of a hybrid parallel genetic algorithm was produced using best fitness after the application

In order to evaluate the performance of the suggested algorithm for the integer nonlinear RRAP, this study performed experiments on 33 variations of Fyffe et. al. (1968), as suggested by Nakagawa & Miyazaki (1981). In this problem, the series–parallel system is connected by

Mutation rate(Pm)

The number of generations

Fig. 7. Neighborhood structure (ring topology)

Population size

**5.1 The reliability-redundancy allocation problem (RRAP)** 

Table 3. Migration parameters

**4.8 Genetic parameters** 

Genetic parameter

Table 4. Genetic parameters

**4.9 Improvement solution** 

of the interchange heuristic.

**5. Numerical experiments** 

Table 4.

Fig. 5. Uniform crossover

#### **4.6 Mutation**

The mutation is a background operator which produces spontaneous random changes in various individuals. The mutation operator applies the uniform mutation in the suggested algorithm as shown in Fig. 6. The steps of the uniform mutation are as follows.



Fig. 6. Uniform mutation

#### **4.7 Migration**

The migration is an exchange operator to change useful information between neighbor subpopulations. Periodically, each sub-population sends its best individuals to its neighbors. When dealing with the migration, the main issues to be considered are migration parameters such as neighborhood structure, the individuals' selection for exchanging, subpopulation size, migration period, and migration rate. In the suggested algorithm, the neighborhood structure uses a ring topology as shown in Fig. 7 and the individuals' selection for exchanging is determined by the application of the fitness function. Other migration parameters are shown in Table 3.

Fig. 7. Neighborhood structure (ring topology)


Table 3. Migration parameters

### **4.8 Genetic parameters**

134 Real-World Applications of Genetic Algorithms

The mutation is a background operator which produces spontaneous random changes in various individuals. The mutation operator applies the uniform mutation in the suggested

**Step 1.** The mutation bits were selected by comparing a random number with the mutation rate after Generating a random number between 0~1 for all individual bits. **Step 2.** The value of the selected bits were substituted with a new value between 0 and the

The migration is an exchange operator to change useful information between neighbor subpopulations. Periodically, each sub-population sends its best individuals to its neighbors. When dealing with the migration, the main issues to be considered are migration parameters such as neighborhood structure, the individuals' selection for exchanging, subpopulation size, migration period, and migration rate. In the suggested algorithm, the neighborhood structure uses a ring topology as shown in Fig. 7 and the individuals' selection for exchanging is determined by the application of the fitness function. Other

algorithm as shown in Fig. 6. The steps of the uniform mutation are as follows.

maximum number of components in each subsystem.

Fig. 5. Uniform crossover

Fig. 6. Uniform mutation

migration parameters are shown in Table 3.

**4.7 Migration** 

**4.6 Mutation** 

The genetic parameters include the population size, crossover rate (Pc), mutation rate (Pm), and the number of generations. It is hard to find the best parametric values, so the following parameters were obtained by repeated experiments. The genetic parameters are shown in Table 4.


Table 4. Genetic parameters

### **4.9 Improvement solution**

The suggested algorithm includes different heuristics such as swap, 2-opt, and interchange (except for RAPCC) for improvement of the solution. The swap heuristic was used to exchange each bit which selected two solutions among the five solutions generated by the parallel genetic algorithm. After applying the swap heuristic, a solution of the parallel genetic algorithm was selected by using best fitness. In a selected solution, the 2-opt heuristic performed the exchanging of two bits to enable improvement. The interchange heuristic was applied to each subsystem to exchanging sequences of bits. Finally, a solution of a hybrid parallel genetic algorithm was produced using best fitness after the application of the interchange heuristic.

### **5. Numerical experiments**

### **5.1 The reliability-redundancy allocation problem (RRAP)**

In order to evaluate the performance of the suggested algorithm for the integer nonlinear RRAP, this study performed experiments on 33 variations of Fyffe et. al. (1968), as suggested by Nakagawa & Miyazaki (1981). In this problem, the series–parallel system is connected by

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 137

<sup>=</sup> , *i n* <sup>=</sup> 1,2, , (18)

≤ ≤ , 1,2, , *i n* <sup>=</sup> (19)

1

*ix x*

*ij i*

*xx x*

The experimental results including component structure, reliability, cost, and weight by

No. W Components structure Reliability Cost Weight 1 191 333, 11, 444, 3333, 222, 22, 111, 1111, 12, 233, 33, 1111, 11, 34 0.9868110 130 191 2 190 333, 11, 444, 3333, 222, 22, 111, 1111, 11, 233, 33, 1111, 12, 34 0.9864161 130 190 3 189 333, 11, 444, 3333, 222, 22, 111, 1111, 23, 233, 13, 1111, 11, 34 0.9859217 130 189 4 188 333, 11, 444, 3333, 222, 22, 111, 1111, 23, 223, 13, 1111, 12, 34 0.9853782 130 188 5 187 333, 11, 444, 3333, 222, 22, 111, 1111, 13, 223, 13, 1111, 22, 34 0.9846881 130 187 6 186 333, 11, 444, 333, 222, 22, 111, 1111, 23, 233, 33, 1111, 22, 34 0.9841755 129 186 7 185 333, 11, 444, 3333, 222, 22, 111, 1111, 23, 223, 13, 1111, 22, 33 0.9835049 130 185 8 184 333, 11, 444, 333, 222, 22, 111, 1111, 33, 233, 33, 1111, 22, 34 0.9829940 130 184 9 183 333, 11, 444, 333, 222, 22, 111, 1111, 33, 223, 33, 1111, 22, 34 0.9822557 129 183 10 182 333, 11, 444, 333, 222, 22, 111, 1111, 33, 333, 33, 1111, 22, 33 0.9815183 130 182 11 181 333, 11, 444, 333, 222, 22, 111, 1111, 33, 233, 33, 1111, 22, 33 0.9810271 129 181 12 180 333, 11, 444, 333, 222, 22, 111, 1111, 33, 223, 33, 1111, 22, 33 0.9802902 128 180 13 179 333, 11, 444, 333, 222, 22, 111, 1111, 33, 223, 13, 1111, 22, 33 0.9795047 126 179 14 178 333, 11, 444, 333, 222, 22, 111, 1111, 33, 222, 13, 1111, 22, 33 0.9784003 125 178 15 177 333, 11, 444, 333, 222, 22, 111, 113, 33, 223, 13, 1111, 22, 33 0.9775953 126 177 16 176 333, 11, 444, 333, 222, 22, 33, 1111, 33, 223, 13, 1111, 22, 33 0.9766905 124 176 17 175 333, 11, 444, 333, 222, 22, 13, 1111, 33, 223, 33, 1111, 22, 33 0.9757079 125 175 18 174 333, 11, 444, 333, 222, 22, 13, 1111, 33, 223, 13, 1111, 22, 33 0.9749261 123 174 19 173 333, 11, 444, 333, 222, 22, 13, 1111, 33, 222, 13, 1111, 22, 33 0.9738268 122 173 20 172 333, 11, 444, 333, 222, 22, 13, 113, 33, 223, 13, 1111, 22, 33 0.9730266 123 172 21 171 333, 11, 444, 333, 222, 22, 13, 113, 33, 222, 13, 1111, 22, 33 0.9719295 122 171 22 170 333, 11, 444, 333, 222, 22, 13, 113, 33, 222, 11, 1111, 22, 33 0.9707604 120 170 23 169 333, 11, 444, 333, 222, 22, 11, 113, 33, 222, 13, 1111, 22, 33 0.9692910 121 169 24 168 333, 11, 444, 333, 222, 22, 11, 113, 33, 222, 11, 1111, 22, 33 0.9681251 119 168 25 167 333, 11, 444, 333, 22, 22, 13, 113, 33, 222, 11, 1111, 22, 33 0.9663351 118 167 26 166 333, 11, 44, 333, 222, 22, 13, 113, 33, 222, 11, 1111, 22, 33 0.9650416 116 166

0 *ij x* ≥ , 1,2, , *i n* = , *j m* = 1,2, , , Integer (20)

1, if of the th component are used for subsystem

*ix x i i im <sup>r</sup>* = − *qq q* , ( ) <sup>1</sup> *im im <sup>x</sup> <sup>x</sup> q r im* = − *im* , *i n* <sup>=</sup> 1,2, ,

*x j i*

(21)

*x u*

<sup>1</sup> *i i i im*

*y*

1

*j*

=

<sup>1</sup> 0

*ij*

where, ( ) 1 2

using a hybrid parallel genetic algorithm are shown in Table 6.

<sup>1</sup> 1 2 ln 1 *i i im*

0, otherwise *i im*

*i im*

=

1

*y*

*ix x*

=

*x x*

*i im*

1 *m*

*u u*

14 parallel subsystems and each has three or four components of choice. The objective is to maximize the reliability of the series–parallel system subject to the cost constraint of 130 and weight constraint ranging from 159 to 190. The maximum number of components is 6 in each subsystem. The component data for testing problems are listed in Table 5.


Table 5. Component data for testing problems

To use CPLEX, this study performed additional steps for transforming the integer nonlinear RRAP into an equivalent binary knapsack problem(Bae et al., 2007; Coit, 2003) as shown in Eqs. (15) to (21).

$$\text{Maximize} \quad \ln R\_S = \sum\_{i=1}^{n} \sum\_{\mathbf{x}\_{i1}=\mathbf{0}}^{\underline{u}\_i} \cdots \sum\_{\mathbf{x}\_{in}}^{\underline{u}\_i} r\_{i\mathbf{x}\_{i1}\ldots\mathbf{x}\_{in}} \cdot y\_{i\mathbf{x}\_{i1}\ldots\mathbf{x}\_{in}} \tag{15}$$

$$\text{Subject to } \sum\_{i=1}^{n} \sum\_{x\_{i1}=0}^{u\_i} \cdots \sum\_{x\_{in}}^{u\_i} \left( \sum\_{j=1}^{m} x\_{ij} \cdot c\_{ij} \right) \cdot y\_{i x\_{i1} \ldots x\_{in}} \le \mathbb{C}\_S \tag{16}$$

$$\sum\_{i=1}^{n} \sum\_{\mathbf{x}\_{i1}=0}^{\boldsymbol{u}\_{i}} \cdots \sum\_{\mathbf{x}\_{in}}^{\boldsymbol{u}\_{i}} \left( \sum\_{j=1}^{m} \boldsymbol{x}\_{ij} \cdot \boldsymbol{w}\_{ij} \right) \cdot \boldsymbol{y}\_{i\mathbf{x}\_{i1}\ldots\mathbf{x}\_{in}} \leq \boldsymbol{W}\_{\mathbf{S}} \tag{17}$$

14 parallel subsystems and each has three or four components of choice. The objective is to maximize the reliability of the series–parallel system subject to the cost constraint of 130 and weight constraint ranging from 159 to 190. The maximum number of components is 6 in

> Component choices Choice 1 Choice 2 Choice 3 Choice 4 R C W R C W R C W R C W

1 0.90 1 3 0.93 1 4 0.91 2 2 0.95 2 5 2 0.95 2 8 0.94 1 10 0.93 1 9 \* \* \* 3 0.85 2 7 0.90 3 5 0.87 1 6 0.92 4 4 4 0.83 3 5 0.87 4 6 0.85 5 4 \* \* \* 5 0.94 2 4 0.93 2 3 0.95 3 5 \* \* \* 6 0.99 3 5 0.98 3 4 0.97 2 5 0.96 2 4 7 0.91 4 7 0.92 4 8 0.94 5 9 \* \* \* 8 0.81 3 4 0.90 5 7 0.91 6 6 \* \* \* 9 0.97 2 8 0.99 3 9 0.96 4 7 0.91 3 8 10 0.83 4 6 0.85 4 5 0.90 5 6 \* \* \* 11 0.94 3 5 0.95 4 6 0.96 5 6 \* \* \* 12 0.79 2 4 0.82 3 5 0.85 4 6 0.90 5 7 13 0.98 2 5 0.99 3 5 0.97 2 6 \* \* \* 14 0.90 4 6 0.92 4 7 0.95 5 6 0.99 6 9

To use CPLEX, this study performed additional steps for transforming the integer nonlinear RRAP into an equivalent binary knapsack problem(Bae et al., 2007; Coit, 2003) as shown in

> <sup>1</sup> 1 0 ln *i i*

= =

*Subject to* 1 <sup>1</sup> 10 1

*i im*

== =

*ix x j*

<sup>1</sup> 10 1

*i im*

== =

*ix x j*

*i i*

*n m u u*

*i i*

*n m u u*

*ix x R ry*

*n u u*

*i im*

⋅⋅ ≤

*S ix x ix x*

*i im i im*

*i im*

(16)

*ij ij ix x S*

(17)

*xc y C*

⋅⋅ ≤

1

*ij ij ix x S*

*xw y W*

*i im*

= ⋅ (15)

each subsystem. The component data for testing problems are listed in Table 5.

Subsystem No.

Eqs. (15) to (21).

Table 5. Component data for testing problems

 *Maximize* 1 1

$$\sum\_{\mathbf{x}\_{i1}=0}^{u\_i} \cdots \sum\_{\mathbf{x}\_{in}}^{u\_i} y\_{i\mathbf{x}\_{i1}\cdots\mathbf{x}\_{in}} = \mathbf{1} \; \text{ } i = 1 \; \text{2} \; \dots \; n \tag{18}$$

$$1 \le \sum\_{j=1}^{m} \chi\_{ij} \le u\_i \quad i = 1, 2, \cdots, n \tag{19}$$

$$\mathbf{x}\_{[i]} \ge \mathbf{0} \,, \ i = \mathbf{1}, \mathbf{2}, \cdots, \mathbf{n} \,, \ j = \mathbf{1}, \mathbf{2}, \cdots, \mathbf{m} \,, \ \text{Integer} \tag{20}$$

$$\begin{cases} \text{y}\_{i x\_{i1} \cdots x\_{in}} = \begin{cases} 1, & \text{if } x\_{ij} \text{ of the } j \text{th component are used for subsystem } i\\ 0, & \text{otherwise} \end{cases} \tag{21}$$

$$\text{where}\\
\text{, } r\_{i\mathbf{x}\_{i1}\ldots\mathbf{x}\_{im}} = \ln\left(1 - q\_{i1}^{\mathbf{x}\_{1i}} q\_{i2}^{\mathbf{x}\_{i2}} \cdots q\_{im}^{\mathbf{x}\_{im}}\right), \; q\_{im}^{\mathbf{x}\_{im}} = \left(1 - r\_{im}\right)^{\mathbf{x}\_{im}}, \; i = 1, 2, \cdots, m$$

The experimental results including component structure, reliability, cost, and weight by using a hybrid parallel genetic algorithm are shown in Table 6.


A Hybrid Parallel Genetic Algorithm for Reliability Optimization 139

16 176 0.9766905 0.9764 0.976690 0.9765 0.9766905 0.9766905 7 / 10 17 175 0.9757079 0.9753 0.975708 0.9757 0.9757079 0.9757079 9 / 10 18 174 0.9749261 0.9744 0.974788 0.9749 0.9746901 0.9749261 6 / 10 19 173 0.9738268 0.9738 0.973827 0.9738 0.9737580 0.9738268 8 / 10 20 172 0.9730266 0.9727 0.973027 0.9730 0.9730266 0.9730266 9 / 10 21 171 0.9719295 0.9719 0.971929 0.9719 0.9719295 0.9719295 7 / 10 22 170 0.9707604 0.9708 0.970760 0.9708 0.9707604 0.9707604 9 / 10 23 169 0.9692910 0.9692 0.969291 0.9693 0.9692910 0.9692910 8 / 10 24 168 0.9681251 0.9681 0.968125 0.9681 0.9681251 0.9681251 8 / 10 25 167 0.9663351 0.9663 0.966335 0.9663 0.9663351 0.9663351 8 / 10 26 166 0.9650416 0.9650 0.965042 0.9650 0.9650416 0.9650416 9 / 10 27 165 0.9637118 0.9637 0.963712 0.9637 0.9637118 0.9637118 7 / 10 28 164 0.9624219 0.9624 0.962422 0.9624 0.9624219 0.9624219 7 / 10 29 163 0.9606424 0.9606 0.959980 0.9606 0.9606424 0.9606424 8 / 10 30 162 0.9591884 0.9591 0.958205 0.9592 0.9591884 0.9591884 6 / 10 31 161 0.9580346 0.9580 0.956922 0.9580 0.9580346 0.9580346 7 / 10 32 160 0.9557144 0.9557 0.955604 0.9557 0.9557144 0.9557144 6 / 10 33 159 0.9545648 0.9546 0.954325 0.9546 0.9545648 0.9545648 8 / 10

Table 7. Comparison of CPLEX, meta-heuristics and HPGA (C=130)

and is shown in Fig. 8.

*L GA* 1( ) : MPI(%) of GA results *L TS* 2( ) : MPI(%) of TS results *L ACO* 3( ) : MPI(%) of ACO results

In order to calculate the improvement of reliability for existing studies and the suggested algorithm, a maximum possible improvement (MPI) was obtained by Eqs. from (22) to (25)

> 1( ) 100 1 *HPGA GA GA*

2( ) 100 1 *HPGA TS TS*

3( ) 100 1

4( ) 100 1 *HPGA IA IA*

*R*

*R*

*R*

*HPGA ACO ACO*

*R*

<sup>−</sup> = × <sup>−</sup> (22)

<sup>−</sup> = × <sup>−</sup> (23)

<sup>−</sup> = × − (24)

<sup>−</sup> = × − (25)

*R R L GA*

*R R L TS*

*R R L ACO*

*R R L IA*


Table 6. Experimental results by using HPGA (C=130)

The experimental results compared to the results of CPLEX and existing meta-heuristics, such as GA (Coit & Smith, 1996), TS (Kulturel-Konak et al., 2003), ACO (Liang & Smith, 2004), and IA (Chen & You, 2005) are shown in Table 6. The comparison of CPLEX, metaheuristics, and the suggested algorithm is shown in Table 7.

The suggested algorithm in all problems showed the optimal solution 6~9 times out 0f 10 runs and obtained same or superior solutions compared to the meta-heuristics. Of the results in Table 7, when compared with the meta-heuristics, 25 solutions are superior to GA, 7 solutions are superior to TS and ACO, and 9 solutions are superior to IA, respectively. The other solutions are the same. The suggested algorithm could paratactically evolve by operating several sub-populations and improve on the solution through swap, 2-opt, and interchange heuristics.


A Hybrid Parallel Genetic Algorithm for Reliability Optimization 139

27 165 333, 11, 444, 333, 22, 22, 11, 113, 33, 222, 11, 1111, 22, 33 0.9637118 117 165 28 164 333, 11, 44, 333, 222, 22, 11, 113, 33, 222, 11, 1111, 22, 33 0.9624219 115 164 29 163 333, 11, 44, 333, 22, 22, 13, 113, 33, 222, 11, 1111, 22, 33 0.9606424 114 163 30 162 333, 11, 44, 333, 22, 22, 11, 113, 33, 222, 13, 1111, 22, 33 0.9591884 115 162 31 161 333, 11, 44, 333, 22, 22, 11, 113, 33, 222, 11, 1111, 22, 33 0.9580346 113 161 32 160 333, 11, 44, 333, 22, 22, 11, 111, 33, 222, 13, 1111, 22, 33 0.9557144 112 160 33 159 333, 11, 44, 333, 22, 22, 11, 111, 33, 222, 11, 1111, 22, 33 0.9545648 110 159

The experimental results compared to the results of CPLEX and existing meta-heuristics, such as GA (Coit & Smith, 1996), TS (Kulturel-Konak et al., 2003), ACO (Liang & Smith, 2004), and IA (Chen & You, 2005) are shown in Table 6. The comparison of CPLEX, meta-

The suggested algorithm in all problems showed the optimal solution 6~9 times out 0f 10 runs and obtained same or superior solutions compared to the meta-heuristics. Of the results in Table 7, when compared with the meta-heuristics, 25 solutions are superior to GA, 7 solutions are superior to TS and ACO, and 9 solutions are superior to IA, respectively. The other solutions are the same. The suggested algorithm could paratactically evolve by operating several sub-populations and improve on the solution through swap, 2-opt, and

1 191 0.9868110 0.9867 0.986811 0.9868 0.9868110 0.9868110 8 / 10 2 190 0.9864161 0.9857 0.986416 0.9859 0.9864161 0.9864161 7 / 10 3 189 0.9859217 0.9856 0.985922 0.9858 0.9859217 0.9859217 9 / 10 4 188 0.9853782 0.9850 0.985378 0.9853 0.9853297 0.9853782 8 / 10 5 187 0.9846881 0.9844 0.984688 0.9847 0.9844495 0.9846881 8 / 10 6 186 0.9841755 0.9836 0.984176 0.9838 0.9841755 0.9841755 9 / 10 7 185 0.9835049 0.9831 0.983505 0.9835 0.9834363 0.9835049 8 / 10 8 184 0.9829940 0.9823 0.982994 0.9830 0.9826980 0.9829940 9 / 10 9 183 0.9822557 0.9819 0.982256 0.9822 0.9822062 0.9822557 7 / 10 10 182 0.9815183 0.9811 0.981518 0.9815 0.9815183 0.9815183 9 / 10 11 181 0.9810271 0.9802 0.981027 0.9807 0.9810271 0.9810271 8 / 10 12 180 0.9802902 0.9797 0.980290 0.9803 0.9802902 0.9802902 9 / 10 13 179 0.9795047 0.9791 0.979505 0.9795 0.9795047 0.9795047 9 / 10 14 178 0.9784003 0.9783 0.978400 0.9784 0.9782085 0.9784003 7 / 10 15 177 0.9775953 0.9772 0.977474 0.9776 0.9772429 0.9775953 8 / 10

Reliability Number

HPGA (This study)

of optimal by HPGA (10 runs)

Table 6. Experimental results by using HPGA (C=130)

interchange heuristics.

No. W

heuristics, and the suggested algorithm is shown in Table 7.

CPLEX GA TS ACO IA



In order to calculate the improvement of reliability for existing studies and the suggested algorithm, a maximum possible improvement (MPI) was obtained by Eqs. from (22) to (25) and is shown in Fig. 8.

$$L1 \text{(GA)} = \frac{R\_{HPGA} - R\_{GA}}{1 - R\_{GA}} \times 100 \tag{22}$$

$$L2\text{(TS)} = \frac{R\_{HPGA} - R\_{TS}}{1 - R\_{TS}} \times 100\tag{23}$$

$$L\mathcal{B}(ACO) = \frac{R\_{HPGA} - R\_{ACO}}{1 - R\_{ACO}} \times 100\tag{24}$$

$$L4(IA) = \frac{R\_{HPGA} - R\_{IA}}{1 - R\_{IA}} \times 100\tag{25}$$

*L GA* 1( ) : MPI(%) of GA results *L TS* 2( ) : MPI(%) of TS results *L ACO* 3( ) : MPI(%) of ACO results

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 141

The suggested algorithm presented optimal solutions in all large-scale problems. For the experimental results of large-scale problems 1~4, the suggested algorithm showed the optimal solutions 4~8 times out of 10 runs. The optimal solution to large-scale problem 5

In order to evaluate the performance of the suggested algorithm for the reliability allocation problem with multiple-choice, this study performed experiments by using the Nahas & Nourelfath (2005) and the Kim et al. (2008) examples in series. The Nahas & Nourelfath (2005) examples consist of four examples: examples 1, 2, and 3 consist of 15 subsystems with 60, 80, and 100 components, respectively, and example 4 consists of 25 subsystems with 166 components. The budgets are \$1,000, \$900, \$1,000, and \$1,400, respectively. Examples by the Kim et al. (2008) consist of two large-scale examples (examples 5 and 6). Large-scale examples are presented through connecting the system data of example 4. Examples 5 and 6 consist of 100 and 200 subsystems with budgets of \$7,200~\$7,650 and \$14,400~\$15,100,

To use CPLEX, this study performed an additional step for transforming the nonlinear

The experimental results of examples 1~4 including component structure, reliability, and

cost by using CPLEX and a hybrid parallel genetic algorithm are shown in Table 9.

1 1 1 1 ln ln ln *n m n m S ij ij ij ij i j i j R rx x r* = = = = = ⋅= ⋅

CPLEX HPGA

∏ (26)

Reliability Component structure Reliability Cost

3-3-3-5-2-3-2-2-3-1-2-3-4-

3-3-3-5-2-3-2-2-3-1-2-3-4-

2-3-3-4-2-3-2-2-3-1-2-3-3-

2-2-2-3-4-3-2 0.857054 990

4-1-2-3-4-3-1 0.915042 900

3-2-2-4-4-4-2 0.965134 995

4-1-2-3-3-4-2-3-2-2-3-1 0.865439 1,395

3-1-3-3-3-4-2-3-2-2-3-1 0.865439 1,395

4-1-3-3-3-5-3-3-2-2-3-1 0.865439 1,400

**5.2 The reliability allocation problem with component choices (RAPCC)** 

objective function into the linear function as shown in Eq. (26).

Number of

4 25 166 1,400 0.865439

components budget

1 15 60 1,000 0.857054 3-4-5-2-3-3-2-3-

2 15 80 900 0.915042 3-3-3-4-2-3-3-2-

3 15 100 1,000 0.965134 3-3-4-4-3-3-2-2-

Table 9. Experimental results of examples 1~4 by using CPLEX and HPGA

As found in Table 9, the suggested algorithm presented the optimal solutions in examples 14 and obtained a new optimal solution (3-3-3-5-2-3-2-2-3-1-2-3-4-3-1-3-3-3-4-2-3-2-2-3-1) in

was obtained by HPGA in 1 out of 10 runs.

respectively.

No. Number of subsystems

example 4.

 *Maximize*

*L IA* 4( ) : MPI(%) of IA results *RHPGA* : system reliability by HPGA

*RGA* : system reliability by GA

*RTS* : system reliability by TS

*RACO* : system reliability by ACO

*RIA* : system reliability by IA

Fig. 8. MPI

The suggested algorithm improved system reliability better than existing studies except for TS in the 1st~20th test problems in which the weight was heavy. In addition, HPGA found superior system reliability compared to TS in the 29th~32th test problems in which the weight was light. The other solutions are almost the same.

Through the experiment, this study found that the performance of HPGA is superior to the existing meta-heuristics. In order to evaluate the performance of HPGA in large-scale problems, 5 more problems are presented through connecting the system data of testing problem 1 (C=190, W=191) in series systems. The large-scale problems consist of 28 subsystems (C=260, W=382), 42 subsystems (C=390, W=573), 56 subsystems (C=520, W=764), 70 subsystems (C=650, W=955) in series systems. After 10 runs using HPGA, the results compared with the optimal solution by CPLEX are shown in Table 8.


Table 8. Experimental results of the large-scale problems

The suggested algorithm improved system reliability better than existing studies except for TS in the 1st~20th test problems in which the weight was heavy. In addition, HPGA found superior system reliability compared to TS in the 29th~32th test problems in which the

Through the experiment, this study found that the performance of HPGA is superior to the existing meta-heuristics. In order to evaluate the performance of HPGA in large-scale problems, 5 more problems are presented through connecting the system data of testing problem 1 (C=190, W=191) in series systems. The large-scale problems consist of 28 subsystems (C=260, W=382), 42 subsystems (C=390, W=573), 56 subsystems (C=520, W=764), 70 subsystems (C=650, W=955) in series systems. After 10 runs using HPGA, the

results compared with the optimal solution by CPLEX are shown in Table 8.

subsystems C W CPLEX HPGA (10 runs)

1 14 130 191 0.9868110 0.9868110 0.000021 8 / 10 2 28 260 382 0.9740720 0.9740720 0.000147 6 / 10 3 42 390 573 0.9612374 0.9612374 0.000564 4 / 10 4 56 520 764 0.9488162 0.9488162 0.001095 1 / 10 5 70 650 955 0.9370413 0.9370413 0.001732 1 / 10

Max S.D. Number of optimal

weight was light. The other solutions are almost the same.

Table 8. Experimental results of the large-scale problems

*L IA* 4( ) : MPI(%) of IA results

Fig. 8. MPI

No. Number of

*RHPGA* : system reliability by HPGA *RGA* : system reliability by GA *RTS* : system reliability by TS *RACO* : system reliability by ACO *RIA* : system reliability by IA

The suggested algorithm presented optimal solutions in all large-scale problems. For the experimental results of large-scale problems 1~4, the suggested algorithm showed the optimal solutions 4~8 times out of 10 runs. The optimal solution to large-scale problem 5 was obtained by HPGA in 1 out of 10 runs.

### **5.2 The reliability allocation problem with component choices (RAPCC)**

In order to evaluate the performance of the suggested algorithm for the reliability allocation problem with multiple-choice, this study performed experiments by using the Nahas & Nourelfath (2005) and the Kim et al. (2008) examples in series. The Nahas & Nourelfath (2005) examples consist of four examples: examples 1, 2, and 3 consist of 15 subsystems with 60, 80, and 100 components, respectively, and example 4 consists of 25 subsystems with 166 components. The budgets are \$1,000, \$900, \$1,000, and \$1,400, respectively. Examples by the Kim et al. (2008) consist of two large-scale examples (examples 5 and 6). Large-scale examples are presented through connecting the system data of example 4. Examples 5 and 6 consist of 100 and 200 subsystems with budgets of \$7,200~\$7,650 and \$14,400~\$15,100, respectively.

To use CPLEX, this study performed an additional step for transforming the nonlinear objective function into the linear function as shown in Eq. (26).

$$\text{Maximize } \cdot \ln R\_S = \ln \left\{ \prod\_{i=1}^{n} \left( \sum\_{j=1}^{m} r\_{ij} \cdot \mathbf{x}\_{ij} \right) \right\} = \sum\_{i=1}^{n} \sum\_{j=1}^{m} \mathbf{x}\_{ij} \cdot \ln r\_{ij} \tag{26}$$

The experimental results of examples 1~4 including component structure, reliability, and cost by using CPLEX and a hybrid parallel genetic algorithm are shown in Table 9.


Table 9. Experimental results of examples 1~4 by using CPLEX and HPGA

As found in Table 9, the suggested algorithm presented the optimal solutions in examples 14 and obtained a new optimal solution (3-3-3-5-2-3-2-2-3-1-2-3-4-3-1-3-3-3-4-2-3-2-2-3-1) in example 4.

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 143

14,400 0.802546 0.802218 0.000425 -0.1661 0.802218 0 -0.1661 0.802364 0.000110 -0.0922

14,450 0.806496 0.806167 0.000396 -0.1700 0.806167 0 -0.1700 0.806251 0.000076 -0.1266

14,550 0.814290 0.813792 0.000352 -0.2682 0.813792 0 -0.2682 0.813792 0.000182 -0.2682

14,600 0.818299 0.817388 0.000391 -0.5014 0.817798 0.000053 -0.2757 0.817984 0.000003 -0.1734

14,700 0.826207 0.824787 0.000709 -0.8171 0.825364 0.000142 -0.4851 0.825774 0.000325 -0.2491

14,800 0.833851 0.833428 0.000891 -0.2546 0.833510 0.000026 -0.2052 0.833604 0.000450 -0.1487

14,650 0.822160 0.821656 0.000891 -0.2834 0.821656 0 -0.2834 0.822160 0 0

14,750 0.830105 0.829263 0.000815 -0.4956 0.829427 0.000026 -0.3991 0.830105 0.000407 0

14,850 0.837614 0.837448 0.000824 -0.1022 0.837614 0 0 0.837614 0 0

14,900 0.841310 0.840805 0.000786 -0.3182 0.841310 0.000107 0 0.841310 0 0

14,950 0.844856 0.844009 0.000506 -0.5459 0.844856 0.000215 0 0.844856 0 0

15,000 0.848500 0.848332 0.000610 -0.1109 0.848500 0.000027 0 0.848500 0 0

15,050 0.852076 0.851991 0.000722 -0.0575 0.852076 0 0 0.852076 0 0

15,100 0.855751 0.855582 0.000679 -0.1172 0.855751 0 0 0.855751 0 0

As shown in Table 11, the result of SA and TS gave the optimal solution 2 and 6 times out of the 10 cases, respectively. The suggested algorithm found the optimal solution 8 times for the same cases and it showed the same or superior MPI compared to that of SA and TS. As the results in Table 12 show that, when compared with TS and the reoptimization procedure (TS+SA), the suggested algorithm gave the optimal solution 9 times out of the 15 cases and showed the same or superior MPI than TS and the reoptimization procedure (TS+SA). This is because the suggested algorithm could parallelly evolve by operating several sub-

Throughout the experiment, this study found that performance of HPGA is superior to existing meta-heuristics. This study has generated one more example, example 7, which is presented through connecting the system data of example 4 in series. Example 7 consists of 1,000 subsystems with \$90,000\$99,000 budgets. After 10 runs using CPLEX and HPGA in example 7, the experimental results including the maximum, standard deviation values, and

Table 12. Experimental results of example 6 by using CPLEX and HPGA (10 runs)

populations and improve the solution through swap and 2-opt heuristics.

maximum possible improvement (MPI) are shown in Table 13.

14,500 0.810301 0.809890 0.000519 -0.2167 0.809970 0 -0.1745 0.810301 0.000094 0

TS TS+SA Reoptimization HPGA (This Study)

Max S.D. %MPI Max S.D. %MPI Max S.D. %MPI

Budget CPLEX

After 10 runs using HPGA in examples 14, the experimental results including maximum, average and standard deviation values were compared with existing meta-heuristics such as ACO (Nahas & Nourelfath, 2005), SA (Kim et al., 2004), and TS (Kim et al., 2008). The comparison of meta-heuristics and the suggested algorithm is shown in Table 10.


Table 10. Experimental results of examples 1~4 by using CPLEX and HPGA

The suggested algorithm in examples 14 generated the optimal solutions without standard deviation and showed the same or superior solution compared to meta-heuristics.

In order to evaluate the performance of HPGA in large-scale problems, this study performed experiments by using examples in series as suggested by the Kim et al. (2008). After 10 runs using CPLEX and HPGA in examples 5 and 6, experimental results including maximum, standard deviation values, and maximum possible improvement (MPI) compared with existing meta-heuristics such as simulated annealing, tabu search, and reoptimization procedure by the Kim et al. (2008) are shown in Tables 11 and 12. The MPI was obtained by Eq. (27).

$$\%MPI = \frac{\{Max - CPLEX\}}{\{1 - CPLEX\}} \times 100\tag{27}$$


Table 11. Experimental results of example 5 by using CPLEX and HPGA (10 runs)

After 10 runs using HPGA in examples 14, the experimental results including maximum, average and standard deviation values were compared with existing meta-heuristics such as ACO (Nahas & Nourelfath, 2005), SA (Kim et al., 2004), and TS (Kim et al., 2008). The

ACO SA TS HPGA

Max Ave. S.D. Max Ave. S.D. Max Ave. S.D. Max Ave. S.D. 1 0.85705 0.85705 0 0.85705 0.85705 0 0.857054 0.857054 0 0.857054 0.857054 0 2 0.91504 0.91504 0 0.91504 0.91504 0 0.915042 0.915042 0 0.915042 0.915042 0 3 0.96512 0.96439 0.00050 0.96513 0.96503 0.00033 0.965134 0.965134 0 0.965134 0.965134 0 4 0.86543 0.86491 0.00038 0.86543 0.86536 0.00025 0.865439 0.865439 0 0.865439 0.865439 0

The suggested algorithm in examples 14 generated the optimal solutions without standard

In order to evaluate the performance of HPGA in large-scale problems, this study performed experiments by using examples in series as suggested by the Kim et al. (2008). After 10 runs using CPLEX and HPGA in examples 5 and 6, experimental results including maximum, standard deviation values, and maximum possible improvement (MPI) compared with existing meta-heuristics such as simulated annealing, tabu search, and reoptimization procedure by the Kim et al. (2008) are shown in Tables 11 and 12. The MPI was obtained by

> ( ) % <sup>100</sup> (1 ) *Max CPLEX MPI*

7,200 0.895758 0.895575 0.001342 -0.1756 0.895758 0.000312 0 0.895758 0.001017 0 7,250 0.900167 0.899438 0.001050 -0.7302 0.899984 0.000305 -0.1833 0.899984 0.000236 -0.1833 7,300 0.904599 0.903866 0.001027 -0.7683 0.904414 0.000390 -0.1939 0.904599 0.000529 0 7,350 0.908866 0.908405 0.001202 -0.5058 0.908866 0.000480 0 0.908866 0.000424 0 7,400 0.913154 0.912601 0.000499 -0.6368 0.913064 0.000337 -0.1036 0.913114 0.000107 -0.0461 7,450 0.917184 0.916815 0.000510 -0.4456 0.917093 0.000494 -0.1099 0.917184 0.000229 0 7,500 0.921141 0.920770 0.000743 -0.4705 0.921141 0.000365 0 0.921141 0.000156 0 7,550 0.925023 0.925023 0.000590 0 0.925023 0.000502 0 0.925023 0.000172 0 7,600 0.929013 0.928269 0.000696 -1.0481 0.929013 0.000445 0 0.929013 0 0 7,650 0.931526 0.931526 0.000388 0 0.931526 0 0 0.931526 0 0

Table 11. Experimental results of example 5 by using CPLEX and HPGA (10 runs)

*CPLEX*

<sup>−</sup> = × − (27)

SA TS HPGA (This Study) Max S.D. %MPI Max S.D. %MPI Max S.D. %MPI

(This study)

comparison of meta-heuristics and the suggested algorithm is shown in Table 10.

Table 10. Experimental results of examples 1~4 by using CPLEX and HPGA

deviation and showed the same or superior solution compared to meta-heuristics.

No.

Eq. (27).

Budget CPLEX


Table 12. Experimental results of example 6 by using CPLEX and HPGA (10 runs)

As shown in Table 11, the result of SA and TS gave the optimal solution 2 and 6 times out of the 10 cases, respectively. The suggested algorithm found the optimal solution 8 times for the same cases and it showed the same or superior MPI compared to that of SA and TS. As the results in Table 12 show that, when compared with TS and the reoptimization procedure (TS+SA), the suggested algorithm gave the optimal solution 9 times out of the 15 cases and showed the same or superior MPI than TS and the reoptimization procedure (TS+SA). This is because the suggested algorithm could parallelly evolve by operating several subpopulations and improve the solution through swap and 2-opt heuristics.

Throughout the experiment, this study found that performance of HPGA is superior to existing meta-heuristics. This study has generated one more example, example 7, which is presented through connecting the system data of example 4 in series. Example 7 consists of 1,000 subsystems with \$90,000\$99,000 budgets. After 10 runs using CPLEX and HPGA in example 7, the experimental results including the maximum, standard deviation values, and maximum possible improvement (MPI) are shown in Table 13.

A Hybrid Parallel Genetic Algorithm for Reliability Optimization 145

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Table 13. Experimental results of example 7 by using CPLEX and HPGA (10 runs)

As shown in Table 13, the suggested algorithm presented the optimal solution in 3 times out of 10 cases. While the budget increased, the suggested algorithm found the near-optimal solution.

### **6. Conclusions**

This study suggested mathematical programming models and a hybrid parallel genetic algorithm for reliability optimization with resource constraints. The experimental results compared HPGA with existing meta-heuristics and CPLEX, and evaluated the performance of the suggested algorithm.

The suggested algorithm presented superior solutions to all problems (including large-scale problems) and found that the performance is superior to existing meta-heuristics. This is because the suggested algorithm could paratactically evolve by operating several subpopulations and improve the solution through swap, 2-opt, and interchange (except for RAPCC) heuristics.

The suggested algorithm would be able to be applied to system design with a reliability goal with resource constraints for large scale reliability optimization problems.

### **7. References**


90,000 0.831082 0.830757 0.000681 -0.1924 91,000 0.847706 0.846918 0.000594 -0.5174 92,000 0.860003 0.859647 0.000317 -0.2543 93,000 0.871659 0.871516 0.000183 -0.2228 94,000 0.883369 0.883275 0.000262 -0.0806 95,000 0.895226 0.895185 0.000208 -0.0391 96,000 0.904832 0.904832 0.000079 0 97,000 0.913836 0.913791 0.000055 -0.0522 98,000 0.920716 0.920716 0.000016 0 99,000 0.924869 0.924869 0 0

Table 13. Experimental results of example 7 by using CPLEX and HPGA (10 runs)

As shown in Table 13, the suggested algorithm presented the optimal solution in 3 times out of 10 cases. While the budget increased, the suggested algorithm found the near-optimal

This study suggested mathematical programming models and a hybrid parallel genetic algorithm for reliability optimization with resource constraints. The experimental results compared HPGA with existing meta-heuristics and CPLEX, and evaluated the performance

The suggested algorithm presented superior solutions to all problems (including large-scale problems) and found that the performance is superior to existing meta-heuristics. This is because the suggested algorithm could paratactically evolve by operating several subpopulations and improve the solution through swap, 2-opt, and interchange (except for

The suggested algorithm would be able to be applied to system design with a reliability goal

Ait-Kadi, D. & Nourelfath, M. (2001). Availability Optimization of Fault-Tolerant Systems,

Bae, C. O.; Kim, H. G.; Kim, J. H.; Son, J. Y. & Yun, W. Y. (2007). Solving the Redundancy

*International Journal of Industrial Engineering*, Special Issue, pp.315-323. Bellman, R. & Dreyfus, S. (1958). Dynamic Programming and the Reliability of Multicomponent Devices, *Operations Research*, Vol.6, No.2, pp.200-206.

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Allocation Problem with Multiple Component Choice using Metaheuristics,

with resource constraints for large scale reliability optimization problems.

*Management (IEPM 2001)*, Quebec, Canada, August, 2001.

HPGA Max S.D. %MPI

Budget CPLEX

solution.

**6. Conclusions** 

RAPCC) heuristics.

**7. References** 

of the suggested algorithm.


**7** 

*Malaysia* 

**Hybrid Genetic Algorithm-Support Vector** 

**Machine Technique for Power Tracing in** 

*Universiti Teknologi Malaysia (UTM), Universiti Malaysia Perlis (UniMAP) and* 

The electric power industry is under deregulation in response to changes in the law, technology, market and competition. The aim of deregulation is to optimize the system welfare by introducing competitive environment, mainly among the power producers. Developing fair and transparent power flow and loss allocation method has been an active

In the last decades, several power flow tracing algorithms have been proposed in literature mainly from physical flow approach (Bialek & Tam, 1996; Wu et al., 2000; Sulaiman et al., 2008; Pantos et al., 2005) and circuit theory approach (Teng, 2005; Wen-Chen et al., 2004; Mustafa et al., 2008a; Lo & Alturki, 2006). The concept of proportional sharing principle has been proposed by (Bialek & Tam, 1996). This approach has a drawback in handling the transmission losses by introducing fictitious nodes on every lossy branch which will causes the expansion of distribution matrix. The graph method was proposed by (Wu et al., 2000) where the method is basically using the searching technique of the paths and routes from a particular generator to a particular load. The adaptation of the methods from (Bialek & Tam, 1996) and (Wu et al., 2000) has been proposed in (Sulaiman et al., 2008) for tracing the power and loss in deregulated power system. However, the main disadvantage of this approach is it cannot be applied for the circular power flow system. Modification of (Bialek & Tam, 1996) has been done in (Pantos et al., 2005) to trace the real and reactive power by introducing decoupled power flow to overcome the lossy system problem. This method introduces equivalent model of a line for reactive power tracing. The effects of line charging to original generators and loads are integrated. Nevertheless, the actual power contribution

The uses of circuit theory in power tracing have been introduced in (Teng, 2005; Wen-Chen et al., 2004; Mustafa et al., 2008a; Lo & Alturki, 2006). In (Teng, 2005), a method that applies

topic of research, with many transactions taking place at any time.

from generators to loads has been ignored.

\* Corresponding Author

**1. Introduction** 

**Deregulated Power Systems** 

*Universiti Kebangsaan Malaysia (UKM)* 

Mohd Wazir Mustafa, Mohd Herwan Sulaiman\*, Saifulnizam Abd. Khalid and Hussain Shareef

