Optimization of series-parallel systems is an important aspect of equipment design strategies. The optimized system characteristics, such as reliability, cost, weight, and volume contribute toward designing the best machine. This approach is challenging because the reliability needs to be maximized whereas the other objective functions need to be minimized. In practice, system reliability optimization is critical, and over the last two decades, considerable effort has been devoted toward the development of reliability criteria for quantifying the nature of generation, transmission, and circulation in composite system frameworks. To improve component reliability and implement redundancy while achieving a trade-off between system performance and resources, reliability design that aims to establish an optimal system-level configuration has long been considered an important advantage in reliability engineering. At present, system reliability is of considerable research significance, as engineering fields involve continual advancements in fixed systems and applications with increasing levels of complexity. Thus, it is imperative for production systems to perform satisfactorily during their expected lifespan. However, failure is an inevitable phenomenon associated with technological advancement of the equipment used in various industries. The reliability-redundancy allocation problem (RRAP) has been studied to optimize system reliability on the basis of the redundancy allocation problem (RAP)  . The RRAP has attracted considerable attention from the viewpoint of developing heuristic optimization algorithms. This paper focuses on an RRAP with the objective of maximizing system reliability under nonlinear constraints, such as system cost, weight, and volume. The RRAP has been shown to be an NP-hard problem, and various optimization approaches have been proposed to solve it. These methods, which are called meta-heuristic methods, have been widely researched and implemented. They can obtain feasible solutions within limited computing time. The main goal of RRAPs is to select the levels of redundancy and component reliability for maximizing and improving system reliability and performance. RRAPs are useful for designing not only systems that are taken together on a large scale but also systems produced in large-scale industrial operation using off-the-shelf components.
2. Literature Review
A reliability-redundancy optimization problem can be formulated using components and levels of redundancy to maximize some objective function, given system-level constraints on reliability, cost, and/or weight. The problem of maximizing system reliability through redundancy and component reliability selection is called the reliability-redundancy allocation problem (RRAP). Reliability optimization has been the subject of several studies by Kuo et al.  ,  ,  . Forsthoffer  , Kundur  , Hejzlar  , and Seebgrets  conducted studies on overspeed protection, such as analysis of the instability of steam turbines and analysis of the reliability of wind turbines. Dhingra  developed an application of the reliability-redundancy optimization problem with regard to overspeed protection by using a multi-objective approach to maximize system reliability and minimize consumption of resources (cost, weight, and volume). This approach involves a goal programming formulation and a goal achievement method for generating Pareto optimal solutions. Control and overspeed protection for a gas turbine are nearly the same as those for a steam turbine. A gas turbine operates at a higher temperature than a steam turbine; hence, it requires closer control, called control sequencing. Sequencing allows automatic control of the gas turbine. Fetanat et al.  proposed an optimal design for control and overspeed protection of gas turbine by means of reliability-redundancy optimization achieved using a new type of harmony search algorithm (HSA) known as the elitism Box-Muller harmony search algorithm (EBMHSA). Dhingra and Rao  used goal programming and goal attainment formulations under fuzziness in a multi-objective reliability apportionment problem subject to several design constraints. Rao proposed three methods for finding the optimal solution of each objective function: a method for determining reliability, a method for minimizing cost, and a method for controlling weight. Rao’s approach has been developed to optimize redundant series-parallel systems, all components of which are time-dependent. The proposed model for simulation is the overspeed control system for a gas turbine engine. This model, proposed by Dhingra, is a combination of mechanical and electrical systems. Overspeed control is the first step against excessive speed. In general, the emergency reset of the system is designed independent of the overspeed control. Hence, high-reliability operation of control valves is considered. In the normal working mode, the control valves are opened sequentially  . Luus  proposed a new non-linear integer programming method that considers the component reliability to be fixed. However, a more general problem is one where the optimal redundancy in each stage is determined to obtain the maximum system reliability. To solve the RRAP, several global optimization methods as well as heuristic and meta-heuristic methods have been proposed in the literature, including the Lagrangian multiplier method, branch and bound method, and linear programming     . These approaches do not guarantee exact optimal solutions but achieve reasonably good solutions for complex problems with relatively short computing time. Heuristic techniques, including genetic algorithms, require derivatives for all non-linear constraint functions, which are not derived easily because of the high computational complexity. Yokota et al.  and Hsieh et al.  applied genetic algorithms (GA) to mixed-integer reliability optimization problems. Zhao et al.  developed a hybrid GA with a flexible allowance technique for solving constrained engineering design optimization problems. Kanagaraj et al.  and Ghodrati and Lofti  developed a hybrid cuckoo search (CS)/GA algorithm to solve reliability-redundancy optimization problems and global optimization problems, respectively. Gen and Yun  developed a soft computing approach for solving various reliability optimization problems. This method combines rough search techniques and local search techniques to prevent premature convergence of the solution. Zou et al.  proposed a global harmony search algorithm for solving bridge and overspeed protection system optimization problems by combining the harmony search algorithm with concepts from particle swarm optimization. Different programming and evolutionary optimization techniques have been adopted to optimize different types of RRAPs, e.g., GA  and a new interpretation and formulation of the RRAP  using a new mixed strategy and a modified version of the genetic algorithm (MVGA), which shows distinct advantages compared to traditional approaches. Afonso et al.  proposed a modified version of the imperialist competitive algorithm (ICA) and demonstrated its capabilities by comparing its results with the best-known results of different benchmarks. Quy  developed a new method to optimize a multi-objective model in certain mechanical systems by using the fuzzy multi-objective method. His approach is based on the algorithm proposed by Rao  , and he applied it to the modeling and analysis of the overspeed control system of a gas turbine engine. All the components of this model are time-dependent. The performance of the algorithm was verified by programmed simulation of the above-mentioned model. In summary, Dhingra  , Rao and Dhingra  , and Quy  developed effective multi-objective fuzzy optimization techniques for engineering design. In particular, they adopted fuzzy programming, which is a powerful technique for solving optimization problems with fuzzy parameters. However, the use of uncertain information for reliability allocation requires further investigation. Moreover, they treated component risk/cost functions as continuous. Thus, no general method for solving the component reinforcement problem with discontinuous risk/cost functions has been proposed thus far. In addition, the three above-mentioned studies did not adopt any random-search-based global optimization methods. In other words, the entire family of meta-heuristics that can efficiently solve highly nonlinear nonconvex mixed-integer optimization problems has been overlooked. The major drawback of these studies is that none of them has developed a practical tool for designing actual components with distinct physical properties, such as cost, weight, volume, and reliability. In this paper, we present a hybrid GA (HGA) approach based on the redundancy allocation problem to find the number of redundant components that either maximize reliability or minimize cost, weight, and volume under various resource constraints. The computational results of our approach are compared with those of previously proposed algorithms.
3. Reliability-Redundancy Allocation Problems (RRAPs)
In this study, a reliability-redundancy allocation problem of minimizing the multi-objective function [−f1, f2, f3] subject to several nonlinear design constraints can be stated as a nonlinear mixed-integer programming model. The multi-objective formulation was obtained by applying cost and weight constraints to an objective function. In other words, the general problem of reliability and redundancy is assigned to each of the subsystems such that the system reliability, cost, and weight are optimized. The problem is overspeed protection of a gas turbine system with a time-related cost function, and the multi-objective RRAP model is as follows:
Many designers have attempted to improve the reliability of manufacturing systems or product components for greater competitiveness in the market. Typical approaches for achieving higher system reliability include increasing the reliability of system components and using redundant components in various subsystems of the system   .
4. Mathematical Formulation of the Problem
The mathematical model of the optimization problem is given by the equations below. The system reliability, cost, weight, and product of weight and volume are constrained by the design. The resulting multi-objective reliability apportionment problem is as follows: find n and r that minimize the multi-objective function [−f1, f2, f3] subject to . Figure 1 shows a typical example of a series-parallel system configuration with k-out-of-n subsystem reliabilities
Figure 1. General series-parallel redundancy system.
ri: Reliability of component in subsystem i;
ni: Number of redundant components in subsystem i;
r, n: Vectors of ri and ni;
Rs: System reliability;
N: Number of subsystems in the system;
f1: Objective function for system reliability;
f2: Objective function for system cost;
f3: Objective function for system weight;
gi: (.): Constraint function #j;
aj: Constraint limit #j;
m: Number of constraints.
5. Methodology Framework
Before introducing the RRAP, we present the following assumptions and notations that have been used throughout the entire paper. The hybrid function allows the optimization algorithm to identify the solution of the redundancy problem that achieves the optimal trade-off between the optimization objectives from several optimal solutions. We performed 10 simulations for every experiment and used the best result among the 10 reliability values obtained. The best configuration of each point corresponding to the largest reliability value is given with the corresponding cost, weight, and weight values.
・ The supply of components is unlimited.
・ The weight and volume of the components are known and deterministic.
・ All the redundant components of individual subsystems have different values, and every branch of the system has a different number of components.
・ The failure rate of the components in each subsystem is constant.
・ Failed components do not damage the system and are not repaired.
・ All redundancies are active: the hazard function is the same regardless of whether it is in use.
・ Failures of individual components are independent of one another but dependent on the number of working elements.
As mentioned above, few studies have reported the use of HGA for reliability allocation optimization with time-dependent reliability. We need to check whether our approach of using only HGA can guarantee the location of the optimal solution and whether the final solution obtained by the proposed HGA is superior to that obtained by existing methods.
Figure 2 shows the flowchart of the proposed algorithm. The HGA procedures that implement our methodology are illustrated. The proposed algorithm involves the following steps:
1) Define the functions of the design problem (Rs, Ws, Vs, and Cs).
2) Define the nonlinear constraints.
3) Define the lower bound and upper bound for ri and ni.
4) Chose the optimization algorithm (fmincon, fminmax, GA, and HGA).
5) Solve the optimization problem.
6) Calculate the optimal values (Rs, Cs, and Ws).
The hybrid GA is a combination of fmincon and GA. GA is used to find the global optima for optimization problems. “Fmincon” uses gradient information to facilitate rapid convergence. “HybridFcn” allows the GA to find the valley containing the global minimum. Then, fmincon is used to rapidly obtain the minimum of this valley. A hybrid function is an optimization function that runs after the GA terminates in order to improve the value of the fitness function. The hybrid function uses the final point from the GA as its initial point.
This study consists of two parts. In the first part, we identify the approach for solving the problem by using MATLAB code and compare the results with previous results  to determine the number of redundant components in stage i and the reliability for each component. The second part involves a novel contribution: we develop a model for the entire system with the desired level of reliability. Specifically, we develop a simulation procedure and implement it with different numbers of components for each stage with different values of each component. We use this novel approach to determine the converged value of system reliability until we obtain the values of ni and ri corresponding to value of the maximum reliability. Toward this end, we need to perform optimization. For the general structure of the network, we fixed the system reliability to a certain level, i.e., greater than or equal to 0.95.
We implemented a single-objective function with nonlinear constraints and tested it using two methods (ni is an integer in our problem). The results are summarized in Table 1 and Table 2. In addition, we implemented a multi-objective function. The initial results were obtained for four functions, and ri and ni were randomly set to evaluate each function.
Figure 2. Flowchart of proposed simulation procedure.
Table 1. Simulation results for single-objective function using fmincon optimization method.
Table 2. Simulation results for single-objective function using fminimax optimization method.
First step: We implemented a multi-objective function, and we defined the general objective function as follows:
; (new definition)
The above-mentioned has three parts: reliability, cost, and weight. This equation maximizes reliability but minimizes cost and weight. It is a normalized form of the objective function because we consider the upper bound of each objective. We penalized the reliability (with a value of 10) for greater emphasis. In addition, we set the upper bounds for Cs and Ws as 400 and 500, respectively. Therefore, if we divide by these values and take the sum, we will always get a number less than one. Thus, we normalized the functions (f1, f2, and f3).
Second step: We used fmincon and fminmax to solve the objective function.
Third step: We used the GA toolbox and applied this algorithm to our single- and multi-objective function problems. The results are summarized in Table 3.
Fourth step: We applied the GA to a new type of multi-objective function and evaluated the results.
Fifth step: We applied the global multi-objective GA to the problem and obtained 70 sets of Pareto optimal solutions.
Last step: We applied HGA optimization to single- and multiple-objective functions on the basis of our first approach. The results are summarized in Table 4.
Our multi-objective function aims to minimize cost and weight in the first approach. The results of our optimization give us ni and ri for each stage as well as for the entire system, as shown in the final result table. In this study, we performed optimization using GA and HGA. We used the same approach as that for obtaining a constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming (fmincon). We used different optimization approaches and finally used HGA. Specifically, we employed GA and fmincon to implement HGA using the first approach by varying ri and ni to achieve the desired system reliability with the objective function. Further, we fixed the system reliability Rs to obtain a system with minimum cost and weight in order to determine the structure of our new design in the second approach, which minimizes the worst-case value of a set of multivariable functions, starting at an initial estimate. The values may be subject to constraints. This is generally referred to as the minimax problem (fminmax).
We also varied the level of system reliability to show how we can select the desired system reliability; accordingly, we can change the structure of the entire system. In this step, we used GA and MATLAB toolbox. Here, we do not maximize the system reliability Rs but we want Rs = A, and we want to determine the system structure for achieving the minimum cost and weight. We assumed that ni is a continuous value. In this case, the first method of optimization using fmincon is summarized in Table 5. In addition, we can see the result of the second approach of optimization, i.e., fminmax. The results of our contribution are summarized in Tables 6-8, which show the different values obtained after we fixed the system reliability.
We tested various algorithms to identify the best ones, which were found to be GA or HGA.
Table 3. Simulation results for single-objective function using GA optimization method.
Table 4. Simulation results for single-objective function using hybrid optimization method.
Table 5. Simulation results using fmincon optimization method when system reliability Rs = A.
Table 6. Simulation results using fminimax optimization method when system reliability Rs = A.
Table 7. Simulation results using GA optimization method when system reliability Rs = A.
Table 8. Simulation results using hybrid optimization method when system reliability Rs = A.
6. Hybrid Genetic Algorithm (HGA) for Multi-Objective Optimization
Most previous studies have focused on several methods for solving redundancy optimization problems. In this study, we develop an approach by considering some aspects that have not been considered previously. The mathematical model represents the multi-objective HGA with a constraint-handling strategy for solving the proposed model. HGA is a meta-heuristic method that is used to solve optimization problems efficiently. In this method, first, an initial set of random potential solutions including a number of particles is created. Each particle represents a solution of the problem and has a position and velocity that change in each iteration so that better solutions can be obtained.
7. A Case Study: Overspeed Protection System for a Gas Turbine
To evaluate the performance of the HGA in reliability optimization problems, overspeed detection continuously provided by the electrical and mechanical systems is considered in a case study. The benchmark considered is an overspeed protection system for a gas turbine. When overspeed occurs, it is necessary to cut off the fuel supply using control valves, i.e., the four valve controllers (V1-V4) must close. The control system is modeled as a four-stage series-parallel system, as shown in Figure 3.
Each stage represents a controller that can be considered as a parallel system. All the components of the system have the same failure rate. The equivalent circuit of the overspeed control system is shown in Figure 4.
Here, vi is the volume of each component in subsystem i, V is the upper limit on the sum of the subsystem products of volume and weight, C is the upper limit on the system cost, and W is the upper limit on the system weight. The parameters αi and βi are constants representing the physical characteristics of each component in stage i. T is the operating time during which a component must not fail. The input parameters of the overspeed protection system for a gas turbine are listed in Table 9.
Figure 3. Block diagram of overspeed protection system for gas turbine with four valves.
Figure 4. Equivalent circuit: four-stage series-parallel system.
Table 9. Design values of different parameters used in overspeed protection system of gas turbine.
8. Computational Results and Discussion
We compared our solutions with those obtained in a previous study  . From Table 10, it is clear that our HGA approach obtains better solutions for the series-parallel system compared to the other approaches presented in the literature. The best fitness and mean fitness of the system cost, system weight, and multi-objective functions are shown in Figures 5-7, respectively.
The mathematical model used for calculating the objective function is employed to define the solution that guarantees an optimal trade-off between the two objectives, and the result is shown in Figure 8. Figure 9 shows the average distance between individuals.
These figures show the number of generations in GA. In addition, the values of each objective function in each iteration are shown. The toolbox is employed to generate these figures, which can be used to determine the most suitable reliability level that minimizes the total cost, weight, and volume subject to various constraints.
The runs of the HGA were continuously monitored throughout the generations (Figures 5-7). These plots show the best and mean fitness values of the fitness functions after 100, 100, and 300 generations, respectively. For Figure 5, the best fitness is in the range of 38.787 and the mean fitness is in the range of 38.795. For Figure 6, the best fitness is in the range of 55.9112 and the mean fitness is in the range of 55.9136. For Figure 7, the best fitness is in the range of 0.462836 and the mean fitness is in the range of 0.462929. From these plots, it can easily be observed that the fitness value converges toward the optimal value from generation to generation.
Table 10. Comparison of simulation results of optimal solutions of single- and multi-objective function for series-parallel system using HGA with other results presented in the literature.
Figure 5. Best fitness and mean fitness of the system cost.
Figure 6. Best fitness and mean fitness of the system weight.
Figure 7. Best fitness and mean fitness of the multi-objective functions.
Figure 8. Overall best Pareto front obtained by multi-objective optimization and HGA: cost vs. weight and distance of individuals.
Figure 9. Average distance between individuals.
The upper plot function in Figure 8 is the HGA Pareto function, which plots the Pareto front (limited to any three objectives) at every generation. This plot shows the trade-off between the two components of f. It is plotted in the objective function space. The lower plot shows the histogram distances of individuals.
The upper plot in Figure 9 shows the average distance between individuals for each objective, which is a good measure of the diversity of the initial population that affects the performance of the HGA. In general, if the diversity is too high or too low, the HGA might not perform well. Here, it is obvious that the distance does not reach extreme values, so it is considered that the performance is good. The lower plot shows the histogram of the parents, which indicates the parents that contribute to each generation of children populated by each individual.
We considered only the case of multi-objective optimization with the HGA technique for our contribution, and we generated/calculated the values of Ws, Cs, and Vs for 19 values of Rs = A (A = 0.9900, 0.9905, 0.9910, 0.9915, …, 0.9980, 0.9985, and 0.9990). The results are summarized in Table 11(a). On the basis of these tables, we plotted the curves ri for each stage and Cs, Ws, and Vs as functions of Rs. Further, we determined the mathematical equation of each of these curves. We used the nonlinear regression technique. If the utility of these equations is good, we can use them to estimate/calculate the values of r1, r2, r3, r4, Cs, Ws, and Vs for any value of Rs (Rs = 0.9900 to 0.9990). Thus, from a practical point of view, these equations are extremely useful. We also obtained the nonlinear regression fitted line plot, which can be used to investigate the relationship between two continuous variables, namely a response variable and a predictor variable. Thus, we can derive a regression equation and plot the regression line. For the copper expansion data, the method determines the type of relationship with these graphs and the line is fitted as per the requirement of data points. Minitab uses the Gauss-Newton algorithm, imposes a maximum of 200 iterations, and employs a tolerance of 0.00001 to achieve convergence. It displays a plot of the data overlaid with a curve illustrating the best-fitting equation based on our expectation function. The plot of the copper expansion data indicates that the specified rational polynomial is a good fit for the data. The points are fairly close to the curve and follow the curve without any systematic deviations from it. If we fit these models, differences will be observed in the desired values as well as in the corresponding points in the graph. This is because the fitted value is given, not the original one. Therefore, it is called the expected value or return of the model. The values of the parameters can be obtained using Minitab. We simply put the values of the parameters in the following regression equation.
With the parameter estimates in Table 12, we obtain r1New, r2New, r3New, r4New, CsNew, WsNew, and VsNew, as with each value of Rs obtained previously. Then, we obtain a scatter plot between Rs and r1New, r2New, r3New, r4New, CsNew, WsNew, and VsNew, as shown in Figure 10. There will be
Figure 10. Scatter plot of r1, r2, r3, r4, Cs, Ws, and Vs vs. Rs - (Rs) = 0.9900 - 0.9990.
Table 12. Explained variable with parameters when Rs = 0.9900 - 0.9990.
non-linear parameters when we fit the models given previously.
The results obtained using multi-objective optimizations with the HGA are summarized in Table 11(a). It can be seen that the number of components ni and the individual component reliability ri in various stages are different. However, in practice, ni must be an integer. Therefore, must approximate the values of ri and adjust the values of ni to integer values. The new results with this approximation are summarized in Table 11(b). It can be seen that when the numbers of components in different stages, ni, are modified to integer values, the cost, weight, and volume of the system are reduced slightly.
In this study, we must ensure that the number of simulations (n) for each time is sufficient to achieve convergence. To this end, we changed the value of n (0, 1, 2, 3, …, 75), and for the simulation with different values of Rs, we can say that for all values of Rs, the simulation process converges at n = 30, and for the case of Rs = 0.9900, the converged values of r1, r2, r3, and r4 are 0.8724, 0.9567, 0.8838, and 0.8668, respectively, as shown in Figure 11. In summary, we have discussed our novel approach, i.e., design of system reliability using the simulation process. The advantage of our approach is that the reliable regression curves have been generated using the proposed simulation process (Figure 10) and the utility of these curves for the system design is that they can help the designer to determine any level of reliability ri of the system components, the corresponding value of cost, weight, and volume depending on the chosen value of Rs.
In this study, we proposed a hybrid genetic algorithm and presented a novel system design for the entire system with the desired level of reliability. Thus, we achieved two objectives. First, we evaluated our approach to determine the robustness of our method by comparing it with another method in the literature. The results indicated that our approach yields better results. Second, we used this approach to develop a new simulation process for system design. We varied Rs and obtained different r1, r2, r3, r4, Cs, Ws, and Vs. Then, we plotted the curves, which are of great practical significance because they enable the designer of the system to determine the values of r1, r2, r3, r4, Cs, Ws, and Vs corresponding to the value of Rs. Using Rs = 0.9904, the designer could directly use the curves to obtain all the required values. Some values converge after several iterations in some cases. The performance and robustness of the proposed approach can easily be evaluated. Rapid convergence can be achieved using our model and approach, as shown in Figure 11. Moreover, robustness can be confirmed on the basis of similar results obtained under different initial conditions, as shown in Figure 11. In addition, Figure 10 illustrates the practical utility of our approach, i.e., the designer can determine the reliability of each component corresponding to any value of system reliability Rs.
Finally, we fixed the system reliability to obtain a satisfactory system with minimum cost and weight. Comparison of the simulation results indicates the superiority of HGA over other algorithms in terms of searching quality and robustness of the solution. The main advantage of the proposed multi-objective approach is that it offers greater flexibility to system designers for testing problems. Our HGA improves the objective function values and gives the best-known solutions for benchmark suites. Thus, to the best of our knowledge, HGA is an effective algorithm for application to the RRAP. It is especially useful when the optimization problem under consideration is complex.
Figure 11. Scatter plot of r1, r2, r3, and r4 versus number of simulations (n).
In the future, we will focus on extending our approach to other algorithms, such as hybrid nonlinear mixed integer programming, to achieve better results.
Abbreviations and Acronyms
MOO: Multi-objective optimization; HGA: Hybrid genetic algorithm; GA: Genetic algorithm; RAP: Redundancy allocation problem; RRAP: Reliability-redundancy allocation problem; HAS: Harmony search algorithm; EBMHSA: Elitism Box-Muller harmony search algorithm; MVGA: Modified version of the genetic algorithm; ICA: Imperialist competitive algorithm; Fmincon: Find minimum of constrained; Fminimax: Solve minimax constraint problem.
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