In this study, we develop a new meta-heuristic-based approach to solve a multi-objective optimization problem, namely the reliability-redundancy allocation problem (RRAP). Further, we develop a new simulation process to generate practical tools for designing reliable series-parallel systems. Because the RRAP is an NP-hard problem, conventional techniques or heuristics cannot be used to find the optimal solution. We propose a genetic algorithm (GA)- based hybrid meta-heuristic algorithm, namely the hybrid genetic algorithm (HGA), to find the optimal solution. A simulation process based on the HGA is developed to obtain different alternative solutions that are required to generate application tools for optimal design of reliable series-parallel systems. Finally, a practical case study regarding security control of a gas turbine in the overspeed state is presented to validate the proposed algorithm.
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) [
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. [
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:
max R s ( r , n ) & min C s ( r , n ) & min W s ( r , n )
Subjectto : g j ( r , n ) ≤ a j , j = 1 , ⋯ , m
1 ≤ n i ≤ 10 , i = 1 , 2 , ⋯ , 4 , n i ∈ Z +
0.5 ≤ r i ≤ 1 − 10 − 6 , r i ∈ r +
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 [
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 g j ( r , n ) ≤ a j , j = 1 , ⋯ , m .
where
f 1 ( r , n ) isthesystemreliability (1)
f 2 ( r , n ) isthetotalsystemcost (2)
f 3 ( r , n ) isthetotalsystemweight (3)
Subject to
g 1 ( r , n ) = ∑ i = 1 4 v i n i 2 ≤ 250 ( Volume ) (4)
g 2 ( r , n ) = ∑ i = 1 4 w i n i exp ( n i 4 ) ≤ 500 ( Weight ) (5)
g 3 ( r , n ) = ∏ i = 1 4 [ 1 − ( 1 − r i ) n i ] ≥ 0.95 ( Reliability ) (6)
g 4 ( r , n ) = ∑ i = 1 4 α i ( − t ln r i ) β i [ n i + exp ( n i 4 ) ] ≤ 400 ( Cost ) (7)
1 ≤ n i ≤ 10 , i = 1 , 2 , 3 , 4 , [ g 5 ( r , n ) ] , n i ∈ Z + (8)
0.5 ≤ r i ≤ 0.999999 , [ g 6 ( r , n ) ] , r i ∈ r + (9)
where
f 1 = m a x R s ( r , n ) = ∏ i = 1 4 [ 1 − ( 1 − r i ) n i ] (10)
f 2 = min C s ( r , n ) = ∑ i = 1 4 α i ( − t ln r i ) β i [ n i + exp ( n i 4 ) ] (11)
f 3 = min W s ( r , n ) = ∑ i = 1 4 w i n i exp ( n i 4 ) (12)
Notation
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.
二级标题字号 10 磅
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.
Assumptions
・ 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.
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 [
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
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Maximize System Reliability | 1 2 3 4 | 0.8998 0.8680 0.9439 0.8728 | 5 6 4 6 | Rs = 0.9999 Cs = 419.2534 Ws = 541.2671 Vs = 217 |
Minimize System Cost | 1 2 3 4 | 0.5846 0.5184 0.6988 0.5252 | 5 6 4 5 | Rs = 0.9439 Cs = 36.0616 Ws = 475.1981 Vs = 195 |
Minimize System Weight | 1 2 3 4 | 0.9534 0.9313 0.9770 0.9351 | 1 2 1 2 | Rs = 0.9232 Cs = 422.7688 Ws = 60.8431 Vs = 20 |
Multi-objective Functions | 1 2 3 4 | 0.8493 0.7980 0.9147 0.8060 | 3 3 2 3 | Rs = 0.9740 Cs = 109.3931 Ws = 147.0485 Vs = 57 |
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Maximize System Reliability | 1 2 3 4 | 0.9001 0.8685 0.9431 0.8732 | 5 6 4 6 | Rs = 0.9999 Cs = 420.2802 Ws = 541.2671 Vs = 217 |
Minimize System Cost | 1 2 3 4 | 0.5846 0.5184 0.6988 0.5252 | 5 6 4 5 | Rs = 0.9439 Cs = 36.0616 Ws = 475.1981 Vs = 195 |
Minimize System Weight | 1 2 3 4 | 0.9534 0.9313 0.9770 0.9351 | 1 2 1 2 | Rs = 0.9232 Cs = 422.7688 Ws = 60.8431 Vs = 20 |
Multi-objective Functions | 1 2 3 4 | 0.8493 0.7980 0.9148 0.8059 | 3 3 2 3 | Rs = 0.9740 Cs = 109.3850 Ws = 147.0485 Vs = 57 |
First step: We implemented a multi-objective function, and we defined the general objective function as follows:
f = 10 f 1 + f 2 / 400 + f 3 / 500 ; (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
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
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
We tested various algorithms to identify the best ones, which were found to be GA or HGA.
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Maximize System Reliability | 1 2 3 4 | 0.8902 0.8603 0.9500 0.8806 | 5 6 4 5 | Rs = 0.9999 Cs = 389.3556 Ws = 475.1981 Vs = 195 |
Minimize System Cost | 1 2 3 4 | 0.6106 0.5550 0.6509 0.5465 | 5 5 5 5 | Rs = 0.9501 Cs = 37.4312 Ws = 471.1963 Vs = 200 |
Minimize System Weight | 1 2 3 4 | 0.8977 0.9537 0.9732 0.8987 | 2 2 1 2 | Rs = 0.9511 Cs = 414.0766 Ws = 72.9236 Vs = 23 |
Multi-objective Functions | 1 2 3 4 | 0.8504 0.7956 0.9167 0.8049 | 3 3 2 3 | Rs = 0.9740 Cs = 109.1462 Ws = 147.0485 Vs = 57 |
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Maximize System Reliability | 1 2 3 4 | 0.8971 0.8659 0.9358 0.8769 | 5 6 4 5 | Rs = 0.9999 Cs = 381.5582 Ws = 475.1981 Vs = 195 |
Minimize System Cost | 1 2 3 4 | 0.7997 0.7896 0.7154 0.8393 | 4 4 5 4 | Rs = 0.9939 Cs = 133.4582 Ws = 346.2031 Vs = 155 |
---|---|---|---|---|
Minimize System Weight | 1 2 3 4 | 0.9668 0.8715 0.9572 0.9382 | 2 2 2 2 | Rs = 0.9769 Cs = 440.5520 Ws = 89.0309 Vs = 32 |
Multi-objective Functions | 1 2 3 4 | 0.8536 0.7977 0.9189 0.8133 | 3 3 2 3 | Rs = 0.9757 Cs = 114.0175 Ws = 147.0485 Vs = 57 |
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Minimize System Cost | 1 2 3 4 | 0.5846 0.5184 0.6988 0.5252 | 5 6 4 5 | Rs = 0.9500 Cs = 36.0616 Ws = 475.1981 Vs = 195 |
Minimize System Weight | 1 2 3 4 | 0.9534 0.9313 0.9770 0.9351 | 1 2 1 2 | Rs = 0.9500 Cs = 422.7688 Ws = 60.8431 Vs = 20 |
Multi-objective Functions (Cost + Weight) | 1 2 3 4 | 0.8326 0.7755 0.9053 0.7840 | 3 3 2 3 | Rs = 0.9500 Cs = 91.7003 Ws = 147.0485 Vs = 57 |
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Minimize System Cost | 1 2 3 4 | 0.5846 0.5184 0.6988 0.5252 | 5 6 4 5 | Rs = 0.9500 Cs = 36.0616 Ws = 475.1981 Vs = 195 |
Minimize System Weight | 1 2 3 4 | 0.9534 0.9313 0.9770 0.9351 | 1 2 1 2 | Rs = 0.9500 Cs = 422.7688 Ws = 60.8431 Vs = 20 |
Multi-objective Functions (Cost + weight) | 1 2 3 4 | 0.8325 0.7755 0.9054 0.7841 | 3 3 2 3 | Rs = 0.9500 Cs = 91.7228 Ws = 147.0485 Vs = 57 |
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Minimize System Cost | 1 2 3 4 | 0.6317 0.5327 0.6800 0.5980 | 5 5 5 4 | Rs = 0.9500 Cs = 37.5962 Ws = 425.1462 Vs = 182 |
Minimize System Weight | 1 2 3 4 | 0.8925 0.9369 0.9720 0.9440 | 2 2 1 2 | Rs = 0.9500 Cs = 426.6653 Ws = 72.9236 Vs = 23 |
---|---|---|---|---|
Multi-objective Functions (Cost + Weight) | 1 2 3 4 | 0.8179 0.7812 0.8894 0.7656 | 3 3 2 3 | Rs = 0.9500 Cs = 83.8740 Ws = 47.0485 Vs = 57 |
Objective | Stage | Reliability | Component | Simulation Result |
---|---|---|---|---|
Minimize System Cost | 1 2 3 4 | 0.5846 0.5184 0.6988 0.5252 | 5 6 4 5 | Rs = 0.9500 Cs = 36.0616 Ws = 475.1981 Vs = 195 |
Minimize System Weight | 1 2 3 4 | 0.9534 0.9313 0.9770 0.9351 | 1 2 1 2 | Rs = 0.9500 Cs = 422.7688 Ws = 60.8431 Vs = 20 |
Multi-objective Functions (Cost + Weight) | 1 2 3 4 | 0.8326 0.7755 0.9053 0.7840 | 3 3 2 3 | Rs = 0.9500 Cs = 91.7003 Ws =147.0485 Vs = 57 |
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.
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
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
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
Number of stages | 4 | |||
---|---|---|---|---|
Lower limit on Rs | 0.95 | |||
Upper limit on cost | 400 | |||
Upper limit on weight | 500 | |||
Upper limit on volume | 250 | |||
Operating time | 1000 hours | |||
Stage | 105 αi | βi | vi | wi |
1 | 1.0 | 1.5 | 1 | 6 |
2 | 2.3 | 1.5 | 2 | 6 |
3 | 0.3 | 1.5 | 3 | 8 |
4 | 2.3 | 1.5 | 2 | 7 |
We compared our solutions with those obtained in a previous study [
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
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
Results given in Ref. [ | Results given by hybrid genetic algorithm | ||||||
---|---|---|---|---|---|---|---|
Objective | Stage | Reliability | Component | Simulation result | Reliability | Component | Simulation result |
Maximize System Reliability | 1 2 3 4 | 0.866288 0.850029 0.918417 0.913049 | 6.0 6.0 4.0 4.0 | Rs = 0.999881 Cs = 381.12183 Ws = 485.77850 Vs = 188.0 | 0.8971 0.8659 0.9358 0.8769 | 5 6 4 5 | Rs = 0.9999 Cs = 381.5582 Ws = 475.1981 Vs = 195 |
Minimize System Cost | 1 2 3 4 | 0.559777 0.599392 0.685273 0.703375 | 6.0 6.0 4.0 4.0 | Rs = 0.971340 Cs = 54.472889 Ws = 485.778504 Vs = 188.0 | 0.7997 0.7896 0.7154 0.8393 | 4 4 5 4 | Rs = 0.9939 Cs = 133.4582 Ws = 346.2031 Vs = 155 |
Minimize System Weight | 1 2 3 4 | 0.864883 0.944821 0.905934 0.880399 | 3.0 2.0 2.0 2.0 | Rs = 0.971597 Cs = 295.029388 Ws = 107.352295 Vs = 370 | 0.9668 0.8715 0.9572 0.9382 | 2 2 2 2 | Rs = 0.9769 Cs = 440.5520 Ws = 89.0309 Vs = 32 |
Multi- Objective Optimization | 1 2 3 4 | 0.820009 0.806433 0.869349 0.865680 | 4.0 3.0 3.0 2.0 | Rs = 0.971641 Cs = 119.04067 Ws = 177.234863 Vs = 69.0 | 0.8536 0.7977 0.9189 0.8133 | 3 3 2 3 | Rs = 0.9757 Cs = 114.0175 Ws = 147.0485 Vs = 57 |
The upper plot function in
The upper plot in
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
Explainedvariable = a + b ∗ R s + c ∗ R s 2 + d ∗ R s 3 .
With the parameter estimates in
Explained variable | r1N | r2N | r3N | r4N | CsN | WsN | VsN |
---|---|---|---|---|---|---|---|
Parameter | |||||||
a | −30601.9 | −47481.2 | −17136.2 | −39862.7 | −180794266 | −212968023 | −88910519 |
b | 92528.1 | 143535 | 51814.3 | 120529 | 546466156 | 643748903 | 268766970 |
c | −93254.8 | −144634 | −52221.3 | −121476 | −550583510 | −648634663 | −270819890 |
d | 31329.5 | 48581.2 | 17544.1 | 40810.6 | 184911867 | 217854114 | 90963580 |
non-linear parameters when we fit the models given previously.
The results obtained using multi-objective optimizations with the HGA are summarized in
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
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
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.
In the future, we will focus on extending our approach to other algorithms, such as hybrid nonlinear mixed integer programming, to achieve better results.
Saleem, E.A.A., Dao, T.-M. and Liu, Z.H. (2018) Multiple-Objective Optimization and Design of Series-Parallel Systems Using Novel Hybrid Genetic Algorithm Meta-Heuristic Approach. World Journal of Engineering and Technology, 6, 532-555. https://doi.org/10.4236/wjet.2018.63032
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.