Availability Optimization Decision Support Design System for Different Repairable n-Stage Mixed Systems

Abstract

This study attempts to propose an availability optimization decision support design system for repairable n-stage mixed systems, in which different combinations of subsystems, such as parallel, standby, and k-out-of-q, are connected in series configuration. The enumeration method, tabu search, simulated annealing, non-equilibrium simulated annealing, and the modified redundancy allocation heuristic combined with a modified genetic algorithm will be proposed to solve the system availability optimization problem and further determine the appropriate system configuration design. Several simulated cases are conducted by following the procedural flow of the proposed availability optimization decision support design system to reach the optimal allocations of the component redundancy amount, the optimal repair rates, and the optimal failure rates of all subsystems to minimize the total system cost under several configuration constraints for different repairable n-stage mixed systems. Simulated results display that the proposed availability optimization decision support design system can definitely take advantage of different component redundancy system designs, including the parallel-series system, n-stage standby system, n-stage k-out-of-q system, and n-stage mixed system, to save a lot of cost and meet the high level of the system availability requirement compared to the n-stage single component series system. Additionally, the results for all proposed combined methods also show that the parallel-series system can obviously reach the same level of system availability requirement with less system total cost, in contrast to the n-stage standby system, by presuming the identical deteriorating probability for both the operating components and the standby components. The performance comparisons of five proposed combined methods for four proposed system configurations are analyzed comprehensively. It can be concluded that the performances of the modified redundancy allocation heuristic method, combined with a modified genetic algorithm on the criteria of the optimal system costs for four proposed system configurations, are not only superior to the other four combined methods, but also to the performances on the criteria of CPU running time for four proposed system configurations.

1. Introduction and Literature Review

System availability generally plays a very important role in evaluating the performance of systems applied in many fields, which include the military force, aerospace and defense industry, electrical power systems, telecommunication systems, and computer network systems. The work in Wang [1] clarifies availability as the possibility that a repairable system keeps the operation at a period of time. The author of Blanchard [2] had noticed that most of the total system life cycle cost for a designed system is contributed to operating, maintenance operations, and logistics demands, which closely relate to system availability. Furthermore, many studies notice that the life cycle cost remarkably leans against the decision made during the initial system establishment epoch. The work in [3] also notes that system availability is an idea tightly associated with reliability, and it indicates the level of evaluating the operation of repairable systems. Regarding the respect of the total life cycle of system operations, availability optimization employed in the preliminary system design and establishment period acts decisively in influencing system reliability, system maintenance designing, logistics demands, and corresponding life cycle costs within the system-designed operating period.

By observing recent existing literature, one of the main trends to discuss the system availability optimization problem is to derive system availability optimization models to distribute the availability for all subsystems or components of each subsystem in order to minimize or maximize the objective function, depending on the system availability requirements or other resource design constraints. These are normally defined as availability allocation problems [3,4,5,6,7,8,9]. In general, the availability allocation problems can be extended from the typical reliability–redundancy allocation problems evolved for decades [10,11,12,13,14,15,16,17,18,19,20,21].

The availability optimization problem developed in this paper needs to concurrently manage one category of discrete decision variables, the number of redundancy components of all subsystems, and two categories of continuous decision variables, including the repair rates and failure rates of the components of all the subsystems. This can be classified as a non-deterministic polynomial-time hard problem [3,7,8,14,18,21,22,23]; therefore, it is very critical to select the appropriate methods to resolve the availability optimization problem proposed in this study. Furthermore, Kayedpour et al. [7] note that most studies of reliability optimization problems and availability optimization problems assume a predetermined configuration strategy, such as parallel, standby, k-out-of-n, or other specific configuration subsystems, and these simplified assumptions are not in line with real-world situations. Therefore, this study intends to propose the decision support design system of the availability optimization problem for repairable n-stage mixed systems, in which different combinations of subsystems, such as parallel, standby, and k-out-of-n, are connected alternatively in series configuration.

Kuo et al. [14] develop an exact approach incorporating the Lagrange multiplier method and the branch-and-bound technique to solve constrained reliability optimization problems, but it needs to take more CPU running time. Recently, heuristic approaches or meta-heuristic approaches have been adopted smoothly to deal with various reliability redundancy allocation problems and availability optimization problems. Regarding reliability redundancy allocation problems, Peiravi et al. [18] formulate a reliability redundancy allocation problem and solves it by applying an efficient genetic algorithm; Ravi et al. [19] propose an improved non-equilibrium simulated annealing method to obtain the optimal amount of redundancy to maximize system reliability or minimize the system cost for two types of intricate systems; Chambari et al. [10] propose an efficient simulated annealing algorithm to select the redundancy strategy, component, and redundancy level for each subsystem in order to maximize the system reliability; and Kim and Jang [12] propose a redundancy allocation problem of complex systems with heterogeneous components and develops a modified tabu search method to conduct an adaptive procedure to produce efficient neighborhood solutions. With regard to availability optimization problems, Elegbede and Adjallah [23] develop a genetic algorithm-based methodology to solve the multi-objective availability optimization problem with respect to parallel–series systems; Martorell et al. [24] propose both the single-objective genetic algorithm and the multi-objective genetic algorithm to handle the multi-objective availability optimization problem; and Chiang and Chen [4] propose the simulated annealing-based multi-objective genetic algorithm to resolve the availability optimization problem concerning parallel-series systems. The work in [3] proposes the knowledge-based decision support system for the genetic algorithm optimization model to implement availability optimization concerning series–parallel systems; Quzineb et al. [17] propose a Tabu search method to resolve the redundancy allocation problem for series–parallel systems in order to determine the minimal cost system configuration under availability-related constraints; Kayedpour et al. [7] propose Markov processes and a non-dominated sorting genetic algorithm II (NSGA-II) to resolve the reliability design problem concerning instantaneous availability, repairable components, and the choice of configuration strategies; and Samanta and Basu [21] develop a multi-objective particle swarm optimization to solve a multi-objective availability optimization problem in a series-parallel system with both repairable and non-repairable components.

According to the above-surveyed literature, the genetic algorithm seems be widely accepted in the area of reliability optimization problems and availability optimization problems to deal with diverse continuous and discrete decision variables and obtain the optimal solution [3,4,7,9,18,23,24,25]. Furthermore, other meta-heuristic approaches, including the simulated annealing method [4,10], non-equilibrium simulated annealing method [8,9], and tabu search approach [12,13,17], are also applied to find the optimal amount of redundancy for reliability and availability optimization problems. In addition, some combined methods have been evolved by various combinations of search methods and heuristic approaches to solve reliability optimization problems or availability optimization problems [4,8,11,15,26,27]. These combined methods are demonstrated to be more effective when applying two proper heuristic or meta-heuristic methods to deal with discrete decision variables and continuous decision variables, respectively.

Because a genetic algorithm uses the chromosome mechanism to evaluate the related fitness function, it is more appropriate to be adopted to deal with continuous decision variables in the typical reliability allocation problem and availability optimization problem. In addition, observing the above-surveyed literature, the simulated annealing approach, non-equilibrium simulated annealing approach, tabu search approach, and redundancy allocation heuristic method are demonstrated to perform well when dealing with discrete decision variables in the typical reliability allocation problem and availability optimization problem. Because system configuration constraints will restrict the number of redundant components when there is a smaller ceiling, the enumeration method and the modified redundancy allocation heuristic method developed in the proposed availability optimization decision support design system should be appropriate to handle discrete decision variables and the number of redundant components of all subsystems.

Therefore, the proposed availability optimization decision support design system will combine an enumeration method, tabu search, simulated annealing method, non-equilibrium simulated annealing method, and modified redundancy allocation heuristic method with a modified genetic algorithm to solve the related availability optimization problems and further determine the appropriate system configuration design. These five proposed combined methods are abbreviated as ENUM–MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA in this study. In these five proposed combined methods, the modified genetic algorithm is adopted to take care of continuous decision variables comprising the component repair rates and component failure rates of all subsystems; the enumeration method, simulated annealing approach, non-equilibrium simulated annealing approach, tabu search approach, and modified redundancy allocation heuristic method are used to deal with the discrete decision variables of the redundancy number of components for all subsystems.

In addition, most of the literature surveyed above only considers a single system configuration to discuss the availability optimization problem. In contrast, this study develops the availability optimization decision support design system to apply five proposed combined methods to solve the system availability optimization problem for a parallel-series system, n-stage standby system, n-stage k-out-of-q system, and n-stage mixed system and further determine the appropriate system configuration design. The main difference between this study and those in the above-surveyed literature is summarized as

Table 1. The main difference between this study and the above-surveyed literature.

	This Study	10	4	23	5	3	7	12	14	15	24	17	18	19	21
Reliability Optimization		V	V					V	V	V			V	V
Availability Optimization	V			V		V	V				V	V			V
Exact approach									V	V
ENUM	V
GA	V		V	V		V	V				V		V
TS	V							V				V
SA	V	V	V
NESA	V													V
PSO															V
Heuristic	V							V		V
Series	V	V
Parallel		V					V
Parallel-series	V	V	V	V					V	V
Series-parallel						V						V			V
n-stage standby system	V						V						V
n-stage k-out-of-q system	V	V
n-stage mixed system	V
Complex system							V	V			V			V

, based on problem type, solving method, and system configuration.

2. Problem Description and Mathematical Model Formulation

This paper will develop the availability optimization decision support design system for repairable n-stage mixed systems, in which different combinations of subsystems include parallel, standby, and k-out-of-q connected in series configuration. The proposed decision support design system will first develop the availability optimization mathematic model to handle three decision variables, including the amount of components ${{\displaystyle q}}_i$ of each subsystem, the component repair rate ${{\displaystyle \mu }}_i$ of each subsystem, and the component failure rate ${{\displaystyle f}}_i$ of each subsystem, and to obtain the optimal system availability design from various aspects in this proposed model. In this mathematic model, the system cost objective function is minimized by complying with various constraints, including system design configurations and system availability requirements.

Referring to the notation section in Appendix A, the template availability optimization mathematical model is delineated as follows:

$$Min{{\displaystyle C}}_s(\overline{Q},\overline{f},\overline{\mu })={\displaystyle \sum _{i=1}^n[{{\displaystyle \alpha }}_i}{({{\displaystyle f}}_i)}^{-{{\displaystyle \beta }}_i}+{{\displaystyle \mu }}_i*{{\displaystyle rc}}_i][q_i+\exp ({{\displaystyle q}}_i/4)]$$

(1)

s.t.

$${\displaystyle \sum _{i=1}^n{w}_i{q}_i\ast \exp ({{\displaystyle q}}_i/4)\le W}$$

(2)

$${\displaystyle \sum _{i=1}^{n}w{v}_i{(q_i)}^2\le WV}$$

(3)

$$A_S(\overline{Q},\overline{f},\overline{\mu })\ge {{\displaystyle A}}_S^L$$

(4)

In the above mathematical formulation Equation (1), the system cost objective function comprises the reliability-related cost, repair-related cost, and component redundancy-related cost. They are depicted in detail as follows:

• The component reliability cost rate: ${\alpha }_i{(f_i)}^{-{\beta }_i}$
• The component repair cost rate: ${\mu }_i*{rc}_i$
• The component redundancy-related cost: $q_i+\exp (q_i/4)$
• Equation (2) is the generalized formula of the system design configuration constraint of weight.
• Equation (3) is the generalized formula of the system design configuration constraint of the product of weight and volume.
• Equation (4) is the generalized formula of the system availability requirement constraint.

As this proposed availability optimization mathematical model considers an n-stage mixed system with different combinations of subsystems, including parallel, standby, and k-out-of-q connected in series configuration, the component availability formula, subsystem availability formula, and system availability formula in Equation (4) are formulated below:

• Component availability

The components comprised in the n-stage mixed system are repairable, and the point availability $a(t)$ and unavailability $\overline{a}(t)$ can be denoted as the probability that the state is normal or defective at time t. The initial states comply with three conditions, including $a(0)$ = 1, $\overline{a}(0)$ = 0, and $a(0)$ + $\overline{a}(0)$ = 1, hence the differential equation of the point availability is acquired by:

$$a(t+\mathsf{\Delta }t)=a(t)-f\mathsf{\Delta }ta(t)+\mu \mathsf{\Delta }t\overline{a}(t)$$

(5)

Rearranging the terms in Equation (5), it is further derived as

$$\frac{a(t+\mathsf{\Delta }t)-a(t)}{\mathsf{\Delta }t}=-(f+\mu )a(t)+\mu$$

(6)

The left of Equation (6) is the derivative on time. It can be reformulated as the later differential equation,

$$\frac{d}{dt}a(t)=-(f+\mu )a(t)+\mu$$

(7)

Both sides of Equation (7) multiply the integrating factor of ${{\displaystyle e}}^{f+\mu }$ and further obtain

$${{\displaystyle e}}^{(f+\mu )t}\frac{d}{dt}a(t)+(f+\mu ){{\displaystyle e}}^{(f+\mu )t}a(t)=\mu {{\displaystyle e}}^{(f+\mu )t}$$

(8)

According to the derivative of ${{\displaystyle e}}^{(f+\mu )t}a(t)$, the equation becomes

$$\frac{d}{dt}({{\displaystyle e}}^{(f+\mu )t}a(t))=\mu {{\displaystyle e}}^{(f+\mu )t}$$

(9)

Synthesizing both sides of Equation (9) leads to ${{\displaystyle e}}^{(f+\mu )t}a(t)={\displaystyle \int \mu {{\displaystyle e}}^{(f+\mu )t}d}t+D$, and in addition, $a(t)=\frac{\mu }{f+\mu }+D{{\displaystyle e}}^{-(f+\mu )t}$. Since $a(0)=1$ holds, it consequently yields the point availability

$$a(t)=\frac{\mu }{f+\mu }+\frac{f}{f+\mu }{{\displaystyle e}}^{-(f+\mu )t}$$

(10)

In the following, the component interval availability is defined as follows:

$${{\displaystyle a}}^{*}(T)=\frac{1}{T}{\displaystyle {\int }_0^T(\frac{\mu }{f+\mu }+\frac{f}{f+\mu }{{\displaystyle e}}^{-(f+\mu )t})dt}=\frac{\mu }{f+\mu }+\frac{f}{{(f+\mu )}^{2}T}(1-{{\displaystyle e}}^{-(f+\mu )T})$$

(11)

Finally, the component asymptotic inherent availability is derived by assuming T infinity for component interval availability, which leads to

$${{\displaystyle a}}^{*}(\infty )=\frac{\mu }{f+\mu }$$

(12)

2.

System availability

The n-stage mixed system applied in the proposed availability optimization mathematical model comprises different combinations of subsystems, including parallel, standby, and k-out-of-n connected in series configuration. Next, parallel subsystem availability, standby subsystem availability, and k-out-of-q subsystem will be derived in detail based on the component interval inherent availability.

(1) 

Parallel subsystem availability and parallel-series system availability

For each parallel subsystem i, it is unavailable only when all ${{\displaystyle q}}_i$ components are unavailable. Therefore, the parallel subsystem availability can be deduced as follows:

$${{\displaystyle A}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)=[1-{(1-\frac{{{\displaystyle \mu }}_i}{{{\displaystyle f}}_i+{{\displaystyle \mu }}_i})}^{{{\displaystyle q}}_i}]$$

(13)

Parallel-series system configuration means that all parallel subsystems are connected in series. The parallel-series system availability can be directly formulated as

$$A_S(\overline{Q},\overline{f},\overline{\mu })={\displaystyle \coprod _{i=1}^n{{\displaystyle A}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)}={\displaystyle \coprod _{i=1}^n[1-{(1-\frac{{{\displaystyle \mu }}_i}{{{\displaystyle f}}_i+{{\displaystyle \mu }}_i})}^{{{\displaystyle q}}_i}]}$$

(14)

(2) 

Standby subsystem availability and n-stage standby system availability

The standby subsystem i consists of ${{\displaystyle q}}_i$ components, in which the primary component is in operation and the other ${{\displaystyle q}}_i-1$ components are backup units. It is assumed that switching failures and failures in the standby units can be ignored. Furthermore, the state transition mechanism of the standby subsystem can be described as follows:

In the state transition mechanism, state j, which ranges from 1 to ${{\displaystyle q}}_i$, means that component j is in operation because previous j − 1 components have failed already, and the other components are still in backup condition. State F means all components are failed in this standby subsystem.

The related Markov differential equations of the state probabilities for the standby subsystem can be derived as follows:

$$\frac{d}{dt}{{\displaystyle P}}_1(t)=-f{{\displaystyle P}}_1(t)+\mu {{\displaystyle P}}_2(t)$$

(15)

$$\frac{d}{dt}{{\displaystyle P}}_j(t)=f{{\displaystyle P}}_{j-1}(t)+\mu {{\displaystyle P}}_{j+1}(t)-(f+\mu ){{\displaystyle P}}_j(t)\mathrm{for}j=1,2,\cdots ,{{\displaystyle q}}_i-1$$

(16)

$$\frac{d}{dt}{{\displaystyle P}}_{{{\displaystyle q}}_i}(t)=f{{\displaystyle P}}_{{{\displaystyle q}}_i-1}(t)+\mu {{\displaystyle P}}_F(t)-(f+\mu ){{\displaystyle P}}_{{{\displaystyle q}}_i}(t)$$

(17)

$$\frac{d}{dt}{{\displaystyle P}}_F(t)=f{{\displaystyle P}}_{{{\displaystyle q}}_i}(t)-\mu {{\displaystyle P}}_F(t)$$

(18)

Since this research focuses the steady-state availability, $A(\infty )$. Note that the derivative on the left-hand side of the above simultaneous equations vanishes when $t$ goes to infinity. The above simultaneous equations can be further derived as follows:

$$-f{{\displaystyle P}}_1(\infty )+\mu {{\displaystyle P}}_2(\infty )=0$$

(19)

$$f{{\displaystyle P}}_{j-1}(\infty )+\mu {{\displaystyle P}}_{j+1}(\infty )-(f+\mu ){{\displaystyle P}}_j(\infty )=0\mathrm{for}j=1,2,\cdots ,{{\displaystyle q}}_i-1$$

(20)

$$f{{\displaystyle P}}_{{{\displaystyle q}}_i-1}(\infty )+\mu {{\displaystyle P}}_F(\infty )-(f+\mu ){{\displaystyle P}}_{{{\displaystyle q}}_i}(\infty )=0$$

(21)

$$f{{\displaystyle P}}_{{{\displaystyle q}}_i}(\infty )-\mu {{\displaystyle P}}_F(\infty )=0$$

(22)

Furthermore, Equation (19) can be rearranged to ${{\displaystyle P}}_2(\infty )=\frac{f}{\mu }{{\displaystyle P}}_1(\infty )$; Equation (20) can be rewritten as ${{\displaystyle P}}_j(\infty )={{\displaystyle (\frac{f}{\mu })}}^{j-1}{{\displaystyle P}}_1(\infty )$ for $j=1,2,\cdots ,{{\displaystyle q}}_i-1$. By induction, Equation (21) can be rewritten as ${{\displaystyle P}}_{{{\displaystyle q}}_i}(\infty )={{\displaystyle (\frac{f}{\mu })}}^{{{\displaystyle q}}_i-1}{{\displaystyle P}}_1(\infty )$, and Equation (22) can also be transformed to ${{\displaystyle P}}_F(\infty )={{\displaystyle (\frac{f}{\mu })}}^{{{\displaystyle q}}_i}{{\displaystyle P}}_1(\infty )$.

According to the basic rule of probability, all of the probabilities must sum to one:

$${\displaystyle \sum _j{{\displaystyle P}}_j(\infty )}=1$$

(23)

By applying the above probability property of Equation (23), it further yields the following form:

$[{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle q}}_i}}{(\frac{f_{}}{{\mu }_{}})}^j]*{{\displaystyle P}}_1(\infty )=1}$. Accordingly, we can obtain

$${{\displaystyle P}}_1(\infty )=[{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle q}}_i}}{(\frac{f}{\mu })}^j{]}^{-1}}$$

(24)

Finally, we can derive all state probabilities as follows:

$${{\displaystyle P}}_j(\infty )=[{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle q}}_i}}{(\frac{f}{\mu })}^j{]}^{-1}}{(\frac{f}{\mu })}^{j-1}\mathrm{for}j=1,2,\cdots ,{{\displaystyle q}}_i$$

(25)

$${{\displaystyle P}}_F(\infty )=[{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle q}}_i}}{(\frac{f}{\mu })}^j{]}^{-1}}{(\frac{f}{\mu })}^{{{\displaystyle q}}_i}$$

(26)

The steady-state availability of the system can be given by

$A(\infty )={\displaystyle \sum _{j\in O}{{\displaystyle P}}_j(\infty )=1-{{\displaystyle P}}_F(\infty )}$, where the sum over the operational states

Accordingly, the steady-state availability of the standby subsystem i can be yielded as follows:

$${{\displaystyle \mathrm{A}}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)=1-[{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle q}}_i}}{(\frac{{{\displaystyle f}}_i}{{{\displaystyle \mu }}_i})}^j{]}^{-1}}{(\frac{{{\displaystyle f}}_i}{{{\displaystyle \mu }}_i})}^{{{\displaystyle q}}_i}$$

(27)

The n-stage standby system configuration indicates that all subsystems are standby configurations and connected in series. The n-stage standby system availability can be directly formulated as

$$A_S(\overline{Q},\overline{f},\overline{\mu })={\displaystyle \coprod _{i=1}^n{{\displaystyle \mathrm{A}}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)}={\displaystyle \coprod _{i=1}^n[1-[{\displaystyle \sum _{j=0}^{{}_{{}_{q_i}}}{(\frac{f_i}{{\mu }_i})}^j{]}^{-1}{(\frac{f_i}{{\mu }_i})}^{q_i}}]}$$

(28)

(3) 

$k_i$-out-of-$q_i$ subsystem availability and n-stage $k_i$-out-of-$q_i$ system availability

For each ${{\displaystyle k}}_i$-out-of-${{\displaystyle q}}_i$ subsystem i, at least ${{\displaystyle k}}_i$ components are operational within all ${{\displaystyle q}}_i$ components, and then the subsystem is operational. Therefore, the ${{\displaystyle k}}_i$-out-of-${{\displaystyle q}}_i$ subsystem availability can lead to the following forms:

$${{\displaystyle \mathrm{A}}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)=1-{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle k}}_i-1}}\left(\frac{{{\displaystyle q}}_i!}{j!({{\displaystyle q}}_i-j)!}\right){(\frac{{{\displaystyle \mu }}_i}{{{\displaystyle \mu }}_i+{{\displaystyle f}}_i})}^j{(1-\frac{{{\displaystyle \mu }}_i}{{{\displaystyle \mu }}_i+{{\displaystyle f}}_i})}^{{{\displaystyle q}}_i-j}}$$

(29)

The n-stage ${{\displaystyle k}}_i$-out-of-${{\displaystyle q}}_i$ system configuration represents that all subsystems are linked in series, and these subsystems are all of ${{\displaystyle k}}_i$-out-of-${{\displaystyle q}}_i$. The n-stage ${{\displaystyle k}}_i$-out-of-${{\displaystyle q}}_i$ system availability can be directly formulated as

$$A_S(\overline{Q},\overline{f},\overline{\mu })={\displaystyle \coprod _{i=1}^n{{\displaystyle \mathrm{A}}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)}=\underset{i=1}{\overset{n}{{\displaystyle \mathsf{\Pi }}}}\left[1-{\displaystyle \sum _{j=0}^{{}_{{{\displaystyle k}}_i-1}}\left(\frac{{{\displaystyle q}}_i!}{j!({{\displaystyle q}}_i-j)!}\right){(\frac{{{\displaystyle \mu }}_i}{{{\displaystyle \mu }}_i+{{\displaystyle f}}_i})}^j{(1-\frac{{{\displaystyle \mu }}_i}{{{\displaystyle \mu }}_i+{{\displaystyle f}}_i})}^{{{\displaystyle q}}_i-j}}\right]$$

(30)

Several assumptions are made to set up the problem as follows:

• Suppose all components have the properties of an electrical component (i.e., R(t) = $e^{-ft}$).
• Suppose the life time of system t = 1000.
• Assume all subsystems are connected in series.
• Assume only one single failure mode.
• Suppose all components used in the same subsystem have the same failure rate.
• Assume all components in the same subsystem have the same repair rate.
• Suppose the component availability adopted in this paper is inherent availability: $a_j=\frac{{\mu }_j}{{{\displaystyle \mu }}_j+{{\displaystyle f}}_j}$
• Assume the repairable n-stage mixed system comprises different combinations of subsystems, including parallel, standby, and k-out-of-q.
• Assume all subsystems and all components comprised in each subsystem are S-independent.

3. Five Proposed Availability Optimization Combined Methods

The initiated availability optimization decision support design system develops five combined methods, including ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA, to resolve the system availability optimization problem with two stages. In the major stage, the enumeration method, the tabu search approach, simulated annealing approach, non-equilibrium simulated annealing approach, and modified redundancy allocation heuristic method are developed to search for the appropriate number of redundant components for all subsystems simultaneously. In the second stage, regarding the number of redundant components selected in the first stage, the modified genetic algorithm is then conduced to obtain the optimal repair rates and failure rates of the components for all subsystems at the same time. The comprehensive procedures for the five initiated combined methods are depicted as the following:

3.1. ENUM-MGA Combined Method

Step 1:

 

(1)

Assume the values of parameters including ${{\displaystyle \alpha }}_i$, ${{\displaystyle \beta }}_i$, ${{\displaystyle rc}}_i$, $p_i$, P, ${{\displaystyle w}}_i$, W

(2)

Assume the low bounds of quantity of components of all subsystems ${{\displaystyle \overline{Q}}}^L=\left[11111\right]$

(3)

According to specification limit on weight-volume combination of product $W{V}_{}$ and specification limit on weight of product ${{\displaystyle W}}_{}$, calculate the upper bounds of quantity of components of all subsystems ${{\displaystyle \overline{Q}}}^u$.

(4)

Assume the initial base point ${{\displaystyle \overline{Q}}}^0=[11111]$.

(5)

According to the initial base point ${{\displaystyle \overline{Q}}}^0=[11111]$, assume the base point ${{\displaystyle \overline{Q}}}^b={{\displaystyle \overline{Q}}}^0$

(6)

Initiate ${{\displaystyle C}}_S^{*}$ equal to a extremely high cost.

Step 2:

Start the enumeration multi-layer for loop from subsystem 1 to subsystem 5 as the following pseudocode and steps:

for ${{\displaystyle \overline{Q}}}^b$[1] = 1 to ${{\displaystyle \overline{Q}}}^u$[1]

    for ${{\displaystyle \overline{Q}}}^b$[2] = 1 to ${{\displaystyle \overline{Q}}}^u$[2]

         for ${{\displaystyle \overline{Q}}}^b$[3] = 1 to ${{\displaystyle \overline{Q}}}^u$[3]

            for ${{\displaystyle \overline{Q}}}^b$[4] = 1 to ${{\displaystyle \overline{Q}}}^u$[4]

                for ${{\displaystyle \overline{Q}}}^b$[5] = 1 to ${{\displaystyle \overline{Q}}}^u$[5]

Step 3:

Generate the basic points, ${\overline{Q}}^b$. Check if ${\overline{Q}}^b$ meet constraints of system weight, system volume, system availability requirements. If yes, proceed to Step 4, or skip this iteration and continue to next iteration.

Step 4:

Pass ${\overline{Q}}^b$ into GA module to obtain the optimal repair rates and the optimal failure rates of components for all subsystems, ${\overline{f}}^{*}\left({{\displaystyle \overline{Q}}}^b\right)$, ${\overline{\mu }}^{*}\left({{\displaystyle \overline{Q}}}^b\right)$, and calculate system total cost ${{\displaystyle C}}_s({\overline{Q}}^b),{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))$.

Step 5:

If ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b)),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b)<{{\displaystyle C}}_s^{*}$, then ${{\displaystyle C}}_s^{*}={{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b)),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b)$, ${\overline{Q}}^{*}$ = ${\overline{Q}}^b$, ${\overline{f}}^{*}\left({{\displaystyle \overline{Q}}}^{*}\right)={\overline{f}}^{*}\left({{\displaystyle \overline{Q}}}^b\right)$, ${\overline{\mu }}^{*}\left({{\displaystyle \overline{Q}}}^{*}\right)={\overline{\mu }}^{*}\left({{\displaystyle \overline{Q}}}^b\right)$.

                end

            end

        end

    end

end

Step 6:

Stop. Set down ${{\displaystyle \overline{Q}}}^{*}$, ${{\displaystyle \overline{f}}}^{*}({{\displaystyle \overline{Q}}}^{*})$, ${{\displaystyle \overline{\mu }}}^{*}({{\displaystyle \overline{Q}}}^{*})$, ${{\displaystyle C}}_s^{*}({\overline{Q}}^{*},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{*}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{*}))={{\displaystyle C}}_s^{*}$

3.2. TS-MGA Combined Method

Step 1:

 

(1)

Initiate the values of ${{\displaystyle \alpha }}_i$, ${{\displaystyle \beta }}_i$, ${{\displaystyle rc}}_i$, $p_i$, P, ${{\displaystyle w}}_j$, W,

(2)

Assume the initial amount of iteration TabuIte equal to 1. Define the appropriate threshold of iteration TabuMaxIte equal to 200.

(3)

Assume the initial base point ${{\displaystyle \overline{Q}}}^0=[11111]$.

(4)

Initiate ${{\displaystyle C}}_S^{*}$ equal to an extremely large cost.

(5)

According to the initial base point ${{\displaystyle \overline{Q}}}^0=[11111]$, assume the base point ${{\displaystyle \overline{Q}}}^b={{\displaystyle \overline{Q}}}^0$

(6)

Initiate Tabulist = $\left[\right[11111][11111][11111][11111][11111][11111][11111]]$

(7)

Assume the aspiration criterion as follows: if the related simulated result is less than the aspiration cost asp_cost, then the aspiration criterion holds.

Step 2:

Start Tabu loop from TabuIte = 1 to TabuMaxIte.

Step 3:

Generate the neighboring candidate points, $N({\overline{Q}}^b)$. Check if $N({\overline{Q}}^b)$ meet constraints of system weight, system volume, system availability requirements. If yes, proceed to Step 4, or repeat Step 3.

Step 4:

Pass $N({\overline{Q}}^b)$ into GA module to obtain the optimal repair rates and the optimal failure rates of components for all subsystems, ${\overline{f}}^{*}(N{{\displaystyle \overline{(Q}}}^b\left)\right)$, ${\overline{\mu }}^{*}({{\displaystyle \overline{(Q}}}^b\left)\right)$, and calculate system total costs for all neighboring points in the candidate list, ${{\displaystyle C}}_s(N({\overline{Q}}^b),{{\displaystyle \overline{f}}}^{*}(N({\overline{Q}}^b)),{{\displaystyle \overline{\mu }}}^{*}(N({\overline{Q}}^b)))$.

Step 5:

Pick the candidate point $\mathrm{N}({{\displaystyle \overline{Q}}}^b\left)\right[1]$ associated with the lowest system total cost ${{\displaystyle C}}_s(\mathrm{N}({\overline{Q}}^b\left)\right[1],{{\displaystyle \overline{f}}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[1]),{{\displaystyle \underset{¯}{\overline{\mu }}}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[1]))$ during the candidate list.

Step 6:

If ${{\displaystyle C}}_s(\mathrm{N}({\overline{Q}}^b\left)\right[1],{{\displaystyle \overline{f}}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[1]),{{\displaystyle \overline{\mu }}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[1]))<{{\displaystyle C}}_s^{*}$, then let ${{\displaystyle \overline{Q}}}^{*}$ = $\mathrm{N}({{\displaystyle \overline{Q}}}^b)$[1], and ${{\displaystyle C}}_s^{*}={{\displaystyle C}}_s(\mathrm{N}({\overline{Q}}^b),{{\displaystyle \overline{f}}}^{*}(N({\overline{Q}}^b)[1]),{{\displaystyle \overline{\mu }}}^{*}(N({\overline{Q}}^b)[1]))$[1]

Step 7:

If all neighboring candidate points, $N({\overline{Q}}^b)$ are not in the Tabu list, update the base point ${{\displaystyle \overline{Q}}}^b=\mathrm{N}({{\displaystyle \overline{Q}}}^b\left)\right[1]$, the system cost ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))={{\displaystyle C}}_s(\mathrm{N}({\overline{Q}}^b\left)\right[1],{{\displaystyle \overline{f}}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[1]),{{\displaystyle \overline{\mu }}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[1]))$ and proceed Step 8, or directly head to Step 9.

Step 8:

Implement the Tabu move, update the Tabulist as the following pseudocode, and directly proceed to Step 10:

       for k=length(Tabulist):2

            Tabulist[k]=Tabulist[k-1]

     end

     Tabulist[1] = $\mathrm{N}({{\displaystyle \overline{Q}}}^b\left)\right[1]$

Step 9:

By conforming the defined aspiration criterion, implement the Tabu move, update the Tabulist as the following pseudocode:

      for k = 1 to length($\mathrm{N}({{\displaystyle \overline{Q}}}^b)$)

      if $\mathrm{N}({{\displaystyle \overline{Q}}}^b\left)\right[\mathrm{k}]$ in Tabulist and ${{\displaystyle C}}_s(\mathrm{N}({\overline{Q}}^b\left)\right[k],{{\displaystyle \overline{f}}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[k]),{{\displaystyle \overline{\mu }}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[k]))$<

          asp_cost

                for kk = length(Tabulist):2

                    Tabulist[kk] = Tabulist[kk-1]

               end

               Tabulist[1] = $\mathrm{N}({{\displaystyle \overline{Q}}}^b\left)\right[\mathrm{k}]$

               ${{\displaystyle \overline{Q}}}^b=\mathrm{N}({{\displaystyle \overline{Q}}}^b\left)\right[\mathrm{k}]$

               ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))={{\displaystyle C}}_s(\mathrm{N}({\overline{Q}}^b\left)\right[k],{{\displaystyle \overline{f}}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[k]),{{\displaystyle \overline{\mu }}}^{*}(\mathrm{N}({\overline{Q}}^b\left)\right[k]))$

               break

          end

     end

Step 10:

If Ite < MaxIte, return to Step 2, or Stop. Set down ${{\displaystyle \overline{Q}}}^{*}$, ${{\displaystyle \overline{f}}}^{*}\left({{\displaystyle \overline{Q}}}^{*}\right)$, ${{\displaystyle \overline{\mu }}}^{*}\left({{\displaystyle \overline{Q}}}^{*}\right)$, ${{\displaystyle \mathrm{C}}}_{\mathrm{s}}^{*}({\overline{Q}}^{*},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{*}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{*}))$

3.3. SA-MGA and NESA-MGA Combined Method

Step 1:

 

(1)

Assume the values of parameters including ${{\displaystyle \alpha }}_i$, ${{\displaystyle \beta }}_i$, ${{\displaystyle rc}}_i$, $p_i$, P, ${{\displaystyle w}}_i$, W

(2)

Assume the low bounds of quantity of components of all subsystems ${{\displaystyle \overline{Q}}}^L=\left[11111\right]$

(3)

According to specification limit on weight-volume combination of product $W{V}_{}$ and specification limit on weight of product ${{\displaystyle W}}_{}$, calculate the upper bounds of quantity of components of all subsystems ${{\displaystyle \overline{Q}}}^u$

(4)

Begin with the initial amount of outloop iteration Outloop = 0. Assume the appropriate maximum threshold of outloop iteration MaxOutloop.

(5)

Assume the appropriate value of temperature parameter ${{\displaystyle T}}_{outloop}$

(6)

Assume the initial amount of inloop iteration Inloop equal to 0. Determine the appropriate maximum threshold of Inloop iteration MaxInloop.

(7)

Initiate ${{\displaystyle C}}_S^{*}$ equal to an extremely high cost.

Step 2:

Start SA and NESA outloop from Outloop = 1 to MaxOutloop.

Step 3:

Start the initial base point of quantity of components of all subsystems ${{\displaystyle \overline{Q}}}^0$, randomly ranging from low bound ${{\displaystyle \overline{Q}}}^L$ to upper bound ${{\displaystyle \overline{Q}}}^u$.

Step 4:

Verify if the initial base point of ${{\displaystyle \overline{Q}}}^0$ fulfills constraints of system weight-volume combination, system weight, system availability requirements. If it is true, proceed to Step 5, else, return to Step 3.

Step 5:

Assume the base point ${{\displaystyle \overline{Q}}}^b$ equal to ${{\displaystyle \overline{Q}}}^0$. Enter ${{\displaystyle Q}}^b$ to GA module to further obtain the optimal repair rates and the optimal failure rates of components for all subsystems,${\overline{f}}^{*}\left({{\displaystyle \overline{Q}}}^b\right)$, ${\overline{\mu }}^{*}{{\displaystyle \overline{(Q}}}^b)$, and compute system total cost ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))$.

Step 6:

Start SA and NESA Inloop from Inloop = 1 to MaxInloop

Step 7:

Randomly pick up the neighboring point by adopting the following procedures:

(1)

${{\displaystyle \overline{Q}}}^{nb}={{\displaystyle \overline{Q}}}^b+({{\displaystyle \overline{Q}}}^U-{{\displaystyle \overline{Q}}}^b)*\left(\mathrm{rand}\right(0,1)-0.5)$

(2)

If ${{\displaystyle \overline{Q}}}^{nb}$ ranges from ${{\displaystyle \overline{Q}}}^L$ to ${{\displaystyle \overline{Q}}}^u$, proceed directly to Step 8, otherwise, repeat Step 7.

Step 8:

Examine if the neighboring point ${\overline{Q}}^{nb}$ is in line with constraints of system weight-volume combination, system weight, system availability requirements. If it holds, proceed to Step 9, or return to Step 7.

Step 9:

Enter ${\overline{Q}}^{nb}$ to GA module to acquire the corresponding optimal repair rates and the corresponding optimal failure rates components for all subsystems, ${\overline{f}}^{*}{{\displaystyle \overline{(Q}}}^{nb})$, ${{\displaystyle \overline{\mu }}}^{*}{{\displaystyle \overline{(Q}}}^{nb})$, and compute related system total cost ${{\displaystyle C}}_s({\overline{Q}}^{nb},{{\displaystyle f}}^{*}({\overline{Q}}^{nb}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{nb}))$.

Step 10:

Compute cost difference between system total cost of the neighboring point ${\overline{Q}}^{nb}$ and the base point ${{\displaystyle \overline{Q}}}^b$, ${{\displaystyle C}}_d={{\displaystyle C}}_s({\overline{Q}}^{nb},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{nb}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{nb}))-{{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))$.

Step 11:

Compute the accepting probability Pa = $\exp (-{{\displaystyle C}}_d/({{\displaystyle T}}_{outloop}))$.

Step 12:

Create a random number rand from 0 to 1. If Pa is great than rand, directly proceed to Step 13, or return to Step 7 to pick up the other neighboring point.

Step 13:

 

SA-MGA module:

Let ${{\displaystyle \overline{Q}}}^b$ = ${{\displaystyle \overline{Q}}}^{nb}$, and ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))={{\displaystyle C}}_s({\overline{Q}}^{nb},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{nb}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{nb}))$. Set Inloop = Inloop+1.

NESA-MGA module:

(1)

Let ${{\displaystyle \overline{Q}}}^b$ = ${{\displaystyle \overline{Q}}}^{nb}$, and ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))={{\displaystyle C}}_s({\overline{Q}}^{nb},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{nb}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{nb}))$.

Set Inloop = Inloop + 1.

(2)

If, ${{\displaystyle C}}_d$ < 0, proceed to Step 15.

Step 14:

If Inloop > MaxInloop, proceed to Step 15, or return to Step 6.

Step 15:

Assume ${{\displaystyle \overline{Q}}}_{outloop}$ = ${{\displaystyle \overline{Q}}}^b$, Outloop = Outloop + 1. ${{\displaystyle T}}_{outloop}=(1-\theta ){{\displaystyle T}}_{outloop}$ where $\theta$ is the temperature adjustment parameter.

Step 16:

If ${{\displaystyle C}}_s({\overline{Q}}^b)<{{\displaystyle C}}_S^{*}$, proceed to Step 17, or head to Step 18.

Step 17:

${{\displaystyle \overline{Q}}}^{*}$ = ${{\displaystyle \overline{Q}}}^b$, ${{\displaystyle C}}_S^{*}={{\displaystyle C}}_S({{\displaystyle \overline{Q}}}^b,{{\displaystyle \overline{f}}}^{*}({{\displaystyle \overline{Q}}}^b),{{\displaystyle \overline{\mu }}}^{*}({{\displaystyle \overline{Q}}}^b))$

Step 18:

If Outloop > MaxOutloop, proceed to Step 19, or return to Step 2.

Step 19:

Stop. Set down ${{\displaystyle \overline{Q}}}^{*}$, ${{\displaystyle \overline{f}}}^{*}({{\displaystyle \overline{Q}}}^{*})$, ${{\displaystyle \overline{\mu }}}^{*}({{\displaystyle \overline{Q}}}^{*})$, ${{\displaystyle C}}_s^{*}({\overline{Q}}^{*},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{*}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{*}))$

3.4. MRAH-MGA Combined Method

Step 1:

Determine the appropriate values of ${{\displaystyle \alpha }}_i$, ${{\displaystyle \beta }}_i$, ${{\displaystyle mc}}_i$, $p_i$, P, ${{\displaystyle w}}_i$, W

Step 2:

Initiate the base point ${{\displaystyle \overline{Q}}}^0=\left[11111\right]$.

Step 3:

Decide the base point ${{\displaystyle \overline{Q}}}^b$ = ${{\displaystyle \overline{Q}}}^0$.

Step 4:

Input ${{\displaystyle \overline{Q}}}^b$ to MGA module to obtain the optimal component repair rates and the optimal component failure rates for all subsystems, ${\overline{f}}^{*}{{\displaystyle \overline{(Q}}}^b)$, $\overline{\mu }{{\displaystyle \overline{(Q}}}^b)$, and compute total system cost ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))$.

Step 5:

Define the sensitive function for all subsystems as follows:

${{\displaystyle S}}_i$ = $\frac{{{\displaystyle C}}_i({{\displaystyle q}}_i)-{{\displaystyle C}}_i({{\displaystyle q}}_i+1)}{\underset{k=1}{\overset{m}{{\displaystyle \Sigma }}}(\frac{{{\displaystyle g}}_{ki}({{\displaystyle q}}_i+1)-{{\displaystyle g}}_{ki}({{\displaystyle q}}_i)}{{{\displaystyle g}}_k^{remain}})}$ where ${{\displaystyle g}}_{ki}({{\displaystyle q}}_i)$ means the consumption quantity of system configuration constraint k when ${{\displaystyle q}}_i$ units of components are utilized in subsystem i, and ${{\displaystyle g}}_k^{remain}$ means the remaining quantity of system configuration design constraint k.

Step 6:

Create ${{\displaystyle S}}_{(i)}$ by ranking ${{\displaystyle S}}_i$ in descending order. ${{\displaystyle S}}_{\left(h\right)}$ = max {${{\displaystyle S}}_i$}. ${{\displaystyle \mathrm{S}}}_{\left(l\right)}$ = min {${{\displaystyle S}}_i$}

Step 7:

In case ${{\displaystyle S}}_{\left(h\right)}$ is greater than 0, proceed to Step 8, or proceed to Step 12.

Step 8:

Let ${{\displaystyle q}}_{\left(h\right)}$ = ${{\displaystyle q}}_{\left(h\right)}$ + 1

Step 9:

In case ${{\displaystyle g}}_k^{remain}$ is greater than 0, modify ${{\displaystyle \overline{Q}}}^b$ and proceed to Step 4, or proceed to Step 10

Step 10:

Let ${{\displaystyle q}}_{\left(h\right)}$ = ${{\displaystyle q}}_{\left(h\right)}$ − 1 and h = h − 1.

Step 11:

In case ${{\displaystyle S}}_{\left(h\right)}$ greater than 0, head to Step 8, or head to Step 12.

Step 12:

Assume ${{\displaystyle q}}_{\left(l\right)}$ = ${{\displaystyle q}}_{\left(l\right)}$ − 1, ${{\displaystyle q}}_{\left(h\right)}$ = ${{\displaystyle q}}_{\left(h\right)}$ + 1, and further modify ${{\displaystyle \overline{Q}}}^b$.

Step 13:

Put ${{\displaystyle \overline{Q}}}^b$ into GA module to obtain the associated optimal component repair rates and the optimal component failure rates for all subsystems, $\overline{\mu }{{\displaystyle \overline{(Q}}}^b)$, ${\overline{f}}^{*}{{\displaystyle \overline{(Q}}}^b)$, and compute system total cost ${{\displaystyle C}}_s({\overline{Q}}^b,{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^b),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^b))$.

Step 14:

Define the sensitive function for all subsystems as

${{\displaystyle S}}_i$ = $\frac{{{\displaystyle C}}_i({{\displaystyle q}}_i)-{{\displaystyle C}}_i({{\displaystyle q}}_i+1)}{\underset{k=1}{\overset{m}{{\displaystyle \Sigma }}}(\frac{{{\displaystyle g}}_{ki}({{\displaystyle q}}_i+1)-{{\displaystyle g}}_{ki}({{\displaystyle q}}_i)}{{{\displaystyle g}}_k^{\prime }})}$ where ${{\displaystyle g}}_{ki}({{\displaystyle q}}_i)$ means the resource consumption amount of system configuration constraint k when ${{\displaystyle q}}_i$ units of components are adopted in subsystem i, and ${{\displaystyle g}}_k^{\prime }$ means the remaining quantity of system configuration design constraint k.

Step 15:

Create ${{\displaystyle S}}_{(i)}$ by sorting ${{\displaystyle S}}_i$ ascendingly. ${{\displaystyle S}}_{(h)}$ = max {${{\displaystyle S}}_i$}. ${{\displaystyle S}}_{(l)}$ = min {${{\displaystyle S}}_i$}

Step 16:

In case ${{\displaystyle S}}_{\left(h\right)}$ > 0, proceed to Step 8, else proceed to Step 17.

Step 17:

Assume ${{\displaystyle q}}_{\left(l\right)}$ = ${{\displaystyle q}}_{\left(l\right)}$ − 1, ${{\displaystyle q}}_{\left(h\right)}$ = ${{\displaystyle q}}_{\left(h\right)}$ + 1, and further modify ${{\displaystyle \overline{Q}}}^b$

Step 18:

In case l = h, proceed to Step 20, else proceed to Step 19.

Step 19:

Update l = l + 1, and head to Step 12.

Step 20:

Stop. Set down ${{\displaystyle \overline{Q}}}^{*}$ = ${{\displaystyle \overline{Q}}}^b$, ${{\displaystyle \overline{f}}}^{*}({{\displaystyle \overline{Q}}}^{*})$, ${{\displaystyle \overline{\mu }}}^{*}({{\displaystyle \overline{Q}}}^{*})$, ${{\displaystyle C}}_s^{*}({\overline{Q}}^{*},{{\displaystyle \overline{f}}}^{*}({\overline{Q}}^{*}),{{\displaystyle \overline{\mu }}}^{*}({\overline{Q}}^{*}))$

3.5. MGA Module

According to the selected amount of redundancy components of subsystems acquired from the ENUM module, TS module, SA module, NESA module, or MRAH module in the first stage, the MGA module is then conducted to obtain the optimal repair rates and the optimal failure rates of the components for all subsystems in the second stage. The complete process of the GA module is as follows:

Step 1:

(1)

Start the initial iteration Ite = 1. According to the convergence situation of simulation result, determine the threshold of iteration MaxIte.

(2)

According to the simulation results, decide the appropriate crossover probability cr-rate, mutation probability mu-rate, and chromosome population.

(3)

Formulate the fitness function as the following equation:

${{\displaystyle C}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)=[{{\displaystyle \alpha }}_i{({{\displaystyle f}}_i)}^{-{{\displaystyle \beta }}_i}+{{\displaystyle \mu }}_i*{{\displaystyle rc}}_i][q_i+\exp ({{\displaystyle q}}_i/4)]$

(4)

Initially assume that the optimal fitness function value cjmin equals an exceedingly large number

Step 2:

Initiate the appropriate amount of chromosomes with the extent of 30 binary genes. All chromosomes adopt their first 15 binary genes to imply the failure rates ${{\displaystyle f}}_i$ ranging from 0.0000001 to 0.001, and adopt their last 15 binary genes to indicate the repair rates ${{\displaystyle \mu }}_i$ varying from 0.0000032 to 0.032.

Step 3:

Decode binary genes of chromosomes into two decision variables, ${{\displaystyle f}}_i$ and ${{\displaystyle \mu }}_i$. The decoding equation is depicted as

${{\displaystyle f}}_i={{\displaystyle fLB}}_i+\frac{{{\displaystyle fU\mathrm{B}}}_i-{{\displaystyle fLB}}_i}{{{\displaystyle 2}}^{15}-1}{\displaystyle \sum _{k=1}^{15}{{\displaystyle b}}_k{{\displaystyle 2}}^{k-1}}$ and ${{\displaystyle \mu }}_i={{\displaystyle \mu LB}}_i+\frac{{{\displaystyle \mu U\mathrm{B}}}_i-{{\displaystyle \mu LB}}_i}{{{\displaystyle 2}}^{15}-1}{\displaystyle \sum _{k=1}^{15}{{\displaystyle b}}_k{{\displaystyle 2}}^{k-1}}$ where

The low bound ${{\displaystyle fLB}}_i$ for the failure rates ${{\displaystyle f}}_i$ is 0.0000001 and ${{\displaystyle \mu LB}}_i$ for the repair rate ${{\displaystyle \mu }}_i$ is 0.0000032

The upper bound ${{\displaystyle fUB}}_i$ for the failure rate ${{\displaystyle f}}_i$ is 0.001 and $\mu {{\displaystyle UB}}_i$ for the repair rate ${{\displaystyle \mu }}_i$ is 0.032

${{\displaystyle b}}_k$ the k-th binary gene.

MATLAB pseudocode for the above decoding equation is as follows:

${{\displaystyle f}}_i$ = ga_bit2num(init_popu(1:15),[0.0000001,0.001]);

${{\displaystyle \mu }}_i$ = ga_bit2num(init_popu(16:30),[0.0000032,0.032]);

Step 4:

 

(1)

Input ${{\displaystyle f}}_i$ and ${{\displaystyle \mu }}_i$ to the fitness function to compute their associated fitness function values, ${{\displaystyle C}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)$

(2)

Examine if the chromosomes is in the line with the system availability requirement constraint: $A_S(\overline{Q},\overline{f},\overline{\mu })$ ≧ ${{\displaystyle A}}_s^L$. In case it is true, directly proceed to Step 5, otherwise, assume ${{\displaystyle C}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)$ equal to a extremely large number and directly proceed to Step 5, or create new chromosomes to update those unqualified chromosomes and return to Step 3.

Step 5:

 

In case $\mathrm{M}in{{\displaystyle C}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)$ < cjmin, assume that the optimal fitness function value cimin equals $\mathrm{M}in{{\displaystyle C}}_i({{\displaystyle q}}_i,{{\displaystyle f}}_i,{{\displaystyle \mu }}_i)$, the optimal failure rate ${{\displaystyle f}}_i^{*}$ equal to the corresponding failure rate ${{\displaystyle f}}_i^{}$, and the optimal repair rate ${{\displaystyle \mu }}_i^{*}$ equal to related repair rate ${{\displaystyle \mu }}_i^{}$.

Step 6:

Rank fitness function values for all chromosomes, and pick chromosomes to the reproduction pool by adopting Roulette wheel selection, which selects those chromosomes with high probability because of their lower fitness function values.

Step 7:

Randomly choose all pairs of chromosomes within the reproduction pool to carry on crossover operations.

Step 8:

Create a crossover probability ${{\displaystyle P}}_{\mathrm{c}}$ according to each pair of chromosomes. In case ${{\displaystyle P}}_{\mathrm{c}}$ is less than cr-rate, directly proceed to the later Step 9, on the contrary, leave for Step 10.

Step 9:

Trigger the two-point crossover operation in the light of probability ${{\displaystyle P}}_{\mathrm{c}}$ for each chosen pair of chromosomes. The first cross point is randomly picked from the first 15 binary genes, which implies the failure rates ${{\displaystyle f}}_i$; the second cross point is randomly selected from the last 15 binary genes, which indicates the repair rates ${{\displaystyle \mu }}_i$. Interchange of the genes between two picked chromosomes is conducted within the interval from the first cross point to the 15-th gene for ${{\displaystyle f}}_i$ and the range from the second cross point to the 30-th gene for ${{\displaystyle \mu }}_i$. MATLAB pseudocode for the above two-point crossover operation is showed as follows:

  for num = 1:n/2

      if rand<= cr_rate

          cr_point1=ceil(rand*15); cr_point2=ceil(rand*15+15);

          for c= cr_point1:15

       k=new_popu(num,c); new_popu(num,c)=new_popu(num+n/2,c);

       new_popu(num+n/2, c)=k;

          end

          for c= cr_point2:30

     k=new_popu(num,c); new_popu(num,c)=new_popu(num+n/2,c);

                    new_popu(num+n/2, c)=k;

          end

     end

  end

Step 10:

Create a mutation probability ${{\displaystyle P}}_m$ for all new chromosomes produced after crossover operations. In case ${{\displaystyle P}}_m$ is less than mu-rate, proceed to Step 11, or head to Step 12.

Step 11:

The mutation operation of two chosen mutation points is started on the basis of probability ${{\displaystyle P}}_m$ for all selected chromosomes. The first mutation point is randomly picked from the first 15 binary genes, which implies the failure rates ${{\displaystyle f}}_i$; the second mutation point is randomly picked from the last 15 binary genes, which indicates the repair rates ${{\displaystyle \mu }}_i$. MATLAB pseudocode for the above two-point mutation operation is showed as follows:

for num = 1:n

               if rand<= mu_rate

                  mu_point1=ceil(rand*15); mu_point2=ceil(rand*15+15);

                  new_popu(num, mu_point1)=1-new_popu(num,mu_point1);

                  new_popu(num, mu_point2)=1-new_popu(num,mu_point2);

               end

          end

Step 12:

In case Ite is less than MaxIte, return to Step 3; Otherwise, stop the process of GA module. Transfer the optimal solution of GA module, which includes cjmin, ${{\displaystyle f}}_i^{*}$, ${{\displaystyle \mu }}_i^{*}$, to the major module.

4. Simulated Study for Availability Optimization System Design

The proposed availability optimization decision support design system in this study attempts to consider the component redundancy number, failure rate, and repair rate in order to decide on an appropriate repairable n-stage mixed system, in which different combinations of subsystems, such as single component, parallel, standby, and k-out-of-q, are connected in series configuration. The enumeration method, the tabu search method, the simulated annealing approach, the non-equilibrium simulated annealing approach, and the modified redundancy allocation heuristic method combined with the modified genetic algorithm will be conducted to run different simulation cases regarding different system configurations and system constraint requirements.

This study will first only adopt the corresponding objective function in Equation (1), the system configuration constraint of system weight in Equation (2), the system configuration constraint of the product of the system weight and system volume in Equation (3), and the system availability requirement constraint in Equation (4) to run most simulation cases according to the procedure flow conducted by the proposed availability optimization decision support design system in Figure 1.

For the following simulated examples,

Table 2. Parameters adopted in the upcoming simulation cases.

Parameters			Subsystem i
	1	2	3	4	5
${{\displaystyle \alpha }}_j$(${{\displaystyle 10}}^{-5}$)	2.33	1.45	0.541	8.05	1.95
${{\displaystyle \beta }}_j$	1.5	1.5	1.5	1.5	1.5
${{\displaystyle mc}}_j$	5000	5000	5000	5000	5000
${{\displaystyle w}}_j$	7	8	8	6	9
$p_j$	1	2	3	4	2
$W{V}_{}$	150
${{\displaystyle W}}_{}$	200
${{\displaystyle A}}_S^L$	0.9

presents the values of all parameters as follows.

For the five proposed ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods, the modified genetic algorithm is adopted to acquire the optimal component repair rates and the optimal component failure rates for all subsystems jointly in the second stage, depending on the redundant component amount determined by ENUM, TS, SA, NESA, and MRAH in the first stage. Before implementing the following simulation cases, the parameter design for the modified genetic algorithm will be conducted first. According to the results of several trial and error tests, it will mostly ameliorate the quality of first final converged solution and the converged time by utilizing a chromosome crossover rate equal to 0.95, a chromosome mutation rate equal to 0.5, a chromosome population equal to 200, and a genetic generation equal to 200 in the five proposed ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods for most simulation cases.

4.1. The Simulation Case for n-Stage Single Component Series System

This study will first conduct the simulation case for the n-stage single component series system, which assumes all subsystems include only one single component, which is the simplest n-stage mixed system. The n-stage single component series system is the easiest system configuration to achieve the availability optimization system design.

As the n-stage single component series system includes only one single component for all subsystems, the related simulation case can be conducted by only adopting the genetic algorithm (GA) to acquire the optimal repair rates and the optimal failure rates of the components for all subsystems. For the simulation case of the n-stage single component series system, the simulation results obtained by the genetic algorithm are shown in

Table 3. The total system costs of 20 simulation runs by GA for the n-stage single component series system.

Total System Cost (CPU Time)
598.52(509.20)	591.117(503.10)	591.117(503.34)	591.117(512.2)	584.827(800.51)	583.807(493)
583.807(510.02)	583.807(536.76)	583.807(550.39)	579.853(538.16)	576.962(497.69)	576.962(511.91)
576.962(540.87)	576.962(542.59)	573.967(507.34)	573.967(509.42)	573.967(510.89)	573.967(577.04)
547.358(521.53)	545.489*(521.152958)
Lowest Total System Cost (CPUTime)     545.489*(521.152958 s)
Avg. Total System Cost (Avg. CPU Time)   578.4171(534.8579 s)

and

Table 4. Optimal Solutions obtained by GA for the n-stage single component series system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.1032	0.1339	0.0520	0.1875	0.0927
${{\displaystyle \overline{\mu }}}^{*}$	0.0039	0.0069	0.0027	0.0064	0.0081
${{\displaystyle \overline{A}}}^{*}$	0.974016	0.981015	0.980774	0.971382	0.988709
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$       545.489
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$         0.900056

. Table 3 demonstrates the total system costs of 20 simulation runs, the related average total system cost equal to 578.4171, and the lowest total system cost equal to 545.49. Furthermore, Table 3 displays the optimal failure rate, the optimal repair rate, the availability for each subsystem, the optimal total system cost, and the optimal system availability for the n-stage single component series system.

Furthermore, the convergence graphs in Figure 2 demonstrate that the total system cost has been ameliorated and finally converged to the lowest total system cost of 545.489, with an average CPU running time of 534.8579 s for the principle n-stage single component series system, by applying the proposed genetic algorithm, which assumes a chromosome crossover rate equal to 0.95, a chromosome mutation rate equal to 0.5, a chromosome population equal to 200, and a genetic generation equal to 200.

4.2. The Simulation Case for n-Stage Standby System

Since the n-stage single component series system assumes all subsystems comprise one single component, the proposed availability optimization decision support design system can only allocate the component failure rate and the component repair rate for each subsystem to achieve the system availability optimization design. However, this kind of system configuration has some disadvantages in reaching the certain level of the system availability requirement. The main disadvantage is that, because all the series connected subsystems only have one single component, the failure of one component would cause the entire system to shut down in order to conduct the repair operation. Furthermore, the system shutdown will result in a huge operation loss cost during the repair. To avoid this kind of system shutdown, the component reliability and component maintainability need to be enhanced dramatically, but it will also cost a lot to meet the high level of the system availability requirement due to technical difficulty. Therefore, the proposed availability optimization decision support design system intends to apply component redundancy to reach the high level of the system availability requirement with less cost. In the sequel, this study will further conduct the simulation cases by the ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods for the n-stage standby system, which assumes all subsystems are standby configurations and are connected in series structure.

Table 5. The total system costs of 20 simulation runs obtained by ENUM–MGA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
245.2(2708.2)	245.07(2785.4)	244.891(2638.2)	244.294(2578)	244.294(2622)	244.294(2690)
244.095(2787.7)	244.038(2544.9)	244.038(2654.1)	244.038(2727.4)	244.015(2647.4)	243.844(2557.0)
243.844(2672.4)	243.844(2687.5)	243.841(2644.3)	243.636(2629.7)	243.606(2636.1)	243.209(2697.3)
243.209(2719.8)	243.209*(2770.7)
Lowest Total System Cost (CPU Time)     243.209*(2770.7)
Avg. Total System Cost (Avg. CPU Time)    244.02545(2669.9273)

and

Table 6. Optimal Solutions obtained by ENUM–MGA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.291248	0.282734	0.148771	0.620876	0.26531
${{\displaystyle \overline{\mu }}}^{*}$	0.00144939	0.000901575	0.000924034	0.00177847	0.00148747
${{\displaystyle \overline{A}}}^{*}$	0.993506	0.978624	0.978161	0.971888	0.973712
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$       243.209
System Availability ${{\displaystyle A}}_S^{*}$          0.9065

show the simulation results produced by the ENUM–MGA combined method. Table 7 and Table 8 display the simulation results produced by the TS–MGA combined method. Table 9 and Table 10 demonstrate the simulation results produced by the SA–MGA combined method. Table 11 and Table 12 show the simulation results produced by the NESA–MGA combined method. Table 13 and Table 14 display the simulation results produced by the MRAH–MGA combined method.

Regarding the simulation case of the n-stage standby system, the simulation outcomes obtained by the ENUM-MGA combined method are displayed in Table 4 and Table 6. Table 5 demonstrates the total system costs of 20 simulation runs, the related average total system cost equal to 244.02545, and the related lowest total system cost equal to 243.209. Table 5 displays the optimal redundancy amounts, the optimal failure rates, the optimal repair rates, availabilities for all subsystems, the optimal total system cost, and the optimal system availability for the principal simulation case of the n-stage standby system. Furthermore, the convergence graphs in the following Figure 3 show that the total system cost can be easily improved and converged to the lowest 243.209 with the average CPU running time of 2669.9273 s for the principle n-stage standby system by adopting the proposed ENUM–GA combined method.

According to the results of Table 3, Table 4, Table 5 and Table 6, the optimal total system cost obtained by the ENUM-MGA combined method for the n-stage standby system is much better than the optimal total system cost obtained by the genetic algorithm for the n-stage single component series system. Moreover, the following additional simulation results conducted by the TS–MGA, SA–MGA, NESA–MGA and MRAH–MGA combined methods for the n-stage standby system obviously show that the proposed availability optimization decision support design system can use the component redundancy design to save a lot of cost and meet the high level of the system availability requirement compared to the n-stage single component series system.

Table 7. The total system costs of 28 simulation runs obtained by TS–MGA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
243.63(50)(1983.4)	242.57(50)(1912.2)	240.86(100)(3949.1)	240.4(100)(3861.67)	237.5(150)(5820.4)
238.27(150)(5752.2)	238.552(200)(7769)	237.988(200)(7817.9)	237.983(200)(7640)	237.838(200)(7648)
237.751(200)(7751)	237.654(200)(7644)	237.587(200)(7645.8)	237.529(200)(7721.7)	237.4(200)(7584.3)
237.396(200)(7624)	237.396(200)(7638)	237.36(200)(7777)	237.306(200)(7672)	237.299(200)(8126)
237.291(200)(7795)	237.272(200)(7740.6)	237.262(200)(7726)	237.012(200)(7911)	236.951(200)(7696)
236.831*(200)(7641)	237(250)(9435.22)	236.97(250)(9555.29)
Lowest Total System Cost (TabuLoop) (CPUTime)         236.831*(200)(7641) Avg. Total System Cost (TabuLoop) (Avg. CPU Time)
243.1(50)(1947.8)	240.6(100)(3905.4)	237.88(150)(5786.3)	237.48(200)(7728.42)	36.99(250)(9495.25)

Table 7. The total system costs of 28 simulation runs obtained by TS–MGA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
243.63(50)(1983.4)	242.57(50)(1912.2)	240.86(100)(3949.1)	240.4(100)(3861.67)	237.5(150)(5820.4)
238.27(150)(5752.2)	238.552(200)(7769)	237.988(200)(7817.9)	237.983(200)(7640)	237.838(200)(7648)
237.751(200)(7751)	237.654(200)(7644)	237.587(200)(7645.8)	237.529(200)(7721.7)	237.4(200)(7584.3)
237.396(200)(7624)	237.396(200)(7638)	237.36(200)(7777)	237.306(200)(7672)	237.299(200)(8126)
237.291(200)(7795)	237.272(200)(7740.6)	237.262(200)(7726)	237.012(200)(7911)	236.951(200)(7696)
236.831*(200)(7641)	237(250)(9435.22)	236.97(250)(9555.29)
Lowest Total System Cost (TabuLoop) (CPUTime)         236.831*(200)(7641) Avg. Total System Cost (TabuLoop) (Avg. CPU Time)
243.1(50)(1947.8)	240.6(100)(3905.4)	237.88(150)(5786.3)	237.48(200)(7728.42)	36.99(250)(9495.25)

By conducting the TS–MGA combined method for the principal n-stage standby system, Table 7 above shows 28 selected simulated results of the total system costs by setting TabuLoop equal to 50 with an average CPU running time of 1947.8 s until 250 with an average CPU running time of 9495.25 s, and it further converges the average total system cost of 243.1 to 236.99 and the total system cost of 243.63 to the lowest 236.831. Since the average total system cost of 237.48 with 200 Tabu loops is a little higher than 236.99 with 250 Tabu loops, and, moreover, the lowest converged total system cost happened at 200 Tabu loops, the simulation process stops running at 200 Tabu loops and finally reaches the lowest converged result of 236.831. Furthermore, for the above lowest converged result of 236.831, which is obtained at 200 Tabu loops, Figure 4 shows that the total system cost and the average total system cost have improved and converged for the principle n-stage standby system by applying the proposed TS–GA combined method.

Table 8 displays the optimal system cost related to the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability, as follows:

Table 8. Optimal Solutions obtained by TS–MGA for the n-stage standby system.

TabuLoop = 200 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3261	0.2749	0.1507	0.5963	0.2555
${{\displaystyle \overline{\mu }}}^{*}$	0.0012	0.0010	0.0010	0.0017	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9852	0.9838	0.9808	0.9710	0.9750
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$      236.831
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$          0.9000

Table 8. Optimal Solutions obtained by TS–MGA for the n-stage standby system.

TabuLoop = 200 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3261	0.2749	0.1507	0.5963	0.2555
${{\displaystyle \overline{\mu }}}^{*}$	0.0012	0.0010	0.0010	0.0017	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9852	0.9838	0.9808	0.9710	0.9750
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$      236.831
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$          0.9000

Table 9. The total system costs of 20 simulation runs obtained by SA–GA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
244.11(50)(4403.51)	243.35(50)(4197.83)	243.07(50)(4204.29)	242.844(50)(4358.55)	242.44(50)(4768)
238.693(50)(4418.54)	241(100)(8680)	240.75(100)(8609.14)	237.89(100)(8817.26)	238.50(150)(12,605)
237.11(150)(12,422)	239.08(200)(17,081)	237.13(200)(16,711)	238.87(250)(21,676)	236.46*(250)(21,233)
237.02(400)(22,524)	237.01(400)(22,061.8)	236.86(400)(22,088)	236.77(500)(47,528.5)	236.76(500)(46,295)
Lowest Total System Cost (SAloop)(CPUTime)    236.46*(250)(21,232.93)
Avg. Total System Cost (SAloop) (Avg. CPUTime)
242.4(50)(4391.8)	239.88(100)(8702.2)	237.8(150)(12,513.5)	238.1(200)(16,896.14)	236.97(400)(22,224.54)
236.766(500)(46,911.73)

Table 9. The total system costs of 20 simulation runs obtained by SA–GA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
244.11(50)(4403.51)	243.35(50)(4197.83)	243.07(50)(4204.29)	242.844(50)(4358.55)	242.44(50)(4768)
238.693(50)(4418.54)	241(100)(8680)	240.75(100)(8609.14)	237.89(100)(8817.26)	238.50(150)(12,605)
237.11(150)(12,422)	239.08(200)(17,081)	237.13(200)(16,711)	238.87(250)(21,676)	236.46*(250)(21,233)
237.02(400)(22,524)	237.01(400)(22,061.8)	236.86(400)(22,088)	236.77(500)(47,528.5)	236.76(500)(46,295)
Lowest Total System Cost (SAloop)(CPUTime)    236.46*(250)(21,232.93)
Avg. Total System Cost (SAloop) (Avg. CPUTime)
242.4(50)(4391.8)	239.88(100)(8702.2)	237.8(150)(12,513.5)	238.1(200)(16,896.14)	236.97(400)(22,224.54)
236.766(500)(46,911.73)

By conducting the SA–MGA combined method for the principal n-stage standby system, Table 9 above shows 20 selected simulated results of the total system costs by setting SAloop equal to 50 with an average CPU running time of 4391.8 s until 400 with an average CPU running time of 22,224.54 s, and it further converges the average total system cost of 242.4 to 236.97 and the total system cost of 244.11 to the lowest 236.46. Since the average total system cost of 236.97 with 400 SA loops is a little higher than 236.766 with 500 SA loops, but with around half of the CPU running time, the simulation process stops running at 400 SA loops and finally reaches the lowest converged result of 236.46. Moreover, Figure 5 also displays that the average total system cost and the total system cost have improved and converged for the principal n-stage standby system by adopting the proposed SA–GA combined method.

For the above lowest converged result of 236.46, which is obtained at 250 SA loops, Table 10 displays the optimal system cost related to the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability, as follows:

Table 10. Optimal Solutions obtained by SA–MGA for the n-stage standby system.

SAloop = 250, cr-rate = 0.95, mu-rate = 0.5, GApopulation = 200, GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3325	0.2738	0.1401	0.6057	0.2560
${{\displaystyle \overline{\mu }}}^{*}$	0.0011	0.0010	0.0010	0.0017	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9823	0.9852	0.9818	0.9718	0.9747
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         236.46
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$            0.9000

Table 10. Optimal Solutions obtained by SA–MGA for the n-stage standby system.

SAloop = 250, cr-rate = 0.95, mu-rate = 0.5, GApopulation = 200, GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3325	0.2738	0.1401	0.6057	0.2560
${{\displaystyle \overline{\mu }}}^{*}$	0.0011	0.0010	0.0010	0.0017	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9823	0.9852	0.9818	0.9718	0.9747
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         236.46
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$            0.9000

Table 11. The total system costs of 20 simulation runs obtained by NESA–MGA for the n-stage standby system.

InLoop = 10, cr-rate = 0.95, mu-rate = 0.5, GApopulation = 200, GAloop = 200
Total System Cost (NESAloop) (CPU Time)
238.953(50)(871)	237.815(50)(879.4)	238.456(50)(956.6)	238.298(100)(1840.4)	237.94(100)(1990.4)
237.98(150)(2830)	237.83(150)(2736)	237.63(200)(3832)	237.76(200)(3869.33)	237.36(250)(4552.24)
237.41(250)(4592.9)	237.58(250)(4517.4)	237.396(250)(4603)	237.8(400)(7029.45)	237.62(400)(7974.95)
237.59(400)(7405.6)	237.35(500)(9305.5)	237.27(500)(9441.5)	237.57(600)(11,486.7)	237.57(600)(11,339.5)
Lowest Total System Cost (NESAloop) (CPU Time)             237.27*(500)(9441.5)
Avg. Total System Cost (NESAloop) (Avg. CPU Time)
	238.408(50)(902)	238.12(100)(1915)	237.9(150)(2783.2)	237.69(200)(3851)
	237.44(250)(4566.45)	237.67(400)(7470)	237.31(500)(9373.45)	237.57(600)(11,413)

Table 11. The total system costs of 20 simulation runs obtained by NESA–MGA for the n-stage standby system.

InLoop = 10, cr-rate = 0.95, mu-rate = 0.5, GApopulation = 200, GAloop = 200
Total System Cost (NESAloop) (CPU Time)
238.953(50)(871)	237.815(50)(879.4)	238.456(50)(956.6)	238.298(100)(1840.4)	237.94(100)(1990.4)
237.98(150)(2830)	237.83(150)(2736)	237.63(200)(3832)	237.76(200)(3869.33)	237.36(250)(4552.24)
237.41(250)(4592.9)	237.58(250)(4517.4)	237.396(250)(4603)	237.8(400)(7029.45)	237.62(400)(7974.95)
237.59(400)(7405.6)	237.35(500)(9305.5)	237.27(500)(9441.5)	237.57(600)(11,486.7)	237.57(600)(11,339.5)
Lowest Total System Cost (NESAloop) (CPU Time)             237.27*(500)(9441.5)
Avg. Total System Cost (NESAloop) (Avg. CPU Time)
	238.408(50)(902)	238.12(100)(1915)	237.9(150)(2783.2)	237.69(200)(3851)
	237.44(250)(4566.45)	237.67(400)(7470)	237.31(500)(9373.45)	237.57(600)(11,413)

By conducting the NESA–MGA combined method regarding the principal n-stage standby system, Table 11 above shows 20 selected simulated results of the total system costs by setting NESAloop equal to 50 with an average CPU running time of 902 s until 500 with an average CPU running time of 9373.45 s, and it further converges the average total system cost of 238.408 to 237.31 and the total system cost of 238.953 to the lowest 237.31. Since the average total system cost of 237.31 with 500 NESA loops is lower than 237.57 with 600 NESA loops, the simulation process stops running at 500 NESA loops and finally reaches lowest converged result of 237.27. Therefore, Figure 6 also clearly illustrates that the average total system cost and the total system cost have improved and at last converged for the principal n-stage standby system by adopting the proposed NESA–MGA combined method.

Additionally, for the lowest converged result of 237.27, Table 12 illustrates the optimal system cost associated with the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability.

Table 12. Optimal Solutions obtained by NESA–MGA for the n-stage standby system.

NESAloop = 500 InLoop = 10 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	3	2	3	3
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2786	0.2873	0.1411	0.5643	0.3147
${{\displaystyle \overline{\mu }}}^{*}$	0.0016	0.0010	0.0010	0.0018	0.0010
${{\displaystyle \overline{A}}}^{*}$	0.9759	0.9809	0.9812	0.9776	0.9802
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         237.27
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$            0.9000

Table 12. Optimal Solutions obtained by NESA–MGA for the n-stage standby system.

NESAloop = 500 InLoop = 10 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	3	2	3	3
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2786	0.2873	0.1411	0.5643	0.3147
${{\displaystyle \overline{\mu }}}^{*}$	0.0016	0.0010	0.0010	0.0018	0.0010
${{\displaystyle \overline{A}}}^{*}$	0.9759	0.9809	0.9812	0.9776	0.9802
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         237.27
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$            0.9000

In Table 12 and Table 14, the simulation results obtained by the MRAH -MGA combined method display the total system costs of 20 simulation runs, the related average total system cost equal to 237.807, and the lowest total system cost equal to 226.450, which is superior to all other combined methods and has an extremely low average CPU running time of around 89 s.

Table 13. The total system costs of 20 simulation runs obtained by MAHA–MGA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (Running Time)
238.067(90.18)	237.87(90.21)	239.598(91.55)	238.514(86.33)	238.916(90.19)	238.53(88.33)	241.376(91.44)
236.094(89.25)	237.921(90.24)	238.342(91.67)	238.875(96.67)	238.603(97.45)	235.253(97)	235.74(96.6)
239.891(98)	239.066(97.4)	239.607(98.4)	238.037(96.7)	239.392(97.4)	226.450*(5.31)
Lowest Total System Cost (CPU Time)     226.450*(5.31)
Avg. Total System Cost (Avg. CPU Time)   237.807(89.0277)

Table 13. The total system costs of 20 simulation runs obtained by MAHA–MGA for the n-stage standby system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (Running Time)
238.067(90.18)	237.87(90.21)	239.598(91.55)	238.514(86.33)	238.916(90.19)	238.53(88.33)	241.376(91.44)
236.094(89.25)	237.921(90.24)	238.342(91.67)	238.875(96.67)	238.603(97.45)	235.253(97)	235.74(96.6)
239.891(98)	239.066(97.4)	239.607(98.4)	238.037(96.7)	239.392(97.4)	226.450*(5.31)
Lowest Total System Cost (CPU Time)     226.450*(5.31)
Avg. Total System Cost (Avg. CPU Time)   237.807(89.0277)

Table 14. Optimal Solutions obtained by MAHA–MGA for the n-stage standby system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	3	3
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3886	0.2331	0.1960	0.7868	0.4314
${{\displaystyle \overline{\mu }}}^{*}$	0.0024	0.0012	0.0013	0.0017	0.0012
${{\displaystyle \overline{A}}}^{*}$	0.9484	0.9796	0.9907	0.9866	0.9948
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         226.450
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$           0.90335

Table 14. Optimal Solutions obtained by MAHA–MGA for the n-stage standby system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	3	3
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3886	0.2331	0.1960	0.7868	0.4314
${{\displaystyle \overline{\mu }}}^{*}$	0.0024	0.0012	0.0013	0.0017	0.0012
${{\displaystyle \overline{A}}}^{*}$	0.9484	0.9796	0.9907	0.9866	0.9948
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         226.450
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$           0.90335

Moreover, the following Figure 7 remarkably demonstrates that the total system cost has been ameliorated and, in the end, converged for the principle n-stage standby system by conducting the proposed MRAH–MGA combined method.

4.3. The Simulation Case for Parallel-Series System

Besides the n-stage standby system, the parallel-series system can also be applied to take advantage of component redundancy to reach the high level of the system availability requirement with less cost. The proposed availability optimization decision support design system will further conduct the simulation cases by the ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MAHA–MGA combined methods for the parallel-series system, which assumes all subsystems are parallel configurations and are connected in series structure.

Table 15. The total system costs of 20 simulation runs obtained by ENUM–MGA for the parallel series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
222.778(2707.3)	222.146(2664.6)	222.086(2641.2)	221.922(2625.9)	221.765(2673.6)	221.53(2678.2)
221.449(2660.6)	221.344(2753.8)	221.327(2702.6)	221.193(2695.1)	221.078(2618.5)	221.034(2646.6)
220.947(2658.4)	220.902(2782.6)	220.861(2677.4)	220.84(2646.9)	220.566(2772.6)	220.543(2661.2)
220.31(5100.7)	220.278(2696.55)
Lowest Total System Cost (CPUTime)       220.278*(2696.550)
Avg. Total System Cost (Avg. CPUTime)      221.245 (2803.223)

and

Table 16. Optimal Solutions obtained by ENUM–MGA for parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3102	0.3113	0.1622	0.6634	0.2483
${{\displaystyle \overline{\mu }}}^{*}$	0.00125	0.000796	0.00091	0.001517	0.001390
${{\displaystyle \overline{A}}}^{*}$	0.992142	0.977773	0.977109	0.971819	0.977018
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           220.278
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$             0.900001

show the simulation results produced by the ENUM–MGA combined method. Table 17 and Table 18 display the simulation results produced by the TA–MGA combined method. Table 19 and Table 20 demonstrate the simulation results produced by the SA–MGA combined method. Table 21 and Table 22 show the simulation results produced by the NESA–MGA combined method. Table 23 and Table 24 display the simulation results produced by the MAHA–MGA combined method.

For the simulation case of the parallel-series system, the simulation results obtained by the ENUM-MGA combined method is expressed in Table 14 and Table 16 and Figure 8. Table 15 shows the total system costs of 20 simulation runs, the related average total system cost of 221.245, and the related lowest total system cost of 220.278. The convergence graphs in Figure 8 distinctly illustrate that the total system cost can likely be improved and converged for the principal parallel-series system by implementing the proposed ENUM–MGA combined method.

Furthermore, Table 15 demonstrates the optimal system cost, the optimal failure rates, the optimal repair rates, the availabilities for all subsystems, and the system availability for the principal simulation case of the parallel-series system. In Table 16, the optimal total system cost obtained by the ENUM-MGA combined method for the parallel-series system is much better than the optimal total system cost obtained by the genetic algorithm for the n-stage single component series system in Table 6. Moreover, other simulation results produced by the TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods for the parallel-series system also show that the proposed availability optimization decision support design system can apply the component redundancy design of the parallel-series system to decrease a lot of the cost and meet the high level of the system availability requirement compared to the n-stage single component series system.

Table 17. The total system costs of 32 simulation runs obtained by TA–MGA for the parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
222.42(50)(1894.79)	221.92(50)(1966.3)	220.05(50)(1873.5)	218(100)(3791.59)	216.825(100)(3799)
216.59(100)(3787.9)	216.64(150)(5670.7)	215.69(150)(5760)	215.51(150)(5656.5)	215.17(200)(7409.4)
215.169(200)(7426.8)	215.167(200)(7364.7)	215.15(200)(7787.3)	215.14(200)(7363)	215.13(200)(7440.6)
215.1(200)(7405.4)	215.09(200)(7432.3)	215.086(200)(7537.2)	215.064(200)(7345.14)	214.993(200)(7394.4)
214.972(200)(7802)	214.972(200)(7621.7)	214.957(200)(7374.8)	214.91(200)(7439.5)	214.88(200)(7470.98)
214.823(200)(7336)	214.779(200)(7455)	214.74(200)(7497.28)	214.74(200)(7425.56)	214.985(250)(9484.38)
214.69(250)(9350.25)	214.62*(250)(9361)
Lowest Total System Cost (TabuLoop)(CPU Time)         214.62*(250)(9361)
Avg. Total System Cost (TabuLoop) (Avg. CPUTime)
221.46(50)(1911.5)	217.14(100)(3793)	215(200)(7468.7)	214.765(250)(9398.55)

Table 17. The total system costs of 32 simulation runs obtained by TA–MGA for the parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
222.42(50)(1894.79)	221.92(50)(1966.3)	220.05(50)(1873.5)	218(100)(3791.59)	216.825(100)(3799)
216.59(100)(3787.9)	216.64(150)(5670.7)	215.69(150)(5760)	215.51(150)(5656.5)	215.17(200)(7409.4)
215.169(200)(7426.8)	215.167(200)(7364.7)	215.15(200)(7787.3)	215.14(200)(7363)	215.13(200)(7440.6)
215.1(200)(7405.4)	215.09(200)(7432.3)	215.086(200)(7537.2)	215.064(200)(7345.14)	214.993(200)(7394.4)
214.972(200)(7802)	214.972(200)(7621.7)	214.957(200)(7374.8)	214.91(200)(7439.5)	214.88(200)(7470.98)
214.823(200)(7336)	214.779(200)(7455)	214.74(200)(7497.28)	214.74(200)(7425.56)	214.985(250)(9484.38)
214.69(250)(9350.25)	214.62*(250)(9361)
Lowest Total System Cost (TabuLoop)(CPU Time)         214.62*(250)(9361)
Avg. Total System Cost (TabuLoop) (Avg. CPUTime)
221.46(50)(1911.5)	217.14(100)(3793)	215(200)(7468.7)	214.765(250)(9398.55)

By applying the TS–MGA combined method for the principal parallel-series system, Table 17 above shows 32 selected simulated outcomes of the total system costs by assuming TabuLoop equal to 50 with an average CPU running time of 1911.5 s until 250 with an average CPU running time of 9398.55 s, and it further converges the average total system cost from 221.46 to 214.765 and the total system cost of 222.42 to the lowest 214.62. Since the average total system cost of 215 with 200 Tabu loops is just a little higher than 214.765 with 250 Tabu loops, the simulation process stops running at 250 Tabu loops and finally achieves the lowest converged result of 214.62. Figure 9 demonstrates that the total system cost and the average total system cost have improved and converged for the principal parallel-series system by adopting the proposed TS–MGA combined method.

Additionally, for the above-mentioned lowest converged total system cost of 214.62, which is obtained at 250 Tabu loops, Table 18 displays the optimal system cost related to the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability, as follows:

Table 18. Optimal Solutions obtained by TA–MGA for the parallel-series system.

TabuLoop = 250 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3688	0.2925	0.1429	0.6483	0.2609
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0009	0.0009	0.0015	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9808	0.9838	0.9816	0.9730	0.9766
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         214.62
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$              0.9000

Table 18. Optimal Solutions obtained by TA–MGA for the parallel-series system.

TabuLoop = 250 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3688	0.2925	0.1429	0.6483	0.2609
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0009	0.0009	0.0015	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9808	0.9838	0.9816	0.9730	0.9766
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         214.62
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$              0.9000

By adopting the SA–MGA combined method to implement the simulation runs for the parallel-series system, the following Table 19 displays 20 selected simulated results of the total system costs by assuming SAloop equal to 50 with an average CPU running time of 4175.08 s until 150 with an average CPU running time of 12,235.82 s, and it further converges the average total system cost of 220.6968 to 215.80295 and the total system cost of 222.13 to the lowest 214.7569. Because the average total system cost of 215.80295 with 150 SA loops is higher than 215.7916 with 100 SA loops, the simulation process can certainly stop running at 100 SA loops and converge to the lowest result of 214.7569.

Table 19. The total system costs of 20 simulation runs obtained by SA–MGA for the parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
222.13(50)(4181.43)	220.93(50)(4194.25)	219.025(50)(4149.57)	217.15(100)(8355.43)
216.7015(100)(8194.88)	216.7011(100)(8157.18)	216.69(100)(8143.85)	216.135(100)(8144.83)
216.11(100)(8105.175)	215.825(100)(8155.55)	215.59(100)(8169.5)	215.54(100)(8179.6)
215.45(100)(8345.35)	215.14(100)(8187.4)	215.11(100)(8179.3)	215.02(100)(8171.8)
214.95(100)(8166.9)	214.7569*(100)(8182.35)	215.9081(150)(12,510.3)	215.6978(150)(12,235.8)
Lowest Total System Cost (SAloop)(CPUTime)          214.7569*(100)(8182.35)
Avg. Total System Cost (SAloop)(CPUTime)
220.6968(50)(4175.08)	215.7916*(100)(8182.2)	215.80295(150)(12,235.81699)

Table 19. The total system costs of 20 simulation runs obtained by SA–MGA for the parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
222.13(50)(4181.43)	220.93(50)(4194.25)	219.025(50)(4149.57)	217.15(100)(8355.43)
216.7015(100)(8194.88)	216.7011(100)(8157.18)	216.69(100)(8143.85)	216.135(100)(8144.83)
216.11(100)(8105.175)	215.825(100)(8155.55)	215.59(100)(8169.5)	215.54(100)(8179.6)
215.45(100)(8345.35)	215.14(100)(8187.4)	215.11(100)(8179.3)	215.02(100)(8171.8)
214.95(100)(8166.9)	214.7569*(100)(8182.35)	215.9081(150)(12,510.3)	215.6978(150)(12,235.8)
Lowest Total System Cost (SAloop)(CPUTime)          214.7569*(100)(8182.35)
Avg. Total System Cost (SAloop)(CPUTime)
220.6968(50)(4175.08)	215.7916*(100)(8182.2)	215.80295(150)(12,235.81699)

In the following, Figure 10 displays that the total system cost and the average total system cost have improved and converged for the principal parallel-series system by employing the proposed SA–MGA combined method.

Furthermore, for the above lowest converged result of 214.7569, which is obtained at 100 SA loops, Table 20 shows the optimal system cost related to the optimal failure rate, the optimal repair rate, the availability of each subsystem, and the system availability, as follows:

Table 20. Optimal Solutions obtained by SA–MGA for the parallel-series system.

SAloop = 100 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3552	0.2914	0.1431	0.6854	0.2495
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0009	0.0010	0.0015	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9810	0.9843	0.9853	0.9685	0.9768
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         214.7569
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$            0.9000

Table 20. Optimal Solutions obtained by SA–MGA for the parallel-series system.

SAloop = 100 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3552	0.2914	0.1431	0.6854	0.2495
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0009	0.0010	0.0015	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9810	0.9843	0.9853	0.9685	0.9768
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         214.7569
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$            0.9000

By utilizing the NESA–MGA combined method to conduct the simulation cases for the parallel-series system, the following Table 21 displays 22 selected simulated results of the total system costs by assuming NESAloop equal to 50 with an average CPU running time of 493.3975588 s until 1000 with an average CPU running time of 12,787.95585 s, and it further converges the average total system cost of 218.12722 to 215.3146 and the total system cost of 218.9289 to the lowest 214.9928. Since the average total system cost of 216.01448 with 300 NESA loops is just a little higher than 215.3146 with 1000 NESA loops, the simulation process ceases running at 1000 NESA loops and finally reaches the lowest converged result of 214.9928.

Table 21. The total system costs of 20 simulation runs obtained by NESA–MGA for the parallel-series system.

InLoop = 10 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (NESAloop) (CPU Time)
218.9289(50)(488.32)	218.7698(50)(495.24)	218.3487(50)(495.17)	218.087(50)(478.486)	216.502(50)(509.77)
218.06(100)(996.86)	217.459(100)(951)	217.274(100)(941.2)	217.24(100)(942.9)	216.11(100)(968.27)
217.45(200)(1988.4)	216.87(200)(1966.1)	216.85(200)(1892)	216.48(200)(2000)	215.7231(200)(1998.7)
216.3836(300)(3047)	216.0787(300)(3059.9)	216.0624(300)(2890)	216.0151(300)(2845.8)	215.5326(300)(2894)
215.6364(1000)(12,711.696)	214.9928*(1000)(12,864.215)
Avg. Total System Cost(NESAloop)(CPUTime)
218.12722(50)(493.3975588)	217.22942(100)(960.0457)	216.6767(200)(1969.03851)	216.01448(300)(2947.507125)	215.3146(1000)(12,787.95585)
Lowest Total System Cost (NESAloop)(CPUTime)214.9928*(1000)(12,864.215)

Table 21. The total system costs of 20 simulation runs obtained by NESA–MGA for the parallel-series system.

InLoop = 10 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (NESAloop) (CPU Time)
218.9289(50)(488.32)	218.7698(50)(495.24)	218.3487(50)(495.17)	218.087(50)(478.486)	216.502(50)(509.77)
218.06(100)(996.86)	217.459(100)(951)	217.274(100)(941.2)	217.24(100)(942.9)	216.11(100)(968.27)
217.45(200)(1988.4)	216.87(200)(1966.1)	216.85(200)(1892)	216.48(200)(2000)	215.7231(200)(1998.7)
216.3836(300)(3047)	216.0787(300)(3059.9)	216.0624(300)(2890)	216.0151(300)(2845.8)	215.5326(300)(2894)
215.6364(1000)(12,711.696)	214.9928*(1000)(12,864.215)
Avg. Total System Cost(NESAloop)(CPUTime)
218.12722(50)(493.3975588)	217.22942(100)(960.0457)	216.6767(200)(1969.03851)	216.01448(300)(2947.507125)	215.3146(1000)(12,787.95585)
Lowest Total System Cost (NESAloop)(CPUTime)214.9928*(1000)(12,864.215)

In the following, Figure 11 illustrates that the total system cost and the average system cost have improved and converged for the principle parallel-series system by introducing the proposed NESA–MGA combined method.

Furthermore, for the lowest converged result of 214.9928, Table 22 illustrates the optimal system cost associated with the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability.

Table 22. Optimal Solutions obtained by NESA–MGA for the parallel-series system.

NESAloop = 1000 InLoop = 10 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3584	0.295	0.1303	0.6676	0.2533
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0009	0.0009	0.0014	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9796	0.9843	0.9851	0.9680	0.9789
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         214.9928
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$          0.9000

Table 22. Optimal Solutions obtained by NESA–MGA for the parallel-series system.

NESAloop = 1000 InLoop = 10 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3584	0.295	0.1303	0.6676	0.2533
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0009	0.0009	0.0014	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9796	0.9843	0.9851	0.9680	0.9789
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$         214.9928
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$          0.9000

Table 22 and Table 24 display the simulation consequences acquired by the MAHA-MGA combined method for the total system costs of 20 simulation runs, the related average total system cost equal to 215.7925, and the lowest total system cost equal to 208.9659, which is superior to all other combined methods and has an extremely low average CPU running time of around 88.455 s.

Table 23. The total system costs of 20 simulation runs obtained by MRAH–MGA for the parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
218.4072(88.5)	218.2744(87.895)	218.0376(87.42)	217.6616(90.248)	217.5924(90.046)	216.8774(89.3642)
216.1849(89.5495)	216.0018(87.3153)	215.9571(85.0562)	215.9565(91.17)	215.7456(85.56)	215.635(90.1)
215.6152(84.49)	215.4144(94.57)	215.35(88.73)	215.13(84.82)	214.99(86.16)	214.0611(84.58)
213.99(85.17)	208.9659*(98.64)
Lowest Total System Cost (CPUTime))         208.9659*(98.64)
Avg. Total System Cost (Avg. CPU Time)     215.7925(88.455)

Table 23. The total system costs of 20 simulation runs obtained by MRAH–MGA for the parallel-series system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
218.4072(88.5)	218.2744(87.895)	218.0376(87.42)	217.6616(90.248)	217.5924(90.046)	216.8774(89.3642)
216.1849(89.5495)	216.0018(87.3153)	215.9571(85.0562)	215.9565(91.17)	215.7456(85.56)	215.635(90.1)
215.6152(84.49)	215.4144(94.57)	215.35(88.73)	215.13(84.82)	214.99(86.16)	214.0611(84.58)
213.99(85.17)	208.9659*(98.64)
Lowest Total System Cost (CPUTime))         208.9659*(98.64)
Avg. Total System Cost (Avg. CPU Time)     215.7925(88.455)

Table 24. Optimal Solutions obtained by MRAH–MGA for the parallel-series system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	3	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.4079	0.3744	0.2456	0.5615	0.3082
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0010	0.0010	0.0020	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9880	0.9796	0.9963	0.9630	0.9710
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$        208.9659
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$             0.901614

Table 24. Optimal Solutions obtained by MRAH–MGA for the parallel-series system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	3	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.4079	0.3744	0.2456	0.5615	0.3082
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0010	0.0010	0.0020	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9880	0.9796	0.9963	0.9630	0.9710
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$        208.9659
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$             0.901614

In addition, the following Figure 12 remarkably demonstrates that the total system cost has decreased and eventually converged for the principle parallel-series system by implementing the proposed MRAH–MGA combined method.

4.4. The Simulation Case for n-Stage k-out-of-q System

In the previous three sections, the proposed availability optimization decision support design system has designed three systems, including the n-stage single component system, n-stage standby system, and parallel-series system, which subsystems are assumed to be operational when at least one component is operational. However, there are some special cases in the real world where more than one component of a subsystem is required to be operational to make sure the whole system is working. In the sequel, the proposed availability optimization decision support design system intends to develop some n-stage systems, which require at least more than one component operational within their subsystems. For the q/2-out-of-q subsystem, at least q/2 components are operational within all q components, and then the subsystem is operational. Therefore, this study will further perform the simulation cases by the ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods for the n-stage q/2-out-of-q system, which assumes all subsystems are q/2-out-of-q configurations and are connected in series structure.

Table 25. The total system costs of 20 simulation runs obtained by ENUM–MGA for the n-stage q/2-out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
261.061(4773.19)	261.061(4884.37)	259.029(4802.89)	258.554(4850.2)	257.92(4776.48)	257.739(4790.7)
257.136(4847.6)	256.942(4802.4)	256.797(4768.56)	256.797(4770.7)	256.489(4887.86)	256.444(4828.93)
256.418(4804.88)	256.373(4839.62)	255.263(4874.39)	255.126(4830.67)	254.983(5118.66)	254.966(4852.91)
254.75(4802.44)	254.75*(4763.558)
Lowest Total System Cost (CPUTime)       254.75*(4763.557856)
Avg. Total System Cost (Avg. CPUTime)     256.9299(4833.551079)

and

Table 26. Optimal Solutions obtained by ENUM–MGA for the n-stage q/2-out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.196681	0.22741	0.147154	0.479162	0.276051
${{\displaystyle \overline{\mu }}}^{*}$	0.00244639	0.00124628	0.000813691	0.00242784	0.0015031
${{\displaystyle \overline{A}}}^{*}$	0.994463	0.976187	0.976545	0.972831	0.975926
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          254.75
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$           0.900053

show the simulation results produced by the ENUM–MGA combined method. Table 27 and Table 28 display the simulation results produced by the TA–MGA combined method. Table 29 and Table 30 demonstrate the simulation results produced by the SA–MGA combined method. Table 31 and Table 32 show the simulation results produced by the NESA–MGA combined method. Table 33 and Table 34 display the simulation results produced by the MRAH–MGA combined method.

For the numerical cases of the n-stage q/2-out-of-q system, the numerical results acquired by the ENUM-MGA combined method are described in Table 25 and Table 26 and Figure 13. Table 25 demonstrates that for the total system costs of 20 simulation runs, the average total system cost equals 256.9299 and the lowest total system cost equals 254.75. The convergence graph in Figure 13 explicitly indicates that the total system cost can likely be ameliorated and converged for the main n-stage q/2-out-of-q system by applying the proposed ENUM–MGA combined method. Furthermore, Table 26 demonstrates the optimal system cost, the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability for the principal simulation case for n-stage q/2-out-of-q system.

In Table 26, the optimal total system cost obtained by the ENUM-MGA combined method for the n-stage q/2-out-of-q system, which is just like the n-stage standby system and parallel-series system, is also much superior to the optimal total system cost obtained by the modified genetic algorithm for the n-stage single component series system in Table 6. Furthermore, other simulation results produced by the TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods for the n-stage q/2-out-of-q system also prove that the proposed availability optimization decision support design system can apply the component redundancy design of the n-stage q/2-out-of-q system to reduce cost and meet the certain level of the system availability requirement compared to the n-stage single component series system.

Table 27. The total system costs of 20 simulation runs obtained by TA–MGA for the n-stage q/2-out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
262.3425(50)(3575.53)	262.2905(50)(3563.52)	262.2645(50)(3557.276)	262.0538(50)(3605.8)
262.0088(50)(3561.249)	261.7917(50)(3634.4527)	262.2589(100)(7246.5046)	260.9296(100)(7143.268)
260.7991(100)(7183.86)	260.74(100)(7156.376)	260.34(100)(7161.29)	259.68(150)(10,747.22)
256.96(150)(10,650.82)	255.3389(150)(10,727.65)	255.7538(200)(14,803)	254.8258(200)(14,318.947)
254.4347(200)(14,404.277)	254.8747(250)(18,055.384)	254.8549(250)(17,686.767)	254.0007*(250)(17,993.076)
Lowest Total System Cost (TabuLoop)(CPUTime)  254.0007*(250)(17,993.076)
Avg. Total System Cost (TabuLoop) (Avg. CPUTime)
262.1253(50)(3582.97)	261.01(100)(7178.26)	257.327(150)(10,708.56)	255.0048(200)(14,508.74)
254.5768(250)(17,911.74)

Table 27. The total system costs of 20 simulation runs obtained by TA–MGA for the n-stage q/2-out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
262.3425(50)(3575.53)	262.2905(50)(3563.52)	262.2645(50)(3557.276)	262.0538(50)(3605.8)
262.0088(50)(3561.249)	261.7917(50)(3634.4527)	262.2589(100)(7246.5046)	260.9296(100)(7143.268)
260.7991(100)(7183.86)	260.74(100)(7156.376)	260.34(100)(7161.29)	259.68(150)(10,747.22)
256.96(150)(10,650.82)	255.3389(150)(10,727.65)	255.7538(200)(14,803)	254.8258(200)(14,318.947)
254.4347(200)(14,404.277)	254.8747(250)(18,055.384)	254.8549(250)(17,686.767)	254.0007*(250)(17,993.076)
Lowest Total System Cost (TabuLoop)(CPUTime)  254.0007*(250)(17,993.076)
Avg. Total System Cost (TabuLoop) (Avg. CPUTime)
262.1253(50)(3582.97)	261.01(100)(7178.26)	257.327(150)(10,708.56)	255.0048(200)(14,508.74)
254.5768(250)(17,911.74)

By applying the TA–MGA combined method for the principal n-stage q/2-out-of-q system, Table 27 above shows 20 selected simulated outcomes of the total system costs by assuming TabuLoop equal to 50 with an average CPU running time of 3582.97 s until 250 with an average CPU running time of 17,911.74 s, and it further converges the average total system cost of 262.1253 to 254.5768 and the total system cost of 262.3425 to the lowest 254.0007. Since the average total system cost of 255.0048 with 200 Tabu loops is just a little higher than 254.5768 with 250 Tabu loops, the simulation process stops running at 250 Tabu loops and finally achieves the lowest converged result of 254.0007. In addition, Figure 14 illustrates how the total system cost has improved and finally converged for the n-stage q/2-out-of-q system by conducting the proposed TA–MGA combined methods.

Additionally, for the above-mentioned lowest converged total system cost of 254.0007, which is obtained at 250 Tabu loops, Table 28 displays the optimal system cost related to the optimal failure rates, the repair rates, the availability of all subsystems, and the system availability, as follows:

Table 28. Optimal Solutions obtained by TA–MGA for the n-stage q/2-out-of-q system.

TabuLoop = 250 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	4	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2417	0.2132	0.1613	0.4915	0.2629
${{\displaystyle \overline{\mu }}}^{*}$	0.0018	0.0014	0.0008	0.0022	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9864	0.9830	0.9810	0.9667	0.9788
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          254.0007
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$                0.9000

Table 28. Optimal Solutions obtained by TA–MGA for the n-stage q/2-out-of-q system.

TabuLoop = 250 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	4	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2417	0.2132	0.1613	0.4915	0.2629
${{\displaystyle \overline{\mu }}}^{*}$	0.0018	0.0014	0.0008	0.0022	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9864	0.9830	0.9810	0.9667	0.9788
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          254.0007
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$                0.9000

Table 29. The total system costs of 20 simulation runs obtained by SA–MGA for the n-stage q/2-out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
283.9753(25)(4042.73)	283.563(25)(4044.1266)	262.1185(25)(4047.668)	261.4488(25)(3961.58)
260.0856(25)(3999.2586)	257.8223(25)(3954.55)	257.4265(25)(3968.1949)	261.1689(50)(8044.036)
257.6876(50)(7999.579)	256.4644(50)(7875.32)	254.2033(50)(7946.35)	253.2374(50)(7989.318)
253.0851(50)(8079.2985)	251.9111(50)(8077.339)	251.0012(100)(16,361.8)	247.9969(100)(16,218.2)
247.0783(100)(16,154.4)	245.7722(100)(15,967.8)	243.5175(150)(23,846.4)	242.8014*(150)(24,025.1)
Lowest Total System Cost (SAloop)(CPUTime)        242.8014*(150)(24,025.1)
Avg. Total System Cost (SAloop)(CPUTime)
266.6342857(25)(4002.6)	255.3939714(50)(8001.6)	247.96215(100)(16,175.5)	243.15943(150)(23,935.8)

Table 29. The total system costs of 20 simulation runs obtained by SA–MGA for the n-stage q/2-out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
283.9753(25)(4042.73)	283.563(25)(4044.1266)	262.1185(25)(4047.668)	261.4488(25)(3961.58)
260.0856(25)(3999.2586)	257.8223(25)(3954.55)	257.4265(25)(3968.1949)	261.1689(50)(8044.036)
257.6876(50)(7999.579)	256.4644(50)(7875.32)	254.2033(50)(7946.35)	253.2374(50)(7989.318)
253.0851(50)(8079.2985)	251.9111(50)(8077.339)	251.0012(100)(16,361.8)	247.9969(100)(16,218.2)
247.0783(100)(16,154.4)	245.7722(100)(15,967.8)	243.5175(150)(23,846.4)	242.8014*(150)(24,025.1)
Lowest Total System Cost (SAloop)(CPUTime)        242.8014*(150)(24,025.1)
Avg. Total System Cost (SAloop)(CPUTime)
266.6342857(25)(4002.6)	255.3939714(50)(8001.6)	247.96215(100)(16,175.5)	243.15943(150)(23,935.8)

By applying the SA–MGA combined method to continue the simulated numerical cases for the n-stage q/2-out-of-q system, Table 29 displays 20 picked simulated outcomes of the total system costs by presuming SAloop equal to 25 with an average CPU running time of 4002.587657 s until 150 with an average CPU running time of 23,935.75 s, and it further converges the average total system cost of 266.6342857 to 243.15943 and the total system cost of 283.9753 to the lowest 242.8014. As the average total system cost of 243.15943 with 150 SA loops is just a little lower than 247.96215 with 100 SA loops, the simulation process ceases at 150 SA loops and converges to the lowest result of 242.8014. Consequently, the convergence graph in Figure 15 also demonstrates the trend that the total system cost has improved and eventually converged for the principle n-stage q/2-out-of-q system by conducting the proposed SA–MGA combined method.

In addition, for the above lowest converged result of 242.8014, which is obtained at 150 SA loops, Table 30 shows the optimal system cost related to the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability, as follows:

Table 30. Optimal Solutions obtained by SA–MGA for the for the n-stage q/2-out-of-q system.

SAloop = 150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2225	0.2136	0.1470	0.5036	0.2538
${{\displaystyle \overline{\mu }}}^{*}$	0.0021	0.0014	0.0009	0.0023	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9904	0.9814	0.9818	0.9672	0.9750
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           242.8014
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$               0.9000

Table 30. Optimal Solutions obtained by SA–MGA for the for the n-stage q/2-out-of-q system.

SAloop = 150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2225	0.2136	0.1470	0.5036	0.2538
${{\displaystyle \overline{\mu }}}^{*}$	0.0021	0.0014	0.0009	0.0023	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9904	0.9814	0.9818	0.9672	0.9750
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           242.8014
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$               0.9000

Conducting the NESA–MGA combined method to simulate numerical cases for the n-stage q/2-out-of-q system, Table 31 demonstrates 20 incited simulated consequences of the total system costs by supposing NESAloop equal to 50, which average CPU running time is 1791.623921 s until 150, which average CPU running time is 5351.647619 s, and it further converges the average total system cost of 246.7436125 to 239.555425 and the total system cost of 267.1363 to the lowest 238.6284. As the average total system cost of 239.555425 with 150 NESA loops is pretty close to 239.7594875 with 100 SA loops, the simulation process halts at 150 NESA loops and converges to the lowest outcome of 238.6284.

Table 31. The total system costs of 20 simulation runs obtained by NESA–MGA for the n-stage q/2 out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (NESAloop) (CPU Time)
267.1363(50)(1815.2488)	266.8671(50)(1847.3193)	240.2965(50)(1748.9829)	240.2547(50)(1761.1064)
240.0845(50)(1815.1721)	240.083(50)(1799.07156)	239.7278(50)(1735.1919)	239.499(50)(1810.89833)
240.3252(100)(3765.378)	239.8942(100)(1716.0388)	239.8309(100)(3499.851)	239.755(100)(3623.509)
239.7221(100)(3639.3678)	239.7076(100)(3751.0742)	239.5324(100)(3512.8106)	239.3085(100)(3507.3925)
240.099(150)(5112.6764)	239.9187(150)(5415.5617)	239.5756(150)(5408.593)	238.6284*(150)(5469.7597)
Lowest Total System Cost (NESAloop)(CPUTime)     238.6284*(150)(5469.7597)
Avg. Total System Cost (NESAloop)(CPUTime)
246.7436125(50)(1791.623921)	239.7594875(100)(3376.927721)	239.555425(150)(5351.647619)

Table 31. The total system costs of 20 simulation runs obtained by NESA–MGA for the n-stage q/2 out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (NESAloop) (CPU Time)
267.1363(50)(1815.2488)	266.8671(50)(1847.3193)	240.2965(50)(1748.9829)	240.2547(50)(1761.1064)
240.0845(50)(1815.1721)	240.083(50)(1799.07156)	239.7278(50)(1735.1919)	239.499(50)(1810.89833)
240.3252(100)(3765.378)	239.8942(100)(1716.0388)	239.8309(100)(3499.851)	239.755(100)(3623.509)
239.7221(100)(3639.3678)	239.7076(100)(3751.0742)	239.5324(100)(3512.8106)	239.3085(100)(3507.3925)
240.099(150)(5112.6764)	239.9187(150)(5415.5617)	239.5756(150)(5408.593)	238.6284*(150)(5469.7597)
Lowest Total System Cost (NESAloop)(CPUTime)     238.6284*(150)(5469.7597)
Avg. Total System Cost (NESAloop)(CPUTime)
246.7436125(50)(1791.623921)	239.7594875(100)(3376.927721)	239.555425(150)(5351.647619)

Therefore, the convergence graph in Figure 16 clearly displays the trend that the total system cost will ameliorate and finally converge for the n-stage q/2-out-of-q system by implementing the proposed NESA–MGA combined method.

Additionally, for the above lowest converged outcome of 238.6284, which is obtained at 150 NESA loops, Table 32 illustrates the optimal system cost related to the optimal failure rate, the optimal repair rate, the availability of each subsystem, and the system availability, as follows:

Table 32. Optimal Solutions obtained by NESA–MGA for the for the n-stage q/2-out-of-q system.

SAloop = 150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2500	0.2303	0.1509	0.4489	0.2487
${{\displaystyle \overline{\mu }}}^{*}$	0.0014	0.0014	0.0010	0.0026	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9760	0.9801	0.9822	0.9776	0.9799
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$        238.6284
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$        0.9000

Table 32. Optimal Solutions obtained by NESA–MGA for the for the n-stage q/2-out-of-q system.

SAloop = 150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.2500	0.2303	0.1509	0.4489	0.2487
${{\displaystyle \overline{\mu }}}^{*}$	0.0014	0.0014	0.0010	0.0026	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9760	0.9801	0.9822	0.9776	0.9799
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$        238.6284
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$        0.9000

Table 33 and Table 34 display the simulation consequences obtained by the MRAH-MGA combined method for the n-stage q/2-out-of-q system, showing the total system costs of 20 simulation runs, the related average total system cost of 239.999455, and the lowest total system cost equal to 226.8125, which is superior to all other combined methods and has an extremely low average CPU running time of around 262.7216 s.

Table 33. The total system costs of 20 simulation runs obtained by MRAH–MGA for the n-stage q/2- out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
243.3626(266.64)	241.8375(261.93)	241.6838(258.7)	241.6701(256.99)	241.43(265.25)	240.9255(263.05)
240.9255(256.75)	240.9154(270.45)	240.8245(255.27)	240.7902(260.678)	240.6668(257)	240.2744(272.2)
239.8724(270.69)	239.872(283.1)	239.872(255.66)	239.6613(262.1)	239.5804(255.9)	239.506(263.87)
239.5061(258.029)	226.8125*(260.12)
Lowest Total System Cost (CPU Time)        226.8125*(260.12)
Avg. Total System Cost (Avg. CPU Time)       239.999455(262.7216)

Table 33. The total system costs of 20 simulation runs obtained by MRAH–MGA for the n-stage q/2- out-of-q system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
243.3626(266.64)	241.8375(261.93)	241.6838(258.7)	241.6701(256.99)	241.43(265.25)	240.9255(263.05)
240.9255(256.75)	240.9154(270.45)	240.8245(255.27)	240.7902(260.678)	240.6668(257)	240.2744(272.2)
239.8724(270.69)	239.872(283.1)	239.872(255.66)	239.6613(262.1)	239.5804(255.9)	239.506(263.87)
239.5061(258.029)	226.8125*(260.12)
Lowest Total System Cost (CPU Time)        226.8125*(260.12)
Avg. Total System Cost (Avg. CPU Time)       239.999455(262.7216)

Table 34. Optimal Solutions obtained by MRAH–MGA for the n-stage q/2-out-of-q system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3184	0.2656	0.2125	0.4842	0.3759
${{\displaystyle \overline{\mu }}}^{*}$	0.0020	0.0015	0.0009	0.0023	0.0018
${{\displaystyle \overline{A}}}^{*}$	0.9922	0.9626	0.9886	0.9635	0.9932
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           226.8125
System Availability ${{\displaystyle A}}_S^{*}$                 0.903544

Table 34. Optimal Solutions obtained by MRAH–MGA for the n-stage q/2-out-of-q system.

Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	2	2	2	2	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3184	0.2656	0.2125	0.4842	0.3759
${{\displaystyle \overline{\mu }}}^{*}$	0.0020	0.0015	0.0009	0.0023	0.0018
${{\displaystyle \overline{A}}}^{*}$	0.9922	0.9626	0.9886	0.9635	0.9932
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           226.8125
System Availability ${{\displaystyle A}}_S^{*}$                 0.903544

Additionally, Figure 17 demonstrates that the total system cost is improved and converged for the main n-stage q/2-out-of-q system by executing the proposed MRAH–MGA combined method.

4.5. The Simulation Case for n-Stage Mixed System

In the previous four sections, the proposed availability optimization decision support design system has designed four individual systems, including an n-stage single component system, n-stage standby system, parallel-series system, and n-stage q/2-out-of-q system, which all have subsystems assumed to be the same configuration. Yet, in the real world, the actual system is generally complicated and comprises various subsystems with different configurations. Next, the availability optimization decision support design system aims to design the n-stage mixed systems with different combinations of subsystems, including a standby, parallel, and q/2-out-of-q connected in series configuration. This study will further conduct the simulation cases by adopting the ENUM-MGA, TA–MGA, SA–MGA, NESA–MGA, and MRAH–MGA combined methods for one specific n-stage mixed system, which comprises two parallel subsystems, two standby subsystems, and one q/2-out-of-q subsystem.

Table 35. The total system costs of 20 simulation runs obtained by ENUM–MGA for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (CPU Time)
229.507(3070.428)	229.507(3141.03)	229.349(3043.59)	229.349(3160.6)	228.871(3062.2)	228.871(3199.52)
228.523(3164.76)	228.494(3133.18)	228.494(3297.81)	228.41(3103.03)	228.343(3035.475)	228.343(3133.74)
228.239(3069.49)	228.239(3162.49)	228.199(3019.39)	228.107(3130.18)	227.762(3012.51)	227.762(3143.59)
227.224(3086.66)	227.164(3072.13)
Lowest Total System Cost (CPUTime)        227.164*(3072.13)
Avg. Total System Cost (Avg. CPUTime)      228.43785(3112.091)

and

Table 36. Optimal Solutions obtained by ENUM–MGA combined method for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	2	2	3	3
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.313036	0.220666	0.153684	0.593748	0.324968
${{\displaystyle \overline{\mu }}}^{*}$	0.0012619	0.00134002	0.00083322	0.00140642	0.00100508
${{\displaystyle \overline{A}}}^{*}$	0.992148	0.977247	0.97575	0.973842	0.976876
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$      227.164
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$      0.900009

show the simulation results of the ENUM–MGA combined method. Table 37 and Table 38 display the simulation outcomes produced by the TA–MGA combined method. Table 39 and Table 40 demonstrate the simulation results generated by the SA–MGA combined method. Table 41 and Table 42 show the simulation outcomes created by the NESA–MGA combined method. Table 43 and Table 44 display the simulation outcomes resulting from the MRAH–MGA combined method.

Regarding the simulation cases of the n-stage mixed system, the numerical results obtained by the ENUM-MGA combined method is illustrated in the following Table 34 and Table 36 and Figure 18. Table 35 shows the total system costs of 20 simulation runs, in which the average total system cost equals 228.43785 and the lowest total system cost equals 227.164. The convergence graph in Figure 18 denotes that the total system cost has improves and converged for the n-stage mixed system by enforcing the proposed ENUM–MGA combined method. Furthermore, Table 35 demonstrates the optimal system cost, the optimal failure rates, the optimal repair rates, the availability of all subsystem, and the system availability for the principal simulation case of the n-stage mixed system.

By adopting the TA–MGA combined method for the n-stage mixed system, 20 picked simulated results of the total system costs are displayed in the following Table 37 by setting TabuLoop equal to 50 with an average CPU running time of 2207.1796 s until 250 with an average CPU running time of 11,108.103 s, and they are further converged from the average total system cost of 228.8267 to 221.294 and from the total system cost of 229.1124 to the lowest 221.1764. Since the average total system cost of 221.9263083 with 200 Tabu loops is just a little higher than 221.294 with 250 Tabu loops, the simulation process stops running at 250 Tabu loops and finally achieves the lowest converged result of 221.1764.

Table 37. The total system costs of 20 simulation runs acquired by TA–MGA combined method for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
229.1124(50)(2227.49)	228.541(50)(2186.86)	225.9619(100)(4345.43)	225.0738(100)(4470.18)
223.2599(150)(6641.27)	222.8208(150)(6655.76)	222.5571(200)(9621.79)	222.1876(200)(9335.25)
221.9989(200)(9412.08)	221.9579(200)(9410.79)	221.9419(200)(9498.078)	221.8878(200)(9814.1)
221.8746(200)(8811.95)	221.8395(200)(9169.73)	221.8395(200)(8834.7)	221.7808(200)(8776.96)
221.6745(200)(10,261.2)	221.5756(200)(8838.7)	221.4116(250)(11,185)	221.1764*(250)(11,031.066)
Lowest Total System Cost(TabuLoop)(CPUTime) 221.1764*(250)(11,031.066)
Avg. Total System Cost (TabuLoop)(CPUTime)
228.8267(50)(2207.1796)	225.51785(100)(4407.8031)	223.04035(150)(6648.5153)	221.9263083(200)(9315.444)
221.294(250)(11,108.103)

Table 37. The total system costs of 20 simulation runs acquired by TA–MGA combined method for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (TabuLoop) (CPU Time)
229.1124(50)(2227.49)	228.541(50)(2186.86)	225.9619(100)(4345.43)	225.0738(100)(4470.18)
223.2599(150)(6641.27)	222.8208(150)(6655.76)	222.5571(200)(9621.79)	222.1876(200)(9335.25)
221.9989(200)(9412.08)	221.9579(200)(9410.79)	221.9419(200)(9498.078)	221.8878(200)(9814.1)
221.8746(200)(8811.95)	221.8395(200)(9169.73)	221.8395(200)(8834.7)	221.7808(200)(8776.96)
221.6745(200)(10,261.2)	221.5756(200)(8838.7)	221.4116(250)(11,185)	221.1764*(250)(11,031.066)
Lowest Total System Cost(TabuLoop)(CPUTime) 221.1764*(250)(11,031.066)
Avg. Total System Cost (TabuLoop)(CPUTime)
228.8267(50)(2207.1796)	225.51785(100)(4407.8031)	223.04035(150)(6648.5153)	221.9263083(200)(9315.444)
221.294(250)(11,108.103)

Moreover, the convergence graph in Figure 19 illustrates how the total system cost has been reformed and at last converged for the n-stage mixed system by implementing the proposed TA–MGA combined method.

Regarding the above lowest optimal total system cost of 221.1764, which is converged at 250 TA loops, Table 38 delineates the detailed information, including the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability.

Table 38. Optimal Solutions obtained by TA–MGA combined method for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3511	0.2778	0.1402	0.6599	0.2538
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0010	0.0009	0.0015	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9838	0.9846	0.9816	0.9716	0.9743
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$       221.1764
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$        0.9000

Table 38. Optimal Solutions obtained by TA–MGA combined method for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3511	0.2778	0.1402	0.6599	0.2538
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0010	0.0009	0.0015	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9838	0.9846	0.9816	0.9716	0.9743
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$       221.1764
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$        0.9000

By employing the SA–MGA combined method to execute the simulated numerical examples for the n-stage mixed system, the following Table 39 displays 21 picked simulated results of the total system costs by supposing SAloop equal to 50 with an average CPU time of 4050.163 s until 150 with an average CPU time of 14,840.777 s, and it further converges the average total system cost of 228.4432 to 222.2532 and the total system cost of 229.2389 to the lowest 222.2532. For the case when the average total system cost of 222.2532 with 150 SA loops is lower than 222.2608 with 200 SA loops, the simulation process ceases at 150 SA loops and converges to the lowest total system cost of 222.2532.

Table 39. The total system costs of 21 simulation runs obtained by SA–MGA for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
229.2389(50)(4970.83)	228.4881(50)(2227.22)	227.6027(50)(4952.435)	225.5595(100)(9425.54)
225.5595(100)(10,604.14)	224.8517(100)(9812.857)	224.7499(100)(9550.08)	224.4633(100)(9920.997)
224.4258(100)(9953.296)	224.3913(100)(9775.858)	224.0892(100)(9659.037)	224.0892(100)(10,342.6)
224.0509(100)(9603.48)	224.0509(100)(10,283.213)	224.0485(100)(9558.37)	224.0485(100)(9420.227)
223.698(100)(9663.32)	223.698(100)(16,025.8)	222.2665(150)(14,807.37)	222.2532*(150)(14,874.18)
222.2608(200)(20,724.566)
Lowest Total System Cost (SAloop)(CPUTime)    222.2532*(150)(14,874.18)
Avg. Total System Cost (SAloop)(CPUTime)
228.4432(50)(4050.163)	224.3849467(100)(224.384947)	222.25985(150)(14,840.777)

Table 39. The total system costs of 21 simulation runs obtained by SA–MGA for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (SAloop) (CPU Time)
229.2389(50)(4970.83)	228.4881(50)(2227.22)	227.6027(50)(4952.435)	225.5595(100)(9425.54)
225.5595(100)(10,604.14)	224.8517(100)(9812.857)	224.7499(100)(9550.08)	224.4633(100)(9920.997)
224.4258(100)(9953.296)	224.3913(100)(9775.858)	224.0892(100)(9659.037)	224.0892(100)(10,342.6)
224.0509(100)(9603.48)	224.0509(100)(10,283.213)	224.0485(100)(9558.37)	224.0485(100)(9420.227)
223.698(100)(9663.32)	223.698(100)(16,025.8)	222.2665(150)(14,807.37)	222.2532*(150)(14,874.18)
222.2608(200)(20,724.566)
Lowest Total System Cost (SAloop)(CPUTime)    222.2532*(150)(14,874.18)
Avg. Total System Cost (SAloop)(CPUTime)
228.4432(50)(4050.163)	224.3849467(100)(224.384947)	222.25985(150)(14,840.777)

Consequently, the convergence graph in Figure 20 demonstrates the trend that the total system cost has improved and in the end converged for the n-stage mixed system by utilizing the proposed SA–MGA combined method.

In addition, for the above lowest converged result of 222.2532, which is acquired at 150 SA loops, Table 40 describes the detailed information, including the optimal system cost, the optimal failure rates, the optimal repair rates, the availability of all subsystems, and the system availability, as follows:

Table 40. Optimal Solutions obtained by SA–MGA for the for the n-stage mixed system.

SAloop =150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3409	0.2825	0.1563	0.6524	0.2629
${{\displaystyle \overline{\mu }}}^{*}$	0.0011	0.0010	0.0010	0.0014	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9872	0.9839	0.9823	0.9668	0.9756
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          222.2532
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$               0.9000

Table 40. Optimal Solutions obtained by SA–MGA for the for the n-stage mixed system.

SAloop =150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3409	0.2825	0.1563	0.6524	0.2629
${{\displaystyle \overline{\mu }}}^{*}$	0.0011	0.0010	0.0010	0.0014	0.0015
${{\displaystyle \overline{A}}}^{*}$	0.9872	0.9839	0.9823	0.9668	0.9756
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          222.2532
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$               0.9000

Employing the NESA–MGA combined method to simulate numerical cases of the n-stage mixed system, Table 41 illustrates 20 chosen simulated outcomes of the total system costs by assuming NESAloop equal to 50, which an average CPU running time is 1104.221 s until 200, which an average CPU running time is 4383.1 s, and it further converges the average total system cost of 224.20025 to 221.49795 and the total system cost of 224.841 to the lowest 221.4456. For the reason that the average total system cost of 221.99705 with 150 NESA loops is pretty close to 221.49795 with 200 NESA loops, the simulation process halts at 200 NESA loops and converges to the lowest outcome of 221.4456.

Table 41. The total system costs of 20 simulation runs obtained by NESA–MGA for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (NESAloop) (CPU Time)
224.841(50)(1132.647)	223.5595(50)(1075.794)	224.1458(100)(2100.519)	224.0577(100)(2051.946)
223.2254(100)(2173.69)	223.2249(100)(2084.81)	223.1785(100)(2073.37)	223.1368(100)(1930.83)
223.1191(100)(2039.28)	223.0354(100)(1963.96)	222.8479(100)(2072.59)	222.7554(100)(2108.03)
222.7554(100)(2108.03)	222.7554(100)(2101.82)	222.6034(100)(2112.59)	222.4549(100)(1987.02)
222.4549(100)(2005.96)	222.5399(150)(3335.04)	221.4542(150)(3330.326)	221.5503(200)(4490.57)
221.4456*(200)(4275.63)
Lowest Total System Cost (NESAloop)(CPUTime)   221.4456*(200)(4275.63)
Avg. Total System Cost (NESAloop)(CPUTime)
224.20025(50)(1104.221)	223.07111(100)(2057.601)	221.99705(150)(3332.682)	221.49795(200)(4383.1)

Table 41. The total system costs of 20 simulation runs obtained by NESA–MGA for the n-stage mixed system.

cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Total System Cost (NESAloop) (CPU Time)
224.841(50)(1132.647)	223.5595(50)(1075.794)	224.1458(100)(2100.519)	224.0577(100)(2051.946)
223.2254(100)(2173.69)	223.2249(100)(2084.81)	223.1785(100)(2073.37)	223.1368(100)(1930.83)
223.1191(100)(2039.28)	223.0354(100)(1963.96)	222.8479(100)(2072.59)	222.7554(100)(2108.03)
222.7554(100)(2108.03)	222.7554(100)(2101.82)	222.6034(100)(2112.59)	222.4549(100)(1987.02)
222.4549(100)(2005.96)	222.5399(150)(3335.04)	221.4542(150)(3330.326)	221.5503(200)(4490.57)
221.4456*(200)(4275.63)
Lowest Total System Cost (NESAloop)(CPUTime)   221.4456*(200)(4275.63)
Avg. Total System Cost (NESAloop)(CPUTime)
224.20025(50)(1104.221)	223.07111(100)(2057.601)	221.99705(150)(3332.682)	221.49795(200)(4383.1)

Accordingly, the convergence graph in Figure 21 displays the trend that the total system cost of the n-stage mixed system can be bettered and eventually converged by utilizing the proposed NESA–MGA combined method.

Regarding the above lowest optimal total system cost of 221.4456, which is obtained at 200 NESA loops, the following Table 42 delineates the complete information, including the optimal total system cost, the optimal failure rates, the optimal repair rates, the availability for all subsystems, and the system availability.

Table 42. Optimal Solutions obtained by NESA–MGA for the for the n-stage mixed system.

SAloop = 150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3670	0.2758	0.1482	0.5969	0.2498
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0010	0.0011	0.0013	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9818	0.9839	0.9855	0.9701	0.9746
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          221.4456
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$              0.9000

Table 42. Optimal Solutions obtained by NESA–MGA for the for the n-stage mixed system.

SAloop = 150 cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3670	0.2758	0.1482	0.5969	0.2498
${{\displaystyle \overline{\mu }}}^{*}$	0.0010	0.0010	0.0011	0.0013	0.0014
${{\displaystyle \overline{A}}}^{*}$	0.9818	0.9839	0.9855	0.9701	0.9746
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$          221.4456
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$              0.9000

According to Table 42, the simulation outcomes acquired by the MRAH-MGA combined method for the n-stage mixed system are classified into two modes based on the different parameter design for the modified genetic algorithm. Mode 1 applies the originally proposed parameter design for the modified genetic algorithm—which assumes a chromosome crossover rate equal to 0.95, a chromosome mutation rate equal to 0.5, a chromosome population equal to 200, and a genetic generation equal to 200—to obtain the total system costs of 20 simulation runs with the average total system cost of 223.72877 and the lowest total system cost equal to 222.0561. The outcomes acquired from mode 1 are worse than the other combined methods, except for the ENUM-MGA and SA-MGA combined methods. Therefore, mode 2 adopts the other parameter design for the modified genetic algorithm—which assumes a chromosome crossover rate equal to 0.95, a chromosome mutation rate equal to 0.05, a chromosome population equal to 50, and a genetic generation equal to 50—to obtain the total system costs of 20 simulation runs with the average total system cost of 234.300425 and the lowest total system cost equal to 216.977. Comparing the performance between mode 1 and mode 2, the lowest total system cost of mode 2 is apparently lower than that of mode 1, but the average total system cost of mode 2 is higher than that of mode 1. The main reason for this is that the variance of the total system cost acquired from mode 2 of 90.67164485 is much bigger than that of mode 1 at 1.346792558. Therefore, even when the average total system cost of mode 2 is bigger than that of mode 1, there still exists the simulation result, in which the lowest total system cost of mode 2 is significantly lower than that of mode 1 and the other four combined methods, and it has an extremely low CPU running time of around 7.691521 s. The convergence graph in Figure 22 further displays the trends of the total system costs of the n-stage mixed system by implementing the proposed mode 1 MRAH–MGA combined method and the mode 2 MRAH–MGA combined method.

Table 43. The total system costs of 40 simulation runs obtained by MRAH–MGA for the n-stage mixed system.

Mode 1: cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Mode 2: cr-rate = 0.95, mu-rate = 0.05, GApopulation = 50, GAloop = 50
Total System Cost (CPU Time)(Mode)
225.6124(117.52)(1)	225.4198(117.01)(1)	225.0842(116.427)(1)	225.0159(116.804)(1)	225.0068(117.214)(1)
224.7567(119.738)(1)	224.4263(111.36)(1)	223.9598(112.564)(1)	223.8361(118.652)(1)	223.7809(110.09)(1)
223.6381(109.987)(1)	223.5414(117.902(1)	223.1572(117.8)(1)	223.0616(117.35)(1)	222.9696(115.826)(1)
222.6717(117.809)(1)	222.4027(112.205)(1)	222.1095(116.968)(1)	222.0686(110.17)(1)	222.0561(114.55)(1)
251.287(7.585734)(2)	246.9526(7.664593)(2)	243.9434(7.705879)(2)	243.6529(7.59)(2)	242.9158(7.652)(2)
242.087(7.599641)(2)	240.1311(7.584795)(2)	239.9994(15.7473)(2)	234.6201(9.6669)(2)	232.8847(7.748)(2)
232.8333(7.847)(2)	232.7708(7.614695)(2)	232.3982(7.707281)(2)	229.2715(7.6298)(2)	228.2998(7.585)(2)
28.2592(7.599764)(2)	227.8516(8.261175)(2)	220.0629(7.555221)(2)	218.8102(7.59)(2)	216.977(7.692)(2)
Lowest Total System Cost(CPU)(mode)    222.0561(114.55)(1)         216.977*(7.691521)(2)
Avg. Total System Cost(CPU)(mode)        223.72877*(115.39779)(1)    234.300425(8.1814052)(2)
Var. Total System Cost(mode)            1.346792558*(1) 90.67164485(2)

Table 43. The total system costs of 40 simulation runs obtained by MRAH–MGA for the n-stage mixed system.

Mode 1: cr-rate = 0.95 mu-rate = 0.5 GApopulation = 200 GAloop = 200
Mode 2: cr-rate = 0.95, mu-rate = 0.05, GApopulation = 50, GAloop = 50
Total System Cost (CPU Time)(Mode)
225.6124(117.52)(1)	225.4198(117.01)(1)	225.0842(116.427)(1)	225.0159(116.804)(1)	225.0068(117.214)(1)
224.7567(119.738)(1)	224.4263(111.36)(1)	223.9598(112.564)(1)	223.8361(118.652)(1)	223.7809(110.09)(1)
223.6381(109.987)(1)	223.5414(117.902(1)	223.1572(117.8)(1)	223.0616(117.35)(1)	222.9696(115.826)(1)
222.6717(117.809)(1)	222.4027(112.205)(1)	222.1095(116.968)(1)	222.0686(110.17)(1)	222.0561(114.55)(1)
251.287(7.585734)(2)	246.9526(7.664593)(2)	243.9434(7.705879)(2)	243.6529(7.59)(2)	242.9158(7.652)(2)
242.087(7.599641)(2)	240.1311(7.584795)(2)	239.9994(15.7473)(2)	234.6201(9.6669)(2)	232.8847(7.748)(2)
232.8333(7.847)(2)	232.7708(7.614695)(2)	232.3982(7.707281)(2)	229.2715(7.6298)(2)	228.2998(7.585)(2)
28.2592(7.599764)(2)	227.8516(8.261175)(2)	220.0629(7.555221)(2)	218.8102(7.59)(2)	216.977(7.692)(2)
Lowest Total System Cost(CPU)(mode)    222.0561(114.55)(1)         216.977*(7.691521)(2)
Avg. Total System Cost(CPU)(mode)        223.72877*(115.39779)(1)    234.300425(8.1814052)(2)
Var. Total System Cost(mode)            1.346792558*(1) 90.67164485(2)

In light of the lowest converged result of 216.977, which is obtained by mode 2 of the MRAH-MGA combined method, Table 44 delineates the complete information comprising the optimal component redundancy amounts, the optimal failure rates, the optimal repair rates, the availability for all subsystems, the system availability, and the optimal total system cost, as follows:

Table 44. Optimal Solutions obtained by mode 2 of MRAH–MGA for the for the n-stage mixed system.

cr-rate = 0.95, mu-rate = 0.05, GApopulation = 50, GAloop = 50
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3515	0.3115	0.2880	0.5268	0.3496
${{\displaystyle \overline{\mu }}}^{*}$	0.0009	0.0010	0.0016	0.0012	0.0017
${{\displaystyle \overline{A}}}^{*}$	0.9805	0.9951	0.9898	0.9594	0.9800
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           216.977
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$             0.90806

Table 44. Optimal Solutions obtained by mode 2 of MRAH–MGA for the for the n-stage mixed system.

cr-rate = 0.95, mu-rate = 0.05, GApopulation = 50, GAloop = 50
Decision Variables			Subsystem(i)
	1	2	3	4	5
${{\displaystyle \overline{Q}}}^{*}$	3	3	2	3	2
${{\displaystyle \overline{f}}}^{*}$(10−3)	0.3515	0.3115	0.2880	0.5268	0.3496
${{\displaystyle \overline{\mu }}}^{*}$	0.0009	0.0010	0.0016	0.0012	0.0017
${{\displaystyle \overline{A}}}^{*}$	0.9805	0.9951	0.9898	0.9594	0.9800
Optimal Total System Cost ${{\displaystyle C}}_S^{*}$           216.977
$\mathrm{System}\mathrm{Availability}{{\displaystyle A}}_S^{*}$             0.90806

4.6. The Comparison of Five Proposed System Configurations

According to the results from Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25, Table 26, Table 27, Table 28, Table 29, Table 30, Table 31, Table 32, Table 33, Table 34, Table 35, Table 36, Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43 and Table 44, the optimal total system cost and CPU running time based on the four proposed system configurations and the five proposed combined methods are further collected and rearranged as the following

Table 45. The optimal total system cost and CPU running time based on four system configurations and five combined methods.

	ENUM-MGA	TA-MGA	SA-MGA	NESA-MGA	MRAH-MGA
n-stage standby	243.209(2697.3)	236.831(7641)	236.46(21,232.93)	237.27(9441.5)	226.450*(5.31)
parallel-series	220.278(2696.55)	214.62(9361)	214.7569(8182.35)	214.9928(12,864)	208.9659*(98.64)
n-stage q/2-out-of-q	254.75(4764)	254(17,993)	242.8014(24,025)	238.6284(5469.76)	226.8125*(260)
n-stage mixed	227.164(3072)	221.1764(11,031)	222.2532(14,874)	221.4456(4275.6)	216.977*(7.69)
GA n-stage single component series system      545.489(521.152958)

.

According to the results of Table 45, all optimal total system costs obtained by the ENUM-MGA, TS-MGA, SA-MGA, NESA-MGA, and MRAH-MGA combined methods for the n-stage standby system, parallel-series system, n-stage q/2-out-of-q system, and n-stage mixed system are comprehensively much better than the optimal total system cost of 545.489 obtained by the genetic algorithm for the n-stage single component series system. Therefore, it is demonstrated clearly that the proposed availability optimization decision support design system can easily apply different component redundancy system designs to save a lot of cost and reach the high standard of the system availability requirement compared to the n-stage single component series system.

Additionally, from the outcomes in Table 45, all optimal total system costs obtained by the ENUM-MGA, TA-MGA, SA-MGA, NESA-MGA, and MRAH-MGA combined methods for the parallel-series system are apparently lower than those obtained for the n-stage standby system. Notably, the difference between the standby subsystem and the parallel subsystem is that for the parallel subsystem, all of the components are in operational condition, but for the standby subsystem, only one component is in operational condition and all of the other components are in standby condition. Since this study assumes the same deteriorating probability for operating components and for standby components, the parallel subsystem can clearly meet the same level of availability requirement with less cost compared to the standby subsystem. This is why in this study, the optimal total system cost for the parallel-series system is lower than the n-stage standby system. However, in the real world, the deteriorating situation of the standby component should be better than the operating component. This could lead to the cost saving of the standby subsystem to cover the original advantage of the parallel subsystem in order to easily reach high-level availability.

Moreover, Table 45 also shows that all of the optimal total system costs obtained by the ENUM-MGA, TA-MGA, SA-MGA, NESA-MGA, and MRAH-MGA combined methods for the n-stage q/2-out-of-q system are higher than those obtained for the n-stage standby system and parallel-series system, which can be verified easily from the aspect of system configuration structure and system availability theory.

Finally, the proposed n-stage mixed system in this study comprises two parallel subsystems, two standby subsystems, and one q/2-out-of-q subsystem. Therefore, Table 45 clearly displays that all of the optimal total system costs acquired by the ENUM-MGA, TA-MGA, SA-MGA, NESA-MGA, and MRAH-MGA combined methods for this n-stage mixed system are higher than the parallel-series system, but lower than the n-stage standby system and the n-stage q/2-out-of-q system.

4.7. The Performance Comparison of Five Proposed Combined Methods

Based on the outcomes from Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25, Table 26, Table 27, Table 28, Table 29, Table 30, Table 31, Table 32, Table 33, Table 34, Table 35, Table 36, Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43, Table 44 and Table 45, the performance comparisons of the five proposed combined methods for the four proposed system configurations are illustrated and analyzed completely as follows.

Comparing the performance of the five combined methods for the n-stage standby system, the optimal system cost is 243.2092 for the ENUM-MGA combined method, 236.831 for the TS-MGA combined method, 236.463 for the SA-MGA combined method, 237.27 for the NESA-MGA combined method, and 226.45 for the MRAH -MGA combined method. The performance ranking regarding the criteria of the optimal total system cost has the sequence MRAH-MGA, SA-MGA, TS-MGA, NESA-MGA, ENUM-MGA. Moreover, the CPU running time is 2697.3 s for the ENUM-MGA combined method, 7641 s for the TA–MGA combined method, 21,232.9 s for the SA–MGA combined method, 9441.45 s for the NESA–MGA combined method, and 5.31 s for the MRAH–MGA combined method. Therefore, the performance ranking regarding the criteria of CPU running time has the order MRAH-MGA, ENUM-MGA, TA-MGA, NESA-MGA, SA-MGA. Consequently, for the n-stage standby system, the optimal total system cost obtained by conducting the MRAH-MGA combined method is superior to all of the other combined methods, and it also has an extreme and lowest CPU running time of around 5.31 s.

To compare the performance of the five combined methods for the parallel-series system, the optimal total system cost is 220.278 for the ENUM-MGA combined method, 214.62 for the TS-MGA combined method, 214.7569 for the SA-MGA combined method, 214.9928 for the NESA-MGA combined method, and 208.9659 for the MRAH-MGA combined method. The performance ranking regarding the criteria of the optimal total system cost is ranked as the sequence MRAH-MGA, TS-MGA, SA-MGA, NESA-MGA, ENUM-MGA. Moreover, the CPU running time is 2696.55 s for the ENUM-MGA combined method, 9361 s for the TS-MGA combined method, 8182.35 s for the SA–MGA combined method, 12,864.215 s for the NESA-MGA combined method, and 98.64 s for the MRAH–MGA combined method. Therefore, the performance ranking regarding the criteria of CPU running time has the order MRAH-MGA, ENUM-MGA, SA-MGA, TS-MGA, NESA-MGA. Consequently, for the parallel-series system, the optimal total system cost obtained by conducting the MAHA-GA combined method is superior to all of the other combined methods, and it also has an extreme and lowest CPU running time of around 98.64 s.

Concerning the performance of the five different combined methods for the n-stage q/2-out-of-q system, the optimal system cost is 254.75 for the ENUM-MGA combined method, 254.0007 for the TA–MGA combined method, 242.8014 for the SA–MGA combined method, 238.6284 for the NESA–MGA combined method, and 226.8125 for the MRAH–MGA combined method. As a result, the performance ranking concerning the criteria of the optimal total system cost has the order MRAH-MGA, NESA-MGA, SA-MGA, TS-MGA, ENUM-MGA. Additionally, the CPU running time is about 4763.557856 s for the ENUM-MGA combined method, 17,993.076 s for the TA–MGA combined method, 24,025.088 s for the SA–MGA combined method, 5469.7597 s for the NESA–MGA combined method, and 260.12 s for the MRAH-MGA combined method. Therefore, the performance ranking for the criteria of CPU running time has the sequence MRAH-MGA, ENUM-MGA, NESA-MGA, TA-MGA, SA-MGA. Accordingly, with regard to the n-stage q/2-out-of-q system, the optimal total system cost obtained by conducting the MRAH-MGA combined method is evidently better than all of the other combined methods, and it also has the lowest CPU running time of 98.64 s.

Regarding the performance of the five different combined methods for the n-stage mixed system, the optimal system cost is 227.164 for the ENUM-MGA combined method, 221.1764 for the TS-MGA combined method, 222.2532 for the SA-MGA combined method, 221.4456 for the NESA-MGA combined method, and 216.977 for mode 1 of the MRAH-MGA combined method. Therefore, the performance ranking concerning the criteria of the optimal total system cost has the ranked mode 1 sequence MRAH-MGA, TS-MGA, NESA-MGA, SA-MGA, ENUM-MGA. In addition, the CPU running time is about 3072.13 s for the ENUM-MGA combined method, 11,031.066 s for the TS-MGA combined method, 14,874.18 s for the SA-MGA combined method, 4275.63 s for the NESA-MGA combined method, and 7.691521 s for the MRAH-MGA combined method. Therefore, the performance ranking for the criteria of CPU running time has the sequence MRAH-MGA, ENUM-MGA, NESA-MGA, TS-MGA, SA-MGA. Accordingly, regarding the n-stage mixed system, the optimal total system cost obtained by conducting mode 1 of the MRAH-MGA combined method is significantly better than all of the other combined methods, it also has the lowest CPU running time of 7.691521 s.

Concluding the above comparison results, the following

Table 46. The performance ranking of the optimal total system cost and CPU running time for five combined methods based on four system configurations.

	ENUM-MGA	TA-MGA	SA-MGA	NESA-MGA	MRAH-MGA
(Ranking of the Optimal Total System Cost)(Ranking of CPU Running Time)
n-stage standby	(5)(2)	(3)(3)	(2)(5)	(4)(4)	(1)(1)
parallel-series	(5)(2)	(2)(4)	(3)(3)	(4)(5)	(1)(1)
n-stage q/2-out-of-q	(5)(2)	(4)(4)	(3)(5)	(2)(3)	(1)(1)
n-stage mixed	(5)(2)	(2)(4)	(4)(5)	(3)(3)	(1)(1)
Total ranking index	(20)(8)	(11)(15)	(12)(18)	(13)(15)	(4)(4)

and

Table 47. The relative comparison indices for the optimal total system cost and CPU running time.

	ENUM-MGA	TA-MGA	SA-MGA	NESA-MGA	MRAH-MGA
(Index of the Optimal Total System Cost)(Index of CPU Running Time)
n-stage standby	(1.074)(507.966)	(1.0458)(1438.983)	(1.0442)(3998.669)	(1.0478)(1778.06)	(1)(1)
parallel-series	(1.0541)(27.337)	(1.027)(94.9006)	(1.0277)(82.952)	(1.0288)(130.4136)	(1)(1)
n-stage q/2-out-of-q	(1.1232)(18.323)	(1.1199)(69.2038)	(1.0705)(92.4038)	(1.0521)(21.0375)	(1)(1)
n-stage mixed	(1.0469)(399.4798)	(1.0193)(1434.46)	(1.0243)(1934.2)	(1.0206)(55.995)	(1)(1)
Avg.	(1.0746)(238.276)	(1.053)(759.387)	(1.0417)(1527.0562)	(1.0373)(496.3765)	(1)(1)

demonstrate the detailed performance ranking and the relative comparison indices of the optimal total system cost and CPU running time for the five proposed combined methods based on the four proposed system configurations.

Summarizing the performances of the optimal total system cost in Table 46, the MRAH-MGA combined method ranks in the top spot for all proposed four system configurations; the TA-MGA combined method ranks in the second spot for two proposed system configurations, the third spot for one proposed system configuration, and the fourth spot for one proposed system configuration; the SA-MGA combined method ranks in the second spot for one proposed system configuration, the third spot for two system configurations, the second spot for one proposed system configuration, the third spot for two proposed system configurations, and the fourth spot for one system configuration; the NESA-MGA combined method ranks in the second spot for one system configuration, the third spot for one system configuration, and the fourth spot for two system configurations; and the ENUM-MGA combined method ranks last for all system configurations. Furthermore, from the view of the total performance ranking of the optimal system cost, the MRAH–MGA combined method (4) is in the first spot, the TS–GA combined method (11) is in the second spot, the SA-MGA combined method (12) is in the third spot, the NESA-MGA combined method (13) is in the fourth spot, and the ENUM-MGA combined method (20) is in the last spot. Therefore, it is evident that the performance of the optimal system cost for the MRAH–MGA combined method significantly surpasses the other four combined methods.

In addition, regarding to the performances of the CPU running time in Table 46, the MRAH-MGA combined method ranks in first place for all system configurations; the ENUM-MGA combined method ranks in second place for all system configurations; the NESA-MGA combined method ranks in third place for two system configurations, second place for one system configuration, and fourth place for one system configuration; the TS-MGA combined method ranks in third place for one system configuration and fourth place for three system configurations; and the SA-MGA combined method ranks in third place for one system configuration and the last place for three system configurations. Moreover, in terms of the total performance ranking of CPU running time, the MRAH–MGA combined method (4) is in first place, the ENUM–MGA combined method (8) is in second place, the TS-MGA combined method (15) and NESA-MGA combined method (13) are both in third place, and the SA-MGA combined method (18) is in last place. Therefore, it is evident that the performance of the CPU running time for the MRAH-MGA combined method is largely superior to the other four combined methods.

The relative compared indices for the optimal total system cost and CPU running time is determined as the ratio 1 for the lowest outcome, and the ratios for the other four combined methods are their values divided by the value of the lowest outcome. Table 47 expresses the values for all five proposed combined methods. In Table 47, the average relative compared index for the optimal total system cost acquired by MRAH-MGA is 1, NESA-MGA is 1.0373, SA-MGA is 1.0417, ENUM-MGA is 1.0469, and TS-MGA is 1.053. The performance of the optimal total system cost from the MRAH-MGA combined method is almost 3.73% superior to NESA-MGA, 4.17% superior to SA-MGA, and 4.69% superior to ENUM-MGA. Furthermore, the average relative compared index for the CPU running time from the MRAH–MGA combined method is 1, the ENUM-MGA combined method is 238.276, the NESA-MGA combined method is 496.3765, the TA-MGA combined method is 759.387, and the SA-MGA combined method is 1527.0562. The performance of CPU running time from the MRAH-MGA combined method is approximately 237 times superior to the ENUM-MGA combined method, 497 times superior to the NESA-MGA combined method, 758 times superior to the TA–MGA combined method, and 1526 times superior to the SA–MGA combined method. Therefore, it can be concluded that the performance of the MRAH-MGA combined method in acquiring the optimal system cost is not only superior to the other four combined methods, but the performance of the MRAH-MGA combined method in CPU running time is also superior compared to the other four proposed combined methods. The main reason for this is that system configuration constraints restrict the number of redundancy components in cases of a smaller ceiling, thus the proposed well-designed heuristic approach of MRAH has no difficulty finding the optimal redundant component amount to reach the optimal total system cost, even with much less CPU running time compared to TS, SA, NESA, and ENUM.

5. Conclusions

This paper proposes an optimization decision support design system for repairable n-stage mixed systems, in which different combinations of subsystems include parallel, standby, and k-out-of-q connected in series configuration. This proposed decision support design system first develops the available optimization mathematic model to handle three decision variables, including the number of redundancy components of all subsystems, the component failure rates of all subsystems, and the component repair rates of all subsystems, in order to obtain the optimal system availability design from the various aspects of this proposed model. In this mathematic model, the system cost objective function is formulated by complying with various constraints, such as system design configurations and system availability requirements. Component availability, parallel-series system availability, n-stage standby system availability, and n-stage k-out-of-q system availability are further derived in detail based on component interval inherent availability and the structures of different system configurations.

For solving the proposed mathematic model, the detailed procedures of five proposed combined methods—ENUM-MGA, TS–MGA, SA–MGA, NESA–MGA, and MRAH–MGA—are derived to obtain the number of optimal redundancy components of all subsystems, the optimal component failure rates of all subsystems, the optimal component repair rates of all subsystems, and the related optimal total system cost.

Several simulated cases for the system availability optimization design are conducted by following the procedure flow of the proposed availability optimization decision support design system to decide the appropriate repairable n-stage mixed system, in which different combinations of subsystems, such as single component, parallel, standby, and k-out-of-q, are connected in series configuration. These simulated results display that the proposed availability optimization decision support design system can definitely take advantage of different component redundancy system designs, including parallel-series system, n-stage standby system, n-stage k-out-of-q system, and n-stage mixed system, to save a lot of cost and meet the high level of the system availability requirement compared to the n-stage single component series system. Moreover, it is also demonstrated that the parallel-series system can clearly meet the same level of availability requirement with less cost compared to the n-stage standby system by assuming the same deteriorating probability for operating components and for standby components. On the contrary, assuming the deteriorating situation of the standby component is better than the operating component may lead to the cost savings of the n-stage standby system in order to cover the original advantage of the parallel-series system in order to easily reach high level availability.

Based on the outcomes of the simulated cases conducted by the proposed availability optimization decision support design system, the performance comparisons of five proposed combined methods for four proposed system configurations are analyzed completely. It can be concluded that the performance of MRAH-MGA in finding the optimal system costs for four proposed system configurations are not only better than the other four combined methods, but also the performances of the MRAH-MGA combined method in CPU running time for four proposed system configurations are all superior compared to the other four proposed combined methods. It can be easily proved that because the system configuration constraints of weight and volume restrict the number of redundancy components in a smaller amount ceiling, the proposed well-designed heuristic approach of MRAH has no difficulty finding the optimal redundancy component amount to reach the optimal total system cost, even with much less CPU running time compared to TS, SA, NESA, and ENUM.