Numerical Design Structure Matrix–Genetic Algorithm-Based Optimization Method for Design Process of Complex Civil Aircraft Systems

Abstract

In the requirement-driven forward design process of civil aircraft, the large number of design tasks of complex systems with varying difficulty and the complex relationships between design tasks lead to unnecessary repetitive design iterations. In order to solve the above problems, the concept of overlap coefficient is proposed to further sort out the forward and backward logical relationships between design tasks and the civil aircraft system design process optimization model based on a numerical design structure matrix. The algorithm NSGA-II is improved and verified with the flight control system design as a case study. The results show that the proposed method can effectively improve the efficiency of complex system design and provide technical support for the optimization of the design process of complex civil aircraft systems.

1. Introduction

Over the last decades, Model-Based Systems Engineering (MBSE) has gained increasing importance in industry and academia [1]. The International Council on Systems Engineering defines MBSE as a key element in the further development of systems engineering [2,3]. Compared to traditional text-based methodology, MBSE shows great engineering value in terms of increased communication efficiency, time and cost savings, improved consistency, optimized system design and representation, and established model traceability [4,5].

In the era of information technology development, many industries, such as aerospace, defense, energy, etc., are facing the problem of explosive growth of system development complexity [6]. As a typical representative of complex systems, civil aircraft systems have been driven by the MBSE methodology [7] and toolchain [8], and their design granularity has become more and more refined [9,10,11]. Design tasks of different design difficulty have sequential or parallel relationships with each other, and coupling often occurs when designing in multiple iterations. It is not possible to identify and decouple design tasks with complex coupling relationships by simple human combing. The inability to identify and decouple design tasks with complex coupling relationships through simple human combing reduces the efficiency of civil aircraft design under the guidance of the MBSE methodology, resulting in higher design costs and longer design cycles [6].

Therefore, there is an urgent need to sort out the task flow of civil aircraft system design, decompose the design process, and establish an optimization model, so as to reduce the loss caused by non-essential iterations.

The innovations of the proposed method can be summarized as the following two points:

(1) The complex coupling relationships in the system design process are fully considered, the task relationships in the complex system design process are identified and characterized based on a numerical design structure matrix (NDSM), and the civil aircraft system design process optimization model is established.

(2) Focusing on the discrete characteristics of the tasks in the design flow optimization and the problem of the diversity and poor distribution of the solution set of the traditional NSGA-II, a selection operator based on the K-nearest neighbors’ mean centroid is proposed, which makes the distribution of the solution set more uniform and take different optimization objectives into account more adequately.

The structure of the paper is as follows. Section 2 describes the work related to the research of this paper. Section 3 presents a description of the design process optimization problem for civil aircraft systems, establishes a design process optimization model based on a numerical DSM and improves the design of the algorithm. Section 4 validates and analyzes the model and algorithm developed in this paper using a civil aircraft flight control system as a case study. Section 5 summarizes the whole paper and the future outlook.

2. Related Works

Process optimization is systematic work aimed at improving process efficiency. By analyzing the correlation relationship and influence degree of each task/activity in the process, establishing a scientific and reasonable process optimization model and solving it, the tasks/activities are re-planned to achieve realistic purposes such as shortening the process cycle, reducing the cost, and reducing unnecessary iterative cycles [12].

2.1. Process Modeling

Process modeling is the key to conducting process optimization research. As an abstract representation of the process model of complex activities, process modeling is able to sort out different types and characteristics of process tasks, and at the same time, it is able to describe the problems in the form of formal or mathematical models [13]. To date, a wealth of research results have been achieved in the field of process modeling for design activities, and the mainstream process modeling methods and comparisons are shown in

Table 1. Mainstream process modeling methods and comparison.

	DSM [14,15]	Petri Net	IDEFx	CPM and ERT [16]
Ability to describe the process	Strong	Strong	Strong	Strong
Ability to analyze iterations	Strong	Normal	Normal	Normal
Reusability of model	Normal	Strong	Normal	weak
Performance of dynamic representation	Normal	Normal	Weak	Weak
Support for time performance analysis	No	Yes	No	Yes
Support for simulation software	Yes	Yes	No	Yes

.

Due to the complexity of processes, there does not exist a universal model that can describe all types of processes. When oriented to real-world problems, there are also large differences in the models created by different process modeling approaches. Among the above modeling methods, a DSM is a square matrix composed of row and column elements with the same order of arrangement, which is used to represent various types of activities and their serial, parallel, and coupling relationships. Compared with other modeling methods, the DSM is inadequate in time performance analysis and dynamic representation [17]. However, its powerful process description capability and iterative analysis [14,15,17] can appear extremely effective for modeling complex system design processes involving a long life cycle.

Reference [18] extracted the data information in the model file during the design process based on the existing open-source MBSE software and automatically generated a design structure matrix in xml format. Reference [19] characterized the interaction relationship of a civil engine system based on a DSM and analyzed and optimized the design process by establishing a mapping matrix between designers and the actual system. Reference [20] proposed a design process modeling approach based on Object Process Methodology (OPM) and a design structure matrix; after modeling the process using OPM, a DSM was established based on the relationships between design processes, and the process optimization model was solved using an improved Idicula–Gutierrez–Thebeau Algorithm (IGTA+). The Boeing Company developed a process description for investigating the effects of a noisy aerodynamic acoustic installation on a jet airplane. The design and analysis activities interacted through a modest number of feedforward and feedback links, and the process activities were represented in a DSM. A clearer view was provided to the designer about the sensitivity of each activity to a given change [21]. Reference [22] analyzed the possibility of propagation between two connected elements and the impact on the element when a change occurs based on a DSM, using Westland’s helicopter EH101 as a case study. Mathematical models were developed to predict the impact of risk propagation due to changes in the design process. Lockheed Martin used a DSM to optimize the design of the activities of the avionics upgrade development process for the F-16 Multi-Purpose Jet Fighter in order to avoid and manage additional process iterations as well as cost increases due to process delays [23]. Reference [24] oriented the structural design of a trainer with the amount of information connectivity between design elements as the optimization objective and solved it by a genetic algorithm. The results demonstrated that the DSM could effectively reconfigure the tasks in the design phase and greatly reduce the complexity of the design.

Existing studies have shown that the DSM can efficiently visualize complex processes characterized by a large number of iterations [17,25] and can succinctly and clearly describe the feedback and coupling relationships among the activities of a development process. The DSM can provide support for the qualitative and quantitative analysis of processes (e.g., process visualization, calculation of process execution time, etc.) when modeling task-based design processes.

2.2. Algorithm for Process Optimization

Solving process optimization problems is likewise an important foundation for process optimization research. Many important advances have been made in the study of process optimization algorithms, which can be broadly classified into two categories: exact algorithms and heuristic algorithms.

Exact algorithms include integer programming, enumeration algorithms, and branch-and-bound (BB). Exact algorithms can solve the optimal solution of a specific problem, but the solution efficiency is low, and they are basically unable to do anything for large-scale process optimization problems containing 60 tasks or more.

For more complex process optimization problems, scholars usually adopt meta-heuristic algorithms or intelligent optimization algorithms, such as the ant colony algorithm [26], forbidden search algorithm [27,28], simulated annealing algorithms, genetic algorithm [29,30], etc. Among them, genetic algorithms are favored by a large number of scholars because of their superior algorithmic performance and good inclusion ability. A large number of studies also showed that genetic algorithms and their improved algorithms (NSGA, NSGA-II, etc.) had good results for solving process optimization problems.

In addition, many process optimization problems are accompanied by the requirement of optimizing multiple objectives, and Non-dominated Sorting Genetic Algorithm (NSGA-II) with an elite reservation strategy is a common algorithm used to solve such problems. However, the standard NSGA-II algorithm suffers from poor algorithmic convergence and population diversity when solving different real problems.

In summary, the NSGA-II algorithm has higher efficiency than other existing algorithms in solving process optimization problems. The NSGA-II algorithm performs well in solving process optimization problems involving multiple objectives and is more suitable for solving design activity optimization problems for complex systems such as civil aircraft systems. However, the algorithm needs to be improved and optimized according to the actual problems to achieve better optimization results.

3. Description of the System Design Process Optimization Problem

In this section, the methodology is described in detail. Firstly, the NDSM is used to describe the correlation relationships between design tasks in complex system design. Secondly, based on the NDSM, a civil aircraft complex system design process optimization model is established. Thirdly, the K-nearest neighbors’ mean centroid is introduced to improve the NSGA-II algorithm. Finally, the solution flow of the improved algorithm is introduced. The roadmap of the civil aircraft system design process optimization is shown in Figure 1.

3.1. Numerical DSM-Based Modeling for Design Process Optimization

Suppose the civil aircraft system design process is initially divided into m design tasks. There are complex pre–post relationships between some of the design tasks, implying that there exists a certain type of task that needs to be carried out on the basis of full or partial completion of other tasks. Obviously, the design tasks related to system verification must be executed after the design tasks related to system requirements analysis. In order to express and analyze the logical relationships between many design tasks more intuitively, the numerical DSM is used to characterize the relationships between the design tasks of the civil aircraft system. Three typical relationships in the DSM are shown in Figure 2. The established numerical DSM is a square matrix with the same order as the number of design tasks, denoted as $D=\left\{d_{ij}\right\}$, $d_{ij}$ denotes the element in the i-th row and j-th column of the matrix, which represents the dependency between task j and task i, and its value range satisfies $0\le d_{ij}\le 1$.

When $i=j$, $d_{ij}$ is the element on the main diagonal of the DSM, which is recorded as a null value, indicating that there is no sequential relationship between the design task and itself in terms of execution order. $d_{ij}=0$ represents that there is no dependency between the two tasks, so there is no strict sequential execution order between the two tasks.

When $i>j$, $d_{ij}$ is located below the main diagonal of the DSM, indicating a forward execution relationship between the two tasks. $d_{ij}=1$ represents that the two tasks are executed in a strict order, implying that the execution of task i can only start after the complete execution of task j. If $0<d_{ij}<1$, it means that there is a partial overlap between the two tasks. The start time of task i is advanced relative to the progress of task j. The degree of overlap is represented by the value of $d_{ij}$, where $1-d_{ij}$ denotes the ratio of the overlapping execution activity time to the execution time of task i. An example of sequential execution between two tasks is shown in Figure 3.

When $i<j$, $d_{ij}$ is located above the main diagonal of the DSM, indicating that there is feedback between the two tasks in the execution order. $d_{ij}=1$ represents that there is a strict feedback relationship between the two tasks, implying that the execution of task i can only start after the complete execution of task j. If $0<d_{ij}<1$, it means that there is a partial feedback overlap between the two tasks, and the start time of task i is somewhat earlier than the start time of task j. The degree of overlap is represented by the value of $d_{ij}$, where $1-d_{ij}$ denotes the ratio of the overlapping execution activity time to the execution time of task i. An example of feedback execution between two tasks is shown in Figure 4.

If there is a feedback relationship between tasks, the sequence of task execution between the start of the execution of a task and the next execution of that task is called a feedback loop. The length of a feedback loop is defined as the distance between the tasks at the head and tail of the feedback loop, measured in terms of the number of tasks involved in the feedback loop. The longer the feedback length, the longer the task dependency chain is, and the complexity of the system and the difficulty of iterative adjustment increase. In addition, when there is an overlap between two different feedback loops, then the overlap is referred to as a feedback crossover point, denoted c.

3.2. Analysis and Establishment of Design Process Optimization Model

After constructing the NDSM of a civil aircraft system design process, it is necessary to consider the efficiency and complexity of the civil aircraft design process from multiple perspectives. A reasonable civil aircraft system design process optimization model needs to take into account the total design cycle and the coupling complexity relationship in the design. At the same time, the explicit and implicit constraints in the design process optimization need to be explored and satisfied.

Define the decision variable $x_{ij}$. If $x_{ij}$ takes the value of 1, it means that task i is executed before task j; if $x_{ij}$ takes the value of 0, it means that task i is executed after task j.

Define $T_t$ as the total design time of the civil aircraft system, as shown in Equation (1):

$$T_t=\underset{0\le i\le m}{\max }\left(f_i\right)$$

(1)

where $f_i$ denotes the end time of execution of task i.

Define F as the total feedback length of the civil aircraft system design process, as shown in Equation (2):

$$F=\sum _{j=1}^m\sum _{j=1,j\ne i}^m{d}_{ij}{x}_{ij}\left(\sum _{k=1,k\ne j}^m{x}_{kj}-\sum _{k=1,k\ne i}^m{x}_{ki}\right)$$

(2)

Define $C_r$ as the total number of feedback crossover points of the civil aircraft system design process, as shown in Equation (3):

$$C_r=\sum c$$

(3)

Subsequently, the constraint relationships between task processes need to be considered and mined to further define the feasible domain boundaries of the process optimization problem.

For any two tasks i and j with dependencies, there must be a sequential execution order between them, so the constraint shown in Equation (4) needs to be satisfied.

$$x_{ij}+x_{ji}=1,1\le i,j\le m$$

(4)

If task i is executed before task j and task j is executed before task k, then it must satisfy that task i is executed before task k. The constraint is shown in (5):

$$x_{ij}+x_{jk}+x_{ki}\le 2,1\le i,j,k\le m,s_i\le s_j,s_j\le s_k$$

(5)

where $s_i$ denotes the start time of task i.

If there is a certain overlap between any two tasks i and j, it is necessary to ensure that the start time of the downstream task be executed after the upstream task has been executed to a certain extent. Let task i be the upstream task and task j be the downstream task, then the start time of the downstream task should satisfy the constraint shown in Equation (6):

$$s_j\ge \underset{i}{max}\left\{f_i-t_i\times d_{ji}\right\},i\in \left\{k\mid 0<d_{jk}<1\right\}$$

(6)

where $s_i$ denotes the start time of task i, $f_i$ denotes the end time of task i, and $t_i$ denotes the duration of the task and satisfies $f_i=s_i+t_i$.

For tasks i and j that need to be executed iteratively, the number of their iterations needs to be specified at each iteration. It is also guaranteed that their execution order in the same iteration satisfies the constraint. Assuming that task i is an upstream task and task j is a downstream task, the start time of the downstream task should satisfy the constraint shown in Equation (7).

$$s_j^r\ge \underset{i}{max}\left\{f_i^r\right\},i\in \left\{k\mid d_{jk}>0,j>k\right\}$$

(7)

where r represents the number of iterations, $s_j^r$ denotes the start time of task j at the rth iteration, $f_i^r$ denotes the end time of task j at the rth iteration.

For tasks that are not involved in an iteration, the constraint of the upstream task i on the start time of the downstream task j is given by Equation (8):

$$s_j\ge \underset{i}{max}\left\{f_i\right\},i\in \left\{k\mid d_{jk}>0,j<k\right\}$$

(8)

In order to be able to complete the full design task as quickly as possible, while minimizing the complexity and risk implications of iterative design, it is necessary to keep the total design time of the civil aircraft system $T_t$ as short as possible while keeping the total feedback length F and the total number of feedback crossover points $C_r$ in the design process as small as possible. Therefore, the civil aircraft system design process optimization model is established as shown in Equation (9).

$$\begin{array}{cc}\hfill \mathrm{Minimize}& T_t,F,C_r\hfill \\ \hfill \mathrm{subject}\mathrm{to}& x_{ij}+x_{ji}=1,1\le i,j\le m\hfill \\ \hfill & x_{ij}+x_{jk}+x_{ki}\le 2,1\le i,j,k\le m,s_i\le s_j,s_j\le s_k\hfill \\ \hfill & s_j\ge \underset{i}{max}\left\{f_i-t_i\times d_{ji}\right\},i\in \left\{k\mid 0<d_{jk}<1\right\}\hfill \\ \hfill & s_j^r\ge \underset{i}{max}\left\{f_i^r\right\},i\in \left\{k\mid d_{jk}>0,j>k\right\}\hfill \\ \hfill & s_j\ge \underset{i}{max}\left\{f_i\right\},i\in \left\{k\mid d_{jk}>0,j<k\right\}\hfill \end{array}$$

(9)

3.3. Optimization of NSGA-II Based on K-Nearest Neighbors’ Mean Centroid

In the standard NSGA-II, the crowding distance calculation relies only on the distance between neighboring individuals in the population. However, not all individuals in the population make it to the next generation, so there are a significant number of individuals in the population that are invalid for the crowding distance calculation. The inclusion of these individuals in the calculation may result in the resulting crowding degree not accurately reflecting the true sparsity of the current solution. In addition, evaluating the crowding distance only based on the distance between two neighboring individuals may result in inaccurate results, which cannot fully reflect the global distribution of the solution set in the target space. The above limitations may result in the standard NSGA-II not being able to efficiently select uniformly distributed individuals at the next generation during the selection process, affecting the algorithm solution speed and the quality of the solution set. The improved NSGA-II flowchart is shown in Figure 5.

A crowding distance calculation method based on the K-nearest neighbors’ mean centroid is proposed as a mechanism to improve the crowding distance calculation. This mechanism pays more attention to the uniformity of the distribution of the individuals to be selected in the set of solutions that have entered the offspring. It is more conducive to selecting individuals with a uniform distribution at the next generation to accelerate the evolutionary process and enhance the diversity and distribution uniformity of the population.

Firstly, in order to eliminate the possible influence of numerical scale differences between different objective functions on the optimization results, it is necessary to normalize the objective function as shown in Equation (10):

$$f_{t,n}^{\prime }={\displaystyle \frac{f_{t,n}-f_{t,n}^{min}}{f_{t,n}^{max}-f_{t,n}^{min}}}$$

(10)

where $f_{t,n}$ is the value to be normalized in the nth objective function value at the tth generation in the population, $f_{t,n}^{max}$ and $f_{t,n}^{min}$ are the maximum and minimum values in the nth objective function value at the tth generation in the population, and $f_{t,n}^{\prime }$ is the value of the normalized objective function.

Secondly, for the individuals $p_i$ in the population to be selected to calculate the crowding distance, select the K-nearest solutions in the target space that are closest to them in the population that has entered the next generation, denoted as $N_i=\left\{{\mathit{p}}_{i1},{\mathit{p}}_{i2},\cdots ,{\mathit{p}}_{iK}\right\}$, where K is an algorithm parameter and is a pre-set number of nearest neighbor individuals.

Thirdly, calculate the K-nearest neighbors’ mean centroid of individual $p_i$. For set $N_i$, the central position ${\overline{\mathit{p}}}_{\mathit{i}}$ of these individuals in the target space is calculated as shown in Equation (11):

$${\overline{\mathit{p}}}_{\mathit{i}}={\displaystyle \frac{1}{K}}\sum _{j=1}^K{\mathit{p}}_{ij}$$

(11)

where ${\mathit{p}}_{ij}$ is the vector form of the jth individual solution in set $N_i$, and ${\overline{\mathit{p}}}_{\mathit{i}}$ is the K-nearest neighbors’ mean centroid of individual $p_i$.

Lastly, the crowding distance is defined as the distance between an individual $p_i$ and the mean centroid ${\overline{\mathit{p}}}_{\mathit{i}}$ of its set of K immediate neighbors. The crowding distance is calculated as shown in Equation (12):

$${CD}_i=∥{\mathit{p}}_{\mathit{i}}-{\overline{\mathit{p}}}_{\mathit{i}}∥$$

(12)

where ${CD}_i$ denotes the crowding distance of individual $p_i$, and $∥·∥$ denotes the Euclidean distance in the target space.

A schematic diagram of the crowding distance calculation based on the K-nearest neighbors’ mean centroid is shown in Figure 6, where rank 1, rank 2, and rank 3 denote the dominance level at which the solution is located, respectively.

The improved crowding distance calculation method narrows the scope of consideration from the entire population to the set of levels in the population that have entered the next generation when assessing individual crowding distances. Individuals that fail to enter the next generation are avoided from interfering with the crowding distance calculation. In this way, the role of individuals in the population evolution can be assessed more accurately and a more informative crowding distance metric can be derived. In addition, extending the selection of nearest-neighboring individuals from the two adjacent ones to K can reflect the actual sparsity of individuals in the solution set more comprehensively, reducing the impact of an uneven distribution of individuals on the assessment of crowding in special cases.

The key to the crossover operator in the problem of optimizing the design process of a civil aircraft system is that the good genes in the chromosomes of both parents are mainly reflected in the order of the gene sequences. The offspring generated by crossbreeding must satisfy the genetic coding requirements, implying that each design task occurs only once in the chromosome. A position-based crossover operator is utilized in our work. The operation to realize the crossover operator is as follows:

(1) Select a gene segment in parent 2 whose position and length are randomly determined.

(2) Pass the selected gene segment in parent 2 directly to its offspring.

(3) Remove the gene from parent 1 that was passed on to the offspring in the second step.

(4) Fill in the remaining genes in parent 1 in the order in which they are listed in parent 1 to generate a new offspring.

An example of a chromosome crossover with a length of 10 is shown in Figure 7. The yellow segment is a randomly selected segment of a gene from parent 2 that was passed directly to the offspring. The genes contained in this segment were subsequently removed from parent 1 (8,1,9). The remaining genes are labeled in blue and filled into the offspring in the order in which they were in parent 1.

In the civil aircraft system design process optimization problem, the mutation of genes is reflected in the random alignment of gene positions on chromosomes. A mutation operator based on position swapping is applied as follows: two genes are randomly selected from the parent chromosome and their positions are swapped to generate the offspring chromosome. The position swapping is equivalent to changing the execution order of two different tasks in the task list. After the exchange is completed, it is necessary to check whether the task sequence of the offspring chromosome still satisfies the constraint relationships between the tasks. If the exchange does not satisfy the constraints, the gene is restored to the original position, the next mutation operation is performed, and the mutation is stopped when the mutated chromosome satisfies the constraints. The proposed mutation method can not only effectively increase the diversity of the population, but also ensure that the generated offspring solution set is optimized under the premise of satisfying the task constraint relationships.

An example of a chromosome mutation with a length of 10 is shown in Figure 8. The green segments (1,7) are two gene loci randomly selected from the paternal generation. Their positions are exchanged, keeping the rest of the gene loci unchanged to populate the offspring chromosome.

4. Validation and Analysis of Flight Control System Design Case

The civil aircraft flight control system involves input and output of information, energy, materials and other aspects, and plays a role in maintaining or changing the flight attitude of the aircraft during the operation of the civil aircraft. The design of the flight control system of a civil aircraft is of great importance to the flight safety of a civil aircraft. In this section, the proposed model and algorithm are verified for the design process of the flight control system of a civil aircraft. The validation results are analyzed to further illustrate the stability of the model and the robustness of the algorithm.

4.1. Study Case

As a typical complex system, the design complexity and the number of design tasks of the flight control system are representative of the same system level. The design process for a civil flight control system for an aircraft was initially divided to include 21 design tasks, and the numbering of each design task as well as the estimated completion time is shown in

Table 2. List of tasks and unit time required.

Task No.	Task Name	Units of Time Required
1	Requirements analysis and system specification	22.0
2	System architecture design	36.0
3	Hardware platform selection	36.0
4	Software architecture design	59.7
5	System dynamics modeling	36.0
6	Control module control law design	21.0
7	Automatic control system logic hierarchy	18.0
8	Attitude control module design	36.0
9	Auto-throttle design	20.0
10	Control law optimization	36.3
11	Control module evaluation	48.0
12	Implementation system architecture design	42.0
13	Actuator selection	30.5
14	Automatic control module control law design	32.0
15	System integration testing	30.0
16	Input/output recognition	49.0
17	Communications protocol specification	22.3
18	Interface design	37.5
19	Flight data acquisition and analysis	64.0
20	Flight performance verification	30.0
21	Technology assessment and feedback loops	22.0

. The 21 design tasks in Table 2 are the result of the initial decomposition of the design process of the flight control design system, and the design tasks are filled with complex relationships between them. From the point of view of design complexity, it is already a complex problem that is difficult to optimize the process by human efforts in a short period of time. In the actual design cycle, the design tasks of the complex system are far more than 21, but the tasks should be decomposed at the same level to ensure the orthogonality of the design tasks, to avoid the occurrence of too many design tasks at the same level of development. The existence of too many tasks at the same level greatly increases the complexity of the inter-task relationship but inadvertently increases the difficulty of solving the problem.

As shown in Figure 9, the NDSM of the design tasks was established by analyzing the relationships between them.

4.2. Analysis and Discussion

Let the population size $N_p$ be 100, the iteration limit T be 200, the crossover probability $P_c$ be 0.80, the mutation probability $P_m$ be 0.05, and the number of nearest-neighbor individuals K selected for calculating the crowding distance be four. Denote the above algorithm parameters as parameter tuple $\mathbf{P}$, where $\mathbf{P}=\left(N_p,T,P_c,P_m,K\right)$.

When optimizing the design process of a civil aircraft system, the most important goal is to find the superior solutions in the feasible space. It is also necessary to take into account the three optimization objectives such as total design time, total feedback length, and total number of feedback crossover points. Therefore, it is necessary to search for a better set of solutions with better distribution and diversity in the feasible space as much as possible, so as to provide designers with more comprehensive decision-making assistance in the top-level planning stage of the design. In order to visualize the effect of the improved algorithm, the optimization performance of the algorithm was verified in terms of convergence and distribution. The optimization performance of the algorithm was evaluated using the two indicators of $SP$ (Spacing) and $HV$ (Hypervolume). $SP$ was used to measure the standard deviation of the minimum distance from each solution to the other solutions and was used to measure the extent to which the design process solution accommodated multiple optimization objectives. $HV$ was used to measure the volume of the region in the objective space enclosed by the set of non-dominated solutions obtained by the algorithm and the reference point, representing the superiority of the proposed solution.

The $SP$ was calculated as shown in Equation (13):

$$SP=\sqrt{{\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{(\overline{d}-d_i)}^2}$$

(13)

where $d_i=\underset{j}{min}\left({\sum }_{k=1}^m\left|f_k^i\left(x\right)-f_k^j\left(x\right)\right|\right)\left(i,j=1,2,···,n\right)$, denotes the minimum value of the Manhattan distance between individual i and the other individuals. $\overline{d}$ is the mean of $d_i$, n is the number of individuals in the solution set, and k denotes the dimension of the objective functions. The smaller the value of $SP$, the more evenly distributed the solution set is. If $SP=0$, it means that the individuals in the population are completely evenly distributed.

$HV$ was calculated as shown in Equation (14):

$$HV=\delta \left({\cup }_{i=1}^{\left|S\right|}{\nu }_i\right)$$

(14)

where $\delta \left(\right)$ denotes the Lebesgue measure, which is used to measure the generalized volume. $\left|S\right|$ denotes the number of non-dominated solution sets, and $v_i$ denotes the hypervolume formed by the reference point and the ith solution in the solution set. The larger the value of $HV$, the better the convergence and distribution of solutions.

The process optimization problem in the case was solved under the above parameters $\mathbf{P}=\left(100,200,0.80,0.05,4\right)$, and the Gantt charts for original and optimized civil aircraft flight control system design process are shown in Figure 10a and Figure 10b, respectively.

In order to emphasize the improvement effect of the proposed algorithm in terms of population diversity and convergence, the SP and HV indicators of the initial and final populations were used as the quantitative values before and after the optimization of the design process, respectively.

Table 3. The quantitative comparison table of original and optimized design process.

	${\mathit{T}}_{\mathit{t}}$	F	${\mathit{C}}_{\mathit{r}}$	$\mathit{SP}$	$\mathit{HV}$
Original design	701	196	46	0.1696	0.1408
Optimized design	515	121	31	0.0378	0.6103
Optimization effect	−26.53%	−38.27%	−32.61%	−77.71%	333.45%

shows the quantitative comparison table of original and optimized design process in detail.

From Table 3, it can be found that each optimal objective was significantly improved after the optimization, and the diversity and convergence of the population were greatly improved as follows.

Total design time $T_t$: in the original design process, the total design time required was 701 units of time; after optimization, the total design time required was 515 units of time, a reduction of 26.53%.

Total feedback length F: The design process coupling relationships were complex, and there were many unnecessary design iterations in the original design process. The total design feedback length was 196 before the optimization, and the total feedback length was reduced to 121 after the optimization, a reduction of 38.27%.

Total number of feedback crossover points $C_r$: in the original design process, the total number of feedback crossover points was 46, and the number of design crossover points was reduced to 31 after the optimization, a reduction of 32.61%.

In addition, the two indicators, SP and HV, improved, as shown in Figure 11. The significant decrease in SP value indicated that the population was well diversified, and more feasible solutions that took into account the three optimal objectives emerged after the optimization. The increase in HV value represents the forward movement of the Pareto frontier of the solution set, which is a strong proof of the convergence performance of the algorithm. This means that the algorithm searched enough good solutions in the feasible domain, which could reduce unnecessary design iterations and improve the design efficiency.

The optimized design process of the flight control system corresponding to the solution with a total design time of 515 units of time, a total feedback length of 121, and a total number of feedback crossover points of 31 is illustrated in Figure 12. From Figure 12, it can be seen that the proposed process optimization model with the improved NSGA-II could effectively solve the sequence-based design process optimization problem for civil aircraft flight control systems. Although there is some influence of human subjective factors in modeling, the establishment of the NDSM only needs to consider the relationships between two tasks. This system engineering concept, which essentially decomposes the design process, greatly reduces the subjective influence of design engineers when they are directly confronted with complex design tasks.

4.2.1. Discussion on the Number of Nearest Neighbors K

The civil aircraft flight control system process optimization problem is a typical discrete problem, and the solution in the objective space is often directly related to the actual problem to be solved. In order to further verify the improvement effect of the proposed K-nearest neighbors’ mean centroids for crowding distance calculation, different numbers of nearest neighbors were selected for a comparative analysis in parameter setting. The curve of variation in the value of $HV$ for each generation of the population is shown in Figure 13 for a number of nearest neighbors K varying from two to six.

From Figure 13, it can be seen that when $K=4$, the value of $HV$ was maximum and could reach the stable state faster. When K took the value of three or five, its optimization effect was slightly worse than when K took the value of four, but it could also reach the stable state faster. When K was set to two, the algorithm took a longer time to achieve similar optimization results as the other values, and the oscillation of the curve was more obvious. The reason for this is that the number of nearest neighbor individuals considered was too small, resulting in the tendency to cause a large error in the calculation of the crowding distance of the individuals to be selected, which affected the evolution of the population. The optimization effect of the algorithm was also good when the value of K was six, but the value of HV suddenly decreased when the number of iterations was about 180. This is due to the fact that when the number of nearest neighbor individuals considered is too large, it tends to lose sensitivity to the crowding distance and thus tends to enter the local optimum earlier. Although there may be a certain probability of jumping out of the local optimum through mutation, it also brings more instability to the evolution of the population.

4.2.2. Comparison with Other Algorithms

In addition, the improved NSGA-II was compared with the standard NSGA-II and SPEA-II (Strength Pareto Evolutionary Algorithm II) through parameter changes to further verify the superiority of the improved algorithm. All three algorithms are genetic algorithms that optimize based on individual dominance relationships but differ in their optimization strategies. SPEA-II records the set of non-dominated solutions through an external archive mechanism, evaluates individual fitness for individual optimization using strength values based on dominance relationships, and incorporates the Euclidean distance for description in terms of maintaining the diversity of solutions. NSGA-II adopts the fast non-dominated sorting and crowding distance measurement strategies, and effectively balances the diversity of solutions with the computational efficiency through the precise grading and elite retention strategies, which has strong applicability and flexibility. Both two algorithms have their own advantages in different application scenarios, and can meet diverse optimization needs for specific problems, but the SPEA-II algorithm is more outstanding in dealing with complex high-dimensional problems.

Considering the practical problems of the design process optimization of civil aircraft flight control system, the comparison was mainly carried out in the following two aspects:

(1) When there are numerous design tasks, the general choice is to increase the population size to find the better individuals faster, thus speeding up the convergence of the algorithm. Therefore, when the population size decreases, the algorithm that can converge faster is better.

When the population sizes were set to 300, 200 and 100, 10 solutions were derived using the standard NSGA-II, SPEA-II, and modified NSGA-II algorithms. The histograms of SP and HV indexes of the final population at different population sizes were drawn as shown in Figure 14.

From Figure 14, it can be seen that the improved NSGA-II performed significantly better than the other two algorithms when the population size was gradually decreased. On the one hand, even though the population size decreased, the SP value of the improved NSGA-II remained stable, and the value was much lower than that of the other two algorithms. On the other hand, as the population size decreased, the HV value of the improved NSGA-II decreased slightly, but it was significantly higher than that the other two algorithms under the same simulation conditions. The above comparison shows that the improved NSGA-II algorithm is more convergent and has a better distribution of solutions under the same conditions. It means that it can have excellent results in solving more complex civil aircraft system design process optimization problems.

(2) The time required to complete a design task usually changes due to technological advances, requirement changes in the design, design iterations, etc. Therefore, an algorithm with better robustness is better when the time required to complete the design task changes.

In order to simulate variations in the time required for design tasks, different levels of disturbances were set in order to mimic the scenarios of accelerated work progress or delayed work progress that occur in real-world situations. The robustness of the algorithm was tested by randomly selecting the number of tasks $N_m$ and by randomly setting the average degree of disturbances $D_d$. The disturbance parameter was denoted as $\Delta =\left(N_m,D_d\right)$.

Three groups of different levels of disturbances were given to validate the algorithm. The first group was the slight disturbance case, where 4 tasks out of 21 design tasks were randomly selected to be disturbed, and the average disturbance in the design time was 5%. The second group was the normal disturbance case, where 7 tasks out of 21 design tasks were randomly selected to be disturbed, and the average disturbance in the design time was 8%. The third group was the serious disturbance case, where 14 tasks out of 21 design tasks were randomly selected to be disturbed, and the average disturbance in the design time was 10%.

The three algorithms were employed to solve the above disturbance cases separately, and the histograms of SP and HV for different disturbance cases are shown in Figure 15.

The following conclusions can be drawn from Figure 15:

The improved NSGA-II maintained a low SP value under various disturbances, while the SP value of the solution set of the improved NSGA-II was much lower than that of the other two algorithms under the same disturbances. All of the above advantages were attributed to the K-nearest neighbors’ mean center point proposed in the algorithm, which effectively improved the evenness of the solution set.

The standard NSGA-II performed poorly under different perturbations, while the HV values of the improved NSGA-II solution set were almost always higher than those of the other two algorithms.

Overall, the improved NSGA-II had better optimization results in all types of cases and had significant advantages in terms of distribution, convergence, and diversity of solutions.

5. Conclusions and Future Work

Aiming at the existing MBSE-based civil aircraft system design process, which has practical engineering problems such as the large number of design tasks with different difficulties and the complex relationships between design tasks, this paper sorted out the types of relationships between civil aircraft system design tasks and characterized them by a numerical DSM. The constraints in the design process of civil aircraft system were mined and summarized, and a multi-objective optimization model for the design process of civil aircraft system was developed based on the established NDSM. The K-nearest neighbors’ mean centroid was introduced to improve the NSGA-II algorithm, which effectively improved the population evolution strategy. The effectiveness of the model and algorithm was verified by taking the flight control system of civil aircraft as an example. The validation results showed that the established model could effectively characterize the complex relationships between civil aircraft system design tasks, and the improved algorithm was better than the other algorithms and could effectively solve the problem of optimizing the design process of the civil aircraft system.

The utilization of this method in the field of civil aircraft complex system design can effectively complement the design work of domain engineering under the guidance of the current MBSE methodology. This method can also effectively solve the complex system design process optimization problem with multiple design tasks and has better applicability in the complex system design optimization problem with complex timing design constraints.

This paper provides a framework for process optimization of design tasks for complex systems. Similar to the flight control system, other systems of an aircraft can be designed by taking the relationships between system design tasks, the required unit time, and the time constraints of the design tasks as inputs, and following the method in the paper, engineers can generate a design process optimization plan that satisfies the constraints, which can provide them with auxiliary decision-making in carrying out the design activities.

Driven by the model-based systems engineering approach, the granularity of civil aircraft design is bound to become more refined. In our future work, we will continue to consider the hierarchical structure of tasks, model and express the task flow relationships through hypernetworks, and provide support and assisted decision-making in task planning for civil aircraft design engineers to carry out civil aircraft design under the guidance of the MBSE methodology.