An axiomatic system engineering design method based on NSGA-II algorithm applied to complex systems

Abstract

This study proposes an integrated framework that combines Axiomatic Design with Model-Based Systems Engineering, incorporating NSGA-II as an intelligent optimization engine. The framework constructs a multi-level traceability matrix linking requirements, functions, behaviors, and structural elements, and leverages NSGA-II to dynamically optimize the sequencing of the Design Structure Matrix, thereby enhancing the explicitness and visualization of coupling relationships. Rather than directly eliminating dependencies, the approach aims to support a more systematic modularity-oriented analysis. Implemented using SysML, the framework establishes formal mappings among the user, functional, behavioral, and physical domains to ensure bidirectional traceability. In parallel, coupling degrees are quantitatively assessed based on axiomatic design principles, and a multi-objective intelligent search is employed to accelerate convergence within large-scale design spaces. A case study on an active reflector control system demonstrates that the proposed NSGA-II–enhanced framework significantly reduces reliance on expert knowledge in DSM ordering, and exhibits clear advantages in coupling identification and modularity analysis for complex systems.

Introduction

In the broader context of design theory, the ability to effectively design complex products, represent design information in a precise and systematic manner, and ensure its accuracy, completeness, and consistency throughout the entire design process is a critical determinant of product development success. Over the past decades, a variety of domain-oriented frameworks have been proposed to support this objective. Notable examples include Alexander’s¹ minimum-coupling decomposition method, Gero’s Function–Behavior–Structure (FBS) framework², Pimmler and Eppinger’s³ extended Design Structure Matrix (DSM) method, Andreasen’s chromosome model⁴, and the Enhanced Function–Means (EFM) approach⁵. In the field of product architecture, Erens and Verhulst⁶ and Jiao⁷ have further developed serialization-based modeling methodologies. These frameworks have made significant contributions in supporting conceptual design, functional reasoning, and design synthesis. Nevertheless, they provide limited attention to the explicit quantification of coupling, the systematic management of interdependencies in complex engineering systems, and the optimization of DSM sequencing.

To address these challenges, it is essential to revisit the methodological foundations that underpin the study of coupling, dependency management, and system-level design optimization. Different design theories provide complementary perspectives on this issue. Axiomatic Design (AD), for instance, establishes a formalized framework for requirement mapping and decoupling through the independence and information axioms. Model-Based Systems Engineering (MBSE), in contrast, enables a digital modeling environment that translates design logic and decision criteria into executable and verifiable system models. DSM techniques, on the other hand, serve as a practical tool for explicitly representing and optimizing interdependencies. Reviewing these methodologies not only clarifies the theoretical underpinnings of this study but also positions the contribution of the present work within a broader methodological landscape.

Axiomatic Design, initially introduced by Suh⁸, defines the relationship between functional requirements and design parameters through two fundamental axioms and thereby supports the identification of optimal design solutions. Previous research has shown that AD effectively reduces implicit dependencies and facilitates modularization (e.g., Suh⁸; Kulak⁹; Hillström¹⁰; Gonçalves et al.¹¹; Rauch¹²; Zhang¹³). However, most applications of AD remain concentrated in product and software design domains (Kulak¹⁴), with limited capability in supporting cross-domain modeling and capturing dynamic behaviors at the system level. In parallel, MBSE has emerged as a promising paradigm for the digital design of complex systems. Estefan¹⁵ emphasized that MBSE enhances traceability across requirements, functions, behaviors, and structures, while SysML-based approaches have been increasingly adopted in systems engineering practice (Russell¹⁶; Feng¹⁷; Wang¹⁸; Kuelper¹⁹; Moers²⁰; Estefan²¹). These studies collectively highlight the advantages of MBSE in enabling digital and model-driven system design. Nonetheless, current MBSE practices lack robust methodologies for the quantitative evaluation and optimization of system-level coupling, leaving a significant gap for further research.

This study aims to overcome the limitations of existing approaches in coupling quantification, dependency management, and DSM sequencing optimization. The integration of Axiomatic Design (AD) and Model-Based Systems Engineering (MBSE) enables a paradigm shift from “experience-driven design” to “logic- and model-driven design.” The combination of AD and MBSE forms a top-down decomposition process, where a Design Structure Matrix (DSM) is constructed in each design iteration based on the mapping process. However, the decomposition typically yields a large number of functional requirements (FRs) and design parameters (DPs), resulting in a complex design matrix. It is often difficult to visually identify the matrix structure or determine whether the independence axiom is satisfied. To address this challenge, redundant FRs and DPs that do not affect the judgment of independence must be eliminated.

Numerous DSM modularization and sequencing algorithms have been reported in the literature. For instance, Otto²² proposed a clustering-based approach that groups highly coupled elements into modules, while Holtta-Otto and De Weck²³ developed sequencing and tearing algorithms to minimize feedback loops and improve modularity. Foundational work in this domain includes Pimmler and Eppinger’s³ interaction-based clustering method and the comprehensive reviews of DSM partitioning, sequencing, and clustering techniques provided by Browning²⁴ and Yassine²⁵. The common objective of these methods is to reduce coupling through DSM reordering or clustering. Compared with these traditional DSM clustering and sequencing methods, the Non-dominated Sorting Genetic Algorithm II (NSGA-II), with its capabilities in multi-objective optimization, global search, and handling of complex constraints, offers a more suitable solution for DSM optimization within an MBSE environment.

In light of the aforementioned limitations in coupling quantification, dependency management, and DSM optimization, this study integrates AD with MBSE and introduces the NSGA-II multi-objective optimization algorithm. The proposed approach enables systematic DSM extraction, quantitative evaluation of coupling, and dynamic optimization of matrix sequencing.

The novelty of this research lies in three aspects:

1. The systematic extraction of DSM through multi-layer traceability matrices constructed in SysML;

2. The quantitative evaluation of coupling relationships based on the independence axiom;

3. The dynamic optimization of DSM using the NSGA-II multi-objective evolutionary algorithm, thereby enhancing the explicitness and analyzability of system dependencies.

A case study of an active reflector surface control system demonstrates that the proposed method significantly improves the clarity of coupling relationships, reduces reliance on expert experience, and provides a scalable solution for modularity analysis in complex systems.

Methodologies

An axiomatic design-based approach to MBSE modeling

The basic concepts in AD are the user domain, the functional domain, the physical domain, the design axioms, and the “Z” mapping between the design domains. The main activities of MBSE include requirements analysis, functional analysis and decomposition, design synthesis, and feedback loops. AD emphasizes the axiom of independence and the axiom of minimization of information to guide the design process to ensure that the design meets the requirements and is as simple as possible to avoid coupling. AD emphasizes guiding the design process through the axioms of independence and information minimization to ensure that the design meets the requirements and is as simple as possible to avoid coupling, while MBSE manages the system requirements, design, analysis, and validation through a model rather than a document to improve traceability, collaboration, and consistency. Therefore, AD and MBSE can be related through the following three correspondences, as shown in Fig. 1.

Fig. 1

The relationship between AD and MBSE.

(1) The mapping from Functional Requirements (FRs) to Design Parameters (DPs) in AD forms the core logical foundation of the requirements–design model within MBSE, while the behavioral analysis in MBSE is employed to describe the interactions between system functions and design parameters as defined by AD.

(2) AD provides a set of design quality assessment principles to MBSE through the Independence Axiom and the Information Axiom. Using MBSE tools, user requirements, performance metrics, and design constraints can be formalized into SysML requirement diagrams, ensuring the structured representation of requirements.

(3) The FR–DP matrix in AD constitutes the logical basis for the DSM in MBSE, where interface relationships captured in the model are inherited and quantified in the DSM to reflect design coupling. The optimized DSM can then be integrated back into the MBSE model, thereby constraining and updating the FR–DP matrix and enabling an iterative closed-loop alignment between the design and structural matrices throughout the system lifecycle.

Through FBS-based integration, AD and MBSE jointly construct the DSM during each design decomposition, where the DSM serves to represent dependencies among system requirements and functions. This matrix-based representation makes system-level coupling relationships explicit, providing a means for analyzing inter-module interactions. To further enhance the clarity of module identification and coupling analysis, this study proposes a method that combines NSGA-II with the AD–MBSE framework to optimize DSM sequencing and representation. Importantly, this optimization process focuses on improving the visualization and ordering of relationships, rather than altering the actual dependencies among design parameters or directly modifying the system architecture.

The optimization model of AD-MBSE based on NSGA-II

As illustrated in Fig. 2, the proposed framework is built around three core elements: AD, MBSE, and the NSGA-II algorithm. Within this framework, AD provides the conceptual foundation for MBSE modeling. Starting from user requirements, AD systematically considers the interactions between adjacent domains and progressively decomposes the system from a top-level perspective down to detailed solutions that satisfy the design objectives. MBSE complements this process by digitizing the system through formal models that capture the functional requirements and design parameters defined by AD. The construction and analysis of the design matrix are essential steps for identifying coupling within the system. In this context, the axioms of AD serve as the criteria for evaluating the degree of coupling. To further minimize coupling, the NSGA-II algorithm is applied to optimize the sequencing of modules within the design matrix, thereby enhancing the clarity and structure of system representations.

Fig. 2

NSGA-II based AD-MBSE design framework.

It should be emphasized that in this study the combined AD–MBSE framework is employed solely for the construction of the initial DSM and for the definition of design constraints. Once the DSM is generated, the subsequent module sequencing and reordering process is entirely driven by the NSGA-II algorithm-corresponding to the blue highlighted section in Fig. 2-with no manual intervention. All evolutionary operations, including selection, crossover, mutation, and Pareto-front exploration, are executed automatically within the algorithm. The principal model elements of the axiomatic design–based systems engineering framework are formally defined as a set , as expressed in Equation(1), with the detailed description of each element summarized in Table 1.

$$\begin{aligned} M = \bigcup \limits _{i = 0}^n {CR_{i}} + \bigcup \limits _{j = 0}^m {\bigcup \limits _{k = 1}^{K_{j}} {(FR_{k} + EX_{k} + DP_{k}) + \bigcup \limits _{j = 0}^m\bigcup \limits _{l = 1}^{L_{j}} {Maps} } } \end{aligned}$$

(1)

where,

$$\begin{aligned} \bigcup \limits _{i = 0}^n {CR_{i}} = \bigcup \limits _{i = 1}^k {(FR_{i}} + P_{i}) + \bigcup \limits _{i = 1}^l {NFR_{i},\{ k + 1 = n\} } \end{aligned}$$

(2)

In the formula, m represents the number of design iterations, $K_{j}$ is the number of functional requirements in each design level j, and $L_{j}$ is the number of association mappings in each level.

Table 1 Description of the elements required for the design.

Design structure matrix construction

The design process of function and structure is to establish the mapping relationship between two design domains, the functional domain and the structural domain, and the overall process is divided into two levels of mutual mapping relationship between “the same level” and “between different levels”. For the functional and physical domains, the mapping relationship is accurately expressed by a judgment matrix:

$$\begin{aligned} \left\{ {FR} \right\} = |A|\left\{ {DP} \right\} \end{aligned}$$

(3)

Where: FR denotes the design function set, and its element is $FR_{i}(i=1,2,\ldots n)$; DP denotes the structural parameter ensemble, and its element is $DP_{j}(j=1,2,\ldots n)$; A is the design matrix, which expresses the mapping relationship between the design function set and the structural parameter ensemble. Expand Eq.(3) as:

$$\begin{aligned} \left\{ \begin{array}{l} FR1\\ FR2\\ \vdots \\ FRi \end{array} \right\} = \left| {\begin{array}{*{20}{c}} {A11}& {A12}& {\ldots }& {A1j}\\ {A21}& {A22}& {\ldots }& {A2j}\\ \vdots & \vdots & \ddots & \vdots \\ {Ai1}& {Ai2}& {\ldots }& {Aij} \end{array}} \right| \left\{ \begin{array}{l} DP1\\ DP2\\ \vdots \\ DPj \end{array} \right\} \end{aligned}$$

(4)

In the design matrix, $A_{ij}$ denotes the correlation between the characteristic vectors in the functional domain and those in the structural domain. When $A_{ij}$=0, there is no relationship between $FR_{i}$ and $DP_{j}$; conversely, $A_{ij}$=X indicates a direct association between them.

Based on the form of the DSM, the current design can be evaluated with respect to the independence axiom. In the DSM, an “X” denotes the presence of coupling between a FR and a DP. Such couplings are inferred from direct or indirect relationships among SysML model elements, primarily considering:

1. Dependencies between functions and parameters;

2. Interactions or information flows among components;

3. Structural connections between elements.

From the perspective of the independence axiom, system designs can be classified into uncoupled, decoupled, and coupled designs. To satisfy the independence axiom, the DSM should exhibit either an uncoupled or a decoupled form. When the DSM shows a diagonal structure, it indicates an uncoupled design, meaning that FRs and DPs can be directly decomposed for subsequent iterative optimization. In contrast, a triangular DSM corresponds to a decoupled design, where the independence of FRs can be established by reordering the DPs. A fully populated DSM, in which most or all cells are non-zero, violates the independence axiom and implies strong coupling among FRs. In such cases, the system must be redesigned by selecting alternative DPs and adjusting their influence paths on FRs until the DSM approaches a diagonal or triangular structure. This hierarchical decoupling strategy effectively ensures the independence of FRs and enables system optimization through a continuous iterative process.

In the optimization of the DSM, the NSGA-II algorithm improves system modularity and reduces system coupling by optimizing the sequencing and clustering of design parameters. The specific optimization objectives are as follows:

1. Minimization of system coupling – quantified by counting the number of non-zero entries in the upper triangular part of the DSM after reordering. A smaller value indicates fewer coupling relationships between functional requirements and design parameters.

2. Maximization of modularity – optimize parameter ordering so that strongly coupled parameters are concentrated as close to the diagonal as possible

Figure 3 illustrates the axiomatic systems engineering design process based on the NSGA-II algorithm. Initially, the functional requirement hierarchy is established according to the task requirements of the active reflector control system. On this basis, design parameters are constructed, followed by the generation of the design matrix (The specific process is shown in Fig. 4.). Finally, the relationship matrix, constraint conditions, and NSGA-II algorithm parameters are input to conduct a rationality analysis of the proposed design solution.

Fig. 3

Axiomatic system engineering design flow chart based on NSGA-II algorithm.

Fig. 4

Rule-based SysML $\rightarrow$ DSM extraction process.

Design coupling definition

Coupling degree is a key metric used to evaluate the dependency relationships among design parameters, functional requirements, and behavioral elements within the active reflector control system. Excessive coupling can lead to increased system design complexity, thereby hindering future system expansion and maintenance. Consequently, minimizing the coupling degree becomes a core objective in the optimization of the design matrix. Define the global coupling degree G :

$$\begin{aligned} G = \frac{{\sum \nolimits _{i = 1}^N {\sum \nolimits _{j = 1}^N {Aij} } }}{{N \times N}} \end{aligned}$$

(5)

where,

$$\begin{aligned} A_{ij} = {\left\{ \begin{array}{ll} 1, & \text {There is coupling between modules } i \text { and } j \\ 0, & \text {No coupling between modules } i \text { and } j \end{array}\right. } \end{aligned}$$

(6)

In the formula, $A_{ij}$ represents the relationship between modules i and j, $N\times N$ represents the number of fully coupled connections of matrix A, and the value range of G is $\left[ 0,1\right]$ . The higher the value of G, the higher the coupling degree between parameters, which needs to be optimized.

Modularity degree

The degree of modularity M is a measure of the optimisation effect between the modules of the system design parameters, the higher the value of M represents the more independent modules between the system, the smaller the complexity of the system, the smaller the difficulty of the subsequent maintenance, the stronger the scalability. Modularity degree calculation formula:

$$\begin{aligned} Q = \frac{{\sum \nolimits _{i = 1}^N {\sum \nolimits _{j = 1}^N {(A_{ij}} - } \frac{{k_{i}k_{j}}}{{2a}})P_{ij}}}{{2a}} \end{aligned}$$

(7)

where,

$$\begin{aligned} P_{ij}= & {\left\{ \begin{array}{ll} {1,}& i \text { and }j \text { belong to the same module}\\ {0,}& i \text { and }j \text { do not belong to the same module} \end{array}\right. } \end{aligned}$$

(8)

$$\begin{aligned} k_{i}= & \sum \nolimits _{j = 1}^N {A_{ij}} \end{aligned}$$

(9)

$$\begin{aligned} a= & \frac{1}{2}\sum \nolimits _{i = 1}^N {k_{i}} \end{aligned}$$

(10)

In the formula, $P_{ij}$ is a matrix, $k_{i}$ is the number of connections between modules i, and a is the total number of connections in the design matrix. The greater the Q value, the smaller the dependence between modules.

Results and analysis

To evaluate the performance of different optimization algorithms, design structure matrices of sizes $10\times 10$, $15\times 15$, $20\times 20$, $25\times 25$, and were randomly generated. For each matrix size, three optimization strategies-namely the conventional sequencing algorithm, a GA, and the multi-objective evolutionary algorithm NSGA-II-were applied in multiple independent runs, and the corresponding Pareto front results were obtained (Figs. 5, 6, 7, 8 and 9). The optimization performance was quantitatively assessed using two key metrics:

(1) the number of coupled modules located in the upper triangular region of the matrix;

(2) the degree of modularity in the lower triangular region.

The detailed parameter configurations for each optimization algorithm are summarized in Table 2.

Table 2 Comparison of optimization methods, objectives, algorithm parameters, and characteristics.

Fig. 5

$5\times 5$ matrix optimization and result comparison chart.

Fig. 6

$10\times 10$ matrix optimization and result comparison chart.

Fig. 7

$15\times 15$ matrix optimization and result comparison chart.

Fig. 8

$20\times 20$ matrix optimization and result comparison chart.

Fig. 9

$25\times 25$ matrix optimization and result comparison chart.

As shown in Figs. 5, 6, 7, 8 and 9, the problem complexity increases significantly with the enlargement of the matrix scale, and the distribution characteristics of the solution sets obtained by different algorithms exhibit distinct differences. Overall, NSGA-II consistently produces a richer and more uniformly distributed Pareto front across all matrix sizes. In the Pareto front results, the blue points represent the complete set of Pareto-optimal solutions, among which the best decoupling solution, the best modularization solution, and the balanced solution are specifically highlighted. These results indicate that NSGA-II achieves a superior trade-off between reducing the number of coupled modules in the upper-triangular region and improving the modularization degree in the lower-triangular region.

By comparison, the conventional genetic algorithm is still able to obtain solutions close to the optimal front for small-scale matrices (10$\times$10 and 15$\times$15). However, as the matrix dimension increases to 20$\times$20 and beyond, the distribution of its solution sets becomes significantly contracted, revealing an evident lack of diversity. The traditional reduction-based algorithm performs the most limited optimization, with results concentrated in only a small portion of the solution space, making it difficult to simultaneously balance the two performance objectives. To further validate these observations, a detailed case study is presented in the following section.

Case study

To evaluate the feasibility and applicability of this method, a scaled-down model of an active reflector control system with a 5-meter aperture (1:100 of China’s Five-hundred-meter Aperture Spherical Telescope, FAST) is used as a case study.

The active reflector control system adjusts the surface of the reflector by driving actuators installed on the active reflector platform. Based on the general framework of the active reflector platform and the design requirements of the laboratory team, a hierarchical system architecture of the active reflector platform is established, as shown in Fig. 10. This architecture follows a layered design pattern, comprising four layers: the physical entity layer, the model layer, the data layer, and the measurement and control hardware layer. This layered architecture is a common design pattern in system architecture, aiming to enhance system decoupling, scalability, and maintainability.

1. Physical Entity Layer: This layer forms the foundation of the entire system and is composed of key components such as the active reflector, actuators, sensors, and lifting platforms. As the core layer responsible for achieving precise surface deformation, the performance of these physical entities directly affects the overall performance of the platform.

2. Model Layer: Built on the physical entity layer, the model layer serves as an abstraction layer that includes the mathematical model of the active reflector and a series of functions derived from the reflector model. These functions encompass a theoretical deformation model to describe the reflector’s deformation behavior under different control instructions, as well as fault diagnosis algorithms to detect potential failures in the reflector or system components based on monitoring data. The model layer provides theoretical support and decision-making capabilities, which are critical for achieving precise control and fault prevention.

3. Data Layer: Acting as the data processing hub of the platform, the data layer is responsible for receiving monitoring data from the physical entity layer and control instructions from the model layer. The processed data supports the model layer’s calculations and decision-making processes and is transmitted to external devices or users through the measurement and control hardware layer. Additionally, the data layer facilitates communication and enables data exchange and sharing with other systems.

4. Measurement and Control Hardware Layer: This layer serves as the interface between the platform and the physical entity layer and consists of four modules: data acquisition, data communication, drive units, and fault detection. The data acquisition module collects real-time data from the physical entity layer, while the communication module transmits processed instructions from the data layer to the drive unit. The drive unit, in turn, controls actuators, motors, and sensors based on the received instructions. The fault detection module leverages the fault diagnosis algorithms from the model layer to monitor the system’s status in real time, triggering alarms and implementing countermeasures if any anomalies are detected.

Fig. 10

Design framework of 5m aperture active reflector platform.

Based on the active reflector platform, the control system is designed to fulfill two core tasks: (1) dynamically transforming the reflector surface from a spherical to a parabolic configuration by translating observational commands into actuator-driven hardware control signals for real-time shape morphing, and (2) executing debugging commands to position actuators at predefined locations. As per the stipulations of the overarching task designated FR0, a comprehensive evaluation of the user requirements, functional requirements, performance parameters and system constraints has been conducted for the active reflective surface control system. This evaluation aims to facilitate the modelling of the user domain, functional domain, behavioural domain, physical domain and design axioms. The culmination of this process is the generation of the design matrix, which is presented in Fig. 11 as a $24\times 24$ scale. Additionally, the user requirements model elements are outlined in Table 3.

Fig. 11

Model organization diagram of 5m active reflector control system.

Table 3 The user domain model elements.

For the 24$\times$24 design structure matrix, three optimization algorithms are used to optimize the ranking, and each algorithm is run independently for 6 times, and the optimization results are shown in Figs. 12a, 13a, 14a, 15a, 16a and 17a and the corresponding Pareto frontier distributions are shown in Figs. 12b, 13b, 14b, 15b, 16b and17b. The comparison results of the two quantitative indexes are listed in Table 4 and Table 5.

Fig. 12

Results of the first run.

Fig. 13

Results of the second run.

Fig. 14

Results of the third run.

Fig. 15

Results of the 4th run.

Fig. 16

Results of the 5th run.

Fig. 17

Results of the 6th operation.

Table 4 Number of coupled modules in the upper triangular region for different algorithms.

Table 5 Modularity scores obtained by different algorithms.

Integrating the results of Tables 4 and 5, it is evident that the reduction-based algorithm exhibits almost identical outcomes across six independent runs: the number of coupled modules in the upper triangular region remains fixed at 47, and the modularity score remains constant at 606, indicating virtually no improvement to the original matrix structure. In contrast, the GA-based optimization approach demonstrates noticeable enhancements in both metrics: the number of coupled modules is reduced from 30 to a range of 6–13, while the modularity score increases from 606 to 642–767. These results suggest a certain optimization capability, albeit with considerable variability and limited stability. NSGA-II, however, outperforms GA on both objectives. The number of coupled modules consistently falls within 4–5, and the modularity score ranges from 613 to 784. Although some variability is present, the overall performance is superior to both the reduction-based algorithm and GA, achieving lower coupling while maintaining a high degree of modularity. Overall, NSGA-II exhibits stronger stability and optimization capability in design structure matrix refinement, resulting in clearer coupling relationships and facilitating subsequent system architecture analysis and improvement.

Sensitivity analysis

To further verify the robustness of the proposed method, a sensitivity analysis was conducted to evaluate the stability of the three algorithms over six independent runs. The reduction-based algorithm produced identical results across all runs, indicating insensitivity to operating conditions; however, this also reflects its limited optimization capability. In contrast, the GA exhibited considerable variability: the number of coupled modules fluctuated between 6 and 13, while the modularity index ranged from 642 to 676. These results suggest that GA is highly sensitive to initial conditions and genetic operations, prone to premature convergence, and thus lacks stability.

NSGA-II, on the other hand, demonstrated markedly higher stability. The number of coupled modules consistently remained within a narrow range of 4–5, while the modularity index varied between 613 and 784. Although some fluctuations were observed, the overall dispersion was significantly lower than that of GA, indicating that NSGA-II achieves a more stable balance between reducing coupling and enhancing modularity.

These findings confirm that the proposed NSGA-II–enhanced framework maintains high stability and consistency across multiple optimization runs, thereby validating its reliability and applicability in DSM optimization for complex systems.

Conclusion

To address the challenge that manual sequencing and conventional reduction algorithms are often ineffective in identifying and representing coupling relationships in large-scale DSMs for complex system design, this study proposes an AD-MBSE optimization model based on NSGA-II. The proposed approach employs SysML modeling to establish formalized mappings across the requirement, functional, behavioral, and physical design domains, ensuring bidirectional traceability and thereby maintaining the consistency and integrity of the system design. In parallel, a quantitative assessment mechanism grounded in the axiomatic design principle is incorporated to evaluate coupling relationships within the DSM and perform ordering optimization. This enhances the explicitness and visualization of interdependencies among system components.

The advantages of the proposed DSM optimization method become particularly significant under constrained design margins. When sufficient design margins are available, prior studies (Erens⁶; Brahma et al.²⁶) have shown that the benefits of strict modularization tend to diminish, as the additional flexibility introduced by design margins can offset some of the negative effects caused by coupling. Future work will incorporate design margins as an explicit parameter in the DSM optimization framework to better balance modular and integrated design, thereby improving accuracy and efficiency.

Although this study has been presented in a traditional top-down systems engineering context, the iterative nature of the proposed modeling and optimization process also makes it compatible with more agile SE practices, where models and coupling structures are continuously refined as requirements evolve. Regarding scalability, the method can in principle handle DSMs with hundreds of rows and columns; however, due to the computational cost of the metaheuristic optimization algorithm, the runtime becomes significant as the DSM size grows. Future research will address this limitation by integrating hierarchical decomposition, surrogate modeling, and parallelization techniques, enabling broader applicability to complex, large-scale system architectures.