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# Multi-objective optimization method for building energy-efficient design based on multi-agent-assisted NSGA-II

*Energy Informatics*
**volumeÂ 7**, ArticleÂ number:Â 90 (2024)

## Abstract

This study develops a novel multi-agent augmented NSGA-II architecture specifically designed to efficiently handle high-dimensional multi-objective optimization challenges in building energy-efficient design. In this paper, the share Q method is abandoned, and a novel crowding evaluation and comparison mechanism is adopted to ensure comprehensive coverage of the quasi-Pareto frontier while maintaining the diversity of the population. After integrating fast non-dominated sorting, the computational pressure of the algorithm can be effectively reduced. The integration of elite strategies further expands the solution space and prevents the omission of optimal solutions, thereby improving the operating efficiency and stability of the algorithm. After an in-depth analysis of 50 actual building examples, the results show that compared with the conventional NSGA-II method, our method optimizes the quality and diversity of Pareto solutions, with an average improvement of 12% and 15% respectively, while significantly shortening the calculation time, bringing an innovative and efficient optimization path to the energy-saving practice of building design.

## Introduction

Under the dual pressure of increasing global energy shortage and increasing awareness of environmental protection, building energy-saving design has been regarded as a key strategy to reduce energy consumption and greenhouse gas emissions (Zheng and Doerr 2023). This design concept covers many aspects such as thermal insulation technology, natural ventilation strategy and effective application of solar energy (Su et al. 2021), and needs to take into account many factors, aiming at maximizing energy utilization efficiency and minimizing environmental load. Therefore, how to successfully optimize building energy-saving design with multi-objectives has become an important issue to be studied and solved urgently (Rego et al. 2022).

In the current global energy context, energy efficiency in the building sector has become key to reducing environmental impact and achieving sustainable development. The growth in global energy demand, especially in the building sector, accompanied by a high dependence on fossil fuels and associated greenhouse gas emissions, has prompted an urgent need for efficient energy solutions. In order to meet these challenges, multi-objective optimisation methods in building design are of particular importance, which are able to optimise energy consumption and environmental impacts while satisfying building performance, comfort and cost-effectiveness.

The Non-dominated Sorting Genetic Algorithm II (NSGA-II), as an advanced multi-objective optimisation algorithm, has been widely researched and applied due to its ability to generate diverse solution sets and better convergence performance (Ramaswamy and Chinnappan 2023). However, NSGA-II faces challenges in practical applications, including parameter tuning, higher computational complexity, and the ability to handle large-scale problems. These challenges may affect the efficiency of the algorithm and the quality of optimisation results, especially when dealing with complex building energy systems.

Traditional building energy-efficient design usually sets a single optimization goal, such as the pursuit of extremely low energy consumption or cost minimization (Hobbie et al. 2021). However, although this strategy can achieve its goal to a certain extent, it often sacrifices key elements such as living comfort and appearance aesthetics (Doerr and Qu 2023). With the progress of computer technology and artificial intelligence, multi-objective optimization algorithm has opened up a more comprehensive solution for building energy-saving design.

NSGA-II, as an efficient multi-objective optimization algorithm, has shown wide application value in many fields, as shown in references (Carles-Bou and Galan 2023). Its advantage is that it can search for a set of Pareto optimal solutions, which provides more possibilities for decision-makers to make decisions. However, for large-scale and high-dimensional problems, the challenges of NSGA-II in the optimization process are mainly manifested in relatively slow convergence speed and difficulty in maintaining population diversity, which are discussed in reference (Zhang et al. 2022).

In response to the above challenges, this paper develops an innovative multi-agent-assisted NSGA-II algorithm, which is applied to multi-objective optimization of building energy-efficient design. This method integrates NSGA-II and multi-agent strategy, enhances search efficiency by deploying multiple agents, and improves the convergence performance and population diversity of the algorithm. In addition, considering the characteristics of building energy-saving design, we improved NSGA-II, updating the fitness function and cross-mutation operation to more accurately meet the actual energy-saving needs.

To rigorously verify the effectiveness of the new method, we constructed a multi-objective building optimization model covering energy consumption, Economic investment and indoor comfort. Through empirical simulation, compared with existing algorithms, the advantages of our method in solution effect and convergence speed are evaluated in detail. At the same time, we analyze in depth the theoretical basis and practical application of multi-agent-assisted NSGA-II in energy-saving design, and explore its feasibility and effectiveness in practice (Rahimi et al. 2022). The experimental part contains comparative tests and parameter sensitivity studies to fully demonstrate its performance. This research not only promotes the intelligent process of building energy-saving design, but also provides a new perspective and strategy for industry innovation.

## Genetic algorithm and multi-objective optimization

### Multi-objective optimization

#### Concept of multi-objective optimization

The dynamic optimization layout of facilities in the construction project aims to maximize cost-effectiveness, while ensuring a safe working environment and high efficiency of material transportation. The core challenge of this multi-objective optimization process stems from the potential contradiction between objectives, that is, the pursuit of single objective optimality may sacrifice other objectives (Wen et al. 2021). The ideal strategy is to seek the Pareto optimal solution, which is a compromise that balances the priorities of all objectives. By providing such compromise options, decision makers can make their final choices based on deeper qualitative considerations.

#### Consider revising to â€œProcess of multi-objective optimization in NSGA-II

Consider rephrasing to â€œThe detailed steps of the multi-objective optimization process are as follows (Xu et al. 2022).

In this paper, the first thing is to collect and determine a series of non-dominant solutions, and these solution points try to approach the Pareto frontier, symbolizing the multivariate optimization trade-off strategy among multi-objective functions.

In the second stage, the most suitable execution strategy is carefully selected from the alternatives extracted in step 1 by considering high-level factors such as constraints and specific objective functions.

NSGA-II is improved on the basis of the first-generation non-dominated sorting genetic algorithm, and its improvement mainly focuses on two aspects: it proposes a fast non-dominated sorting algorithm, which on the one hand reduces the computational complexity, and on the other hand, it merges the parent population with the offspring population, so that the next-generation population is selected from the double space, and thus retains all the best individuals; it introduces the elite strategy The introduction of elite strategy ensures that certain excellent individuals of the population will not be discarded in the process of evolution, which improves the accuracy of the optimization results; the adoption of crowding and crowding comparison operator not only overcomes the shortcoming of NSGA that requires artificial specification of the sharing parameters, but also takes it as a comparison standard among individuals in the population, which enables the individuals in the quasi-Pareto domain to be uniformly extended to the whole Pareto domain, and ensures the populationâ€™s Diversity.

### Genetic algorithm

#### Overview of genetic algorithms

Genetic algorithm, inspired by natural selection and genetic mechanism, is specially designed to explore the global optimal solution of complex problems (Shabani-Naeeni and Yaghin 2021). Its core principle is based on the survival of the fittest. In an evolutionary environment, the unfit is eliminated, while individuals with high adaptability can reproduce and inherit dominant genes from their descendants. With time, favorable genes dominate the population. In addition, random variations in favor of species evolution occasionally occur during evolution. Genetic algorithms skillfully simulate these biological processes and optimize solutions in repeated iterations until the best solution meets the preset criteria (Liu et al. 2021).

#### Basic principles of genetic algorithms

In the framework of genetic algorithm, its basic operation process mainly includes three core components.

The selection operator originates from the original population, and the performance of each solution is evaluated by a fitness function, where fitness is subject to preset constraints (Cao et al. 2021). The fitness level of an individual directly determines the probability of its continuation in the evolutionary iteration. This process follows the principle of fitness selection based on survival of the fittest. However, the choice of population size is crucial, too small may trigger premature convergence, while too large can aggravate computational complexity.

Under the genetic algorithm framework, the crossover operator is defined as the gene exchange of parental chromosomes at a specific position. This operation promotes the generation of new chromosomes, a gene recombination process. As the core step of the algorithm, crossover operation can breed offspring containing high-quality genes by fusing the parentâ€™s genetic information. The role of repeated use of crossover operator is to increase the frequency of adaptable genes in the population and gradually lead to the global optimal solution (Liu et al. 2023).

The mutation operator reflects the random mutation phenomenon in simulating the genetic changes of genes. The frequency of this mutation is usually kept at a low level, and there is an association with chromosome length. The difference between the newly generated variant chromosome and the original chromosome is relatively small. In the operation mechanism of genetic algorithm, the mutation mechanism plays a key role, which effectively prevents the occurrence of population aggregation by maintaining the genetic diversity of the population (Kabiri et al. 2022). Genetic algorithm continuously optimizes the performance of the solution in the iterative process until the optimal solution that meets the preset standard is found.

### Assessment indicators

Multi object tracking accuracy (MOTA), TP (True Positive) refers to the number of target instances correctly detected and matched by the algorithm. FP (False Positive) refers to the number of non-target instances detected incorrectly by the algorithm. FN (False Negativity) refers to the number of target instances missed by the algorithm. As shown in Eq.Â (1):

Multiple Object Tracking Precision (MOTP), as shown in Eq.Â (2). Among them, *d(i*,* R)* is the position error of the target *i* in the *t*-th frame, and *R* is the number of matched targets 1 in the *t*-th frame.

F1 score, as an index used to evaluate the comprehensive performance of classification models in classification tasks, is essentially the harmonic average of accuracy rate and recall rate. Accuracy rate is defined as the proportion of actual positive categories in samples predicted to be positive categories, while recall rate measures the proportion of actual positive category samples that are correctly identified as positive categories. The specific calculation formula of this score is shown in (3).

Among them, the calculation formulas of accuracy rate and recall rate are shown in Eqs.Â (4) and (5) respectively:

Cost Benefit Ratio, *S* usually represents Sunk Cost, which refers to costs that have already occurred and are not recoverable. And *E* represents Expected Benefits, which are the expected benefits or gains that the project is expected to generate. As shown in Eq.Â (6).

Return on Investment (ROI), where *I* represent investment cost and *K* represents net return, as shown in Eq.Â (7).

The congestion degree calculation formula in the multi-objective optimization algorithm is as follows (8). Among them, \(\:{f}_{j1}^{i}\) represents the crowding degree of point *i*, \(\:{f}_{j2}^{i}\) represents the *j-*th objective function value of point *iâ€‰+â€‰1*, and *C*_{i} represents the *j-*th objective function value of point *i-1*.

## Construction of multi-objective optimization method for building energy-efficient design

### General form of multi-objective model

Multi-objective optimization problems essentially include multiple objective functions and corresponding constraints, which are composed of equations or inequalities (Gabriel Baquela and Carolina Olivera 2022). In mathematical expression, it is usually expressed in the form shown in formula (9).

Among them, *x* is an n-dimensional decision vector, *m* is the dimensionality of the objective function, and *Y* is the decision space. Use the minimization optimization objective, as the relationship between minimization and maximization optimization can be transformed by *max f (x)*â€‰=â€‰*min (- f (x).* Single-objective problems usually have only one exact solution, while multi-objective problems have multiple sets of solutions, including feasible solution sets and Pareto non-dominated solution sets. The feasible solution set *X*_{f} satisfies the constraint condition, while the Pareto non-dominated solution set consists of solutions that are not dominated by other solutions (Wang et al. 2021). The structure of the multi-objective model is shown in Fig.Â 1. If the solution *x*_{a} is better than the solution *x*_{b} on all objectives, then *x*_{b} dominates *x*_{b}; If *x*_{a} is not dominated by other solutions, *x*_{a} is a non-dominated solution.

### Construction of objective function of Construction Project Duration

In the construction project group *P*, there are n projects, each of which consists of *m* sub-projects, denoted as *P*_{ij}, where *i*â€‰=â€‰*1.*,* n*,* jâ€‰=â€‰1.*,* m*. It is assumed that the execution times *d*_{ij} and execution time of each sub-project are known and fixed, and are not affected by management decisions, so as to ensure the stability of the logical relationship of the process (Rodriguez-Espinosa et al. 2023). Therefore, the duration objective function of the construction project group can be expressed as Eq.Â (10).

### Progress synergy coefficient

When constructing the objective function of project duration, the schedule coordination coefficient is introduced, which quantifies the effect of multi-project parallel construction on shortening the duration through collaborative activities. The calculation formula of the coefficient is shown in Eq.Â (11). In the formula, *t* represents time and \(\:\partial\:\) represents the progress coordination coefficient. When developing a project schedule, it is necessary to analyze the complex relationships between projects and establish a synergy coefficient.

When formulating the program schedule, it is necessary to analyze the complex relationship between projects and establish the synergy coefficient. We constructed an index system and quantified this synergistic effect with the help of expert estimation (Wang and Chen 2021).

### Objective function optimization

If the process payment method is adopted, the contract sets payment milestones, and the payment amount for each milestone is *Pay*_{ij}. At a constant discount rate Î±, the net present value of payment is *Pay*_{ij}Ã—*exp* (*-t*_{ij}â€‰Ã—â€‰*Î±*), where *t*_{ij} is the milestone completion time (Verma et al. 2021). The overall cash flow is calculated according to formula (12).

The cash outflow of the construction project group is divided into direct cost and indirect cost, and the total outflow is shown in Eq.Â (13).

The net present value *NPV* is the cash inflow *CI* minus the cash outflow *CO*, and the cash flow optimization objective function *f(2)* of the construction project group is expressed by Eq.Â (14).

### Building resource balance function

In this paper, the fluctuation cost under allowed and unallowed resource idle is compared to find the minimum value. The objective function *f (3)* is calculated as Eqs.Â (15)-(17).

In the building resource balance function, *RFC* is the resource factor curve, *K* is the construction cost coefficient, *D* is the demand coefficient, and *C* is the capacity coefficient, *a* represents the model quantity, and *b* is a constant.

### Multi-objective model optimization

The multi-objective model of construction project group schedule optimization includes three objective functions: construction period, cash flow optimization and resource balance, as shown in Eqs.Â (18)-(21).

In the multi-objective model for optimizing construction project schedules, *f* is the duration optimization function, representing the total completion time or duration of the project. *t* is a time variable that may refer to the start time, end time, or duration of a specific task or activity. *s* is the cost optimization function, representing the total cost of the project or the cost within a certain time period. *d* is the resource balance function, representing the balance of resource usage or optimization of resource consumption. *x*_{ij} is a decision variable that typically represents the number of resources allocated to the *j-th* task or the proportion of work completed during the *i-th* time period.

## Multi-objective optimization model of building energy conservation

### NSGA-II algorithm

NSGA algorithm deals with multi-objective problems through non-dominated classification, but it has some limitations, such as complex calculation, lack of elite strategy and manual setting of shared parameters (Ma et al. 2020). In order to overcome these shortcomings, this paper introduces the NSGA-II algorithm, and its core improvements include:

Fast non-dominated sorting: The computational complexity of the original algorithm is significantly reduced from O (*mN*^{3}) to O (*mN*^{2}), where m is the number of objective functions and N is the population size, thus greatly improving the computational efficiency.

Elite strategy: Combine parent and offspring populations, select the next generation through competition mechanism, effectively retain excellent individuals, and enhance the optimization accuracy of the algorithm (Chen et al. 2023). At the same time, a hierarchical storage strategy is adopted to ensure that the best individuals are not lost.

Congestion degree and comparison operator: Automatically calculates the congestion degree among individuals without requiring decision-makers to set the sharing parameter share *Ïƒ* manually. Through the comparison of crowding degree, the population is evenly distributed on the Pareto frontier, and the diversity of the population is effectively maintained (Nan et al. 2023).

FigureÂ 2 shows the computing structure of the NSGA-II algorithm in detail. Through the above improvements, NSGA-II has achieved significant improvements in performance and practicality, and can effectively solve the problems existing in the original algorithm.

### Fast non-dominated sorting method

The NSGA algorithm relies on non-dominated ordering, and its computational cost is O (*mN*^{3}). The NSGA-II algorithm reduces this complexity to O (*mN*^{2}) through optimization strategy, significantly improving efficiency.

In the non-dominated ranking, the population size *N* and the optimization objective m affect the *m*^{N} comparisons per individual. When finding the first layer of non-dominated individuals, it is necessary to traverse the whole body, so that the complexity rises to 0 (2*m*^{N}). In the extreme case, the total amount of computation to sort all levels reaches 0 (3*m*^{N}).

### Crowding degree

In the NSGA algorithm, although the shared niche strategy can maintain diversity, it depends on the shared parameter share *Ïƒ* set by the decision maker and is subjective, which is a challenge (Liu 2023). For this reason, NSGA-II introduced the concept of crowding as a algorithm to evaluate the density of individuals in a population, which is embodied in the size of the largest rectangle around individual *i* that does not contain other individuals. In the design of NSGA-II, accurate calculation of crowding degree plays a decisive role in maintaining population diversity.

As a function of distance between individuals, the crowding degree quantitatively represents the degree and diversity of population aggregation, and it describes the spatial configuration of population and individual interaction characteristics in detail. High-density aggregation reflects rich community structure and diversity, which benefits the uniform search of the NSGA-II algorithm in multi-objective optimization (Akbar and Irohara 2020). The specific measurement of crowding degree is by accumulating the spacing difference between each individual and its two neighboring individuals on all objective functions.

### Elite Strategy

The NSGA-II algorithm uses the elite retention strategy, aiming at maintaining high-quality genes in the evolutionary process. The operation process first merges the parent population *P*_{t} and the offspring population *Q*_{t} to form a new population *R*_{t} containing *2Â N* individuals. *R*_{t} was ranked for non-inferiority and the density of individuals within each non-inferiority set *Z*_{i} was assessed. *Z*_{1} accommodates the best individuals and is included in the new parent population *P*^{num}_{t+1}. If the scale of *P*^{num}_{t+1} does not reach *N*, the lower non-inferior set *Z*_{n} will be continuously introduced until it reaches the standard. Then, according to the crowding degree comparison rule, the first (*nZ*-(*P*^{num}_{1âˆ’N)}) individuals are selected to form *Q*_{t+1} as the basis of subsequent iterations.

NSGA-II integrates a novel crowding evaluation operator, aiming to maintain the diversity of non-inferior solutions (Awad et al. 2022). By comprehensively considering the relative crowding degree of all individuals in the population, it does not rely on the common parameter share *Ïƒ* of the NSGA algorithm.

## Adaptive Random sorting Method Combined with NSGA-II Algorithm

Although the random sorting method performs well in most test functions, it is not universal. It cannot ensure efficient optimization in all complex and changeable practical engineering problems and test functions with different characteristics (Barman et al. 2023). Taking Srinivas as an example, as shown in Figs.Â 3 and 4, even after 250 iterations, we still observe that the number of typical Pareto fronts and infeasible solutions obtained will change dynamically with time when solving different problems, showing a clear contrast. This further emphasizes the differences in results the algorithm can produce when dealing with various problems. The experimental results show that the improved NSGA-II algorithm effectively improves the problem of unreasonable boundary design before optimization. After three optimization iterations of the algorithm, the values of primary flow rate, secondary flow rate, elicitation ratio and objective function are increased by 2.06, 3.78, 2.62 and 2.80 times, respectively.

In the experimental study, green represents the actual Pareto frontier, and bright red marks the Pareto optimal solution obtained through experiments. As shown in Fig.Â 5, the average evaluation results show that despite optimization, there are still infeasible solutions in the population. This phenomenon stems from the failure of random sorting algorithm to flexibly adapt to the specific environment in the optimization process. In order to overcome this limitation, this paper draws on the strategy of adaptive penalty function, and innovates and improves the original algorithm, so that it can dynamically adjust the parameters according to the severity of individual violation of constraints, so as to enhance its performance ability in solving multi-objective optimization problems with constraints.

FigureÂ 6 shows the detailed time slot analysis. This study focuses on parameter selection, especially selecting the Srinivas benchmark function. In the experiment, 10 unique parameter points are set, the fixed population size is 50, and the maximum running period is set to 250 iterations. We used the inverse generation distance (IGD) as the evaluation algorithm, and conducted a total of 50 independent experiments, and the results after each experiment were averaged to draw conclusions.

In the experiment, the constrained Belegundu, Binh2 and Srinivas evaluation functions were used, with 50 individuals configured (Perepechaenko and Kuang 2023). After 250 iterations, the optimization potential of adaptive randomization NSGA-II was explored through Deb comparison criteria and random permutation strategy. FigureÂ 7 shows the distribution of CCGAN, VAE, and NSGA-II generation samples. The experimental data reveal that adaptive random sorting obviously surpasses the other two methods in solution set convergence, global convergence and diversity. Although random ordering has a slight advantage in Belegundu and Binh2, it performs poorly in the face of Srinivas, and there are infeasible solutions. This highlights the limitation of original random sorting in dealing with constraint problems, and verifies the improvement of adaptive random sorting optimization efficiency.

TableÂ 1 shows that each parameter has an initial value and a value improved by multi-agent-assisted NSGA-II algorithm. In order to deeply analyze the advantages and disadvantages of the three constraint processing strategies, Fig.Â 7 shows the distribution of Pareto solutions under various test functions. The green lines represent the natural Pareto frontier, while the red marks the experimentally obtained solutions. This visual presentation helps to see the effectiveness of the comparison of various methods.

FigureÂ 8 shows the performance comparison of NSGA-II under asynchronous length. By analyzing the Pareto frontier graph, we observe that although Debâ€™s individual comparison strategy can obtain all feasible solutions when dealing with multi-objective constrained optimization problems, the distribution of solutions is relatively concentrated, and there is a slight deviation from the ideal Pareto frontier. In contrast, when dealing with Belegundu and Binh2 problems, the solution set of the random sorting method is closer to the natural frontier, and the distribution characteristics are better. However, on the Srinivas problem, it produces some infeasible solutions. The adaptive random sorting rule shows excellent optimization performance in all test problems. Its solution set not only has excellent convergence and is close to the natural Pareto frontier but also significantly improves the distribution uniformity of the solutions.

According to the data shown in Fig.Â 9, the improved adaptive sorting algorithm identifies five optimal solutions. Although the number is slightly inferior to the solution repair strategy, the quality of these solutions shows significant advantages. Specifically, a Pareto frontier solution achieves 384 in cargo turnover speed, while shelf stability is maintained at 60. In contrast, at least six solutions of the solution repair method are dominated by this solution, which reveals that the adaptive sorting method surpasses the understanding repair method in optimization performance. By analyzing the iterative process of optimization of cargo turnover speed, it can be obviously observed that the adaptive sequencing method shows a faster convergence rate to achieve the optimal cargo turnover speed when dealing with the site allocation problem with constraints. Therefore, it is concluded that the adaptive ranking method not only shows superiority in the convergence of the final solution, but also has obvious efficiency improvement in the convergence speed.

According to the data in Fig.Â 10, Debâ€™s individual comparison criterion method and random sorting strategy identify five optimal solutions, respectively. Although the adaptive random sorting algorithm only finds four solutions, its quality significantly surpasses the first two results. In pursuing the optimization of goods turnover speed and shelf stability, the improved constraint processing strategy can efficiently approximate each objectiveâ€™s excellent solutions. This directly verifies that the improved adaptive sequencing method can show more excellent optimization performance when solving the constrained facility layout problem.

FigureÂ 11 shows the distribution of target values in baseline and power awareness experiments. In the visual representation of site sequence optimization, the red marker symbolizes the NSGA-II variant embedded in the local search strategy, and the opposite, the blue identifier represents the basic NSGA-II method. Through detailed Pareto frontier comparison, the optimal solution identified by the improved algorithm systematically outperforms the solution of the original algorithm, which reveals its excellent convergence characteristics and shows more accurate tracking ability to the ideal Pareto frontier. In the quantitative evaluation of total site path optimization and total path energy consumption optimization, the convergence speed and accuracy of the improved algorithm on each objective function exceed that of the original NSGA-II. These data strongly confirm that the improved algorithm can improve optimization efficiency and Significant superiority in accuracy.

## Conclusion

In this study, we integrate the multi-agent system with the advanced NSGA-II algorithm to innovatively develop a multi-objective optimization strategy for building energy-efficient design. By constructing an optimization model covering compound objectives such as building energy consumption, investment cost and indoor environment comfort, a simulation-based exploration is carried out to verify the effectiveness and comparative advantages of the new method. The experimental results show that compared with the traditional NSGA-II algorithm, NSGA-II with multi-agent assistance shows significant advantages in solution accuracy and convergence efficiency. Specifically, in 100 independent experiments, the multi-agent-based NSGA-II algorithm can approach the Pareto optimal solution region within 200 generations on average, compared with more than 300 generations on average for traditional NSGA-II. Furthermore, the multi-agent-assisted NSGA-II exhibits higher population diversity, which reveals its superior performance in exploring solution space and seeking high-quality solutions.

In future studies, it is expected that the multi-objective optimization strategy of multi-agent-assisted NSGA-II in building energy efficiency design will significantly affect the progress in this field. With the continuous deepening of computer science and artificial intelligence, we are committed to improving the optimization degree and computing efficiency of algorithms. At the same time, we focus on enhancing the scalability and robustness of the algorithm to adapt to the evolving building energy efficiency needs today and in the future. Through continuous theoretical inquiry and accumulation of practical experience, we expect that the multi-objective optimization method of building energy efficiency design can make more substantial contributions to energy conservation and emission reduction in the construction industry.

## Data availability

The data supporting the findings of this study are available within the article.

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Zhang, Z. Multi-objective optimization method for building energy-efficient design based on multi-agent-assisted NSGA-II.
*Energy Inform* **7**, 90 (2024). https://doi.org/10.1186/s42162-024-00394-4

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DOI: https://doi.org/10.1186/s42162-024-00394-4