Enhancing energy-efficient building design: a multi-agent-assisted MOEA/D approach for multi-objective optimization

Abstract

Energy-efficient building design is often challenged by multiple optimization problems due to contradictory objectives that are often hard to balance, so an effective optimization method should be thoroughly considered. Accordingly, a multi-objective evolutionary algorithm is then proposed. Firstly, the multi-agent auxiliary objective evolutionary algorithm for building energy efficiency model is established. According to model result analysis, the proposed algorithm runs fastest for 1640s with the average running time of 1710s in a single-room building, comparing to the least running time of 1680s for the multi-objective particle swarm optimization algorithm. In multi-room buildings, the proposed algorithm runs from 3350s to 3650s, with the average running time of 3500s. In conclusion, the model proposed in this study can comprehensively consider multiple objectives such as energy consumption, cost, comfort, etc. No matter in single-room or multi-room buildings, the model demonstrates superior performance and stability to realize comprehensive optimization of energy conservation design.

Introduction

As global energy crisis and environmental protection awareness amount, energy-efficient building design has gained attention in the construction industry (Sangiuliano 2018; Minnich et al. 2022). Traditional design methods often aims to optimize a single objective, hardly covering energy consumption, cost, comfort and other aspects together (Yang et al. 2018; Akhtar et al. 2023). Therefore, it is particularly important to develop a optimized design method that can comprehensively consider multiple objectives. Multi-objective Evolutionary Algorithm (MOEA) is a search algorithm based on evolutionary computation, specifically for multi-objective optimization problems. This algorithm imitates the natural evolution process and improves individual fitness through evolutionary operations such as selection, crossover and mutation, so as to gradually approach the Pareto optimal solution set of multi-objective optimization problems (Brooks et al. 2021; Kiveks et al. 2019). In a multi-objective optimization problem, maximizing all the objectives together is impossible due to objective conflicts. To solve it, MOEA can effectively explore multi-objective space through well-designed representation and evolution strategies to characterize accurate multi-objective optimal solutions (Chen et al. 2020; Liang et al. 2020). However, MOEA is still challenged by problem of energy-efficient building design (Ozkan and Bulkan 2022). In this field, considering multiple contradictory design objectives, MOEA often takes longer calculation time to generate the optimal solution. Apart from computational complexity and calculation time, it faces potential solution failures within limited time. Therefore, this study introduces multi-agent system and multi-objective evolutionary algorithm MOEA/D to build a new multi-objective optimization method for energy-efficient building design to solve above-mentioned limitations. In summary, this study contributes to optimized population initialization strategy, an efficient cross-mutation operator design and the introduction of an adaptive diversity preservation mechanism, effective solutions to slow convergence and unequal distribution of solution sets by traditional MOEA facing high-dimensional and multi-conflict targets, together with significantly higher algorithm performance and efficiency.

The research has four major parts. The first part is a literature review, which summarizes global researches on multi-objective optimization of energy-efficient building design, and provides theoretical support and research direction for this study. The next is the model development of multi-objective evolutionary algorithm based on MOEA/D, specifically describing the principle and characteristics of MOEA/D algorithm, and corresponding application. Thirdly, building energy efficiency results are analyzed based on multi-agent-assisted MOEA/D to verify the effectiveness of the research model. Finally, major results together with the application prospect and development direction of this proposed method are summarized and concluded.

Related works

The construction and use of buildings will surely have an environmental impact. In this aspect supported by relevant research, BECD can optimize the design of insulation, ventilation, lighting, etc., thereby reducing energy loss and mitigating climate change. That is, optimized BECD scheme can reduce construction costs for higher economic benefits. To improve building energy efficiency, Liu et al. proposed a prediction method for building energy consumption based on building envelope design. The radio frequency model is utilized to rank the importance of each parameter, and evaluate the corresponding correlation using Pearson function. As a result, the radio frequency model has superior advantages in predicting architecture energy consumption comparing to other relevant models (Liu et al. 2019). For higher energy efficiency of existing buildings, Pasichnyi’s team proposed a new data-driven strategic planning method for building energy efficiency (BEE) renovation, and conducted an empirical analysis of this method. With this method, energy efficiency programs became more targeted and better coordinated with potential rich urban energy datasets and data science technologies for better decision-making and strategic planning (Pasichnyi et al. 2019). In addition, Roth and Rajagopal proposed a new benchmark system of building energy use to improve current practices, which is expressed as the score distribution of a building to its construction. A more standardized and robust benchmark model for this study can enhance resource allocation for energy efficiency projects, virtuous competition among building designs and insight into building energy drivers (Roth and Rajagopal 2018). To improve the BEE of building envelope in cold regions, Zhang and Ren proposed a corresponding model based on sensitivity analysis method. The empirical analysis of the high model showed that the sensitivity coefficients of annual cumulative load on building exterior Windows, exterior walls and roof were 7.93%, 5.76% and 5.75%, respectively. It is highly pragmatic to study the influence of envelope structure on BEE in cold regions to reduce building energy consumption (Zhang and Ren 2022).

MOEA/D refers to a decomposition-based MOEA, which divides complex MOO problems into a series of single-objective optimization sub-problems that can be optimized by evolutionary algorithm. This method is proved to be efficient against complex MOO problems. Some researchers have already made outstanding achievements in MOEA/D algorithm. Ho-Huu et al. proposed iMOEA/D to solve MOO problems, especially for applications with complex Pareto fronts. The reliability, efficiency and applicability of iMOEA/D are studied by using 7 benchmark test functions with complex pf and 3 truss structure optimization design problems. Finally, iMOEA/D is superior to MOEA/D and NSGA-II in both function and practical application (Ho-Huu et al. 2018). To solve the problem that most parameter control methods are based on experience or intuition, Gao et al. proposed an operator adaptive selection method named the double adaptive selection DAS strategy, and conducted simulation experiments on this strategy. The effectiveness and superiority of MOEA/D-DAS are verified by its good convergence and distribution (Gao et al. 2019). To satisfy the thermal management of gallium nitride high electron mobility transistor devices, Wang et al. proposed a micro-channel parameter model that integrated computational fluid dynamics and MOEA algorithm, and conducted an empirical analysis of the model. The relationship between minimum thermal resistance and pressure drop is given in this study model, which provides basic guidance for designing cooling micro-channels for GaN HEMT devices (Wang et al. 2021). Moreover, for better parameter strategy in the MOO algorithm, Bo and Gu developed an improved two-population co-evolutionary MOEA/D algorithm. Compared with the other three comparison algorithms, the proposed two-population co-evolutionary MOEA/D algorithm showed significantly superior IGD and HV indicators (Bo and Gu 2022) along with the effectiveness verified in the application of LTE base station power distribution model.

Despite remarkable progress in building energy efficiency, existing methods still have limitations. For example, the energy consumption prediction method based on building envelope design proposed by Liu et al. has accurate energy consumption prediction by radio frequency model, but its prediction ability in dealing with complex building forms and dynamic climate conditions has yet to be verified. The Pasichnyi team’s data-driven transformation strategy plan is more targeted, but may be limited in practice by the integrity and real-time data acquisition. Roth and Rajagopal’s benchmark system provides a standardized evaluation framework for energy efficiency projects, but there is still a challenge to ensure the universal applicability and dynamic adjustment of benchmarks. Based on existing results, an innovative SMOPSO/D method is then proposed. Compared with the RF model, SMOPSO/D can generate more accurate simulations of building energy consumption by comprehensively considering multiple factors such as buildings’ physical characteristics, climate conditions and user behaviors, together with good generalization ability under different building types and climate conditions from it algorithm architecture. While the Pasichnyi team’s approach focuses on strategic planning, SMOPSO/D is directly involved in the design optimization process, providing designers with immediate feedback and an optimal solution set with significantly higher efficiency and effectiveness. While Roth and Rajagopal’s benchmark system is groundbreaking, the deep integration of decomposition strategies with PSO enables SMOPSO/D to build a more robust and standardized optimization framework that better adapts to dynamic different building characteristics and optimization objectives. In a word, this study breaks the limitations of the previous qualitative research and shows strong potential value.

MOO method modeling of BEE design based on multi-agent assisted MOEA/D

This study deeply integrates multi-agent technology with MOEA/D algorithm for an efficient and intelligent optimization method to address complex multi-objective problems in building energy efficiency. The working flow chart is shown in Fig. 1.

Fig. 1

Working flow chart of the research method

According to Fig. 1, this study uses the distributed characteristics of multi-agent technology to divide complex optimization problem of building energy efficiency into multiple single-objective sub-problems, each of which is handled by an independent agent. Among multiple agents, MOEA/D algorithm as the global optimization framework adopts evolutionary algorithms to explore and maintain a set of Pareto optimal solutions in the solution space. In each agent, particle swarm optimization algorithm is introduced as a local search strategy to make each particle move towards the optimal solution continuously in the solution space. Under the framework of fusion, agents exchange information of excellent solutions through efficient communication mechanism, so as to deeply integrate and organically coordinate global search and local search, which provides an efficient, intelligent and promising optimization method for complex multi-objective problems in building energy conservation.

Multi-objective evolutionary optimization algorithm based on improved multi-agent assistance

MOO problem is also called vector optimization problem or multi-criteria optimization problem (Libotte et al. 2022). Simply speaking, multiple goals in a certain scenario should be met simultaneously despite always conflicts with each other towards failures of simultaneous realization. Therefore, the MOO problem needs to be coordinated and compromised for overall objective optimization as possible. MOO problem is displayed in Fig. 2.

Fig. 2

Multi-objective optimization problem

From Fig. 2, the MOO problem involves determining a vector of decision variables within the feasible domain that satisfies all constraints and optimizes the vector of multiple objective functions. Usually, there are conflicts between these objective functions, so “optimization” here refers to finding a solution vector or a set of solution vectors to realize all the objective functions in the target vector following the designer’s requirements. However, it is impossible to optimize multiple objective functions at the same time due to such conflicts. Instead, there needs to be coordination and compromise to optimize each goal as much as possible. The MOO problem can be defined by Eq. (1).

$${X^ * }=\left[ {x_{1}^{ * },x_{2}^{ * },...,x_{n}^{ * }} \right]$$

(1)

In Eq. (1), $X$ is the decision variable. $J$ Eq. or inequality constraint can be expressed by Eq. (2).

$${h_j}(X)=(\leqslant)0,j=1,2,...,J$$

(2)

To minimize the objective vector function $F(X)$, the expression is shown as Eq. (3).

$$F(X)=({f_1}(X),{f_2}(X),...,{f_M}(X))$$

(3)

In Eq. (3), $M$ is the quantity of functions. For MOO problems, all components of decision variable $X$ have their boundary values, which constitute the decision space of the problem. The output value of each objective function forms the objective space of the problem $Z$. That is, the optimization means to find a solution in the decision space, so that the objective function $Z$ in the objective space can be optimized. Pareto domination refers to that a solution is not inferior to another on all objective functions, but is superior to another solution on at least one objective function (Datta et al. 2023). It is an important concept in MOO problems to compare the advantages and disadvantages of different solutions. The value range dominated by Pareto is expressed as Eq. (4).

$$\begin{gathered} \forall i \in \left\{ {1,2,.,M} \right\},{x_i} \leqslant {y_i} \hfill \\ \exists j \in \left\{ {1,2,.,M} \right\},{x_i} \leqslant {y_i} \hfill \\ \end{gathered}$$

(4)

Pareto optimality refers to an ideal state of resource allocation, proposed by Italian economist Vilfredo Pareto. In this state, it is assumed that the group of people and resources can be redistributed in a way that benefits at least one person without harming anyone else. The optimal Pareto value range is expressed as Eq. (5).

$$\begin{gathered} \forall i \in \left\{ {1,2,...,M} \right\},{f_i}(X^{\prime}) \leqslant {f_i}(X) \hfill \\ \exists j \in \left\{ {1,2,...,M} \right\},{f_i}(X^{\prime}) \leqslant {f_i}{(X)_i} \hfill \\ \end{gathered}$$

(5)

Following this equation, all solutions to the MOO problem that satisfy Pareto domination relations are grouped into a Pareto optimal solution set. Particularly, considering MOO problems, local optimal solutions are often more diverse than single-objective optimization problems, so a more accurate search method is needed to find the global optimal solution. Particle swarm optimization (PSO) is based on swarm intelligence, which simulates the behavior law of birds, fish and other biological groups to optimize search. The PSO algorithm is known for versatility and ability to handle a wide range of objective functions and constraints. In addition, this algorithm can also be easily integrated with traditional optimization methods to enhance its effectiveness by overcoming its limitations (Bahrami-Novin et al. 2022). The MOEO algorithm framework based on improved multi-agent assistance is shown in Fig. 3.

Fig. 3

Multi-objective evolutionary optimization algorithm framework based on improved multi-agent assistance

From Fig. 3, in a PSO algorithm, each particle represents a possible solution, and the velocity and position of the particle are renewed by updating the expression. Through continuous iteration, the particles gradually approach to the optimal solution, finally reaching a Pareto optimal solution. Therefore, by applying PSO algorithm to MOO problems, multiple Pareto optimal solutions can be found effectively, and a more comprehensive optimization scheme can be available for decision makers. The particle renewal rate ${x_i}(t+1)$ is shown as Eq. (6).

$${x_i}(t+1)=w{v_i}(t)+{c_1}{r_1}(p{b_i}(t) - {x_{i,j}}(t)+{c_2}{r_2}(g{b_{}}(t) - {x_i}(t))$$

(6)

In Eq. (6), $p{b_i}(t)$ refers to the individual best advantage of the $i$ particle and $g{b_{}}(t)$ represents the global best advantage of the $i$ particle. ${r_1}$ and ${r_2}$ are the uniform random number between [0, 1], ${c_1}$ and ${c_2}$ refer to the learning factor, and $w$ represents the inertia factor. Particle renewal rate is expressed as Eq. (7).

$${x_i}(t+1)={x_i}(t)+{v_i}(t+1)$$

(7)

In Eq. (3.7), ${v_i}$ represents the location and speed of particle $i$ at time $t$.

$${x_{i,j}}(t+1)=\left\{ {\begin{array}{*{20}{c}} {N\left({\frac{{{r_3} \times p{b_{i,j}}(t)+(1 - {r_3}) \times g{b_{i,j}}(t)}}{2},\left| {p{b_{i,j}}(t) - g{b_{i,j}}(t)} \right|+{\beta _j}} \right),ifU(0,1)<0.5} \\ {g{b_{i,j}}(t)\begin{array}{*{20}{c}} {}&{otherwise} \end{array}} \end{array}} \right\}$$

(8)

According to Eq. (8), the complexity of the designed algorithm mainly covers the evaluation of the real target value, the development and updating of the proxy model, and other operators required in the process of population renewal. Its expression is shown as Eq. (9).

$$F=RFEs \cdot {F_R}+{F_S}+{t_{Max}} \cdot {F_{Other}}$$

(9)

In Eq. (9), $RFEs$ shows the statistics of the number of times that Energy Plus software is used to truly evaluate individuals, ${t_{Max}}$ represents the total iterations of the algorithm, ${F_R}$ is the evaluation of the real target value, ${F_S}$ refers to the construction and updating of the proxy model, and ${F_{Other}}$ represents other operators required in the population updating process.

BEE model construction based on multi-agent assisted MOEA/D

In this study, a multi-agent model SMOPSO/D based on the object division feature of MOEA/D was built by introducing PSO algorithm. Using PSO algorithm to address multiple sub-objectives from the original problem, this mechanism can effectively find out the Pareto optimal solution to the MOO problem. Moreover, such mechanism also implements the management and coordination of multiple agent models to ensure joint work and overall optimum. The schematic diagram of BEE model based on multi-agent auxiliary MOEA/D is shown in Fig. 4.

Fig. 4

Schematic diagram of BEE model based on multi-agent assisted MOEA/D

According to Fig. 4, when selecting representative individuals, the accuracy of the proxy model, the convergence and the population diversity should be fully considered. The fill sample generation method selects representative samples from the existing data set for bigger data set and better model performance. When selecting filler samples, sample diversity and quality are always needed to cover the different characteristics and distributions of the data-set. By selecting representative samples, the sample representativeness and proxy model accuracy can be effectively improved. For an individual, its uncertainty degree is expressed in Eq. (10).

$$u(X)=\frac{1}{M}\sum\limits_{{i=1}}^{M} {\sum\limits_{{j=1}}^{{\tau +1}} {\sqrt {\frac{{{{\left({\hat {f}_{i}^{j}\left(X \right) - {{\bar {f}}_i}\left(X \right)} \right)}^2}}}{{\tau +1}}} } }$$

(10)

In Eq. (10), $\hat {f}_{i}^{j}\left(X \right)$ represents the proxy model evaluation value of individual variable $X$, variable $\tau$ represents the neighborhood size of the individual, and ${\bar {f}_i}\left(X \right)$ represents the average value of individual variable $X$ on the target $i$. The flow chart of BEE model based on multi-agent assisted MOEA/D is shown in Fig. 5.

Fig. 5

Flow chart of BEE model based on multi-agent assisted MOEA/D

From Fig. 5, a new individual evaluation mechanism is introduced in this study for the sub-optimization problem. This mechanism is called the individual evaluation mechanism based on the aggregation of neighboring agents, which can predict the objective function values of individuals in the population. By aggregating the prediction results of neighboring agents, this mechanism adopts a weighted average method to generate the predicted value of individual objective function values, thus generating more accurate and reliable predictions. The expression of the objective function after introducing the individual evaluation mechanism of adjacent agent aggregation is shown in Eq. (11).

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} _m}\left({{X_i}} \right)=\sum\nolimits_{{j=0}}^{\tau } {{\varpi _j}} \times {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} _m}\left({{X_i}\left| {SM\_\gamma _{i}^{j}} \right.} \right),m=1,2,...,M$$

(11)

In Eq. (11), ${\varpi _j}$ represents the weight of $SM\_\gamma _{i}^{j}$. If setting the value of ${\varpi _0}$ at 0.5, the weight ${\varpi _j}$ can be expressed as Eq. (12).

$${\varpi _j}=0.5\frac{{{{\left| {\gamma _{i}^{0} - \gamma _{i}^{j}} \right|}^{ - 1}}}}{{\sum\nolimits_{{j=0}}^{\tau } {{{\left| {\gamma _{i}^{0} - \gamma _{i}^{q}} \right|}^{ - 1}}} }}$$

(12)

In Eq. (12), ${\left| {\gamma _{i}^{0} - \gamma _{i}^{j}} \right|^{ - 1}}$ represents the reciprocal of the distance between $\gamma _{i}^{0}$ and $\gamma _{i}^{j}$. With shorter distance between $\gamma _{i}^{0}$ and $\gamma _{i}^{j}$, the sub-optimization problems of $SM\_\gamma _{i}^{j}$ and $SM\_\gamma _{i}^{0}$ are more similar, then $SM\_\gamma _{i}^{j}$ will play a greater role in predicting the objective function value, and the weight will be larger. In the MOEA/D algorithm, the choice of reference point ${Z^ * }$ has a crucial influence on the quality and distribution of the generated Pareto front end. Different from traditional numerical optimization methods, MOEA/D algorithm has two kinds of objective function values: real target value and predicted target value. The effectiveness analysis diagram of reference point selection is shown in Fig. 6.

Fig. 6

Effectiveness analysis of reference point selection

The expression of the new reference point is shown in Eq. (313).

$$z_{m}^{ * }=\left\{ {\begin{array}{*{20}{c}} {f_{m}^{{\hbox{min} }}\begin{array}{*{20}{c}} {}&{}&{f_{m}^{{\hbox{min} }}<\hat {f}_{m}^{{\hbox{min} }}} \end{array}} \\ {\frac{t}{{{\text{T}_{\text{max} }}}} \times f_{m}^{{\hbox{min} }}+\left({1 - \frac{t}{{{\text{T}_{\text{max} }}}}} \right) \times \hat {f}_{m}^{{\hbox{min} }},otherwise} \end{array}} \right\}$$

(13)

In Eq. (13), $f_{m}^{{\hbox{min} }}$ and $\hat {f}_{m}^{{\hbox{min} }}$ are the true and predicted minimum of the new reference point, respectively. ${T_{\hbox{max} }}$ represents the max-value in the sample set T data. To evaluate the model performance, Hyper-volume (HV) was taken as an evaluation index. The over-volume index is an effective unitary quality measure, which is strictly monotonic in Pareto domination. The rule is that larger value means better performance of the corresponding algorithm. The hyper-volume expression is shown in Eq. (14).

$$HV=\rho (\cup _{{i=1}}^{{\left| s \right|}}{v_i})$$

(14)

In Eq. (14), $\rho$ represents the Lebesgue measure, $\left| S \right|$ is the number of non-dominant solution sets. ${v_i}$ means the hypercube formed by the $i$ solution. The SC measure usually used in MOO is selected to compare algorithm convergence. SC measure is a common convergence index to judge the convergence speed and stability by comparing the change of the objective function value in the iterative process. The Eq. (15) shows the calculation formula of SC measure.

$$Sc(A,B)=\frac{{\left| {\left\{ {b \in B:\exists a \in A,a \prec b} \right\}} \right|}}{{\left| B \right|}}$$

(15)

In Eq. (15), $Sc(A,B)=1$ and $Sc(A,B)=0$ mean that all the solutions and none solutions in $B$ are dominated by the solutions in $A$, respectively.

Analysis of BEE Research results based on Multi-agent assisted MOEA/D

To verify the MOO method’s superiority for BEE design based on multi-agent-assisted MOEA/D, the proposed SMOPSO/D method was compared with similar models NSGA-II, MOPSO and ParEGO. In addition, SMOPSO/D method is also superior in better balanced diversity and concentration of solutions when dealing with complex MOO problems, together with higher optimization efficiency. The MOO method based on multi-agent-assisted MOEA/D is highly pragmatic to provide an effective optimization tool for BECD.

Performance analysis based on multi-agent assisted MOEA/D algorithm

In this paper, EnergyPlus is adopted to simulate building energy consumption behavior, and Matlab is used to realize the proposed multi-objective particle swarm optimization algorithm. Each time the optimization algorithm generates a new potential solution, the communication interface between Matlab and EnergyPlus is then built using Visual C++, and the solution is automatically delivered to the EnergyPlus simulation system. Then the received solutions are used to set the corresponding parameters and initiate a simulation process to evaluate key performance indicators such as building energy consumption and user discomfort hours. After simulations, these performance indicators will be fed back to the Matlab algorithm program through the interface program, and used to update the objective function value of the current solution. To verify the performance superiority of the established algorithm, SMOPSO/D was compared with MOPSO. As the algorithm performance is easily affected by various random factors, the results of a single experiment may not be enough to accurately reflect the average performance and performance distribution of a certain algorithm. Thus to ensure the evaluation reliability, these two algorithms were run for 20 times each with a different number of building rooms for more robust performance estimates. In addition, the hyper-volume HV value of each run was recorded. The results are shown in Fig. 7.

Fig. 7

HV values of the two algorithms in buildings with different number of rooms

Figure 7 (a) shows the HV value of the two algorithms in a single-room building. With more iterations, the HV value of the research algorithm gradually outperforms the MOPSO. When iterated for 15 times, the HV values between the two algorithms show the largest difference. In the whole iteration process, the research algorithm always maintains a better state than the MOPSO until termination. In addition, Fig. 7. (b) shows the HV values of the two algorithms in multiple-room buildings. As the iterations increase from 10 to 15 times, the HV value of MOPSO is superior to that of the research algorithm. When the proposed algorithm is terminated, the HV value obtained is greater than that of the MOPSO. To further verify the performance difference between these two algorithms, t-test was used. This is a statistical hypothesis testing method, mainly to test whether there are significant differences in the mean values of two groups of independent samples or the same group of samples under different conditions. The t statistic is calculated based on sample mean, sample variance, sample size and population variance. According to corresponding results, conclusions of performance differences can be made, so their optimization effects can be evaluated accordingly. Pareto frontiers of these two algorithms in buildings of different numbers of rooms are shown in Fig. 8.

Fig. 8

Pareto frontiers of the two algorithms in different building numbers

Figure 8 (a) is the Pareto frontier from these two algorithms operated in single-room building scenario. With higher total energy consumption, the number of uncomfortable hours of SMOPSO/D algorithm gradually decreases. The uncomfortable hours becomes 0 when the total energy consumption is 8 for the proposed algorithm and 12 for the MOPSO algorithm. In addition, Fig. 8 (b) shows the Pareto frontier obtained by two algorithms in multi-room buildings. As the TEC increases, the uncomfortable hours of SMOPSO/D algorithm also gradually decreases. When the TEC is 49, the number of uncomfortable hours reaches the lowest 2600 for the proposed algorithm and 2700 for the MOPSO algorithm. That is, SMOPSO/D algorithm shows better optimization performance in both single-room and multi-room buildings. To further verify the anisotropy of SMOPSO/D algorithm, a running time test is also conducted The running time of the two algorithms in buildings with different numbers of rooms is shown in Fig. 9.

Fig. 9

Running time of the two algorithms in buildings with different numbers of rooms

Figure 9 (a) shows the running time of the two algorithms in a single-room building. As iterations amount, the SMOPSO/D algorithm runs for at least 1640s, and its average running time is 1710s. In comparison, the MOPSO algorithm’s fastest running time was only 1680s. Figure 9 (b) shows the running time of the two algorithms in a multi-room building. With more iterations, the running time of SMOPSO/D algorithm is least of 3350s, with the longest running time of 3650s and the average running time of 3500s, while the MOPSO algorithm demonstrates worse running time performance. To verify the performance superiority of SMOPSO/D algorithm, MOEA, MOPSO and MOEA/D are used as comparison algorithms, and corresponding experimental results are shown in Table 1.

Table 1 Comparisons of performance indicators of different algorithms

From Table 1, the convergence speed, computation time and robustness of SMOPSO/D algorithm in a single-room building are 200 times, 50 s and 98% respectively, but are of 300 times, 100 s and 96% respectively in room construction scenario. The MOEA algorithm before optimization performs worst. In single-room buildings, the convergence speed, computation time and robustness of SMOPSO/D algorithm are 500 times, 100 s and 90%, respectively. In the room construction, the convergence speed, computation time and robustness of SMOPSO/D algorithm are 1000 times, 250 s and 85%, respectively. Therefore, SMOPSO/D algorithm performs best comparing with the other three types of comparison algorithms.

Modeling effect analysis of BEE based on multi-agent assisted MOEA/D

The feasibility of the proposed model is verified by the architectural design of Beijing, the cold region of China, as an example. Beijing is located in 39.80° north latitude, 116.47° East longitude, which is a typical cold region featured by high-rise office buildings. To ensure the fairness and accuracy of the study, when comparing SMOPSO/D model with NSGA-II model, MOPSO model and ParEGO model, the population size N was uniformly set at 50. The selection of these model parameters is based on the optimal configuration recommended in the literature of each comparison model. SMOPSO/D, and the parameter Settings of the comparison model are shown in Table 2.

Table 2 Parameter settings of SMOPSO/D and comparison models

Firstly, the Python program collects data of the location and dimensions of the building model’s walls, windows, roofs, etc., along with the physical properties of the materials, and local climate data files. The Python script then delivers prepared data to the Honeybee plug-in through the Grasshopper platform’s file exchange mechanism. The Honeybee plug-in further invokes EnergyPlus for building energy simulation. After the simulation, the output data from EnergyPlus and Radiance is captured by the Honeybee plug-in and converted into an easy-to-understand format. The study uses a large amount of historical building data and simulation results to train the model, so as to learn the relationship between design parameters and performance indicators. During the training phase, this study selected 1000 different building models as datasets to ensure sufficiently wide and diverse range. In the test phase, to evaluate the performance of the trained model, this study selected a completely independent data set of 200 building models, which were used in simulation tests to verify accurate predictions of the relationship between design parameters and performance of unknown buildings. In the verification phase, this study used actual construction projects or experimental data to check the pragmatic effectiveness and reliability of the optimization algorithm. By testing five real construction projects, its feasibility and practicality in actual operation can be verified. Further to comprehensively evaluate the model performance, the optimal models NSGA-II, MOPSO and ParEGO were compared with the SMOPSO/D model in the experiment, and the two-dimensional structure of the non-dominated solution in the target space was finally obtained in the experiment, as shown in Fig. 10.

Fig. 10

Two-dimensional distribution of non-dominated solutions in target space

Figure 10 (a) is the cost diagram of building energy consumption and cladding structure for different models. The position of SMOPSO/D model and Pareto reference point basically coincide, showing better optimization performance. Comparing with other models, the SMOPSO/D model has the lowest building energy consumption and better envelope cost. In addition, Fig. 10 (b) shows the proportion diagram of building energy consumption-sunshine adequacy of different models. The positions of SMOPSO/D model and Pareto reference points also basically coincide, indicating a good balance between building energy consumption and the proportion of sufficient sunlight. Comparing with other models, the SMOPSO/D model has the lowest energy consumption and a higher percentage of full sunshine. This further demonstrates SMOPSO/D superiority in optimizing building performance. To evaluate the pros and cons of different models and select models with better performance for practical problems, 100 iterations conducted in this study can ensure enough time for models to converge to the region near the optimal solution. The changes of the optimal values of different models are shown in Fig. 11.

Fig. 11

The optimal values of different models change with the optimization process

Figure 11 (a) is the building energy consumption diagram of different models. As the iterations amount to 20, the building energy consumption curve of SMOPSO/D model becomes stable. Comparing with other models, for example, the iteration number is 98 times for stable MOABC model. In addition, Fig. 11 (b) shows the sunshine adequacy ratio of different models. As the iteration times increase to 16, the line chart of sunshine sufficiency ratio of SMOPSO/D model keeps stable. However, the NSGA-II model tends to be stable when the number of iterations is 32. In summary, SMOPSO/D model shows good performance and stability in the optimization process of building energy consumption and sunshine adequacy ratio. The feasible solution ratio of different models changes with the optimization process is shown in Fig. 12.

Fig. 12

Change of feasible solution ratio of different models with optimization process

Figure 12 shows that the number of feasible solutions for different models changes when iterating for 0 to 100 times. The line chart of the amount of feasible solutions of moves around 40 for SMOPSO/D model and around 32 for NSGA-II model, both showing a relatively stable fluctuation trend. However, such indicator fluctuates in a relatively large range around 20 for the MOPSO model and relatively small range of 13 for the MOABC model. In conclusion, the SMOPSO/D model has good stability and performance in finding feasible solutions. The table of abbreviations and nomenclature used in this study is shown in Table 3.

Table 3 Glossary of abbreviations and nomenclature

Discussion

Considering global energy shortage and environmental protection, no one will deny the importance of energy-efficient building design. However, traditional design method has limitations that cannot meet the multi-performance index optimization of modern buildings. Therefore, a multi-objective optimization method for energy-efficient building design based on multi-agent-assisted MOEA/D is proposed. The multi-agent system can simulate the decision-making process in human society, and divide multiple relevant performance indicators into multiple sub-goals that can be jointly optimized through the interaction and cooperation between agents. In summary, MOEA/D algorithm can effectively deal with multi-objective optimization problems due to its parallel optimization characteristics (Dong et al. 2020).

Firstly, the performance of the research algorithm and MOPSO algorithm in office buildings is compared. The experimental results show that in buildings of multiple rooms, although MOPSO algorithm performs well in the initial iteration stage, the research algorithm still obtains higher HV value in the end, which is consistent with Aboud’s conclusion in the research of emergency evacuee classification system based on fuzzy Mopso algorithm, that is, the stability and superiority of the research algorithm in dealing with complex problems (Aboud et al. 2022). The research algorithm demonstrates better performance mainly because it integrated the distributed computing capability of multi-agent systems and the high efficiency of MOEA/D algorithm in dividing multi-objective optimization problems, so as to realize excellent balance between global search and local optimization. Moreover considering building energy conservation and living comfort, SMOPSO/D algorithm can effectively balance these two factors. When the total energy consumption reaches 8, the algorithm has no discomfort hours, which is much better than that of MOPSO algorithm. Besides the balance of energy consumption and comfort, it also reflects the practicability in energy conservation design. This is consistent with the results obtained by Banadkooki and Haghigh in their research on multi-objective optimization of groundwater table modeling based on a hybrid artificial intelligence approach (Banadkooki and Haghighi 2024).

In addition, the study compared the performance of different models with factors such as building energy consumption, perimeter structure cost and sunlight adequacy ratio. According to the results, the SMOPSO/D model has good optimization performance in many performance indexes, and the optimization results basically coincide with the Pareto reference points. It verifies the research algorithm’s superiority in optimizing building performance and efficiency in finding a feasible solution set close to the optimal solution, which is similar to the results of Zhang et al. (Zhang et al. 2021). Finally, in the aspect of the number changes of feasible solutions during the iterative process, the line chart of the number of feasible solutions of the SMOPSO/D model fluctuates around 40, reflecting the high robustness of this model in the search process. Therefore, the SMOPSO/D model is more reliable and efficient in solving the multi-objective optimization problem of energy-efficient building design. Such result is consistent with the results obtained by Ghanbari et al. when using the improved Kerre method to solve fuzzy linear programming problems, which further verifies the applicability and advantages of this model in similar optimization problems (Ghanbari et al. 2021). In addition, the SMOPSO/D model effectively expands the coverage of the search space through the parallel search of multiple agents, ensuring that the algorithm can explore a wider range of potential solution sets. Meanwhile, together with the MOEA/D, the model can divide the complex multi-objective problem into a series of single-objective sub-problems. Then, each sub-problem can be optimized by an agent, which greatly simplifies the original problem with higher solution efficiency. Integrating these two advantages enables SMOPSO/D model to quickly approach Pareto frontier and figure out a feasible solution set close to the optimal solution while maintaining the diversity of solution sets.

To further expand the application scope of SMOPSO/D model, the research provides the following three suggestions. The first is to develop customized optimization strategies for different building types, such as residential buildings, commercial complexes, hospitals, etc., to meet their specific energy saving and comfort needs, Next is to introduce machine learning technology. So that the model can automatically learn and adapt to the law of building energy consumption under different climate conditions, finally improving the model’s applicability, Finally, deep integration of the model and the Internet of Things, big data and other technologies should be explored, so as to achieve real-time monitoring and optimization of building energy consumption and comfort. Moreover, it can also provide more intelligent and efficient solutions for buildings’ energy conservation design.

Conclusion

By integrating multi-agent technology and MOEA/D algorithm, this study aims to solve several objective optimization conflicts in energy-efficient building design. According to experiments, the proposed research algorithm shows significant advantages in the energy efficiency evaluation of single-room buildings. With more iterations, its HV value gradually exceeds that of the MOPSO algorithm. Particularly when iterating for 15 times, the gap between these two reaches the maximum, which clearly verifies the superiority of the research algorithm. However, in multiple-room buildings, although MOPSO algorithm temporarily takes the lead in the initial iteration phase, the research algorithm finally surpasses MOPSO algorithm through longer iteration and optimization, verifying its effectiveness in this aspect. Comparing with three typical multi-objective optimization algorithms, SMOPSO/D model shows outstanding performance. Its optimization results are highly consistent with Pareto reference points, which realizes both minimized building energy consumption and good performance in the cost of external envelope structure, fully proving its high efficiency and accuracy in the multi-objective optimization of energy-conservation building design. After further analysis, the number of feasible solutions of SMOPSO/D model fluctuates steadily around 40 from 0 to 100 iterations, which reflects stability and reliability of its optimization. In summary, SMOPSO/D model has excellent performance of high stability, efficiency and accuracy when dealing with multi-objective optimization problems of energy-efficient building design. In terms of the complex and dynamic building environment, future research should focus on enhancing the model’s adaptability to different building forms, diverse material properties and dynamic external conditions. Specifically, more refined building performance simulation can be achieved by integrating advanced building physics models. Moreover, a dynamic climate data interface can be built and environmental interaction simulation technology can be introduced to capture the interaction between the building and the environment, so as to ensure real-time response of the model to environmental changes.