Weighted-Sum Optimization of Shopping-Mall Atrium Morphology Using a Field-Based Visual Satisfaction Proxy and sDA-Based Daylight Sufficiency

Abstract

Daylighting in shopping-mall atriums affects both perceived visual quality and the availability of daylight for reducing electric-light use under daylight-responsive control. However, early-stage atrium design still lacks a reproducible workflow that connects field-based user-response evidence with parametric daylight simulation in retail settings. This study develops a weighted-sum design-search workflow for shopping-mall atrium morphology by combining a luminance-based visual satisfaction proxy with spatial daylight autonomy (sDA_300/50%) as an annual daylight-sufficiency indicator. Morphological parameter ranges were derived from a survey of 150 atriums in 26 shopping malls in China, and three prevalent atrium types were retained for simulation-based testing. ClimateStudio/Radiance simulations were used to calculate scene mean luminance and sDA_300/50%, and particle swarm optimization was used as a practical search algorithm under a fixed baseline weighting scenario. The reported high-scoring solutions reached sDA saturation, indicating that the final score differences were mainly governed by the luminance-based satisfaction term within the high-daylight-sufficiency region. This study contributes a retail-oriented, reproducible design-search workflow for early-stage atrium morphology decisions, while the results should be interpreted as scenario-based preferred solutions rather than statistically confirmed global optima or a complete visual-risk/energy-performance assessment.

1. Introduction

Atrium daylight performance is closely tied to geometric proportions, roof and façade openings, and climate context [1,2,3,4]. Reviews and simulation studies have established the value of climate-based daylight metrics and have shown that daylight availability in atriums is governed by plan form, well proportions, and glazing design [1,2,3,4]. In parallel, field and perception studies in shopping environments indicate that daylight quality shapes how users appraise public indoor space, and recent survey evidence from shopping malls suggests that customers place particularly high value on daylight quality and visual connection to the outside [5,6]. For retail-oriented atriums, therefore, daylighting design is not only a technical problem of light provision but also a user-centered problem of perceived environmental quality.

A second strand of the literature concerns optimization. Studies that combine visual and energy-related objectives have been reported for other building types such as offices and classrooms [7,8]. More recent work has extended parametric optimization to atrium buildings, commercial atria, and shopping-mall configurations [9,10,11,12]. Xue and Liu identified key design parameters for commercial atrium daylight optimization in cold regions through a Pareto-based workflow [10], and Adel and Sheta showed that shopping mall atrium and skylight configurations can be optimized in hot climates using ClimateStudio and glare-oriented visual comfort metrics [9]. Further recent studies have broadened atrium optimization to jointly consider daylight, energy, and thermal objectives in non-retail atrium buildings [11,12]. These contributions confirm that atrium optimization is an active and maturing field rather than an unexplored one.

The present study addresses a narrower problem within this landscape. Existing atrium optimization studies rely mainly on objective daylight, thermal, or glare indicators. A shopping-mall-oriented workflow that directly embeds a field-derived L_mean–satisfaction relationship into morphology design search remains insufficiently developed [8,9,10,11,12]. Against this background, the present study addresses the following research questions: (1) How can a field-derived relationship between scene mean luminance and user-reported lighting satisfaction be incorporated into a parametric daylighting workflow for shopping-mall atriums? (2) How do typical atrium morphologies perform when a luminance-based satisfaction proxy is combined with sDA-based daylight sufficiency under a weighted scalar formulation? (3) What parameter tendencies can be identified for prevalent shopping-mall atrium types, and what are the methodological limits of interpreting these tendencies as design recommendations?

2. Literature Review

2.1. Visual Comfort of Daylighting in Shopping Malls

Atrium daylighting plays a central role in the environmental quality of shopping malls. Atriums can enrich social interaction and support occupants’ psychological well-being [1]. Hourani and Hammad reported that daylighting quality shapes the dynamics of architectural space, reinforces its aesthetic and emotional dimensions, and enriches occupants’ experience and their sense of interest and pleasure [13]. Jin and Li found that increased daylight admission in atriums improves subjective lighting satisfaction [14]. Sachdev et al. further showed that window position and shape influence perceived satisfaction with lighting quality [15].

Daylighting also has positive effects on sales, customer retention, and dwell time in shopping malls. Heschong et al. reported that customers and employees viewed stores with overhead daylighting favorably, and that skylighting was strongly correlated with higher sales [16], which suggests a tangible commercial value of daylight in retail centers. The Lighting Research Center (LRC) examined organizational outcomes associated with daylight and found that windowless commercial properties commanded lower rents than those with windows [17]. Recent survey evidence from shopping malls further indicates that customers place greater importance on daylight quality and visual connection to the outside than on energy saving alone, which underscores the need to treat perceived lighting satisfaction as an explicit design criterion in retail atrium design [6].

2.2. Energy Performance in Atriums

Electric lighting accounts for a substantial share of electricity use in commercial buildings, and this share can be meaningfully reduced by exploiting daylight [18,19]. Retail sales have also been observed to rise with longer annual daylight availability and with improved daylighting design [20]. Among architectural strategies, the atrium remains one of the most effective means of delivering daylight into shopping malls.

To use this daylight efficiently, the relationships between daylight metrics and atrium morphological parameters have been studied extensively. Field measurements, scale models, and simulations have been used to examine the effects of geometry and surface reflectance on the daylight factor (DF) in atriums and adjacent spaces under overcast sky conditions [21,22,23,24,25,26,27,28,29,30]. Other studies have addressed the influence of atrium type, spatial proportion, roof opening design, and reflector configuration on daylight metrics within climate-based daylight modeling frameworks [2,4,31]. Spatial daylight autonomy (sDA) quantifies whether a space receives sufficient daylight on a horizontal work plane during standard operating hours over a year [32,33], and a simplified method for estimating daylight autonomy (DA) in atriums from spatial parameters has been proposed to support rapid early-design evaluation [3].

2.3. Scope of Daylight-Performance Indicators Used in This Study

The indicators used in this study have different scopes and should not be treated as interchangeable measures. Scene mean luminance (L_mean) is used here as an image-based variable linked to a previously established field regression between luminous conditions and subjective lighting satisfaction in shopping-mall public spaces. It is therefore interpreted as a satisfaction-oriented proxy rather than as a complete visual comfort or glare-risk metric. Spatial daylight autonomy (sDA_300/50%) describes the percentage of analysis points that receive at least 300 lx from daylight for at least 50% of the occupied hours, and is used here to represent annual daylight sufficiency. Under an ideal daylight-responsive lighting-control assumption, high sDA indicates potential for reduced electric-light use, but it does not directly quantify operational lighting energy consumption. Annual sunlight exposure (ASE), daylight glare probability (DGP), and related glare or overexposure metrics remain important for complete annual visual-risk assessments, but they are outside the retained objective terms of the present weighted design-search formulation.

2.4. Optimization Approaches for Daylighting Performance

Recent atrium optimization studies show clear methodological divergence according to research scope. Xue and Liu optimized commercial atrium daylighting in cold regions through a Pareto-based parametric workflow centered on daylight quantity [10], whereas Adel and Sheta examined shopping mall atrium and skylight configurations in a hot climate using ClimateStudio together with glare-oriented metrics [9]. Other recent atrium studies have broadened the objective space to include daylight, energy, and thermal performance in non-retail building types [11,12]. Wagiman et al. demonstrated that visual comfort and energy efficiency can be jointly optimized within a PSO-family framework, although their study addressed office lighting systems and used a different visual-performance metric [8]. Together, these contributions confirm that atrium optimization is already an active field, and the niche of the present study is correspondingly narrow: a weighted-sum formulation is used to embed a field-derived L_mean–satisfaction relationship into the morphology search of shopping mall atriums, while acknowledging that glare-oriented and annual-risk metrics remain valuable for other daylight assessment tasks.

Daylight optimization studies have adopted different algorithmic strategies depending on how the design problem is framed. NSGA-II and MOPSO have been used when the aim is to preserve trade-offs between comfort and energy-related objectives [7,8], while recent atrium studies have often relied on Pareto-oriented tools such as Octopus or GA to explore alternative daylight, energy, or thermal solutions [9,10,11,12]. PSO has likewise been applied in simulation-based building optimization: Futrell et al. showed that a PSO variant could identify high-quality daylighting solutions in a classroom setting, although its convergence was slower than that of the hybrid alternative tested in the same study [34]. In the present work, PSO was chosen not because it is universally superior to GA, NSGA-II, or Octopus, but because the two criteria are combined into a single weighted fitness function; the task therefore reduces to searching for high-scoring solutions within bounded atrium parameter ranges. Wagiman et al. [8] are methodologically relevant in this respect, as they jointly optimized visual comfort and energy efficiency with a PSO-family method, although their framework targeted office lighting systems and used a different visual-performance metric. Within the present formulation, PSO therefore functions as a practical search engine for a retail-atrium-specific workflow rather than as the main contribution of this study. Because the two criteria are collapsed into a weighted scalar objective rather than explored as a Pareto front, PSO is used purely as a practical search algorithm under that formulation.

2.5. Research Objectives

The main objective of this study was to identify favorable shopping mall atrium morphologies under a weighted two-criteria formulation that combines subjective visual satisfaction with daylight sufficiency linked to potential reductions in electric-light use. Rather than estimating actual operational energy use, the framework evaluates daylight-linked lighting energy-saving potential under an ideal daylight-responsive control assumption.

The two daylighting criteria were integrated into a weighted scalar objective. The first criterion is a satisfaction-oriented visual term derived from a previously established relationship between scene mean luminance and subjective lighting satisfaction in shopping-mall public spaces [5]. The second criterion is sDA_300/50%, used here as an annual daylight sufficiency metric and as a proxy for daylight-linked lighting energy-saving potential, rather than as a direct measure of operational lighting energy consumption. This formulation was designed for the present shopping-mall atrium workflow and should not be read as a replacement for glare-oriented or annual visual-risk metrics in other assessment tasks. Although sDA and ASE are commonly treated as complementary annual daylight metrics [2,35], ASE was not incorporated into the present weighted formulation, and the optimization results should therefore not be interpreted as a complete annual visual-risk assessment. The baseline 0.55/0.45 weighting scenario was informed by the revised 12-expert AHP evidence and further examined through weighting-scenario sensitivity analysis, as reported in Section 3.3.3 and Appendix A.

3. Methodology

The methodology comprised four steps: (1) construction of an atrium reference dataset and extraction of common shopping-mall atrium typologies; (2) simulation of luminance maps and sDA_300/50% values using ClimateStudio/Radiance; (3) formulation of a weighted scalar objective combining a luminance-based visual satisfaction proxy with sDA-based daylight sufficiency; and (4) PSO-based design search for preferred parameter combinations under the adopted baseline weighting scenario. Figure 1 gives an overview of the workflow. The simulation workflow was implemented in Rhinoceros v7.3 and its built-in Grasshopper (Robert McNeel & Associates, Seattle, WA, USA), ClimateStudio v1.1 (Solemma LLC, Cambridge, MA, USA), and MATLAB v2020b (The MathWorks, Inc., Natick, MA, USA). ClimateStudio is a daylight and energy simulation plug-in developed by Solemma [36], and the daylight simulation procedure follows Radiance-based daylight simulation practice [37].

3.1. Atrium Reference Dataset and Typology Extraction

3.1.1. Mall Survey and Recorded Parameters

To anchor the morphology design search in empirically observed retail-atrium conditions, an atrium reference dataset was first compiled from 26 shopping malls in China. The case investigation was carried out in 2018 through on-site surveys and architectural drawing review. A mall was included if it contained at least one atrium, was in operation at the time of the survey, and was willing to provide drawings and permit on-site inspection. The 26 selected malls spanned seven major regions of China (Northeast, North, Central, East, South, Southwest, and Northwest). Within each mall, all atriums with access to daylight were counted. For every included atrium, the recorded attributes were plan form, well width, well length, corridor width, atrium location within the mall plan, and daylight-opening pattern. In total, 150 atriums were documented and served as the empirical basis for plan-form prevalence analysis and for defining the initial morphological parameter space.

As shown in

Table 1. Planar parameter statistics for atriums.

Plan Form	Number of Atriums	Percentage	Role in Subsequent Modeling
Rectangular	106	70.67%	selected as base plan form
Oval	20	13.33%	not retained as base plan form
Circular	15	10.00%	not retained as base plan form
Irregular	9	6.00%	not retained as base plan form

, rectangular atriums made up 70.67% of the dataset, while each of the other plan forms accounted for 13.3% or less. Rectangular atriums were therefore adopted as the base plan form in the parametric modeling stage on the basis of prevalence and design applicability, rather than as a claim of universally superior daylight performance. Mall-level provenance, source composition, and screening information are provided in Supplementary Table S1.

3.1.2. Screening Procedure and Retained Typologies

A second screening step was then applied to the compiled dataset to identify daylighting atriums whose spatial location and daylight-opening configuration could be clearly classified. Atriums lacking accurate architectural drawings were excluded first, followed by those with overly complex daylighting conditions or plan forms that were difficult to categorize. This yielded 44 usable atriums for typology extraction. This screening step was used only for typology extraction, whereas the broader 150-atrium dataset remained the reference basis for plan-form prevalence and general morphological parameter information. Based on location within the mall plan, these atriums were grouped as central, semi-enclosed, corner, or linear; their daylighting patterns were coded as top-daylit, vertical-skylit, side-daylit, combined-daylit, two-sided-daylit, or upper-and-side-daylit. The resulting typology distribution is summarized in

Table 2. Location and daylighting patterns of atriums.

Location	Central Atrium		Semi-Enclosed Atrium		Corner Atrium		Linear Atrium
Daylighting pattern
	Top- daylit	Vertical skylit	Side- daylit	Combined-daylit	Two-sided daylit	Upper and twoside daylit	Upper and side daylit
Proportion	47.7%	6.8%	20.5%	9.2%	6.8%	4.5%	4.5%

.

Among the screened atriums, the three most prevalent typologies were the central top-daylit atrium, the semi-enclosed side-daylit atrium, and the semi-enclosed combined-daylit atrium. These three were retained for the subsequent optimization study because they represented the dominant daylighting configurations within the screened dataset. The typology-specific counts, roles, and observed parameter ranges of the 44 retained atriums are provided in Supplementary Table S2.

3.1.3. Parameter Bounds Used in the Design Search

The parameter bounds used in the design search were derived from observed values in the atrium reference dataset rather than assigned arbitrarily. The extracted variables were well width, length-to-width ratio, corridor width, floor height, and number of floors. To allow for cross-typology comparison under a consistent morphological parameter space, the same broad search bounds were adopted for the three simulated atrium typologies, as listed in

Table 3. Common search bounds adopted for the three simulated atrium typologies.

Type	w_Range	lwrate_Range	Zlw_Range	h_Range	F_Range
Common bounds	6–27	1–3	4–10	5–6	1–8

. The implications of using common rather than type-specific bounds are discussed as a limitation.

3.2. Daylight Simulation Setup

Simulation models were generated in Rhinoceros v7.3 and its built-in Grasshopper, after which the luminance maps and sDA_300/50% values were computed in ClimateStudio v1.1.

Artificial lighting in the surrounding retail areas was configured to match typical retail visual requirements. The luminaire used was a linear suspended strip with a diffuse lens, 1.2 m in length, 100 W in rated power, and 10,240 lm in luminous flux. The distance from the luminaires to the boundary was 4 m, and the spacing between adjacent luminaires was 7.5–8.5 m. The resulting artificial-lighting luminance was approximately 40 cd/m², close to the average level measured in the previous field study.

3.2.1. Weather Data, Climate Basis, and Simulation Periods

The daylight simulations used an EnergyPlus Weather (EPW) file representing the climate of Beijing, which was selected to match the climatic context of the modeled shopping-mall atrium cases. Annual sDA calculations were conducted using the hourly weather data contained in the EPW file over the 9:00–17:00 operating period, corresponding to 2920 occupied hours per year. For luminance-map simulations, representative dates were selected to capture seasonal solar-altitude conditions: 21 March, 21 June, 23 September, and 22 December. Because the March and September equinox conditions are similar in solar geometry, the 23 September results were used to represent both equinox conditions in the averaged visual-satisfaction calculation. The daily luminance-map simulation window was 9:00–17:00, with HDR images generated at 9:00, 13:00, and 17:00 to balance simulation duration and result resolution. An earlier study showed that horizontal HDR and Perez skies produced comparably accurate luminance maps; Perez skies were therefore adopted here [38]. Nine HDR images were simulated per viewpoint per floor, and the resulting fitness-function values were averaged.

3.2.2. Luminance-Map Simulation

A simulated luminance map stores the usual red, green, and blue channels for each pixel, as in any conventional image format, together with an absolute luminance value per pixel that fixes the absolute brightness. The default high-quality preset in ClimateStudio v1.1 was used, with ab = 6, ad = 1024, lw = 0.01. A fisheye projection—an ultra-wide-angle panoramic projection—was selected to capture the luminous environment around the atrium; the field of view was 360°. Four samples per pixel were used, and the maps were rendered at 128 × 128 px. The viewpoint was placed at the middle of the corridor on the short side of the atrium (Figure 2), 1.5 m above the floor, and was oriented toward the atrium. Representative simulated luminance maps for the central atrium are shown in Figure 3.

3.2.3. sDA Simulation and Lighting-Standard Basis

Shopping-mall atriums accommodate a range of activities, including circulation, retail, exhibition, and leisure. The target illuminance of 300 lx and the 0.75 m horizontal analysis plane were selected with reference to the lighting standard value for general retail spaces in the Chinese Standard for Lighting Design of Buildings (GB 50034-2013) [39]. The sDA_300/50% criterion follows the IES LM-83 daylight-assessment convention [35], in which an analysis point is considered daylit when it receives at least 300 lx for at least 50% of the occupied hours. As noted in Section 3.3.2, sDA is usually reported together with ASE in annual daylight assessment as a complementary indicator of potential overexposure risk [2,35].

The sDA values were simulated in ClimateStudio v1.1 for Grasshopper, which supports several illuminance-based daylight availability metrics. For an accurate calculation, the spacing of the daylight sensor grid should not exceed 2 ft (610 mm) [40]. A sensitivity check in which the grid spacing was varied from 100 mm to 610 mm in 100 mm steps showed no meaningful change in sDA. To balance observation clarity with simulation speed, a grid spacing of 500 mm was adopted in this study. The calculation period covered 9:00–17:00, accumulating 2920 h across the year. Figure 4 shows the simulated sDA_300/50% values for the different atrium types.

3.3. Weighted Objective Formulation

3.3.1. Luminance-Based Satisfaction Proxy

In addition to objective photometric metrics, subjective evaluation has been widely used in prior lighting-appraisal studies to assess users’ responses to the luminous environment [41,42]. In the authors’ earlier field study of public spaces in shopping centers, a quadratic regression was established between the scene mean luminance (L_mean) and subjective satisfaction under mixed lighting conditions, that is, when daylight and electric lighting were present simultaneously [5]. This relationship was obtained from the regression shown in “Figure 2 in [5]”, based on survey data from 61 respondents with R² = 0.7944. In the present study, the predicted output of this regression is denoted SE_visual and is used for the subsequent design search, as given in Equation (1).

$${SE}_{\mathrm{visual}}=-1.0284L_{\mathrm{mean}}^2+2.055L_{\mathrm{mean}}+3.0034$$

(1)

where SE_visual is the model-predicted subjective satisfaction with the lighting environment, and L_mean is the scene mean luminance in kcd/m².

In the earlier study [5], the original subjective evaluation was defined on a 1–5 scale. Within the calibrated L_mean range of 0.028–1.120 kcd/m² adopted here, however, the predicted SE_visual values fall mainly between about 3 and 4.03. In principle, the interval 3–4.03 could itself be used for normalization, but doing so would map the lowest predicted values in the current model domain directly to zero and could over-penalize low-luminance conditions. For this reason, the present study retains 1—the theoretical lower bound of the original evaluation system—as the lower limit of normalization, and adopts 4.03, the maximum value predicted by Equation (1) within the adopted calibration range, as the upper limit. Accordingly, Equation (2) maps 1–4.03 onto 0–100. This treatment should be read as a modeling choice made for the present fitness-function construction, not as the only defensible normalization strategy; alternative intervals such as 1–5 or 3–4.03 would yield different absolute scales of f₁ and could further shift its relative weight in the integrated objective function.

$$f_1=100({SE}_{\mathit{visual}}- 1)/(4.03-1)=-33.941L_{\mathit{mean}}^2+67.822L_{\mathit{mean}}+66.119$$

(2)

where f₁ is the normalized luminance-based satisfaction proxy derived from the model-predicted SE_visual. The interval 1–4.03 was chosen so as to retain the theoretical lower bound of the original rating system while avoiding direct mapping of the current model-domain minimum onto zero.

The L_mean interval used in the present study, 0.028–1.120 kcd/m², corresponds to the range of L_mean values produced by the PSO-driven simulations and lies within the calibration domain of the original regression in [5]. As shown in Figure 5, the normalized luminance-based satisfaction proxy reaches its maximum at L_mean ≈ 1.0 kcd/m². Below this value the rating rises with increasing L_mean, whereas above it the rating decreases as L_mean continues to grow.

3.3.2. sDA-Based Daylight Sufficiency Term

sDA_300/50% was used here as a daylight-sufficiency metric and, under an ideal daylight-responsive control assumption, as a proxy for daylight-linked lighting energy-saving potential. A higher sDA_300/50% indicates that a larger proportion of the analyzed area can reach the target illuminance from daylight alone during occupied hours, and therefore has greater potential to reduce electric-light use. This interpretation should be read as a proxy-based assessment rather than as a direct estimate of measured or simulated operational lighting energy consumption.

In annual daylight assessment frameworks, sDA is commonly paired with ASE as a complementary metric for potential overexposure risk [2,35]. ASE was not included in the present objective formulation because the current workflow was built to combine a field-derived L_mean–satisfaction model with an annual daylight sufficiency indicator for early-stage morphology screening. Accordingly, the second criterion in this study should be interpreted as daylight sufficiency and daylight-linked lighting energy-saving potential, not as a complete annual visual-risk or energy-performance assessment. Because sDA_300/50% is expressed as a percentage, the normalized second criterion is given by Equation (3).

$$f_2={\mathit{sDA}}_{300/50\%}$$

(3)

where f₂ is the normalized daylight-sufficiency-based proxy for daylight-linked lighting energy-saving potential, and sDA_300/50% is the percentage of analysis points that meet or exceed 300 lx for at least 50% of the analysis period.

The L_mean-based term is used here specifically to operationalize the adopted field-derived satisfaction model within a morphology-oriented design-search workflow. It is therefore treated as a scene-based, satisfaction-oriented visual indicator for the present task, and is not intended as a substitute for glare- or annual-risk metrics that may be required in other daylight assessments. Figure 6 illustrates the normalized sDA-based daylight sufficiency term, which increases linearly with sDA_300/50% and is used in this study as a proxy for daylight-linked lighting energy-saving potential under an ideal daylight-responsive control assumption.

3.3.3. Baseline Weighting Scenario and Sensitivity Analysis

The two retained objective terms in Equation (4) are a luminance-based visual satisfaction proxy (f₁) and an sDA-based daylight-sufficiency term (f₂). The AHP questionnaire covered four perceived daylight-performance criteria: uniformity, softness, subjective lighting satisfaction, and visual comfort rating. Because these four questionnaire criteria are not identical to the two retained objective terms, and because f₂ is an sDA-based daylight-sufficiency term rather than a subjective satisfaction variable, the AHP results are not interpreted as direct objective weights for f₁ and f₂. Instead, they are retained as supporting expert-preference evidence indicating that satisfaction-oriented daylight appraisal was prioritized by the consulted expert panel.

The AHP questionnaire was distributed by email in 2025 and completed by 12 experts, all of whom held doctoral degrees in architecture and had research backgrounds in architectural design, building physical environment, green building, and building energy efficiency. Each expert judgment matrix was reconstructed as a reciprocal matrix before aggregation, a step that also resolved one non-reciprocal pairwise entry in a single questionnaire. The 12 reconstructed matrices were then aggregated using the geometric mean of pairwise comparisons to form a group judgment matrix, and the group priority vector was obtained by the principal eigenvector method. The resulting group matrix showed acceptable consistency, with a consistency ratio (CR) of 0.076. The four global weights were 0.093 for uniformity, 0.082 for softness, 0.368 for subjective lighting satisfaction, and 0.458 for visual comfort rating.

For the weighted design-search demonstration, Equation (4) is formulated as a baseline weighting scenario rather than as a direct AHP-derived objective-weight assignment. The baseline scenario uses w₁ = 0.55 for the luminance-based satisfaction term and w₂ = 0.45 for the sDA-based daylight-sufficiency term. These values were selected to place a slightly higher emphasis on the satisfaction-oriented term while still assigning substantial weight to daylight sufficiency. To examine the influence of this assumption, a sensitivity analysis was conducted by varying w₁ from 0.30 to 0.70, with w₂ = 1 − w₁. Expert-level results, the group judgment matrix, and the weighting-scenario sensitivity analysis are provided in Appendix A.

$$f =w_1·f_1+w_2·f_2$$

(4)

where f₁ is the normalized luminance-based satisfaction proxy, f₂ is the normalized sDA-based daylight-sufficiency term, and w₁ and w₂ are the scenario weights assigned to the two retained terms. In the baseline scenario used for the main analysis, w₁ = 0.55 and w₂ = 0.45. These weights should be interpreted as a scenario setting for the present design-search workflow rather than as a universal or directly AHP-derived measure of objective importance. Because the second term is based on sDA_300/50% alone, f should be interpreted as combining a luminance-based satisfaction proxy with daylight sufficiency and the potential for reduced electric-light use under ideal daylight-responsive control, rather than with actual annual lighting energy consumption. ASE was not embedded in Equation (4) and should therefore be understood as an omitted complementary daylight-risk metric, not as an implicitly satisfied condition.

3.4. PSO-Based Design-Search Implementation

Particle swarm optimization (PSO), originally introduced by Kennedy and Eberhart [43], was used as the practical search algorithm for the weighted scalar objective [44]. In the present implementation, each particle encoded a candidate combination of well width, length-to-width ratio, corridor width, floor height, and number of floors. The length-to-width ratio was used to constrain well proportions so that the simulated atrium remained within conventional architectural ranges. The initial particles for each variable were randomly generated within the permitted value ranges and then used to build the corresponding Rhino–Grasshopper 3D geometry models.

In this framework, the five bounded morphological variables were evaluated through repeated external simulations, and each candidate solution returned a single scalar fitness value after the visual and daylight-sufficiency criteria had been weighted and combined. For this type of black-box, continuous-parameter search, PSO offers a relatively direct coupling strategy between MATLAB v2020b and the Rhino–Grasshopper v7.3/ClimateStudio v1.1 environment, and it has been used previously in building performance optimization [34,45]. The intent was therefore to obtain one preferred solution for each atrium type under a fixed weighting assumption, rather than to perform a Pareto-based multi-objective comparison.

This choice should not be read as evidence that PSO is generally the best algorithm for atrium daylighting optimization. Recent atrium studies have often adopted Pareto-oriented tools such as Octopus or GA [9,10,11,12], and such methods are better suited when the objective is to compare trade-offs explicitly across competing criteria. PSO can also be sensitive to population size and control parameters, and may converge prematurely or require careful tuning. PSO is therefore used here as a practical algorithmic component of a weighted search workflow, with the acknowledgment that alternative algorithms may produce different search behavior or different sets of preferred solutions.

A flowchart of the daylighting-simulation-based design-search approach is given in Figure 7. The Grasshopper definition files, MATLAB v2020b scripts, and workflow instructions used in this study are described in the Code Availability Statement.

Rhino–Grasshopper v7.3 and ClimateStudio v1.1 were first installed on the host computer. The README file at the provided link was then consulted and the corresponding paths were configured. Finally, the relevant .gh files were opened in Grasshopper and the .m files in MATLAB v2020b, as shown in Figure 8 and Figure 9, respectively. The calculation itself consisted of three steps—entering parameters, running the calculation, and viewing the results (Figure 9).

Step 1—Entering parameters. The model parameters were entered into the input file. For a more complex model, variables can be modified or added by editing the program code.

Step 2—Running the calculation. Once the setup was complete, the run button was pressed and the automated design search began. During this process the randomly generated parameters were passed to the Grasshopper program, which performed the geometry construction and the L_mean and sDA_300/50% simulations. The simulation outputs were then fed back to compute the integrated evaluation value. PSO was initialized with a swarm of random particles and searched for high-scoring solutions by updating the particles across generations.

Step 3—Viewing the results. Once the stopping criteria were met, the calculation terminated, and the highest-scoring solution identified in the run could be inspected in the command window.

4. Case Studies

4.1. Description of Building Models

The developed method was applied to three test models representing the central top-daylit atrium, the semi-enclosed side-daylit atrium, and the semi-enclosed combined-daylit atrium. All three models used the common search bounds defined in Table 3 so that the typologies could be compared within the same morphological parameter space. A schematic of the atrium morphological parameters is shown in Figure 10. The long side of the central atrium was aligned east–west, and the side window of the semi-enclosed atrium faced east.

4.2. Setting of the Swarm Optimization

The population size was set to 24 and the maximum number of epochs to 50. The initial particle matrix for each variable was built by combining the design-parameter ranges. Both C₁ and C₂ were set to 2, and the inertial weight was set to 0.4. PSO control parameters followed those in [45] and were refined through preliminary trials conducted by the authors. Evolution of the population continued until either the maximum iteration count was reached or the best individual remained unchanged for ten consecutive generations.

Independent repeated-run statistics across alternative random seeds were not collected, and cross-run stability was therefore not evaluated in this study. Accordingly, the reported parameter combinations should be interpreted as the highest-scoring candidates identified under the present implementation settings rather than as statistically confirmed global optima.

5. Results

The results are presented in four parts. Section 5.1 summarizes the behavior of the two retained objective terms, namely the luminance-based satisfaction proxy f₁ and the sDA-based daylight-sufficiency term f₂. Section 5.2 presents the distribution of evaluated candidate solutions in the f₁–f₂ plane and explains the saturation of the sDA-based term. Section 5.3 reports the preferred parameter combinations identified for the three atrium typologies under the adopted baseline weighting scenario. Section 5.4 compares the three reported solutions and discusses their practical interpretation.

Because independent repeated runs were not evaluated, the configurations should be interpreted as preferred solutions under the present search settings, not as statistically stable global optima.

5.1. Behavior of the Two Retained Objective Terms

The two retained objective terms showed different interpretive roles in the design-search results. The first term, f₁, represents the normalized luminance-based satisfaction proxy derived from the L_mean–satisfaction regression. It varies with the simulated scene mean luminance and is therefore used to differentiate candidate solutions in terms of the adopted satisfaction-oriented visual indicator. The second term, f₂, represents sDA-based daylight sufficiency. Because f₂ is calculated from sDA_300/50%, it increases with the percentage of analysis points that meet the 300 lx threshold for at least 50% of the occupied hours.

All results reported below are based on the baseline weighting scenario described in Section 3.3.3. The reported score should therefore be interpreted as a scenario-based weighted score combining f₁ and f₂, not as a universal measure of visual comfort or as a measure of actual operational lighting-energy performance.

5.2. Candidate-Solution Distribution and sDA Saturation

To examine whether the weighted search produced a broad two-criterion trade-off or a saturated daylight-sufficiency region, all evaluated candidate solutions recorded during the PSO runs were plotted in the f₁–f₂ plane (Figure 11). Each point represents one simulated candidate solution, and the reported preferred solution for each atrium typology is marked separately.

Figure 11 shows that a substantial proportion of high-scoring candidates reached or approached the upper bound of the sDA-based term. This pattern indicates that the daylight-sufficiency term became saturated for many high-performing candidates, especially near the reported preferred solutions. Therefore, within this region, the weighted score no longer reflects a strong trade-off between f₁ and f₂. Instead, the remaining score differences are mainly associated with the luminance-based satisfaction term f₁. The following preferred solutions should consequently be interpreted as high-scoring configurations within an sDA-saturated design region rather than as evidence of a broad visual–daylight trade-off across the full search space.

5.3. Preferred Solutions for the Three Atrium Typologies

Using the PSO-based design-search procedure, one high-scoring parameter combination was identified for each atrium typology under the adopted baseline weighting scenario. Figure 12 shows representative fitness-improvement histories across epochs for the three atrium types. Within the present search settings, most score improvements occurred during the early and middle stages of the runs, whereas later epochs mainly refined the best-so-far solutions. These plots should be read as representative run histories under the current implementation, rather than as evidence of global convergence or cross-run stability.

Table 4. Preferred parameter combinations identified by the PSO-based design search under the baseline weighting scenario and current search settings.

Type	Central Atrium with Top-Daylit	Semi-Enclosed Atrium with Side-Daylit	Semi-Enclosed Atrium with Combined-Daylit
Width	15.50	22.81	6.07
Length-width ratio	1.85	1.02	1.83
Width of corridor	8.27	5.14	4.51
Floor height	5.85	5.15	5.71
Number of floors	3	2	8
f1	85.29	88.61	86.00
f2	100	100	100
Score	91.91	93.73	92.30
Plan and Section view

reports the preferred parameter combinations identified under the adopted baseline weighting scenario, together with the corresponding f₁, f₂, and weighted scores. These configurations are the highest-scoring solutions obtained under the present parameter bounds, PSO settings, and stopping criteria. They are therefore reported as preferred solutions under the current design-search setting, not as universally optimal atrium forms.

All three reported preferred solutions reached f₂ = 100, indicating that they lie in a high-daylight-sufficiency region where the sDA-based term is saturated. Under this condition, the weighted score no longer reflects a strong trade-off between the two retained terms; instead, differences among the reported high-scoring solutions are mainly governed by the luminance-based satisfaction term. The results should therefore be interpreted as identifying preferred morphologies within an sDA-saturated design region rather than as demonstrating a broad visual–daylight trade-off across the full search space.

The semi-enclosed combined-daylit solution should be interpreted with particular caution. Its small well width, eight-floor height, floor area of only 68 m², and low width-to-height ratio indicate a boundary-near mathematical solution generated under the present objective and constraint set. It is therefore not presented as a directly recommended architectural configuration. Additional functional, circulation, evacuation, and constructability constraints would be required before such a geometry could be considered a practically buildable atrium design.

5.4. Cross-Case Comparison and Practical Interpretation

The three reported solutions show different practical implications across the retained atrium typologies. The preferred central top-daylit atrium and the preferred semi-enclosed side-daylit atrium both had floor areas between 400 m² and 800 m², which corresponds to the conventional main-atrium size category [46]. By contrast, the preferred semi-enclosed combined-daylit atrium had a much smaller floor area of 68 m², which reinforces the need to interpret this solution as a boundary-near mathematical outcome rather than as a direct design recommendation. It should also be noted that the reported preferred solutions were generated within the common cross-typology search bounds defined in Table 3, rather than within the narrower type-specific observed ranges summarized in Supplementary Table S2. Therefore, candidates that fall outside the observed range of a specific retained subtype should be interpreted as exploratory outcomes of the common bounded design-search space, not as direct interpolations from that subtype’s empirical sample.

The cross-sectional proportions also differed substantially. The central top-daylit atrium had a width-to-height ratio of 0.87, representing a moderate proportion. The semi-enclosed side-daylit atrium had a ratio of 2.19, indicating a more spacious section. The semi-enclosed combined-daylit atrium had a ratio of 0.15, indicating a very narrow and deep section [47]. These differences suggest that comparable weighted scores can be obtained through different morphological routes, but that the practical plausibility of each route depends on typology-specific functional and constructability constraints.

Overall, the results indicate three main findings. First, within the adopted search space, all reported preferred solutions satisfied the sDA-based daylight-sufficiency term, leading to saturation of f₂. Second, once this saturation occurred, the luminance-based satisfaction term became the main differentiating component among the high-scoring candidates. Third, the practical interpretation of the preferred solutions was typology-dependent: the central top-daylit and semi-enclosed side-daylit solutions showed more conventional spatial proportions, whereas the semi-enclosed combined-daylit solution required additional architectural constraints before it could be considered practically applicable.

6. Discussion

6.1. Interpretation of the Retained Metrics and Weighting Scenario

The two retained terms in the present workflow have different scopes and should not be interpreted as a complete visual-comfort or energy-performance assessment. The first term, f₁, is a luminance-based visual satisfaction proxy derived from the adopted regression between scene mean luminance (L_mean) and subjective lighting satisfaction in shopping-mall public spaces [5]. It is used here because it provides a direct link between simulated luminance conditions and field-based user-response evidence. However, it does not represent spatial variation across multiple user positions, directional discomfort, short-duration glare, or a complete annual visual-risk assessment.

The second term, f₂, is based on sDA_300/50% and represents annual daylight sufficiency. Under an ideal daylight-responsive lighting-control assumption, higher sDA indicates greater potential for reducing electric-light use, but it does not directly quantify measured or simulated operational lighting energy consumption. Therefore, the weighted score used in this study should be interpreted as a scenario-based design-search score combining a luminance-based satisfaction proxy and sDA-based daylight sufficiency, rather than as a universal index of visual comfort or actual lighting-energy performance.

This metric choice is task-specific rather than universal. L_mean was retained because it can be embedded transparently into the adopted field-derived satisfaction model. This does not imply that L_mean replaces glare-oriented or annual visual-risk metrics in other daylight assessment tasks. Daylight glare probability (DGP) and simplified daylight glare probability (SDG) remain important when discomfort glare or directional visual disturbance is the primary concern, and annual sunlight exposure (ASE) remains important where potential annual overexposure risk must be assessed [2,35]. The present workflow should therefore be read as a satisfaction-oriented morphology-screening approach, not as a complete annual visual-risk assessment framework.

The weighting scheme should also be interpreted within the adopted scenario. In the present formulation, the 0.55/0.45 split is not presented as a direct AHP-derived measurement of the relative importance of f₁ and f₂. Instead, it is used as a baseline weighting scenario informed by expert-preference evidence. The four-criterion AHP results are retained as supporting evidence that satisfaction-oriented daylight appraisal received relatively high priority in the consulted expert panel, while the final weighted objective is treated as a scenario setting for the present design-search demonstration. Different expert panels, decision contexts, or weighting scenarios may lead to different preferred solutions.

6.2. Trade-Off Behavior and sDA Saturation

The results show that the daylight-sufficiency term reached saturation in the reported preferred solutions. As reported in Table 4, f₂ reached 100 for all three atrium typologies. This indicates that the preferred solutions were located in a high-daylight-sufficiency region where the sDA-based term had limited remaining ability to differentiate among neighboring high-scoring candidates.

This behavior affects how the weighted results should be interpreted. Once f₂ is saturated, the weighted score no longer demonstrates a strong trade-off between the two retained terms. Instead, differences among the reported high-scoring solutions are mainly shaped by the luminance-based satisfaction term f₁. Therefore, the reported solutions should be interpreted as preferred morphologies within an sDA-saturated design region, rather than as evidence of a broad visual–daylight trade-off across the full search space.

This finding does not invalidate the proposed workflow, but it narrows the interpretation of the results. For early-stage design search, the workflow can still help distinguish among daylight-sufficient candidate morphologies according to the adopted luminance-based satisfaction proxy. However, if the research aim is to preserve and compare the full trade-off structure between competing objectives, future work should adopt a Pareto-based formulation or reformulate the problem as maximizing the satisfaction-oriented visual term under an explicit daylight-sufficiency constraint.

6.3. Relationship with Recent Atrium Optimization Studies

The present study should be positioned as a narrow, retail-specific design-search workflow rather than as a broadly new optimization paradigm. Recent studies have already advanced atrium daylighting and optimization in several directions. Xue and Liu [10] optimized commercial atrium daylighting in cold regions through a Pareto-based parametric workflow, while Adel and Sheta [9] examined shopping-mall atrium and skylight configurations in a hot climate using ClimateStudio and glare-oriented visual-comfort metrics. Other recent atrium studies have further expanded the objective space to include daylight, energy, and thermal performance in non-retail building types [11,12]. These studies show that atrium optimization is an active and mature research area.

Within this context, the contribution of the present study is more narrowly defined. Rather than proposing a new general-purpose optimization algorithm or a complete visual-risk/energy-performance framework, this study assembles a shopping-mall-oriented weighted design-search workflow that connects three elements: survey-derived atrium morphology ranges, a field-derived L_mean–satisfaction relationship, and sDA-based annual daylight sufficiency. The workflow therefore adds value by embedding user-response evidence into early-stage morphology exploration for retail atriums.

Compared with Pareto-oriented atrium studies, the present weighted-sum formulation is simpler and more decision-specific. It does not preserve the full trade-off frontier, but it can provide a practical ranked score under a defined weighting scenario. This makes the workflow more suitable for early-stage screening and comparison of bounded morphology options than for final performance certification or comprehensive daylight-risk assessment.

6.4. Limitations and Future Work

Several limitations should be acknowledged. First, the visual term is based on one fisheye viewpoint located at the middle of the corridor on the short side of the atrium, at a height of 1.5 m, and on selected representative time points. This setup was suitable for constructing a controlled and repeatable satisfaction-oriented workflow, but it cannot represent all possible user positions, viewing directions, short-duration extreme brightness conditions, or discomfort-glare risks. Future work should include multiple viewpoints, annual glare metrics such as DGP or SDG, and annual overexposure indicators such as ASE.

Second, the daylight-related term is represented by sDA_300/50%. This metric captures annual daylight sufficiency and can indicate daylight-linked lighting energy-saving potential under an ideal daylight-responsive control assumption, but it does not simulate actual operational lighting energy consumption. Future studies should couple daylight simulation with lighting-control schedules and operational energy simulation to quantify actual lighting-energy savings more directly.

Third, the reported PSO-derived solutions should be interpreted with caution. The present study used one set of PSO control parameters and did not report independent repeated-run or multi-start statistics. Therefore, the reported parameter combinations should be understood as preferred high-scoring solutions under the present implementation settings, not as statistically stable global optima. Future work should test repeated runs with different random seeds and compare PSO with Pareto-based or evolutionary algorithms such as NSGA-II, GA, MOPSO, or Octopus.

Fourth, the weighting scheme reflects a baseline scenario rather than a universal decision rule. Although the expert-preference evidence supports the use of a satisfaction-oriented weighting emphasis in this study, different expert panels, stakeholder groups, or project priorities may lead to different weights and different preferred solutions. Future work should compare multiple stakeholder-specific weighting scenarios or use Pareto-based methods to avoid collapsing the criteria into a single scalar objective.

Fifth, the simulation workflow was not calibrated against case-specific luminance or illuminance measurements for the three modeled atrium types. The L_mean–satisfaction relationship was adopted from an earlier field study of shopping-center public spaces and was not revalidated for every typology, viewpoint, or lighting condition simulated here. The results should therefore be interpreted as model-based design-search evidence rather than as measurement-validated predictions of user satisfaction in specific built atriums.

Finally, the parameter bounds and typology extraction were derived from a documented atrium dataset for Chinese shopping malls rather than from a statistically representative sample of all retail atriums. The semi-enclosed combined-daylit solution in particular appeared near the search boundary, with a very narrow and deep geometry. Additional functional, circulation, evacuation, and constructability constraints would be required before such a geometry could be considered practically buildable. Future studies should incorporate these architectural constraints and test the workflow across other climatic regions, retail formats, and atrium plan forms.

7. Conclusions

This study developed a weighted-sum design-search workflow for shopping-mall atrium morphology by combining a luminance-based visual satisfaction proxy with sDA-based daylight sufficiency. The workflow links survey-derived atrium parameter ranges, ClimateStudio v1.1/Radiance simulation, a baseline weighted scalar objective, and PSO-based search to support early-stage form exploration for three prevalent shopping-mall atrium typologies.

Under the adopted baseline weighting scenario and current search settings, the workflow identified preferred high-scoring parameter combinations for the central top-daylit, semi-enclosed side-daylit, and semi-enclosed combined-daylit atriums. All three reported preferred solutions reached saturation of the sDA-based term; the final score differences within the high-daylight-sufficiency region were therefore mainly shaped by the luminance-based satisfaction term. The semi-enclosed combined-daylit solution should be interpreted with particular caution because it appeared near the search boundary and would require additional functional and constructability assessment before being considered a practical architectural option.

The main contribution of this study is therefore not a universal optimum or a complete visual-comfort/energy-performance assessment, but a reproducible retail-atrium-oriented workflow that connects field-based user-response evidence with parametric daylight simulation. Within the limits of the present weighting scenario, simulation assumptions, parameter bounds, and PSO implementation settings, the workflow can help early-stage design teams narrow the morphology search space and compare daylight-sufficient atrium options. Future research should incorporate glare and annual overexposure indicators, operational lighting-energy simulation, repeated-run or Pareto-based optimization, measurement calibration, and additional architectural constraints.