Rational settings of illuminance and CCT are beneficial to human health and have been proven in medical research. The illuminance adjustment in lighting can alleviate visual fatigue [
18], while the CCT adjustment can ease sleepiness and enhance comfort [
19,
20]. Here, a mathematical model of the effect of light parameters on human feelings is developed and formulated as a multi-objective optimisation problem (MOP) with constraints. Firstly, the MOP is described in
Section 2.1. Secondly, the objective functions are introduced in
Section 2.2, and finally the decision space under the constraint conditions is introduced in
Section 2.3.
2.1. Problem Description
Trends in human perception of visual comfort, alertness, valence, and arousal of mood in the light environment with different illuminance and CCT levels are the prerequisite for the search of the optimal solutions for lighting control. We conducted a series of lighting experiments with the variables of illuminance and CCT. The participants voted for visual comfort, alertness, valence, and arousal of mood under each lighting condition. The mean voting scores were fitted using non-linear and linear regression methods to obtain polynomial functions, and the non-linear/linear functions of humans’ visual comfort, alertness, and mood with illuminance and CCT are used as the objective functions of MOP. Furthermore, light that is conducive to healthy circadian rhythms is considered the constraint to investigating the light parameter conditions. The non-dominated solutions obtained in a feasible domain with continuously varying parameters from the MOP, that satisfy high visual comfort, enhanced alertness, and positively motivated mood, can provide a research basis for the lighting control strategies. Without loss of generality, the mathematical expression of the above problem is shown in Equation (1).
Let
D be the feasible domain of the above multi-objective optimisation problem,
where
is a decision vector,
F(
x) is a
K-dimensional objective vector,
K is the number of the objective functions,
gi(
x) is an inequality constraint, and
M is the number of the constraints.
2.2. Objective Functions
In our MOP, we focus on four dimensions of human perceptions under the light environment, visual comfort, alertness, valence, and arousal of mood. Visual comfort is defined by the European standard EN 12665 as “the subjective condition of visual health caused by the visual environment” [
21]. Alertness is being awake, attentive, and ready to act or react. Mood can be represented by valence and arousal [
22]. Valence is the degree of pleasantness and can be defined from negative to positive. Arousal is a level of autonomous activation that ranges from calm to excited. The averages of the voting scores of visual comfort, alertness, valence, and arousal of mood (see
Table A2 in
Appendix A) were fitted using non-linear and linear regression methods. According to the regression results, the functions of visual comfort
fcom(
x), alertness
fale(
x), and valence
fval(
x) of mood were binary cubic polynomials on illuminance (
x1) and CCT (
x2), which are shown in Equation (3). The arousal of mood function
faro(
x) was a binary linear regression on illuminance (
x1) and CCT (
x2). The specific parameters for the functions are shown in
Table 1.
Details of the objective functions are shown below.
The visual comfort of users in the different light environments was graded as 0 uncomfortable, 1 less comfortable, 2 neutral, 3 a little comfortable, and 4 comfortable. fcom(x) was lowest at 300 lx, 3000 K, i.e., the least comfortable, and increased with both illuminance and CCT, with an extreme value at 673 lx, 4231 K and a maximum of 3. After that, fcom(x) decreased with both illuminance and CCT, but to a lesser extent with CCT. Maximisation was one of the objective functions.
The alertness of users in the different light environments was graded as 0 sleepy, 1 a little sleepy, 2 neutral, 3 a little alert, and 4 alert. fale(x) was lowest at 300 lx, 3000 K, i.e., the most sleepy, and tended to increase with increasing illuminance and CCT, from sleepy to alert, with an extreme value at 733 lx, 5553 K, and a maximum of 3.4. The level of alertness was the same at 700 lx and 800 lx, and the same at 5000 K and 6000 K. Maximisation was one of the objective functions.
The impact of the user’s valence in the different light environments was graded as 0 unpleased, 1 less pleased, 2 neutral, 3 a little pleased, and 4 pleased. fval(x) tended to arch upwards centrally in the two-dimensional effects of illuminance and CCT, from unpleased to a little pleased, with an extreme value at 665 lx, 4753 K, and a maximum of 2.98. After that, the degree of valence decreased with further increases in illuminance and CCT. Maximisation was one of the objective functions.
The arousal dimension was graded as 0 calm, 1 a little calm, 2 neutral, 3 a little excited, and 4 excited. faro(x) tended to increase with both illuminance and CCT, gradually moving from calm to excited. As the goal was to make the arousal level close to neutral, maximisation was one of the objective functions.
2.3. Decision Space with Constraints
The variables involved in the MOP are
x1: illuminance,
x1 ∈ [300, 800] in the units of lux (lx), and
x2: CCT,
x2 ∈ [3000, 6000] in the units of Kelvin (K). Take classroom lighting as an example, according to the EU standard “Light and lighting—Lighting of work places—Part 1: Indoor work places” (EN 12464-1: 2021), the recommended illuminance of the task area is 500 lx [
23]. According to the Chinese “Architectural Lighting Design Standard” (GB 50034-2013), the horizontal illuminance of the working surface should not be lower than 300 lx, and the recommended CCT range is 3000~5500 K [
24]. In summary, we optimise the illuminance range to a minimum of 300 lx, taking into account energy-saving factors, i.e., lighting power density values, the maximum illuminance is 800 lx. The CCT takes values ranging from 3000 K (warm) to 6000 K (cold). When the illuminance and CCT vary, the CRI value is assumed not to be lower than 80, in accordance with the guidelines contained in the EN 12464-1:2021 Standard.
The constraint condition considers light that is conducive to a healthy circadian rhythm. Light plays a synchronising role in the body’s biological clock. In addition to changing the phase of circadian rhythms, it also regulates the timing and quality of human sleep. The international standard CIE S 026:2018 (CIE 2018) defined spectral sensitivity functions and quantification methods to describe the non-visual effects (melatonin suppression aspect) induced by stimulation of intrinsically photosensitive retinal ganglion cells (ipRGCs) containing melanopsin through retinal-mediated light radiation [
25]. It is worth noting that the spectral characteristics at the eye level play an important role from the point of view of the influence of light on non-visual effects, and the spectral characteristics of the light source used in the lighting experiments should be taken into account. As more melanopsin-based photoreception during the day contributes to increased alertness, circadian rhythm, and good sleep quality, lower melanopsin-based photoreception at night facilitates faster sleep [
26,
27].
Equivalent Melanopic Lux (EML), which considers only the contribution of ipRGCs in non-visual effects, originated with Enezi’s team [
6] and was subsequently improved by Lucas’ team [
7]. Each equivalent
α-opic illuminance (symbol
Eα; unit: α-opic + lux, lx, lumen per square metre, lm·m
−2) specifies a photometric quantity related to the spectral power distribution of irradiance
Ee,λ(
λ) by the following equation. The absolute sensitivity is identical to photopic illuminance for light with an equal-energy spectral power distribution.
where
λ is the wavelength of the radiation. The maximum spectral luminous efficacy
Km = 683.002 lm/W.
V(
λ) is the spectral luminous efficacy function for photopic vision.
Ee,λ(
λ) is the spectral power distribution.
NZ(
λ) is the α-opic sensitivity curve with arbitrary normalisation. α specifies the retinal photopigment for a given organism. For example, the five human variants are: cyanopic, relating to the s-cone photopigment; chloropic, relating to the m-cone photopigment; erythropic, relating to the l-cone photopigment; rhodopic, relating to rhodopsin; and melanopic, relating to melanopsin. As
= 106.857 and
= 1, Equation (4) can be simplified to an equivalent form to the calculations for photopic lux:
The Circadian Stimulus (CS) was proposed by Rea’s team, taking into account the blue-yellow spectral colour-blocking mechanism in the phototransduction pathway caused by S-type cone cells, and the method included two metrics, Circadian light (CL
A) and CS [
8,
9]. The formulas are available as follows.
where 1548 is the constant, sets the normalization of CL
A so that 2856 K blackbody radiation at 1000 lux has a CL
A value of 1000.
Mc(
λ) is the melanopsin (corrected for crystalline lens transmittance).
E(
λ) is the light source spectral irradiance distribution.
S(
λ) is the S-cone fundamental.
mpλ is the macular pigment transmittance.
V(
λ) and
V′(
λ) are the photopic and scotopic luminous efficiency functions, respectively.
RodSat is the half-saturation constant for bleaching rods = 6.5 W/m
2.
ab−y = 0.70,
arod = 0.70.
In our MOP, the thresholds of EML and CS suggested by relevant studies would be constructed as constraints. Details are shown below.
The WELL Building Standard v2 recommends a minimum value of 150 EML for regularly occupied spaces in the vertical plane at eye level between 9 a.m. and 1 p.m. [
28]. Therefore, EML
150 was one of the constraints.
The lighting research centre proposed that at least one hour in the morning satisfying a CS greater than 0.3 is beneficial for enhancing human health and well-being, improving productivity, and reducing long-term health problems associated with circadian rhythm disturbances [
29]. Therefore, CS
0.3 was one of the constraints.
The functions of EML and CS with illuminance (
x1) and CCT (
x2) can be fitted according to
Table A1 in
Appendix A. Feasibility laws were used to deal with the constraints of circadian lighting, and it can be calculated that the constraints on the decision space are to make
x1 468 by solving the two inequalities. Thus, the lighting multi-objective optimisation problem can be formulated as maximising comfort, alertness, valence, and the gap between neutrality and arousal, and considering circadian lighting with
x1 [468, 800],
x2 [3000, 6000]. The mathematical model of the problem is shown below.