# Comprehensive Optimization of Air Quality in Kitchen Based on Auxiliary Evaluation Indicators
## 2 Description of Numerical Method
### Application of the Auxiliary Evaluation Index
#### 2.3.2 Improved Formulation  
To address the limitations of the initial Q-index (Equation (7)), an improved formulation is proposed. This revision focuses on three key enhancements: first, handling negative PMV values (indicating thermal discomfort) by using its absolute value to ensure consistent representation of discomfort magnitude; second, introducing weighting coefficients (A, B, C, D) to reflect the relative importance of each constituent indicator; and third, reconciling the inconsistent directional impacts of individual indicators on the composite index to ensure uniform interpretation of their influence.  
\[Q=\text{A}\frac{100\%-\text{ADPI}}{20\%}+B\frac{|PMV|}{1}+C\frac{CO_{2}}{700} +D\frac{COFP}{50} \tag{8}\]  
where A + B + C + D = 1.  
(1) Determining Weighting Coefficients  
The weights A, B, C, D are derived based on the contribution of each indicator's variation range to the total variability of Q. This ensures balanced influence across metrics. For each indicator \(i\), compute its normalized contribution \(K_{i}\).  
\[K_{i}=\frac{|Q_{imax}-Q_{imin}|}{\sum|Q_{imax}-Q_{imin}|},\ \ i\in\{CO_{2}, COFP,ADPI,PMV\} \tag{9}\]  
Here, \(Q_{imax}\) and \(Q_{imin}\) are the maximum and minimum values of Q when only indicator \(i\) varies within its standard range, while others are fixed at baseline values (per international standards).  
To equalize the contributions of all indicators to \(Q\), the weights A, B, C, D satisfy:  
\[A\times K_{CO_{2}}=B\times K_{ADPI}=C\times K_{PMV}=D\times K_{COFP} \tag{10}\]  
This ensures uniform sensitivity of Q to changes in each indicator.  
(2) Taking the ADPI calculation as an example, according to international standards, ADPI has a maximum value of 100% and a minimum of 0. When determining ADPI's contribution to Q, all other parameters adopt baseline values specified in international standards. Consequently, the contribution of ADPI to Q is calculated as follows:\[Q_{max}=\left(\frac{100\%-0}{20\%}+\frac{1}{1}+\frac{700}{700}+\frac{50}{50} \right)=8\] \[Q_{min}=\left(\frac{100\%-100\%}{20\%}+\frac{1}{1}+\frac{700}{7 00}+\frac{50}{50}\right)=3 \tag{11}\] \[|Q_{max}-Q_{min}|=8-3=5\]  
(3) Proceeding further, after calculating the contribution values K of the four metrics to Q, adjustment coefficients A, B, C, and D are introduced to ensure uniformity in their contributions to Q. Finally, the improved formula for Q is obtained and can be expressed as follows (keep two decimal places):  
\[Q=0.15\;\frac{100\%-ADPI}{20\%}+0.25\;\frac{|PMV|}{1}+0.35\;\frac{CO_{2}}{7 00}+0.25\;\frac{COFP}{50} \tag{12}\]  
The proposed weighting strategy ensures that the improved Q-index (Equation (12)) responds uniformly to the variability of each constituent indicator. In contrast to generic composite metrics (e.g., LEED IAQ credits) or single-dimensional indices (e.g., PMV, ADPI), this index offers a physics-informed, multi-criteria framework explicitly designed for kitchen environments. Departing from recent studies focused on hybrid ventilation strategies [19; 23] or AI-driven optimization for generic spaces [25], this work establishes a deterministic, weighting-based methodology that integrates COFP dynamics--a critical pollutant in kitchen settings--with thermal comfort and airflow distribution.  
The framework enables reproducible ventilation design without reliance on complex sensing systems or computational algorithms, rendering it particularly suited to cost-sensitive residential applications. Practically, a lower Q value denotes superior indoor air quality, characterized by reduced pollutant concentrations and improved occupant comfort, while elevated values indicate performance deficiencies requiring intervention. This index not only advances the quantitative assessment of kitchen air quality but also lays a robust theoretical foundation for the experimental and numerical validations presented in subsequent sections.