*5.1. New Weighting Method Applied on IAQindex and IEQindex Estimation for a CaseStudy Building and Comparison of Its Results with Old Crude Weighting SystemResults*

The previous approach by their weighting scheme (Table 13) was based on the treatment of all main pollutantsin ununiform way. The weights were previously set as equal (for pollutants with disproportionately different concentrations) or as dependent only on the energy expenditure on ventilation (for pollutants with similar) concentrations. The new IAQ and IEQ index results based on MDMA-developed weights werecompared to the old method. The system of objectified weights used for a numerical example of IAQ and IEQ building assessment leads to important conclusions. As expected, the use of objectified weights means that the results of the IAQ and IEQ indices weredifferent than those with subjective weights. For example, at low CO<sup>2</sup> indoorconcentrations, the role of this parameter in the IAQindex assessment was overstated. This is the case for most BMS systems controlling building ventilation, where currently the number of air exchanges (ACH) depends mainly on CO<sup>2</sup> concentration measurements. At relatively low CO<sup>2</sup> air concentrations <1500 ppm, the CO<sup>2</sup> priority compared to HCHO or TVOC attributes is much lower. The cumulative weight of the HCHO and TVOC arguments compared to the "old" IAQ calculation method increased by 48%. This is important from the point of view of planning and managing ventilation systems. According to calculations, the number of users satisfied with the level of CO<sup>2</sup> (85% satisfied) wasa good result. Nevertheless, at the same time, the theoretical number of users satisfied with TVOC concentration wasonly 52%. With novel objective weights for the TVOC argument, the number of users satisfied with IAQ will decrease. In the case of the present studyand the new weight system, the value of IAQindex decreased about 10% compared to previous calculations using the old method. Further, if one considers the existence of additional emissions from the building, finishing materials and furniture, the value of satisfied users will decrease by another 5.5%.

The authors strongly believe that the new approach to weights gives objective and more adequate results for the IAQ and IEQ models. Wider use of the new weighting method can be effective upon supplementing databases with the subjective weights of health risks for various volatile substances in the air.

#### *5.2. The WeightingProblem in Indoor Air Quality Component Sets Framework*

The weights in any decision-making model should not be arbitrary [53]. Some researchers still use weights in modelling without justification. A solution to the weight problem in indoor air quality modelling is proposed. Global weights are not only measures of the importance of attributes in the human–indoor building environment relationship, health risks and possible disturbances of the quasi-stable state due to correlations between attributes, but they are also measures of the potential load of ventilation impacted by emissions and excess concentrations of additional costs for energy use. In this context, the adoption of objectified weights in the IAQ model (also any other model, where the proposed method can be transferred) has significant practical contribution. It is suggested, knowing the specific air pollution values (attributes) for the analyzed building, to calculate the weights for a new alternative created by the user in accordance with the method proposed in article. For less insightful research, the authors suggest using a set of weights developed by us (Table 12) for scenarios/alternatives that aremost similar to the scenario under consideration. According to the adopted MADM method for each analyzed alternative (a1–a5 scenarios for the indoor environment described by concentration/emission levels for attributes or types of pollutants adopted in the IAQ

model), global weights for each attribute (CO<sup>2</sup> or TVOC or HCHO concentration) are calculated separately as the numerical coefficients of Equations (4) or (6). Thisis because the component weights (e.g., CRITIC correlation weight) of global weights depend on the values of the attributes (air pollution concentrations and emissions). Table 12 shows the calculated global weights *wij*,global for each attribute for each alternative a1–a5. For alternative a1 (measured concentrations in thecase study building with an assumption of no additional emissionsform construction products and athropogenic CO2), global weights of the IAQindex model are 0.0194279 for CO2, 0.478148 for TVOC and 0.5024236 for HCHO. For alternative a2 (minimum concentration levels with minimum emission rates), global weights are 0.0466775 for CO2, 0.938422 for TVOC and 0.0149017 for HCHO. For alternative a2-2 (case study result concentrations with minimum emission rates), global weights are 0.0349865 for CO2, 0.935155 for TVOC and 0.0298593 for HCHO. For alternative a3 (minimum concentration level; (e) minimum + 25% range of emission rate), global weights are 0.0422678 for CO2, 0.9426554 for TVOC and 0.0150769 for HCHO. For alternative a4 (minimum concentration level; (e) minimum +25% range of TVOC and HCHO emission rates), global weights are 0.07795658 for CO2, 0.9075486 for TVOC and 0.014494798 for HCHO. For alternative a5 (minimum concentration level; emission minimum +25% range of CO<sup>2</sup> emission rate), global weights are 0.0422712976 for CO2, 0.942738165 for TVOC and 0.014990546 for HCHO. The obtained global weight results clearly indicate which pollutant attributes have priority. In ranges up to about 1500 ppm CO<sup>2</sup> in indoor air, the weight of this attribute in the IAQ model is very small. This means that the ventilation system should take into account, in this respect, the removal of excess TVOC concentration first and, in the second place, excess HCHO. For all alteratives, the weight of the TVOC attribute is significant (for a2–a5, weight is over 90%). The weights for TVOC and HCHO are significantly higher than for CO<sup>2</sup> (one order of magnitude). A 25% increase in CO<sup>2</sup> emissions will not increase its weight in the model.

In this article, the authors have described in detail a procedure for the subjective and objective weight determination for a combined ΣIAQ model, which is composed of the following basic steps:


In our case study building (Table 13), the IAQ model determines global weight values for the attributes of "pollution concentrations" in the form of relative global weights adjusted at all alternative levels. The measures of the significance of the other attributes (additional emissions) are determined by the CRITIC method and are given by correlation weights that additionally impact the global weights of the "concentration" attributes. Therefore, although entropy weights are calculated just like CRITIC weights for all attributes, taking advantage of the fact that they are weights characteristic for given attributes, they can be "selected" and adjusted in the range of the first "concentration" attributes in order to calculate relative global weights for all levels of alternatives.

Thus, the determined global weights are valid within the IAQquality model (4) treated as a sub-expression in the IEQ equation model described by Equation (7). It seems that the approach presented in this study may easily be replicated by other similar IAQ models.

In our future research, the authors will try to unambiguously link the proposed IAQ model and weight system with the demand for ventilation air for various alternatives and the ventilation systems used in case study buildings.

## **6. Conclusions**

Currently, the issue of comfort assessement, based on user satisfaction models, is discussed in numerous research works, but in most of them indoor comfort is analyzed in terms of thermal comfort only [54–56].Fortunately, the recently published european standrd EN 16798-1: 2018 [7] included a wider number of indoor comfort criteria. This standard provides an approach to the classification of IAQ and IEQ models and supportscertification of buildings taking into account specific components of the environmental comfort (IEQ sub-models) including P IAQ (which is also composed of sub-components in functions of pollutant concentrations). This standard and any relatedreseach does not provide practical guidance on how to combine P IAQ model components into one indicator that could be used to classify indoor air conditions and does not provide information of how to aggregate components by an objective weighting system.With the research challenge in mind, the authors presented a relevant IAQ model in [4].In this paper, the authors provided astep-by-step procedure for determining the combined weighting scheme for the IAQ index equation, using the MADMdecision model including calculation of the objective attribute weights by the entropy method, calculation or adoption of subjective health-connected weights from the reference database, use of the CRITIC method in the case of mutually correlating attributes and calculation of objective weights for ventilation energy expenditure to various IAQ sub-components (pollutants). The assumption of the decision model is adequate for the task of rating indoor air ventilation to determine the assumed air quality (our actual work), and setting IAQ model alternatives so that they can meet the criteria taking into account their importance ranking and possibility of ongoing diagnostics of alternatives connected to the perceived actual indoor air situation. The presented weigthing scheme approach for the IAQ model may be regarded as objective, and it generates weighted criteria values (for each analysed attribute and indoor scenario) directly from the variations in criteria values. The provided method eliminates the problems of IAQ model weights with using the subjectivepoint of viewas well as the incompetence orabsence of a responsible decision-maker. The MADM approach, as shown in thiscase study, is based on simple mathematics and can, therefore, be used effectively for choosing the best IEQ and ventilation strategies.

**Author Contributions:** Conceptualisation, M.P. and K.K.; methodology, M.P. and K.K.; software, M.P. and K.K.; validation, M.P. and K.K.; formal analysis, M.P.; investigationtest, resources, M.P.; data curation, M.P.; writing—Original draft preparation, M.P. and K.K.; writing—Review and editing, M.P. and K.K.; visualisation, M.P.; supervision, M.P.; project administration, M.P.; funding acquisition, M.P. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research did not receive external funding.

**Conflicts of Interest:** The authors do not declare any conflicts of interest.

