*2.5. Calculation of the Objective Attribute Weights by the Entropy Method*

#### Step 1. Normalisation of indicators

Since the analysed indicators of indoor air contamination may be provided in different units, authors normalised them as a first step. Normalised decisions, however, depend on the attribute's nature. In the entropy weights procedure, all attributes must be beneficial (positive values), but in the decision matrix all attributes are of a non-beneficial type. Therefore, the conversion of normalised indicators of "cost" type into indicators of "profit" requires building a matrix of "profit type". As such, the higher their indicator values are, the higher the assessment of the decision variant will be. The indicators for which the criteria are based on the assessment of current air pollutants, and the assessment of the emission rate belong to the cost type indicators, are designated by non-beneficial or cost sets. In order to determine weights using the Shannon entropy method, a specific data normalisation method is used [20]. We built a matrix Y = (*yij*), in which all criteria will be of "profit type". Then, the elements of the matrix are determined from Equation (13):

$$y\_{i\bar{j}} = \mathbf{1}/\mathbf{x}\_{i\bar{j}} \tag{13}$$

The normalised decision matrix Z = [*zij*]n×*<sup>m</sup>* with the elements *zij* for determining entropy weights is determined with Equation (14):

$$z\_{i\circ} = \frac{y\_{i\circ}}{\sum\_{i=1}^{n} y\_{i\circ}} = \frac{1/\chi\_{i\circ}}{\sum\_{i=1}^{n} 1/\chi\_{ij}} \tag{14}$$

for various alternatives (more or less complex indoor air quality models) and their respective attribute sets (selected IAQ sub-models to assess various pollutants and emission rates).

Step 2. Calculation of the entropy measure

The entropy vector *e<sup>j</sup>* for a given attribute *j* from the sum of components *zij* and ln(*zij*) calculated for all alternatives *i* = 1 . . . *n* is determined as

$$e\_j = -\frac{1}{\ln n} \Sigma\_{i=1}^n z\_{ij} \ln z\_{ij} \tag{15}$$

where *zij*·ln*zij* is equal to 0 for *zij* = 0 and the variation(divergence) level vector d<sup>j</sup> calculated for each attribute as

$$\mathbf{d}\_{\mathbf{j}} = \mathbf{1} - \mathbf{e}\_{\mathbf{j}} \tag{16}$$

Step 3. Defining the objective weight of each *j* as follows, based on the entropy concept

$$w\_{jentropy} = \frac{d\_j}{\sum\_{j=1}^{m} d\_j} \tag{17}$$

#### *2.6. Calculation of Subjective Attribute Weights Depending on the Toxicity of Air Pollution to Humans*

The weights of IAQindex models must also contain, in addition to the objective entropy weights, the subjective attribute weights that describe factors related to health human toxicity from indoor air pollution.

Variant 1. *S<sup>j</sup> health* subjective weights from databases for criteria of "air pollution specified chemical compound" with tested chemical properties, toxicity, and "health level" and "health scores" assigned to air concentrations of this compound are treated as reference data taken directly from the literature [36] and [40–44]. For example, Yuan [40] adopted the following weights for selected pollutants as part of the research program:"French permanent survey on Indoor Air Quality" [43], US-EPA and WHO databases (Table 4).In some cases when detectinga larger number of indoor air contamination compounds with air concentration >5 µg/m<sup>3</sup> , additive health effects are assumed, and the subjective weight S<sup>j</sup> health can be created with the EU-LCI approach.

**Table 4.** Subjective weights of ten pollutants (plus hot and humid air enthalpy).


\* Based on the recommendation of the European Collaborative Action expert group. This group focuseson the harmonisation ofindoor emission assessments by means of developoing Lowest Concentration Interest (LCI) values (since 2011). This expert group developed a list of relevant substances and described the procedure of LCI calculation. For assessing combination effects of substances, the R-value as the health hazard index was proposed. The R<sup>i</sup> value was established as R<sup>i</sup> = C<sup>i</sup> /LCI<sup>i</sup> , where C<sup>i</sup> is the concentration of compound in a test chamber i. For R<sup>i</sup> < 1, it is assumed that there will be no health effects. R = sum of all R<sup>i</sup> = sum of all ratios (C<sup>i</sup> /LCI<sup>i</sup> ) ≤1.

Variant 2. The subjective s<sup>j</sup> health weights determined by health experts depend on the toxicity of pollutants based on a number of expert opinions r greater than the number of experts k. In the case of not listing compounds in databases, the relative value (signification) of the s<sup>j</sup> health weight for the jth criterion (with a total number of m criteria) can be calculated by the SWARA method presented by Kersuliene et al. [45] and Zavadskas [46] using weight estimation procedures in several steps as follows.

Step 1. The relative importance of the criterion is determined by reference to the weight of s<sup>j</sup> and the weight of the previous criterion sj-1. Then, the weights of s<sup>j</sup> health based on the opinions of all experts are calculated. An example of a simple algorithm for determining the average subjective weight of attributes s<sup>j</sup> is based on the average of opinions (weight proposals) of attributes collected from k experts, i.e., from the formulas given by [46]. The average value of expert attribute reviews is calculated using Equation(18):

$$
\overline{t}\_j = \frac{\sum\_{k=1}^r t\_{jk}}{r} \tag{18}
$$

where *tjk* is the ranking of attribute *j* according to respondent *k*, and *r* is the number of respondents.

Step 2. The subjective weights can be adjusted by dividing the average values of each attribute by the sum of the average values of the attributes (19):

$$s\_j^{\text{head}th} = \frac{\overline{t}\_j}{\sum\_{j=1}^{m} \overline{t}\_j} \tag{19}$$

Step 3. Having determined the average values of attribute reviews from Equation (18) and knowing the established ranking of these opinions, you can calculate the standard deviation of attribute *j* in the ranking of opinions ("dispersion of expert ranking values") by Equation (20):

$$
\sigma\_j^2 = \frac{1}{r-1} \cdot \sum\_{k=1}^r \left( t\_{jk} - \overline{t}\_j \right)^2 \tag{20}
$$

Value σ*<sup>j</sup> 2* is useful for calculating the weights of the effects of attribute correlation wj,CRITIC formula.

#### *2.7. Calculation of the Objective Weights of Correlation between Attributes by the CRITIC Method*

Use of the CRITIC (Criteria Importance Through Inter-criteria Correlation) method in the case of mutually correlating attributes (e.g., attributes constituting "quasi-fixed" impurities and attributes constituting their emissions) to estimate the correlation weights between the criteria is justified. Additionaly to the intensity contrast of dataset attributes in the decision matrix, there was another concept more recently taken into consideration by researchers using MCDM. Diakoulaki's team [47] found that the greater the level of inter-dependency between attributes, the higher the error ranking outcome. CRITIC was analysed in [47] as an objective and new weighting system that took into consideration criteria correlations. The approach included contrast intensities (by means of SD criteria) and combined them with correlations weights. All material selection studies assume that the criteria are independent of each other, while they are also likely to influence each other. Some approaches proposed in recent years for weighting criteria when choosing material not only bypassed this approach, but they did not take into account the interdependence of criteria and proposed methods of objective weighting, which have relevant shortcomings. When correlation between criteria is suspected, the validity of the criteria can be adjusted by entering correlation weights. Calculating the weights of correlations between the criteria of a decision model using Pearson's correlation coefficients is recommended [9,11]. The CRITIC weights marked wj,CRITIC specify a correlation between the criteria, which can include all criteria (or attributes) or only criteria that are influenced. The CRITIC method by Diakoulaki [47], Jahan et al. [12] and Ardakani et al. [10] includes estimations of differences in the "intensity" of criteria values (expressed by the value of their standard deviations) and combines them with weights based on correlation coefficients (step 4). Alternatively, the modified model in Jahan [12] uses only Pearson's correlation coefficients and does not include standard deviation criteria in the equations. As justification, it was stated that the differences resulting from changes in the "intensity" of the criteria were already taken into account by the entropy method, and the correlation weights were calculated accordingly. Therefore, the values of the correlation weights between the criteria are calculated by [10,12] using the

values of Pearson coefficients Rjk, which represent the correlation between the two selected criteria *j* and *k* (correlated criteria) and which can be calculated easily, even in MS Excel. *Rjk* values are read from the symmetrical linear criteria correlation matrix or calculated from Equation (21):

$$R\_{jk} = \frac{\sum\_{i=1}^{n} (\mathbf{x}\_{ij} - \overline{\mathbf{x}}\_{j})(\mathbf{x}\_{ik} - \overline{\mathbf{x}}\_{k})}{\sqrt{\sum\_{i=1}^{n} (\mathbf{x}\_{ij} - \overline{\mathbf{x}}\_{j})^2 \cdot \sum\_{i=1}^{n} (\mathbf{x}\_{ik} - \overline{\mathbf{x}}\_{k})^2}} \quad j \text{ and } k = 1 \ldots m \tag{21}$$

where *xij* and *xik* are *n* total number of alternatives (alternative IAQ models) and matrix elements (Table 1) *xij* (with a peak) of average values of the *j* criterion and *xij* of average values of *k* criteria (emission criteria k1..3 correlating with other criteria, e.g., emission rate of pollutants, which include selected criteria pollutant concentrations). If *Rjk* is close to +1 or −1, this means that the criterion is correlated highly, while *Rjk* close to 0 means there is no correlation. Then, the correlation weights are calculated by the CRITIC method using *Rjk* determined by Equation (21). The effects of the correlation impact of all criteria on other criteria can be calcualted, or the impact of selected *k* criteria (e.g., emissions) on related criteria of the decision model j (e.g., polluting emissions), and only then are selected correlation coefficients used. A common form of the formula for correlation weights using the CRITIC method is Equation (22) [10]:

$$w\_{j, \text{CRITIC}} = \frac{\sum\_{k=1}^{m} (1 - \beta \cdot R\_{jk})}{\sum\_{j=1}^{m} \left(\sum\_{k=1}^{m} (1 - \beta \cdot R\_{jk})\right)} \quad j \text{ and } k = 1 \ldots m \tag{22}$$

where β is equal to +1 if two objective criteria are of the same type (either greater is better) or β = −1 otherwise. In Equation (22), Equation (23) denotes a conflict of measure created by criterion k in relation to the decision:

$$Conflect = \sum\_{k=1}^{m} \left(1 - R\_{jk}\right) \tag{23}$$

According to Zawadskas [30], the sum (23) ought to be multiplied by σ*<sup>j</sup>* <sup>−</sup>, − the standard deviation (SD) of the *j*th criterion. The inclusion of SD follows a similar approach to the entropy approach, assigning a weight to the attribute if it has a corresponding attribute value across alternatives. A "symmetrical matrix" is developed with dimensions *n* × *m* and a generic element *Rjk*. The higher the divergence in the scores of alternatives in criteria *j* and *k*, the lower the value of *Rjk*. This step determines the conflict and then the weight of each decision criterion. The sum provided below denotes the measure of conflict created by criterion *j* with respect to the decision.

The modified and simplified CRITIC method, which has not been used in this study, calculates the correlation weights between criteria without calculating the symmetric linear correlation matrix *Rij*. The method determines the objective weights *wj,CRITIC* on a different principle. It determines the objective weights of the attributes by using contrast intensities for each measure, considered as SD, conflict and correlation coefficient between criteria by Zavadsas et al. [30], Vujicic et al. [31] and Adali et al. [48].

Step 1. Calculation of the normalised decision matrix for wj,CRITIC

For each criterion *xij*, function *rij* which transforms all the values of criteria into the interval [0, 1], is provided:

$$r\_{ij} = \frac{\mathbf{x}\_{ij} - \min \mathbf{x}\_{ij}}{\max \mathbf{x}\_{ij} - \min \mathbf{x}\_{ij}} \tag{24}$$

The basis of the transformation uses the concept of an ideal point. The initial matrix is converted into generic element matrix *rij*. Here, it should be noted that normalisation does not take account of the type of criteria.

Step 2. Calculation of standard deviations in attribute column *j* for all alternatives 1 . . . *n*.

Step 3. Calculation of *C<sup>j</sup>* specifying the amount of information contained in the given attribute *j*. In Equation (25), SD represents the deviation of various values for a given criterion of a mean value. The information of *C<sup>j</sup>* contained in criterion *j* is calculated by Equation (25):

$$\mathbf{C}\_{j} = \sigma\_{j} \sum\_{j=1}^{m} \left( 1 - \mathbf{R}\_{jk} \right) \tag{25}$$

where σ*<sup>j</sup>* is the SD of the *j*th criterion or alternative, and the expression *rjk* is considered equivalent to the correlation coefficient between the two attributes: selected attribute *j* and correlation attribute *k*.

Step 4. Calculation of the weight of the correlation of the criteria correlating *k* with a given criterion *j* is carried out using Equation(26):

$$\left| w \right|\_{j, \text{CRTIC}} = \mathbb{C}\_j / \sum\_{k=1}^m \mathbb{C}\_k \quad j = 1, \dots, m \tag{26}$$

Calculation of correlation coefficients should not be ignored; therefore, this study adopted a modified method of data normalisation and calculation of correlation weights between attributes based on the symmetric linear correlation matrix using the full CRITIC method. Calculation of correlation weights between criteria based on the symmetric linear correlation matrix by the CRITIC method uses a modified data normalisation method. In summary, the calculations were based on the determined correlation coefficients for attribute pairs related to the same impurities wj,CRITIC by using the following steps.

Step 1. Normalisation of initial decision matrix (Table 1)

The recommended [6] equation for normalisation of the original decision matrix for cost criteria is Equation (27)

$$r\_{ij} = \frac{\max(\mathbf{x}\_{i\bar{j}}) - \mathbf{x}\_{i\bar{j}}}{\max(\mathbf{x}\_{i\bar{j}}) - \min(\mathbf{x}\_{i\bar{j}})} \tag{27}$$

but the authors used the transformer version [32,33] in the form:

$$r\_{ij} = \frac{\mathbf{x}\_{ij} - \text{max} \ \mathbf{x}\_{ij}}{\text{min} \ \mathbf{x}\_{ij} - \text{max} \ \mathbf{x}\_{ij}} \tag{28}$$

Step 2. Calculation of standard deviations in data sets assigned to attributes *j* from the value *rij* for all alternatives 1 . . . *n*.

Step 3. Calculation of symmetric linear correlation matrix *Rij*.

The primary matrix (Table 1) is converted to the normalized decision matrix and later to symmetric linear correlation matrix of generic *Rij* elements, which are Pearson's correlation coefficients (21) determined for all pairs of normalised sets on data *rij* assigned to selected attributes, at all levels of alternatives *n*. Then, the most likely correlations corresponding to attribute pairs (e.g., concentrations and emissions of the same compound) were selected from the matrix, and weights were calculated for them wj,CRITIC. A coefficient of linear correlation between each pair of measures is estimated using the following formula that allows to quantify the conflict occurring among various criteria. It is obvious that more discordant the scores of the alternatives in both matrix vertical *i* and horizontal *j* sections, the lower the value *Rij*.

Step 4. Calculation of objective weights using the CRITIC method. The weight of the *j*th criterion is obtained using Equation (29) presented as:

$$w\_{j, \text{CRITIC}} = \mathcal{C}\_j / \sum\_{k=1}^{m} \mathcal{C}\_k \tag{29}$$

where

$$\mathbf{C}\_{i} = \sigma\_{j} \cdot \sum\_{k=1}^{m} \left( 1 - R\_{jk} \right) \tag{30}$$

A higher value of *C<sup>j</sup>* indicates a more considerable amount of information contained in a particular criterion. Therefore, its weight is higher. The CRITIC method assigns greater weight to criteria that have high standard deviations and low correlations with other criteria. A higher *C<sup>j</sup>* value means that more information can be obtained from this criterion, which increases the relative importance of the criterion for the decision problem. The study of correlation effects by the CRITIC method is carried out by calculating the weight *wj,CRITIC* for the *j* attribute that requires it due to the anticipated impact of the *k* attribute, e.g., when it is known that the validity of the attribute "TVOC pollution" in a "quasi-stable state" will increase under the influence of the attribute correlating with"TVOC emission".

#### *2.8. Calculation of the Objective Weights for Ventilation Energy Expenditure*

The objective weights for ventilation energy refer to the combined ΣIAQ model described by polynomial (1), whose words have been assigned weights described by Equation (3), based on the concept of "excess concentration", creating the energy expenditure of ventilation. These are the relative weights calculated for each alternative of the IAQ model. The values ∆c<sup>j</sup> for pollutants adopted inthe decision matrix differ by several orders. In the IAQ index model decision matrix (Table 1), it is necessary to normalise the set of values of "excess concentrations" for the first three attributes in the ΣIAQquality system described by Equation (4). The basic problem indata normalisation for determining weights wij,energy based on the concept of "excess concentration" is the selection of formulas that can be useful for this purpose. Referring to Körth et al. [29] and Vojcica [31], it seems that the following equation can be used to normalise datasets of "type cost", which after conversion to attributes of "type profit" (or beneficial) adjust for IAQ systems corresponding to subsequent alternatives a1 to a5. Normalised values ∆c<sup>j</sup> will correspond to "energy load" weights *wij,energy* = *nij*:

$$m\_{ij} = \left[1 - \frac{\mathbf{x}\_{ij}^{\min}}{\mathbf{x}\_{ij}}\right]^{-1} \tag{31}$$

Taking into account the input data of the decision model summarised in Tables 1 and 5, a dimensionless set of values corresponding to excess concentrations according to (2), normalised for weights, is calculated for the alternatives of this model *wij,energy*. for non-zero attributes *j* = 1, 2, 3 in the form of CO2, TVOC and HCHO pollutant concentrations.

#### *2.9. Combining the Weights by Calculation of Global Weights*

The global weights wij,global contain specified weight multiplication products of various types, adjusted to the sum of these products according to the widely used procedure for building weighting development. In the flow chart (Figure 3), the *wj,global* weights include basic entropy weights *wj,entropy*, subjective weights related to health human toxicity determined using "health-related" databases *s<sup>j</sup> health* and correlation weights *wj,CRITIC*, and ventilation energy weights *wij,energy*. At the same time, correlation weights occur only in those alternatives (P IAQ systems) in which decision matrices contain correlation criteria (this criterion is "pollutant emissions" in Table 1). The weights are assigned to individual attributes *j* = 1 . . . 6 of the decision matrix, and the weighting schemes for five alternative IAQ levels *i* = 1 . . . *n* are composed of weights assigned to specific attributes at a given level of alternatives from a1 to a5, which are adjusted to the sum of weights at a given alternative level *i*. A modified integrated weighting method, presented by Deepa et al. [49] and [15], proposed an approach to combine all weights

calculated using different weight assignment approaches into a single set of weights. The equation to calculate the weights of attributes in alternative levels is

$$w\_{ij\text{-}global} = \frac{\prod\_{\mathcal{g}=1}^{n} w\_{\mathcal{g}}}{\sum\_{\mathcal{g}=1}^{n} \left(\prod\_{\mathcal{g}=1}^{n} w\_{\mathcal{g}}\right)} \tag{32}$$

where *wij,global* represents thecomined weights of the criteria, *w<sup>g</sup>* denotes the criteria weights calculated by using one of the weight calculation methods, and *g* = 1, 2, . . . , *n*, where *n* is the number of methods used.

If a decision is made to assign a given attribute to the objective weight *wj*,entropy and the subjective weight associated with the risk to *s<sup>j</sup>* health health, but with adjastmentto the sum of weights at a given alternative level *i*, then the global weight of toxic pollution is

$$w\_{ij,global} = \frac{[s\_j^{\text{healthy}} \cdot w\_{j,entropy}]}{\Sigma\_{j=1}^{m} [s\_j^{\text{healthy}} \cdot w\_{j,entropy}]} \tag{33}$$

When the alternative and the attribute are assigned the relative weight *wij*,energy and when the alternative indicators are compared to the attributes of type *k* correlating with, e.g.,"pollution emission", then the composite weight of the attribute correlating with the attribute "emission" also includes the weight of the correlation effect *wj*, CRITIC and weight *wj*,energy for the given attribute *j* at the level of the i-th alternative:

$$\left[w\_{ij,\text{global}} = \frac{[\mathbf{s}\_{\,j}^{\text{healthy}} \cdot w\_{j,\text{entropy}} \cdot w\_{j,\text{CBITIC}} \cdot w\_{\text{ij,2nergy}}]}{\sum\_{j=1}^{m} [\mathbf{s}\_{j}^{\text{healthy}} \cdot w\_{j,\text{entropy}} \cdot w\_{j,\text{CBITIC}} \cdot w\_{\text{ij,2nergy}}]} \tag{34}$$

The denominators of the formulas for global attribute weights *j* = 1 . . . *m* at alternative levels *i* = 1 . . . *n* are, therefore, calculated as the sum of the products of entropy weights *wj,entropy* multiplied by *s<sup>j</sup> health* and *wj, CRITIC* and other weights, e.g.,*wj,energy* representing the importance of the attribute for the energy load.

It should be noted that the use of the P IAQ model (for calculating the total energy expenditure for ventilation and using the global weight value in the IEQ inEquation (7)) requires to determine the global weight values only for the attributes of "pollution concentrations" (in the case of the IAQquality model, only for *j* = 1, 2, 3) in the form of relative global weights adjusted at all levels. The measures of the significance of the other attributes (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 entropic weights are calculated just like CRITIC weights for all attributes, taking advantage of the fact that they are weights characteristic for given attributes (they are weights calculated in columns *j*, not relative weights), they can be "selected" and adjusted in the range of attributes to calculate relative global weights for all levels of alternatives.

#### *2.10. Characteristics of the Case Study Building*

In order to determine the weights of the IAQ model, the authors present a casestudy building and use the real scenarios inindoor environment quality. Thecasestudy building is a newly built high tower office with 49 floors and a net area of 59,000 m<sup>2</sup> made of steel and concrete materials with a heavy glass facade. The buildingis located in Warsaw. The experimental part (measurement in situ) included analysis of CO2, HCHO and VOCs indoor concentrationsin the building interiors. Provided in [5], datasets on the pollutants supported the authors' decision matrix development including scenarios/alternatives. At the time of the test, the building was in the pre-occupancy phaseandhadempty spaces with no furniture inside. The indoor walls were plaster covered and freshly painted. Suspended ceilings were mounted, and thefloors were coveredwith a synthetic carpet. The building installationswere active during tests. The building was tested threedays after the internal

work was finished. Measurements were made on the 47th floor, covering area of almost 3000 m<sup>2</sup> . The main focus was on the IAQindex for selected open spaces in which alargenumber of usershave to reside, and which represent the largest occupied floor space. ISO and CENinternational standard based analytical methods were engaged to assess"in situ"HCHO, TVOC and CO<sup>2</sup> concentration values. Theair samples for VOCs tests were collected viaactivesampling with anelectronic mass flow controller that controlled the air flow via test probes (10 dm<sup>3</sup> /h for VOCs and 30 dm<sup>3</sup> /h for HCHO). Indoor air samples were taken from selected representative office locationsand tested later in an accredited laboratory in accordance with ISO 16000-3:2011 and ISO 16000-6:2011. VOCs were sampled on the probe tubes filled with Tenax adsorbent. Later, they were thermally desorbed using a TD-20 device (Shimadzu, Kyoto, Japan). The process of separation and analysis of VOCs was achieved with aGC/MS gas chromatograph equipped with a mass spectrometer (model GCMS-QP2010, Shimadzu, Japan). VOCs were identified by comparing the retention times of chromatographic peaks with the retention times of reference compounds, by searching the NIST mass spectral database (USA). Identified VOC compounds were quantified using a relative identification factor from standard solution calibration curves. TVOC concentration was calculated by summing identified and unidentified VOCs eluting between n-hexadecane and n-hexane.

#### *2.11. Building-Related Research Limitations*

Building-related research limitations were also recognized because the ΣIAQ model and provided weighting scheme applied mainly to office buildings with mechanical ventilation. Quasi-static indoor environment conditions and full air mixing in the building wereassumed by the authors (perfect mix of indoor air). The IAQ model is based on sensory curves that were created by examining people's response to a stimulus with analmost neutral influence of other indoor environment factors. This means that the applicability of the presented IAQ model is limited to indoor thermal conditions in the range of 20–26 ◦C and relative humidity of 30–70%. The valid clothing level of occupants was 0.5–0.8 clo, and metabolic rate accepted was 1–1.2 m. The number of air pollutants as sub-components in the IAQ model was limited to the sub-models presented in [4]. In the example case study building, we analysed building pollution situations based on sixattributes from which we built alternative situations (a1–a5) for the building. These attributes are mostcommonly used in commercial environmental building assessment systems like BREEAM or LEED. The range of applicability of the developed weighing system wasvalid up to the maximum pollution values: 1800 mg/m<sup>3</sup> for ∆CO2, 1000 µg/m<sup>3</sup> for TVOC and 160 µg/m<sup>3</sup> for HCHO.

Health risk in the context of human exposure to a given pollutant is an issue usually analysed inoccupational medicine research. In this context, we consciously limited our research scope and didnot focus directly on human health risk.

## **3. Development of** Σ**IAQquality Model Weighting Schemes for the Case Study Building**

#### *3.1. Development of the Initial Decision Matrix Adapted to the Combined* P *IAQquality Model*

This section providesastep-by-step approach for a numerical calculation of combined P IAQquality model sub-element weights presented in the example case study building.

The results of the casestudy building identified pollutants [5] and were the basis inbuilding a decision matrix. Table 5 represents the initial decision matrix (the input data table) for the defined scenario (fitting of the weighting schemes to various IAQquality quasi-stable states). The matrix consisted of six attributes (concentrations and emissions of three indoor air pollutions) and five alternatives, where:

a1—alternative based on the case study specifics presented in a previous report in [4,5]. In the building under investigation, with quasi-stable air quality, for the indoor pollutant concentrations of CO2, TVOC and HCHO, the assumption was made that the additional emissions of these three air pollutants were not measurable;

a2—alternative based on the assumption of minimum air pollution concentration levels changing very little because the emissions rate atthe minimum level does not affect the quasi-stable state of IAQ; for the CO<sup>2</sup> bio-pollution rate and the CO<sup>2</sup> concentration minimum level value, the bio-effluents rate from a single person during one hour was accepted [42];

a3—alternative predicting the measured air pollution quasi-stable concentration at aminimum level, but under the influence of CO<sup>2</sup> (e), TVOC (e) and HCHO (e) emission rates, it did so at a higher level (addition under minimum level waswithin the 0.25 measuring range); the TVOC or HCHO minimum levels of emission rates were calculated as 0.25 of the (max–min) range estimated by the literature on thepollution emission rate;

a4—alternative is similar to a3, but the higher influence of the emission rate of CO<sup>2</sup> (e) does not exist (there is no addition under the minimum level of CO<sup>2</sup> emission rate);

a5—alternative is similar to a3, but the higher influence of emission ratesof TVOC and HCHO does not exist (there is no addition under the minimum level of TVOC and HCHO emission rates).


**Table 5.** Initial decision matrix with the data assigned to various IAQquality component sets and action scenariospresented by individual alternatives.

(1) General conversion of units, e.g., CO<sup>2</sup> from ppm to mg/m<sup>3</sup> in accordance with [50]. (2) Assumptions for the attribute of CO<sup>2</sup> concentration—(1) minimum 1 person max. 4 people; (2) in a room of 20 m<sup>3</sup> per person; (3) the CO<sup>2</sup> concentration caused by occupant exhalation is approximately 400–1700 ppmv,while the average volume is 20–150 m<sup>3</sup> ·person−<sup>1</sup> and the air change ACH is 0.45–0.70 h−<sup>1</sup> [42]. The time for indoor concentration of CO<sup>2</sup> to reach a quasi-steady state is between 6 and 8 happroximately; (4) the mass balance method [42] can be used to link indoor CO<sup>2</sup> emission rate concentration, as shown in equation m = p·c·Q, where m is the emission rate of the target compound, µgs−<sup>1</sup> , p is occupant number, c is the CO<sup>2</sup> concentration in occupant exhaled air (400 ppmv [42]), µg m−<sup>3</sup> , and Q is the respiratory rate of humans, m<sup>3</sup> s −1 , and is assumed as7.5l min−<sup>1</sup> (1.25 <sup>×</sup> <sup>10</sup>−<sup>4</sup> <sup>m</sup><sup>3</sup> s −1 ). (3) TVOC—Hori [38] study ACH is ca 0.5 h−<sup>1</sup> . (4) HCHO—[35] ACH is set to 0.5 h−<sup>1</sup> .

The set of alternatives is used for checking how the weighting scheme changes when the emission rate of a particular air pollutant is absent or increasing.

Simultaneously, the authors calculated all weights for the additional alternative version a2-2, which differs from alternative a2. This is based on the assumption of air pollution concentration levels, taken from the case study, practically not changing the quasi-stable state because the emissions rate is at a minimum level, like in alternative a2. A minimum level for the CO<sup>2</sup> bio-pollution rate, and TVOC (e) and HCHO (e) minimum level emission rates were established. It can, therefore, be argued that the weighting scheme for the IEQ model with the a2-2 alternative will have a screen emission (more probable than for the a2 alternative). The decision model and its weighting schemes with alternative a2-2 and the same set of attributes havedifferent structures than alternative a2. However, as the values of indoor pollution concentrations are the same in this study (with the assumption of minimum levels of emissions alone), version a2-2 can be used for comparingthe impact of the two alternatives a2 and a2-2. Maximum xij levels of ventilation load are based on the research [2,7,32,35,38,42] and Minimum xij levels by [2,35,38,39,42].

#### *3.2. Attribute Weights Obtained by the Entropy Method*

Determination of objective attribute weights (only for attributes 1, 2 and 3, named "pollution concentration" sub-indices in IAQquality) from Equation (4), calculated in attribute columns by all levels of alternatives belonging to the decision matrix in accordance to the entropy approach, is based onuncertain information measurements contained in the decision matrix and directly provides a set of weights for criteriabased on mutual individual criteria values contrastedwithvariants for each criterionalone and then for all criteria at the same moment. The following steps are suggested in this direction.

Step 1. Normalisation of indicators for the entropy method—since the indicators of air pollutants are in different units, and since the attributes are of a non-beneficial type, the most recommended method to normalise them is used first. In particular, the transformation method according to Equation (34) solves this issue:

$$z\_{j} = \frac{y\_{ij}}{\sum\_{i=1}^{n} y\_{ij}} = \frac{1/\chi\_{ij}}{\sum\_{i=1}^{n} 1/\chi\_{ij}}\tag{35}$$

Table A1 (in Appendix A) presents the normalised decision matrix (according to Equation (35)) with dimensionless numbers *z<sup>j</sup>* representing the normalised response of alternative *i* on attribute *j*. The structure of the decision matrix contains the input parameters of atribute *j* = 1, 2, 3 determined on the assumption of an indoor quasi-stable state environment with minimum level of air pollution concentration and minimum emission rates.

Table 6a presents the shortage of normalised decision matrices (according to Equation (35)). The structure of alternative a2-2 (presented in Table 6b) contains the input parameters of atribute *j* = 1, 2, 3 based on the current experiment [5] with the assumption of an indoor quasi-stable state environment with the experimentaly determined level of air pollution concentration and with minimum emission rates.

Step 2. The entropy measure calculation. The measure of the *j*th attribute under indicator *i* (*i* ranges from 1 to n) is calculated using the following Equation (36):

$$z\_j = -k \cdot \sum\_{i=1}^n z\_{ij} \ln z\_{ij} = -\frac{1}{\ln n} \Sigma\_{i=1}^n z\_{ij} \ln z\_{ij} \tag{36}$$

where constant *k* = 1/ln *n* guarantes that *e<sup>j</sup>* (*j* = 1, 2 . . . , *n* ) belongs to the interval [0;1]. A constant *k* in the matrix with five alternatives can be calculated by Equation (37):

$$k = \frac{1}{\ln(n)} = \frac{1}{\ln(5)} = 0.621335\tag{37}$$


**Table 6.** Normalised decision matrices with dimensionless numbers *z<sup>j</sup>* for (**a**) alternative a2 based on the minimum level of air pollution in a quasi-stable state with minimum emission rates for (**b**) alternative a2-2 based on air pollution values from the case study in a quasi-stable state with emissions rates at a minimum.

Therefore, it is possible to calculate the value of *k* for all attributes *j* = 1 . . . 6 and alternatives *n* = 5. With the help of Table 6a,b, it was possible to calculate *ej*, *d<sup>j</sup>* and entropy weights of criteria values *wj,entropy*, which are presented for Variant 1with alternative a2 and Variant 2 with a2-2 in Table 7. The degree of divergence (*d<sup>j</sup>* ), where *d<sup>j</sup>* (*j* = 1, 2 . . . , *m*), is the inherent intensity of criteria contrast *Cj* . The value (*d<sup>j</sup>* ) of the average intrinsic information contained in each criterion is calculated as Equation (38):

$$\mathbf{d}\_{\mathbf{j}} = \mathbf{1} - \mathbf{e}\_{\mathbf{j}} \tag{38}$$

Step 3. Calulation of the entropy weights of attributes

Next, the entropy weights of each *j* are calculated as follows. The final criteria weight, in the third step of the approach, may be calculated by an additive normalisation (39):

$$\mathbf{w}\_{\text{jentropy}} = (\mathbf{d}\_{\text{j}}) / \sum \mathbf{(d}\_{\text{j}}) \tag{39}$$

Table 7 presents the results of calculations using Equations (38) and (39).

**Table 7.** Entropy, degree of divergence and the relative weights of attribute *j*.


#### *3.3. Attribute Weights by the CRITIC Method*

Step 1. Normalisation of indicators for the CRITIC method

For each criterionj and indicator *xij*, membership function *rij*, which transforms the values of attributes into the interval [0,1], is provided(40):

$$r\_{ij} = \frac{\mathbf{x}\_{ij} - \text{min } \mathbf{x}\_{ij}}{\text{max } \mathbf{x}\_{ij} - \text{min } \mathbf{x}\_{ij}} \tag{40}$$

and the transformer version of formulas is [30,31]:

$$r\_{ij} = \frac{\mathbf{x}\_{ij} - \mathbf{max} \,\mathbf{x}\_{ij}}{\min \mathbf{x}\_{ij} - \max \mathbf{x}\_{ij}} \tag{41}$$

The transformation considers the concept of an ideal point. With Equation (41), the initial matrix is converted into the normalised decision matrix with the generic elements *rij* (Table 8a,b).

**Table 8.** Normalised decision matrices. (**a**) (with alternative a2) based on minimum air pollution levels in a quasi-stable state with minimum emission rates; with zero-dimensional generic elements *rij*. (**b**) (with alternative a2-2) based on air pollution test study levels in a quasi-stable state with minimum emission ratesand zero-dimensional generic elements *rij*.


Table A2 (located in Appendix A) presents the normalised decision matrix Equation (35) with dimensionless indicators *rij* representing the normalised response of alternative *i* on attribute *j*.

Step 2. Calculation of symmetric linear correlation matrix *Rjk* based on all indicators of normalised decision matrices (Table 9) and shortages of symmetric linear correlation matrices *Rjk* with alternatives a2 (Table 9a) and a2-2 (Table 9b).

Step 3. Determining criteria weights for "pollutant concentration" attributes using the CRITIC method.

The amount of information necessary to calculate *C<sup>j</sup>* (42) includes measures of conflict *C<sup>j</sup>* determined with Equations (30) or (42). Attribute (criteria) weights *wj*,CRITIC were obtained from Equation (43) (Table 10).

**Table 9.** Symmetric linear correlation matrix *Rij* (with alternative a2) calculated for all correlation possibilities. (**a**) Shortage of symmetric linear correlation matrix *Rij* (with alternative a2) calculated for all correlation possibilities (in this shape of the matrix, it can be assumed that only six attributes are burdened with additional correlation weights, but in the recognised attribute set only the first three will be used). (**b**) Shortage of symmetric linear correlation matrix *Rij* (with alternative a2-2) calculated for all correlation possibilities.


**Table 10.** Results of the CRITIC method application for decision matrix with alternatives a2 and a2-2.


Aconflict measure created by criterion *j* due to the decision defined by the rest of the criteria can be calcualted, which determinesa quantity of the information in relation to each criterion:

$$\mathbf{C}\_{j} = \sigma\_{j} \sum\_{j=1}^{m} \left( \mathbf{1} - \mathbf{R}\_{jk} \right) \tag{42}$$

Objective CRITIC criteria weights for a chosen attribute are obtained by normalising (adjusting) the values of *C<sup>j</sup>* and, in particular, alternative levels (43):

$$\left| w\_{\text{j,CRITIC}} \right\rangle = \mathbb{C}\_{\text{j}} / \sum\_{k=1}^{m} \mathbb{C}\_{k} \quad j = 1, \ldots, m \tag{43}$$

#### *3.4. Relative Weights Obtained by Normalisation of the "Excess Concentration"*

Taking into account the input data of the decision model compiled in the input decision matrix with alternatives (Table 5), alternatives a1 to a5 (with values of excess concentrations ∆c<sup>j</sup> according to (2) [4]) were calculated with weights *wij*,energy for non-zero attributes *j* = 1, 2, 3. In order to determine the "energy load" weight, the following steps are recommended.

Step 1. Assuming that the minimum concentration value cref in the definition of "excess concentration" given in Equation (2) may be the minimum value min (*xij*), then the formula for normalisation (43) can be determined, based on the data for the alternative a1 obtained in real case studies [5] and data from the literature [2,51,52] regarding the minimum values of pollutants (TVOC and HCHO) occurring in a building. Justification for the manner of assuming the minimum and maximum values of pollution in a building with CO<sup>2</sup> bio-pollution is discussed in the notes of Table 5.

Step 2. Normalisation of weights (energy load) *wij*,energy using Equation (9) transformed to (44) for attributes of type "cost". According to [29,31], it seems that the equation can be used to normalise non-beneficial datasets, which after conversion to profit (or beneficial) attributes can be adjusted for IAQ systems corresponding to subsequent alternatives from a1 to a5. Transformation of the excess concentration values, which correspond to the indicators *xij* at the levels of subsequent alternatives a1 to a5 taken from Table 5, gives the weight of "energy load"*wij, energy*:

$$w\_{ij, energy} = \left[1 - \frac{\mathbf{x}\_{ij}^{\min}}{\mathbf{x}\_{ij}}\right]^{-1} \tag{44}$$

Normalisation of weights based on the concept of "excess concentration"— correct applicationof weights of Equation (44) is possible on the assumption that the min (*xij*) value of indoor pollution determined on the basis of insitu measurements and given in the input decision matrix (Table 5) corresponds to the standard value cref from Equations (2) and (3).

Step 3. Calculation of the relative values of "energy load" weights assigned to attributes of CO2, TVOC and HCHO pollutant indoor concentrations (Table 11).

**Table 11.** Normalisation of the energy load relative weights calculated with Equation (44).



**Table 11.** *Cont.*

#### **4. Results**

## *4.1. Global Weights Calculated for the IAQquality Model Alternatives*

The final step of the weight development methodin Section 3 is to aggregate the calculated set of "component" weights and obtain the resultant global weights. Table 12 shows the determination of global weights calculated according to Equation (33), when the attributes do not take part in the emission process (attribute type *j*),and Equation (34) when attributesdo take part in the emission process (attribute type *k*). Weight values adequate for the numerical example are taken for the set of alternatives a2and a2-2.

**Table 12.** Relative global weights in the weighting schemes calculated for five alternatives of the IAQ model (a1 to a5); global weights obtained with Equation (32).


4.1.1. The Results of the New Weighting Method Applied on IAQindex and IEQindex Results for theCaseStudy Building

A numerical example of the combined P IAQquality model with the new weighting scheme, developed according to Figure 3, is presented in Table 13 in the context of the overall Indoor Environmental Quality (IEQ) asessement of thecasestudy office building (IEQ is the predicted percentage of users satisfied with all components of indoor environmental quality). The input indoor parameters for the case study (in order to assess IEQ sub-components) are taken from previous papers [4,5], where the measured pollutant concentrations were *c*CO2 = 450 ppm, *c*TVOC = 787 µg/m<sup>3</sup> and *c*HCHO = 18 µg/m<sup>3</sup> in the case study building a week after completion of the finishing works.

**Table 13.** The physical parameters representing the indoor environment of the case study and the IEQindex results calculated from Equation (7) with sub-component P IAQ calculations (old approach and new); assuming realistic uncertainty of measurements for a case study (old and new weight systems applied and compared).


The uncertainty of user dissatisfaction with pollutant concentration is determined by a combination of two groups of components. The first measurement uncertainty is related to the real uncertainty of parameters biases when there are measured in indoor environments. The second group concerns the uncertainty of votes in the sensory tests (reliability of responses to a given stimulus of the indoor environment are discussed in literature). In practice, a metrological analysis of the IEQ model (where P IAQ index is the sub-model) was provided in our previous paper [8] considering provisions of the

international metrological guideline: ISO BIPM JCGM 100:2009 Guide to the expression of uncertainty in measurement. For all other possible scenarios, the uncertainty for determining the P IAQ index would be up to ±10% to ±30% (depending on each pollution model uncertainty assessment and the number of IAQ model components taken into account).

#### **5. Discussion**
