*1.1. Research Problem*

In literature on the Multi-Attribute Decision Making (MADM )method, the combined Indoor Air Quality (P IAQ) model optimisation problem has not been addressed thus far to set up the weights of model sub-components (in practice due to excess indoor pollutant concentrations and risk to occupants). One of the main criteria for optimising such a decision problem should be, directly or indirectly, the cost of ventilation for the elimination of excess pollution concentrations. For the purposes of the decision problem, it is assumed that the general criterion for optimising such a weighing system is to include a range of IAQ indicators and components, i.e., types of contaminants that are most important due to health and toxicity or the amount of air to be removed by the ventilation system.

The MADM method, as a specific kind of Multi-Criteria Decision Making (MCDM) approach, has emerged as a formal methodology for assessing available information and providing values supporting decision-makers in many areas, including environmental engineering. The published results suggest that, over the past decade, there has been a significant increase in the use of MCDM in technical areas for civil engineering applications, the design of indoor environments and the improvement of outdoor environments [1]. MCDM makes it possible to assess complicated decision problems limited by numerous and divergent criteria based on expert subjective assessments (subjective weights) and decision models, the purposes of which are to calculate objective weights according to the rules of decision theory and order multiple parameters of the analysed problem. Recently, the construction industry has become one of the main stressors to the environment and society globally [2]. In the context of the continuous decrease in energy consumption by new buildings and the fundamental need to provide comfortable and healthy conditions for the users in indoor environments, there is a growing urgency to use IAQ models that meet certain criteria. IAQ models are the main components of building Indoor Environmental Quality models [3] and increasingly used for building design software (such as Building Information Modelling, BIM) or as part of building managing systems (BMSs) controlling heat, cooling and ventilation systems (HVACs).

The basic aim of this study was to develop a new approach which combines the previously developed ΣIAQ model [4], containing loosely connected components (indoor air pollution elements and their impact on user perception), with a new weighting system based on MADM that would produce a practical tool supporting choices of the best possible indoor environment and air control strategy. As a case study, a large office building was used, for which the authors had already assessed the indoor environmental parameters [4,5], and the authors recalculated the ΣIAQ and Indoor Environmental Quality (IEQ) indices using the new weighting scheme. A similar issue for the external environment was presented in [6]. This team built the MCDM decision model to study the impacts of outdoor air pollution on the economic development of selected zones in China.

How humans perceive indoor comfort is a complex phenomenon. Authors have analysed the problem from a holistic point of view following current developments described widely [3–5]. Researchers assessing indoor comfort in buildings have focused primarily on optimizing individual components of the indoor environment (thermal, light and acoustic comfort, and air quality), as these four parameters connected to the human senses are considered by occupants to be the most important in determining their comfort [3]. Our research work is aimed at identifying the relationship between the indoor parameters and the resulting reactions of building occupants (in particular, in the area of IAQ). This multi-component approach to indoor comfort focuses on the 'human stimuli response' and has led to well-known indoor comfort models such as the thermal comfort model developed by Fanger. In recent years, this holistic approach to the indoor environment has been presented in the European standard EN 15251 and later in EN 16798-1: 2018 [7] where, for the first time, a wider number of indoor comfort criteria were considered. These standards suggest an approach to the classification of IAQ and IEQ models and support certification of buildings taking into account specific components of indoor environment, however, they do not provide a practical guidance or an approach on how to combine them into one indicator that could be useful to classify indoor environmental conditions. The authors solved this problem in a previous publication [4]. When the predominant indoor pollutants have different characteristic values, there is a large problem regarding the choice and weighting method of IAQ sub-components. The problem is considered in this article with use of the MADM approach. Taking into consideration the characteristics and correlations of the selected pollutants, IAQ may be characterized by representative indicators. Our studies on Building Research Establishment Environmental Assessment Method (BREEAM)certified buildings [5] and Leadership in Energy and Environmental Design (LEED) system shown out that carbon dioxide, total volatile organic compounds (TVOCs) and HCHO are the main worldwide, independent and representative environmental indicators. They are being used worldwide as an evaluation index of IAQ in buildings (mainly offices with mechanical ventilation). Since each of these indicators

represents a pollutant class with comparable indoor sources and characteristic dissemination and the indexed group avoids unreliable measurements. This is based on the fact that indicators are "too small" because of decreased critical concentrations. A method of data pre-treatment is proposed by us in the procedures, which reflects concentration differences between pollutants and considerstheir influence on the satisfaction of the building users. Taking into consideration the existing knowledge gap, the authors proposed an objective method to determine weights in theIAQ model in this article. The article provides the procedure to calculate global weights for the IAQ model sub-components. The ΣIAQ model considers the impact of air concentrations of selected air pollutants (CO2, TVOCs, HCHO, etc.) on the building users' perceptions. The weighted IAQ model was developed by the authors considering the standard EN 16798-1:2019 [7] provisions with the intention to create it as a main sub-component of the overall Indoor Environmental Quality (IEQ) model [8,9] with assumptions described in references [3,8]. The IAQ model and weight system applies to office buildings with mechanical ventilation. Quasi-static indoor environment conditions and full air mixing in the building are assumed.

#### *1.2. Basic Information on the MADM Approach*

Some analogies to the presented IAQ weighting problem can be found in the literature, and the studies described so far have led to the establishment of rankings of alternative solutions to the decision problem. The MADM method has beenused as an effective tool for choosing the optimal composition of construction materials [10,11], estimating their weights [12] for the selection of the optimal air filtration method to remove pollutants [13], and choosing the optimal set of indoor environment parameters in an office building [14] when the alternatives are differently dated combinations of these parameters (attributes). Additionally, the MADM technique solved the problem of the impact of air pollution on the level of economic development of Chinese cities (for example, the city of Wuhan) [6]. This project is interesting to the authors because it concerns the impact of air pollution. The research involved determining the weights of air pollutants of various types of VOCs, and of the physical parameters affecting these pollutants (attributes of the decision problem), and establishing a ranking of the specific systems of these environmental parameters occurring in different years (alternatives to the decision problem). In these simulations, a modified Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) was used. This approach uses neural networks to determine weights.

MCDM was also proposed inchoosing a strategy to design sustainable buildings [15]. The purpose of this study was to provide a method combiningclimate changeeffects, adaptive thermal satisfaction, life cycle assessment (LCA), life cycle cost (LCC) analysis and multi-criteria decision making in order to set the most valuable design strategy that would improve building object sustainability. For the assessment and selection of the most appropriate design strategies for buildings, attributes were analysed using a multi-disciplinary approach based on sustainability criteria. In order to find the best building design strategy and make decisions based on several criteria, the Complex Proportional Assessment (COPRAS) method was used [16,17]. This method [15] assumes proportional and direct dependence of the significance of the usability degree of the tested versions of buildings (alternatives) in the system of criteria (attributes) described by the parameters: (1) hours of comfort in the room, (2) primary energy demand, (3) CO<sup>2</sup> emissions, and (4) costs). This method was successfully used to deal with the complex selection of problems. Based on this study, by using the COPRAS method, the authors provided a ranking of the most valuable structural solutions for buildings that meet four characteristic parameters/criteria and their weights built within the decision model.

Most often, MCDM decision models have been used by researchers in the process of selecting components of construction materials by methods such as the "preference technique in classification according to the similarity to the ideal solution" (TOPSIS) [18,19], which uses the principles of analytical hierarchy process (AHP) [17] and various versions of Elimination and Choice Expressing the Reality (ELECTRE) [1]. The method using entropic weights, created on the basis of Shannon's information theory [20] for the selection of materials [10,11], was also used. Additionally, the method of material selection from many alternatives based on Jahan's research [12] and the analysis of correlation between Criteria Importance Through Intercriteria Correlation (CRITIC) criteria [12] was used. Alemi-Ardakani [10] analysed the possibilities of optimising the composition of composite materials using the Adjustable Mean Bars (AMB), Modified Digital Logic (MDL), Numeric Logic (NL) and CRITIC [10] methods, as well as the Peng method [21] of creating rankings of preference functions using composition enrichment technology (PROMETHEE—Preference Ranking Organization Method for Enrichment Evaluations) or a preference selection indicator [1].

MCDM methods have now become so popular for extending knowledge of environmental impacts and sustainable construction that several groups of scientists reviewed the applications and development trends of decision models in "environmental sciences" (Cegan, Linkov et al., 2011 [22] and 2017 [23]) and in sustainable construction (Navarro et al. 2019 [1]).MADM methods are a special case of MCDM methods in which decision-making criteria have been replaced by attributes, i.e., features of problems or processes that are evaluated by analysts. Each MADM technique has its own specifics, assumptions and principles. Almost all of the above-mentioned MCDM methods are characterised by the complexity of the mathematics of their moderate to extreme models. Application of these techniques is difficult, requiring advanced mathematical skills and knowledge. Therefore, the creation of undemanding MADM methods is highly desirable. Multi-purpose optimisation based on proportion analysis, named Multi-Objective Optimization by Ratio Analysis(MOORA)was proposed by Brauers and Zavadskas (2006) [24] and is used for the selection of construction materials [25], as it involves uncomplicated mathematics. Therefore, it can be used effortlessly and effectively to choose the best materials or solutions.

To solve the weight problem in the IAQ model, the authors used two methods: entropy and CRITIC. The objective weighting measure was proposed with the Shannon entropy concept provided in [20]. In MCDM, the entropy relates to the diversity degree within the attribute data set. The higher this diversity degree, the higher the attribute weight. The smaller the entropy within the associated data to the attribute, the higher the power of discrimination of the attribute in changing ranks of alternatives. The entropy weights are calculated with only one set of values dedicated to the same attribute j in all alternative levels of the decision matrix. Entropy weight is a typical attribute weight, but it is always introduced to the global weight as a factor in all decision matrix indices. In the entropy method, an attribute with a performance rating that is very different from the others has more importance in the problem because it has a greater impact on ranking outcomes. The attribute has lesser importance if all attributes have comparable performance ratings for that attribute. An objective method to weigh criteria importance through inter-criteria correlation (CRITIC) based on the SD approach was proposed by Diakoulaki and later explored by Jahan [12]. The higher the level of interdependency between attributes, the larger the ranking outcome error. The CRITIC approach is a popular objective method to calculate weights, which solves the problem of interdependencies between the attributes in all combinations by considering the correlation between the sets of variables from various alternative levels, while calculating the weights. It is one of the objective methods of weight determination that belongs to the class of correlation methods. It uses the information contained in the data matrix in the form of the degree of deviation of a variant value from a given mean value of the criteria. The CRITIC method is particularly useful in the IAQ model, as we show later, because the parameters adopted as attributes in the decision model of this problem are the concentrations of selected pollutants and the emissions of the same pollutants; therefore, in the IAQ decision matrix there are correlating attributes.

#### *1.3. Basic Information on the* P *IAQ Model*

The ΣIAQ model, in which the authors are looking for objective weight sub-elements, uses a selected number of indoor air pollutants (P1, . . . j) and their impact on user dissatisfaction (Percentage Dissatisfied as PD = f(c<sup>j</sup> ) in %). The combined P IAQindex (cumulative percentage of satisfied users with indoor air quality, including selected pollutant impacts of user perception) Equation is

$$\sum \text{IAQ}\_{\text{index}} = \text{WP}\_{\text{I}} \cdot \text{IAQ}(\text{P}\_{\text{I}})\_{\text{index}} + \text{WP}\_{\text{I}} \cdot \text{IAQ}(\text{P}\_{\text{2}})\_{\text{index}} \times \dots \times \text{WP}\_{\text{I}} \cdot \text{IAQ}(\text{P}\_{\text{j}})\_{\text{index}} \tag{1}$$

where sub-indices P IAQ(P<sup>j</sup> ) are the percentage of users satisfied with the pollutant concentration; *WP1,* . . . *P<sup>j</sup>* are weights for each IAQ sub-component for groups of air pollutants with comparable concentrations.

There is a difference in ∆*c<sup>j</sup>* concentration between the measured air concentration of pollutant *c<sup>j</sup>* and the recommended "reference" concentration *cref* (reference by the European Commission "safe" values such as cLCI or cELV), which may be below the actual air concentration in the contaminated rooms. Thus, the concentration excess related to the ventilation rate is

$$
\Delta c\_{\circ} = c\_{\circ} - c\_{ref} \tag{2}
$$

The excess concentration weights WP<sup>j</sup> for the IAQ model until now have been calculated based on the arithmetic mean or by adjusting (normalization) all attribute ∆*c<sup>j</sup>* values using Equation (3):

$$\mathcal{W}\_{\mathcal{P},j} = \frac{\Delta \mathcal{E}\_j}{\sum\_{j \text{1\dots7}} \Delta \mathcal{E}\_j \mathcal{I}} \tag{3}$$

where the sum of the adjusted weights *WP,j* of all pollutants to be removed by ventilation should be in unity. This weighting scheme has been used by the authors in previous years, but in this article, we propose a better and more objective approach based on the MADM approach.

Three levels of complexity of the IAQ model were proposed, and in each, pollutants have been defined that were included in the IAQ model. Figure 1 shows the extended IAQquality model with its sub-indices and also with the sub-components of the IAQcomfort/health model type, i.e., indoor air pollutants important to health with an impact on the energy balance of a building using a mechanical ventilation system.

**Figure 1.** Combined Indoor Air Quality (P IAQ) model with weighting scheme; horizontally, the model is bound by the weighing processes of individual sub-models; vertically, the model is bound by a chain of input components (pollutant concentrations in ventilated indoor air) and a chain of output components (elements of the combined IAQindex equation).

The experimental dependencies of the percentage of persons dissatisfied (*%PD*, where *%PD* = *1-IAQ*) and the concentrations values of air pollutants, *c<sup>j</sup>* , perceived in the adequate ranges are of fundamental significance to the sub-components relevant to the IAQ and IEQ models [9]. In the ΣIAQ model proposed by the authors, the accepted common approach is engaged to transform individual pollutant concentrations into sub-components before these are combined into a single index. The aggregated summary of sub-components, however, may lead to a situation in which all components are below the individual health threshold, but the final index shows that the health threshold is exceeded. Conversely, the averaging of partial sub-components may lead to an outcome whereby the overall index shows an acceptable IAQ value, but one or more partial indicators are higher than their individual health threshold. The solution proposed is to use all sub-component maximum values to provide a final form of the P IAQ. Considering this, we proposed the P IAQ model with three possible levels of model complication due to the application potential, as we show in Figure 2:


**Figure 2.** The IAQ model with three complexity levels reflecting the application potential.

The simplest IAQ index level can be used with the main purpose of supporting a sustainable building assessment and certification by using only three sub-components: CO2, HCHO and TVOC. ΣIAQquality model is used in this article to assess the office building (case study).

From the dependencies, expressed as the curves for the percentage of satisfied users with pollutant IAQ(CO2) or IAQ(VOCodorous), the following Equations (4)–(6) of the models are provided:

$$\sum \text{IAQ}\_{\text{quality}} = \text{W}\_1 \cdot \text{IAQ(CO}\_2) + \text{W}\_2 \cdot \text{IAQ(TVOC)} + \text{W}\_3 \cdot \text{IAQ(HCHO)}\tag{4}$$

$$\begin{aligned} \tiny\begin{aligned} \Sigma \text{IAQ}\_{\text{confort}} &= \text{W}\_1 \cdot \text{IAQ}(\text{CO}\_2) + \text{W}\_2 \cdot \text{IAQ}(\text{TVOC}) + \text{W}\_3 \cdot \text{IAQ}(\text{HCHO}) + \\ &\quad \text{W}\_4 \cdot \text{IAQ}(\text{VOC}\_{\text{odorous}}) + \text{W}\_{4\text{a}} \cdot \text{IAQ}(\text{VOC}\_{\text{odorous}}) + \text{W}\_5 \cdot \text{IAQ}(\text{h}) \end{aligned} \tag{5}$$

$$\begin{aligned} \text{\tiny\tiny\text{IAQ}}\_{\text{confort}} & \text{[} \text{W}\_{\text{I}} \text{IAQ}(\text{CO}\_{2}) + \text{W}\_{\text{2}} \text{IAQ}(\text{TVOC}) + \text{W}\_{\text{3}} \text{IAQ}(\text{HCHO}) + \text{K}^{\*} \\ & \text{W}\_{\text{4}} \text{\tiny\text{IAQ}}(\text{VOC}\_{\text{odorous}}) + \text{W}\_{\text{4a}} \text{IAQ}(\text{VOC}\_{\text{odorous}}) + \text{W}\_{\text{5}} \text{IAQ}(\text{h}) + \\ & \text{W}\_{\text{6}} \text{\textgreater IAQ}(\text{PM2.5}, \text{PM10}) + \text{W}\_{\text{7}} \text{IAQ}(\text{VOC}\_{\text{non-odorous}}) + \\ & + \text{W}\_{\text{7a}} \text{IAQ}(\text{VOC}\_{\text{non-odorous}}) + \text{W}\_{\text{8}} \text{IAQ}(\text{CO}) + \text{W}\_{\text{9}} \text{IAQ}(\text{NO}\_{2}) + \text{W}\_{\text{10}} \text{IAQ}(\text{O}\_{2}) \end{aligned} (6)$$

The structure of P IAQcomfort/health is made of seven (or possibly more) components or IAQ sub-models, which are occupants satisfaction functions for the various types of indoor air pollutants: IAQ(TVOC), IAQ(HCHO),IAQ(CO2), IAQ(VOCodorous), IAQ(PM2.5, PM10), IAQ(enthalpy, h) and the selected IAQ(VOCnon-odorous). The IAQ(VOCnon-odorous) and IAQ(VOCodorous) models should be multiplied, depending on the number of dominant VOC substances; hence, the P IAQcomfort/health index may have more than 11 sub-components in practice.

Combined P IAQindex is an element of the IEQ index in Equation (7) [8], where the authors adopted a crude weighting system in which all elements are weighted identically (0.25 for weights *W*1–*W*4):

$$\text{IEQ}\_{\text{index}} = 0.25 \cdot \text{TC}\_{\text{index}} + 0.25 \cdot \Sigma \text{IAQ}\_{\text{index}} + 0.25 \cdot \text{ACc}\_{\text{index}} + 0.25 \cdot \text{I}\_{\text{index}} \tag{7}$$

where TCindex is the thermal comfort index, expressed as the number of users satisfied with an indoor thermal comfort (in %); ACcindex is the acoustic comfort index, i.e., the number of users satisfied with sound level (in %); and Lindex is Daylight Quality, i.e., the number of users satisfied with daylight in the building (in %).

#### **2. Materials and Methods**

#### *2.1. Research Procedure Diagram*

Figure 3 shows the necessary research phases to calculate the global weights of the combined P IAQ model with MADM and further usage of these weights to calculate the occupant satisfaction IAQindex and IEQindex for the case study building.

**Figure 3.** The research phases to calculate the global weights for the ΣIAQ model (based on the case study building).

#### *2.2. MADM Method Applied to the* Σ*IAQ Model*

The basis for applying the MADM method is a decision matrix, as shown in Table 1. The issue of a weighting scheme is a problem related to making decisions with several criteria, which can be analysed using the model [26]. For every type of P IAQ model with Equations (4), (5) or (6), the decision matrix should be created separately.


**Table 1.** Decision matrix scheme in the MADM model for the concordance of economic and human comfort criteria for each P IAQ scheme described by Equations (4)–(6) separately with alternatives (e.g., P IAQcomfort scheme configurations) and with attributes (air pollution components in quasi-stable states and emissions).

In Table 1, alternatives a1, a2 . . . a(n) for P IAQcomfort configurations are defined further (in Methods). Attributes X1, X<sup>2</sup> and X<sup>j</sup> are the indoor air pollution concentrations of CO2, HCHO, TVOC and VOCodorous in a quasi-stable state of the indoor environment. X5(e), X6(e) and X7(e) are indoor emission processes and their rates per emission surface of CO<sup>2</sup> (e), HCH(e), introduced by an thropogenicemissions or directly emitted from construction products. In the decision matrix of the IAQ weighting scheme according to Table 1, the flow chart presented in Figure 3 provides the necessary steps and, therefore, includes MADM elements:

(1) attributes in the number j = 1 . . . 3 . . . , which are indoor air pollution concentrations covered by the models P IAQquality, P IAQcomfort and P IAQcomfort / health described by Equations (4)–(6) and Figure 2; however, the number of attributes beyond j = 7 (e.g., for P IAQcomfort/health) is not limited because model components like VOCodorous or VOCnon-odorous and other pollutants can be multiplied.

(2) alternatives i = 1 . . . n, which are the P IAQ models described by Equations (4)–(6) and Figure 2 assigned to the quasi-stable state, or defined by emissions. The selected decision model alternatives correspond to three combined indoor air quality models in various attribute set configurations (i.e., changes in attribute sets, IAQ sub-models). Alternative models, i.e.,IAQcomfort and IAQcomfort/health models, have more attributes than the IAQquality model; moreover, the number of these attributes is variable in both the pollutant and emission groups. In the group of alternatives, IAQ model systems not disturbed by emissions, e.g., IAQquality, and models distorted by emission attributes, (e) for example, were considered. The number of alternatives (Table 1) cannot be too small because the standard deviations for the set of alternatives assigned to a given attribute are calculated (i.e., for the attribute column in the decision matrix).

The assumptions in the decision model are adequate for the task of measuring the indoor air ventilation rate to obtain the assumed air quality, setting IAQ model alternatives so they meet the criteria and taking into account their importance ranking and possibility for ongoing diagnosis of alternatives to the assumed overall IEQ assessment. The model should provide a set of attributes that are important for maintaining hygienic and efficient ventilation, both of which are connected to variable air pollutants. This is important because the decisions relate to the scope of complex indoor air quality models, while planning the ventilation system depends on the amount of "excess air pollution" including VOC and SVOC compounds.

In the Materials and Methods and Results sections, an example of the implemented IAQquality model is provided, as well as a shortened Table 5 decision matrix with a set of attributes of type (e) (emission), which includes additional sources of bio-emissions (CO<sup>2</sup> emissions), emission sources from additional equipment and finishing products (HCHO emissions), and emissions from building materials (TVOC). The choice of the attributes was facilitated by the fact that the effects of air pollution, in another open environment and system of conditions and tasks, on the economy were provided by Wang et al., 2017 [6]. This team built a Smart MCDM decision model to study the impact of outdoor air pollution on the economic development of selected zones in China. In addition, looking at the goal of the present paper to rank the importance of selection parameters for human comfort, insights could be

derived from already established research procedures for the selection of construction materials [12] or waste disposal (Savic [27]). Therefore, the attributes of the decision matrix (Tables 1 and 2) include the most typical and expected air pollutants belonging to successively developed ΣIAQ systems. The selection of attributes, i.e., types of pollution, consistent with Equations (4)–(6) also corresponds to the choice of Air Quality Index (AQI) [28].

#### *2.3. Flow Chart of Weighting Scheme Determination*

In the analysed IAQ model, the authors considered excess concentrations that can be measured in a quasi-steady state, and emissions from additional and sometimes temporary sources that periodically increase in the ΣIAQ system of excess concentrations. The number of pollution type attributes is as strictly defined as the ΣIAQ specified composition. The number of emission type attributes is variable, adapted to the selected set of decision pollutants, and does not usually exceed the number of pollution type attributes.

**Figure 4.** Chart for subjective and objective weight determinations for the combined P IAQ model.

The process of determining the weight for the IAQindex model, seen in a number of variants of the composition of its sub-indexes, i.e., in various alternatives, involves weighting scheme calculations adequate to the level of a particular alternative. Each weighting scheme contains relative values of global weights. These weights represent the energy expenditure of the ventilation system on the elimination of excess concentration of a given pollutant (excess concentration (2) [4]) in ventilated air. The process for determining the weighting scheme for the ΣIAQ model is shown in the flow chart in Figure 4. It should be noted that the authors considered it appropriate, after specifying the

technical conditions and physics of the building for which the decision model is used, to add the costs of ventilating the anticipated excess pollutant concentrations. Based on the literature of this multi-parameter problem, however, the authors found it advisable not to insert this attribute at the current level of the generalized decision model for selecting the ΣIAQ weighting scheme.

The idea of the correlation effect on weight *wj,CRITIC* stems from the fact that when the criteria correlation with other attributes is significant, it should have lower importance due to the roles of other criteria. An increase in the value of the concentration of a given indoor air pollutant, with a sudden (dramatic) increase in emissions, is accompanied by an increase in global weight because it must include an additional component, which is the correlation weight.

#### *2.4. General Principles of Estimating IAQ Model Weighting Schemes*

In this section, the authors provide a method with a step-by-step procedure for setting priorities in the group of dominant indoor air pollutants, considering technical, economic and health aspects, by global weights for each component of the IAQ model. Developed for the ΣIAQquality model, the MADM decision matrix includes six attributes: carbon dioxide, total VOC (TVOC) and formaldehyde concentrations (HCHO) in the indoor environment as well as additional carbon dioxide bio-emissions and total VOCs and formaldehyde emissions. All six attributes are associated with cost criteria, and in future the decision model can be developed along this trend. The MADM decision matrix of the IAQquality model includes five alternatives (combination of various IAQquality sub-components) for defining indoor air quality in a building. The informationcontained in every attribute is related to the contrast of each criterion. Standard deviation (SD) and entropy are the measures of intensity and ways to present objective criteria weights. In order to calculate the weights of criteria, the authors use objective methods: entropy and CRITIC, presented in Figures 3 and 4.

Adaptation of the IAQ model with the "new" weighting procedure based on the MDMA approach to our office building case study (see Results and Discussion) was performed mainly for illustrative purposes in the context of the presented MDMA method for IAQ model development. The authors did not focus in depth on any technical issues of the building. Other Indoor Environmental Quality (IEQ) sub-components indexes, such as thermal, acoustic and visual satisfaction (in %), are presented in a paper [5] in order to provide background to the actual research problem.

The authors present the order of steps and mathematical operations leading to the determination of weighting schemes of ΣIAQ models (4), (5) and (6) separately. All model decision matrix variants are covered inan overview (Table 2), which presents decision matrices for the three types of IAQ models: ΣIAQquality, ΣIAQcomfort and ΣIAQcomfort/health, covering various types of contaminants.


**Table 2.** Decision matrices for three P IAQ models with attribute data.

The first step for estimating the MADM framework is normalisation of data (henceforth, the decision matrix indicators), which belong to different types and have different excess concentrations of the most important air pollutants and VOC emissions affecting IAQ indoor environments. In an indoor quasi stable state, attribute parameters include the concentration of these pollutants and additional increases in their concentration when the level is disturbed by additional sources of their emission or bio-pollutant emissions (1). Normalisation of attributes in various units (values of pollutant concentration and emission rate) in large ranges (attribute) is performed to obtain dimensionless quantities. Attribute normalisation is possible in several ways; however, for the research problem discussed here, the most common methods are as follows:

(a) Linear normalisation of attributes belonging to the decision matrix carried out according to the pattern of elements in the normalised decision matrix, belonging to the set of "cost attributes" according to Equation (8):

$$n\_{i\bar{j}} = \min \left( \mathbf{x}\_{i\bar{j}} \right) \!/ \mathbf{x}\_{i\bar{j}} \tag{8}$$

where *j* ∈ N<sup>c</sup> and N<sup>c</sup> represents a set of criteria or attributes. The minimum value of the indicator *xij* of the decision matrix, i.e., the normalisation of attributes in the form of the levels of air pollutants or their emissions in the decision matrix of a complex IAQ model, can be more conveniently determined using Equation (9) provided by Körth [29]:

$$m\_{lj} = 1 - \frac{\varkappa\_{lj}}{\varkappa\_j^{\max}}\tag{9}$$

if air pollution is a negative value indicator.

(b) Normalisation of attributes according to Wang's scheme [6] already applied in relation to the description of air pollution. In addition, positive and negative indicator values represent different meanings (10) and (11). Air pollution is a negative value indicator (negative values), while profit from the absence of pollutants would be a positive indicator. Similarly, the values of the emission rate of individual pollutants are also negative indicators, which can be read from the emission curves, both max (*xij*) and min (*xij*) values. Therefore, in order to normalise the attributes, the authors chose negative values according to Equation (10). Positive values (type benefit) are

$$n\_{ij} = \frac{\mathbf{x}\_{ij} - \min(\mathbf{x}\_{ij})}{\max(\mathbf{x}\_{ij}) - \min(\mathbf{x}\_{ij})} \tag{10}$$

Negative values (type costs) are

$$m\_{ij} = \frac{\max(\mathbf{x}\_{ij}) - \mathbf{x}\_{ij}}{\max(\mathbf{x}\_{ij}) - \min(\mathbf{x}\_{ij})} \tag{11}$$

where *xij* represents the original value and *nij* represents the value after normalisation. However, this way of normalising the IAQ decision matrix in the actual system of its application, in which some configurations (alternatives) have some attributes values equal to 0, proved to be impracticable. The same group also uses the transformer version of Equation (10) provided by Zavadskas [30] and Vujcic [31]:

$$r\_{i\bar{j}} = \frac{\mathbf{x}\_{i\bar{j}} - \mathbf{max}\mathbf{x}\_{i\bar{j}}}{\mathbf{min}\mathbf{x}\_{i\bar{j}} - \mathbf{max}\mathbf{x}\_{i\bar{j}}} \tag{12}$$

The stage of data normalisation of the decision matrix is carried out after it is built, and at the same time, the max (*xij*) and min (*xij*) values are determined in the attribute columns j. These max and min values of the *xij* indicators are inserted into the formula for normalisation of the input data. Therefore, prior to data normalisation, the maximum and minimum values *xij* adopted for the attributes in the studied conditions of the IAQ model should be compiled from the literature. These values will be used to test the normalisation of the input dataset adopted in the decision matrix, while the test normalisation will also address the question of whether a given normalisation method (formulas) normalises the input dataset correctly (the normalisation result cannot be negative) [31] and whether the set of alternatives (combinations of attributes) in the decision matrix has been determined correctly. Literature [2,5,7,32–40] on the minimum pollutant values (min *xij*) include the long-term concentrations [32] or *c*LCI concentration values. The maximum values [2,7,36–38] of the max (*xij*) pollutant concentrations are either short-term values or allowable concentration values *c*ELV [32] according to EN 16798-1: 2019, World Health Organization WHO guidelines [33] and EU recommendations [34]. The results of maximum and minimum concentration tests in representative groups of UK, Danish and Polish building tests published by Shrubsole [2] and Johnston [35] were used. Attributes are initially normalised assuming that, for attributes X<sup>1</sup> . . . m, the values from Table 3 can be inserted into Equations (9) and (12) to obtain normalised attributes nij.


**Table 3.** Example data for the ΣIAQ model decision matrix. Attribute indices xij and normalisation results nij with two methods calculated by Equations (9) and (12), as example alternatives.
