**2. Materials and Methods**

## *2.1. MCDM*/*MCDA in Architecture, Urban Planning and Energy-E*ffi*cient Construction: Literature Review*

As a result of the undertaken literature studies, Tables 1 and 2 present examples of previous applications of the MCDM/MCDA methods for the selected decision-making challenges in the field of architecture, urban planning, and energy-efficient constructions. What is important, selected scientific papers which are indexed in the Web of Science and Scopus databases were assumed as the basis for this elaboration. The developed list is the result of a several-stage analysis of scientific papers in the abovementioned databases. In the first stage, a preliminary selection of articles available in databases was made, using keywords: "MCDM/MCDA in architecture," "MCDM/MCDA in urban planning," "MCDM/MCDA in energy-efficient construction." Then, selected scientific papers were analyzed in terms of criteria which result from the main purpose of the work. Only the research in which the research problem and the methods applied were precisely determined and the territorial delimitation of research was defined, was selected.

The research is presented chronologically, divided into examples of applications in architecture and urban planning (Table 1) and in energy-efficient construction (Table 2). The literature review included 33 publications.

Based on the literature review, it can be noticed that MCDM/MCDA methods are universal tools, which is confirmed by the variety of decision-making challenges presented in Tables 1 and 2. In architecture and urban planning, the selected methods are applied—for example—at the stage of selecting urban revitalization scenarios and for renovation of historical buildings. In addition, assessing sustainable development of urban areas is a popular research topic. MCDM/MCDA methods are also gaining popularity in energy-efficient construction, in which the selection of a design solution often requires one to take into consideration many criteria of different nature. In addition to technical criteria, economic and environmental criteria often occur; a phenomenon that stems from the sustainable development paradigm.

It is also worth mentioning the effectiveness of selected MCDM/MCDA methods. Based on the literature review, it can be stated that the selected methods facilitate decomposition of the decision problem, improve the transparency of decision processes, facilitate comparison of various decision alternatives, identify their strengths and weaknesses [19,29]. However, in addition to the advantages, some authors also noted the weaknesses and limitations of selected MCDM/MCDA methods. For example, Moghtadernejad et al. among the limitations of the AHP and TOPSIS methods (method which were used in their case study) indicate the lack of consideration of interactions between various design criteria [43]. In contrast, Zinatizadeh et al. compared three selected methods: SAW, ELECTRE and TOPSIS, where the TOPSIS method came out best. Other methods, SAW and ELECTRE, were rated lower, e.g., in terms of ability in pair comparison or ability to manage low quality input data [21]. In the work of multi-criteria decision support of a thermal renovation project for a masonry building, the authors used the Delphi, Swing, and PROMETHEE methods, indicating their several limitations, e.g., the method did not take into account the various uncertainties (regarding the assessment of criteria or decision makers' preferences) that could have influenced the final ranking of decision variants [39]. More about the pros and cons of selected MCDM/MCDA methods can be found in works [47,48]. It is worth adding that because of the limitations of individual methods, the hybrid approach is becoming increasingly popular, Zavadskas et al. state that "to have comprehensive assessment, it is better to use two or three different MCDM" [40].

MCDM/MCDA methods are not only universal, but also well-known tools, as evidenced by territorial delimitation of the conducted research. Figure 1 presents places where research focused on the subject of multi-criteria decision support was carried out. The major research centers in this field are Europe (primarily Lithuania, followed by Spain, Italy, and Poland), Asia, and both the Middle and the Far East (Iran and China) (compare: [3]).

**Figure 1.** Countries in which selected studies on MCDM/MCDA were carried out with the focus on architecture and urban planning, as well as in the area of energy-efficient construction. Source: author's own work.

In addition, when analyzing the methods used in the selected papers, it can be seen that the most commonly used method was AHP and its extensions. The AHP method appeared in 16 papers included in the literature overview. Importantly, the AHP method was most often used to calculate the weighted criteria, which were then used to rank decision variants using other methods, e.g., TOPSIS. In addition, research conducted to-date shows that MCDM/MCDA methods can be combined by using the so-called hybrid approach, which allows one to increase the efficiency of the adopted methods. The AHP and TOPSIS hybrid is the most popular combination. A conjunction with the COPRAS method is also increasingly applied. It is worth adding that MCDM/MCDA methods can be integrated with other methods and systems, such as the dynamically developing BIM (building information modeling) system or the GIS (geographic information system) system.

The AHP method, which—for many years now—has been subject to significant interest in many areas around the world see for example: [49,50], is also being considered in terms of its advantages and disadvantages. Among the advantages of the AHP method the following can be mentioned: decomposition of a decision problem using a hierarchical structure tree, the possibility of making comparisons through element pairs that are located at given levels of the structure, the possibility of assessing the consistency of the comparisons by using consistency ratio. The AHP algorithm is also subject to criticism, for example: the independence of the analyzed elements, excessive subjectivity, or difficulty of accounting for uncertainties associated with judgments [43,51].

MCDM/MCDA methods are constantly evolving, and that is why fuzzy multi-criteria methods are becoming an important and equally popular group of computational tools. The most commonly used method is Fuzzy AHP, which dates back to 1983 [52]. The next part of this paper presents this method's algorithm that is subsequently used to develop a case study.

## *2.2. Fuzzy AHP—Algorithm and Selected Scales*

The fuzzy sets theory was first presented in the 1960s by Zadeh, as a mathematical method to facilitate the framing of uncertainty and imprecision, which often accompanies human assessments. In the 1970s, the fuzzy sets theory also appeared within the scope of decision-making challenges. As mentioned in the previous section, the most commonly used method is the Fuzzy AHP method (abbreviated FAHP). Among the researchers who contributed to the development of FAHP are: van Laarhoven and Pedrycz, Buckley, and Chang [53,54]. Importantly, this work uses a known algorithm proposed by Buckley. Individual FAHP calculation steps are shown in Figure 2.

**Figure 2.** Main calculation stages of Fuzzy AHP. Source: author's own work based on [54].

Importantly, the literature on the subject matter quotes various scales used in FAHP (e.g., in online output software application, the user has a choice of nine different scales) [59]. In this paper, in addition to the classic Saaty scale, two fuzzy triangular scales (Table 3) were used. Calculation results and comparative analysis of weights obtained by using individual scales are presented in the subsequent part of this paper.


**Table 3.** Classic Saaty scale and selected Fuzzy triangular scales.

Source: [54–58].

### **3. Results**

The subject of this research is a comparative analysis of the weights for the location criteria of residential area with solar installations and obtained via the classical AHP method [60] with the weights of these criteria, calculated by means of the Fuzzy AHP method (two fuzzy scales included). The assessment of the suitability of the area for residential development with solar installations can be considered as a multi-criteria decision problem. Selecting the location for this type of investment is the net sum of not only spatial, environmental, or legal conditions, but the inhabitants' preferences also play a key part in that decision-making process. For that reason, the excessive subjectivity (of which AHP and FAHP methods are often accused), may be its advantage in the case of such decision-making challenges. Importantly, the decision-making criteria used for the case study are the result of research described in the following papers [60,61].
