## 3 Research Methodology, Course and Results
### Mathematical Methods Used to Support the Analysed Decision-Making Processes  
This study also comprised an analysis of the methods for evaluation of variant decisions, applied in all the models. Building investments and civil structures have own specific characteristics, and therefore not all the available methods can be applied in practice. The most important feature expected from decision-support methods employed in the construction business is their capacity for including a large number of criteria of various character. At the stage when variant solutions are compared, it is unnecessary to use a binary assessment; what is actually needed is to determine the degree to which each variant solution satisfies a given parameter. Another essential requirement is to be able to develop descriptive assessment scales that will be readable and easy to use. To use variant evaluation methods in practice, it is also important to have a relatively simple mathematical apparatus and to present the results clearly. Thus, in the construction activity, a certain group of methods tends to prevail. An analysis of an investment for which an evaluation of alternative solutions has been made shows that these are mostly methods where scores are assigned to assessed variants. The formula for calculating an aggregated assessment of variant '\(i\)' is as follows (1):  
\[F_{i}=\sum_{j=1}^{n}c_{ij} \tag{1}\]  
where:  
\(c_{ij}\)--the assessment of a variant expressed in points scored for criterion \(c_{j}\) by variant \(v_{i}\)  
Figure 11: Percentage share of decision models in the research sample.  
Figure 10: Percentage share of examined decisions.  
\(F_{i}\)--total aggregated assessment of variant \(v_{i}\), (\(i=1,2,\,\ldots,\,m\)).  
Another equally popular approach is to make an evaluation of solutions that takes into account the varied importance of criteria. This method is sometimes called the weight-score method or the weighted sum method. Here, every criterion is assigned a weight, which facilitates a more effective evaluation of the variants, which takes into account the specific character of the project submitted to evaluation. The sum of weights should always equal 1.0. The problem can be written as (2):  
\[F_{i}=\sum_{j=1}^{n}w_{j}\,c_{ij} \tag{2}\]  
where:  
\(c_{ij}\)--the assessment expressed in points scored for criterion \(c_{j}\) by variant \(v_{i}\),  
\(w_{j}\)--weights.  
The following formulas can be used when aggregating mid-term evaluations in scalar methods:  
* arithmetic mean (3): \[F_{i}=\frac{1}{n}\,\sum_{j=1}^{n}c_{ij}\] (3)
* arithmetic weighted mean (4): \[F_{i}=\frac{1}{n}\sum_{j=1}^{n}w_{j}c_{ij}\] (4)
* sum of arithmetic weighted mean (5): \[F_{i}=\frac{\sum_{j=1}^{n}w_{j}c_{ij}}{\sum_{j=1}^{n}w_{j}}\] (5)  
Yet another popular method, admittedly more mathematically advanced, is the Analytic Hierarchy Process (AHP) approach, created in the 1980s by [PERSON]. The method owes its popularity to the fact that it approaches the problem of making an assessment by ordering criteria and arranging them in groups of main criteria, after which hierarchical evaluation is performed, which leads to the determination of vectors of the main criteria and subordinate criteria. The mathematical formula relies of a pairwise comparison matrix and proceeds according to the following template:  
1. Calculation of the value of a normalized matrix (6): \[w_{ij}=\frac{a_{ij}}{\sum_{i=1}^{n}a_{ij}}\] (6)
2. Determination of the value of the vector of sub-priorities (7): \[w_{j}=\sum_{j=1}^{n}w_{j}a_{ij}\] (7) where (8): \[w_{j}=\frac{\sum_{i=1}^{n}w_{ij}}{n}\] (8)  
In order to verify whether the above-mentioned procedures has been correct, we determine:  
* The matrix's own maximum value (9): \[\lambda\frac{1}{w_{i}}\sum_{i=1}^{n}a_{ij}w_{jmax}\] (9)  
The results of the calculations after summing up indicate the best-ranked solution.