Performing a quantitative risk analysis for future railway investments requires that individual hazards have a measurable assessment of their significance. When calculating these values, the authors sought to include all the elements of expert assessment that were taken into account in the analyzed questionnaire, and therefore their validity (M), certainty of estimation (SD), and their impact on the time and cost of the planned investment. The point score was made on a scale of 0-10.

To determine the individual weights of each identified hazard, the authors made calculations as described below.

First, the data from and were normalized. The calculations were made on the basis of equations (1, 2, 3):

$$\begin{array}{r}{M}_{zn}=\frac{{M}_{n}-0}{(10-0)}\end{array}$$(1)where:

*M*_{zn} – normalized value of the average rating for the nth hazard,

*M*_{n} – the value of the average rating for a particular hazard, 0, 10 – the grading scale used in the survey.

$$\begin{array}{r}Z{T}_{zn}=\frac{ZTn}{N}\end{array}$$(2)where:

*ZT*_{zn} – normalized nth hazard affecting the deadline,

*Z*_{Tn} – the number of assessments indicating that a given hazard may have an impact on the time of investment implementation,

*N* – number of surveys for a given investment stage (design stage 112 copies, build stage 85 copies).

$$\begin{array}{r}Z{K}_{zn}=\frac{ZKn}{N}\end{array}$$(3)where:

*ZK*_{zn} – normalized n-th hazard affecting the cost of implementation,

*Z*_{Kn} - – number of assessments indicating that a given hazard may affect the cost of investment implementation,

*N* – number of surveys from a given investment stage.

In the second step, individual normalized values were aggregated. The M_{zn} assessment was additionally reinforced with the standard deviation evaluation, obtaining the M_{kn} value. It resulted from the fact that the smaller the standard deviation for a given hazard, the greater the compatibility of experts as to the possibility of its occurrence:

$$\begin{array}{r}{M}_{kn}={M}_{zn}\left[1+\frac{{M}_{zn}\left(\frac{S{D}_{max}-S{D}_{n}}{S{D}_{max}-S{D}_{min}}\right)}{6}\right]\end{array}$$(4)where:

*M*_{kn} – enhanced *M*_{z} value due to standard deviation *SD*_{n},

*SD*_{n} – standard deviation of hazard n,

*SD*_{max}, SD_{min} –maximum and minimum standard deviation value for all hazards from a given investment stage,

*M*_{zn} – normalized value of the average rating for the nth hazard.

The value of the final, non-dominated weight of each Wn hazard was calculated according to the following formula:

$$\begin{array}{r}{W}_{n}={M}_{kn}+(Z{T}_{zn}\cdot {K}_{T})+(Z{K}_{zn}\cdot {K}_{K})\end{array}$$(5)when

$$\begin{array}{r}{K}_{T}+{K}_{K}=1\end{array}$$(6)where:

*K*_{T} and *K*_{K} – common to all hazards in a given matrix partial weight of hazards: time of investment (*K*_{T}) and investment cost (*K*_{K}).

The final normalized weight value of individual hazards was calculated as:

$$\begin{array}{r}{W}_{nz}=\frac{{W}_{n}}{\sum _{i}^{N}{W}_{n}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\sum _{1}^{N}{W}_{nz}=1\end{array}$$(7)As can be seen from formula 5, in the final formula determining the importance of individual hazards, common partial weights of all hazards can be applied separately for time and cost hazards. The decision on the value of these

partial weights was left to the manager of the planned investment, depending on the situation of the project. Naturally, these weights can be the same and then *K*_{T} = *K*_{K} = 0,5 or, for example, extreme, i.e. *K*_{T} = 0; *K*_{K} = 1 or vice versa.

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