3.3. The Fuzzy Mathematical Model for Quantitative and Linguistic Risk Assessments for Environmental Start-Up Air Transport Projects
A mathematical model for risk assessment for environmental start-up projects in the air transport sector is described, based on linguistic input variables. In the first stage, it is necessary to establish the membership rules and the knowledge base in order to reach the resulting term-evaluation
for each group of risk criteria, and to determine the aggregated estimation of certainty
. In the second stage, based on the estimates obtained
and
, we define a project risk assessment for each group of criteria
[
26].
Consider the first stage—the construction of the membership rules that result from the term-evaluation of risk criteria groups.
Analyze an object from
inputs and the following output:
where
is the resulting term-evaluation with a term-set
for a group of criteria
, and
, and are the input linguistic evaluation criteria for the group
.
is the operator that matches the resulting term-evaluation
for a group of criteria, for input variables
(rule of logical output), where
[
41,
42].
Next, an expert (or a group of experts) builds the rules of membership of the resulting terms for everyone. These rules can be constructed as a percentage of the membership of one or another term of the input variable. Formally, the rules of a membership represent a system of logical utterances, “if, then, else” [
43], which associate the values of the input variables
with one of the possible values
,
,
, as shown below.
If ( and and … and ) Or ( and and … and ) Or … Or ( and and … and ) Then Else …
Similarly, all functional dependencies are formed, which embodies the rules of decision-making reduced to the knowledge base in mathematical form.
The following rules of membership were formulated as a result of practical experience in the risk assessment of start-up projects, as done in References [
4,
25,
26]:
Level L—“low risk”: the environmental start-up air transport project receives the resulting term-evaluation L, if the minimum number of criteria with the term “low risk” is not less than 60%, and the remaining 40% of the criteria at the level are not lower than that of “risk below average”.
Level BA—“risk below average”: the environmental start-up air transport project receives the resulting term-evaluation BA, if the minimum number of criteria under “risk below average” is at least 60%, with the remaining 40% at a level not lower than the “average risk”.
Level A—“average risk”: the environmental start-up air transport project receives the resulting term-evaluation A, if the minimum number of criteria with the term “average risk” is at least 60%, and the remaining 40% level is not lower than “risk above average”.
Level AA—“risk above average”: the environmental start-up air transport project receives the resulting term-evaluation AA, if the minimum number of criteria with the term “risk above average“ is at least 60%, and the remaining 40% of the criteria can be deemed “high risk”.
Level H—“high risk”: the environmental start-up project in air transport receives the resulting term-evaluation H, if the minimum number of criteria with the term “high risk” is 60% or more.
Then, based on the rules for membership in the resulting term, the evaluation of risk criteria groups, as well as a fragment of the knowledge base, for example, with the group criteria
and the resulting term-evaluation L, can be given as shown in
Table 3.
Because the expert puts each variable of
authenticity and their reasoning
in the interval [0, 1], then the linguistic variables can be represented in the form of triangular membership functions as done in Reference [
44]. This means that each linguistic variable
can be replaced by the neighbor
with certainty
. This gives the opportunity to polarize the risks within a group of criteria in order to obtain the resulting term-evaluation according to the knowledge base.
The aggregated score certainty
is calculated according to the following formula [
25]:
where
is the estimation of the certainty of those linguistic variables, which coincides with the resulting term-evaluation for the
i-th criterion by
group of risk criteria,
is their number, and
is the environmental start-up project considered in air transport.
Thus, in the first stage, we obtain the resulting term-evaluation, based on the membership rules, for each group of risk criteria, the considered environmental start-up of the air transport project, and an aggregate assessment of its reliability.
In the second stage of problem solution, the approach described below is used to determine the generalized risk assessment environmental start-up project in air transport for each group of criteria , to achieve an aggregated risk assessment, as well as its linguistic interpretation.
Next, consider the following mathematical model [
25]:
where
is the value of a function equal to the numerical interpretation of the resulting term-estimates,
,
is the aggregated assessment of the certainty of the expert’s thoughts,
is a project risk assessment for each group of criteria
,
is the aggregated risk assessment for environmental start-up projects in the air transport sector across all groups of criteria
, and
is its output linguistic interpretation.
is the operator that matches the output variable
for input variables
.
Because the resulting term-assessment has a level of risk content, then its terms can be adequately determined on a percentage scale (0–100%), each of which sets values from interval , for example L [0, 15], BA [15, 30], A [30, 50], AA [50, 80], and H [80, 100]. That is, a value of 85% risk is treated as “high risk”.
We then consider the dependence of the resulting term evaluation
and its certainty
in the form of the
S-shaped membership function as in References [4, 25], which, in our view, appropriately expresses this dependency.
Since the membership function values (aggregated estimation of certainty) and the intervals of numeric values for
are known, then, for each group of criteria
,
is expressed from Equation (4) [
4].
Equation (5) denotes that a higher value signifies a greater risk of a project start-up in the appropriate group of criteria.
For generalized risk assessments of environmental start-up projects in the air transport sector by groups of criteria
g, the normalized values
are obtained, changing the orientation of objectives.
The estimates , are normalized and represent a criterion for each group aggregated risk assessment of the considered environmental start-ups of air transport projects in relation to the resulting thermal ratings and their reliability.
For DMs for each group of criteria, the weight coefficients are denoted as
from some interval. Then, the corresponding weighted coefficients are set accordingly.
Since all the estimates obtained are normalized by the interval [0, 1], then, in order to obtain a final assessment of the risk of financing the environmental start-up of projects in the air transport sector, the approach below can be used. Depending on the size of the investment, DMs can choose one of the following convolutions:
The resulting estimates are normalized and then matched to the output variable to provide the following scale:
= “Insignificant risk of financing the environmental start-up of the air transport project”;
= “Low risk of financing the environmental start-up of the air transport project”;
= “Average risk of financing the environmental start-up of the air transport project”;
= “High risk of financing the environmental start-up of the air transport project”;
= “Critical risk of financing the environmental start-up of the air transport project”.
The linguistic interpretation of the aggregated risk assessment for financing the environmental start-up projects in the air transport sector is as follows: (0.85, 1]—; (0.67, 0.85]—; (0.36, 0.67]—; (0.21, 0.36]—; [0, 0.21]—.
The suggested decision levels are experimentally obtained, and the decision-maker can change them. To improve the accuracy of boundary estimation, one can change the experience of experts in evaluating the environmental start-up projects. Also, depending on the investment opportunities of investors, if necessary, the level of decision-making can also change [
45].
3.4. Generalized Algorithm for Obtaining an Aggregated Risk Assessment for Environmental Start-Up Projects in the Air Transport Sector
Based on the above fuzzy risk assessment model, the environmental start-up of air transport projects can be written as a generalized aggregated estimation algorithm.
Step 1. Determine the resulting term-evaluation.
Based on the data entered, the projects introduced from the start-up, and the built knowledge base, the resulting term-evaluation by Equation (1) for the groups of criteria is determined: ; ; ; ; .
Step 2. Determine the aggregated estimation of the reliability of the expert’s thoughts.
The aggregated evaluation certainty ,, is calculated according to Equation (2).
Step 3. Obtain a generalized risk assessment for projects by groups of criteria .
For each group of criteria , calculate the level of risk , relative to the percentage scale and the resulting term-evaluation , using Equation (5). The generalized evaluation risk start-up projects for each group of criteria is given by Equation (6).
Step 4. Weight coefficients introduced by groups of risk criteria.
For each group of criteria, DMs set weight coefficients after which, according to Equation (7), the normalized weight coefficients are calculated.
Step 5. Aggregated risk assessment calculated for all groups of criteria.
We determine the aggregated risk assessment using one of the convolutions in Equations (8)–(11).
The equate assessment with the output variable is to obtain a linguistic interpretation of the level of risk financing environmental start-up projects in air transport.
In this way, a fuzzy mathematical model was constructed to obtain an aggregated risk assessment for environmental start-up projects in the air transport sector. The model used expert’s knowledge and reasoning to evaluate the various risk criteria and, based on this, there was an aggregation of views according to the groups of criteria in the final evaluation.