The research was conducted on six measuring sections of two-lane roads (main roads of category I) in the territory of Bosnia and Herzegovina, with a total length of the road network of 60.918 km, which were analyzed as a function of longitudinal gradient. The Vrhovi-Šešlije (M-I-103) road section was analyzed for the longitudinal fall of −5.00% and −1.92%, so that a total of six measuring road sections (
D1–
D6) were considered in the case study. Moreover, at each measuring section, the value of the speed limit was visually identified, and the case study synthesized the obtained data on exceeding the speed limit at each of the measuring sections. The obtained data on the deviation of the real speed from the speed limit were synthesized and presented on the basis of the arithmetic mean of a representative sample of the obtained values for light goods vehicles, which is given in
Table 1.
Further in the paper, eight criteria (
C1–
C8) are analyzed. The data for determining the level of risk of road sections are systematized in
Table 1. The determination of the criteria included the criterion of optimality through:
C1—length of the section (m);
C2—ascent/descent at 1000 m in %;
C3—credible deviation of the arithmetic mean of the real speed from the speed limit for light goods vehicles (km/h);
C4—AADT (Annual Average Daily Traffic) (vehicles/day);
C5–
C8—number of accidents with fatalities, with severely injured persons, with slightly injured persons, and with material damage, respectively.
5.1. Determining Criteria Weights by Rough D’LMAW Methodology
In the previous section, eight criteria used to determine the level of risk on five road sections (six measuring sections) have been defined. The criteria are marked with codes C1–C8, as follows: C1—length of the section (m); C2—ascent/descent at 1000 m in %; C3—speed deviation from the limit speed; C4—AADT (Annual Average Daily Traffic) (vehicles/day); C5—number of traffic accidents with fatalities; C6—number of traffic accidents with severely injured persons; C7—number of traffic accidents with slightly injured persons; and C8—number of traffic accidents with material damage. The following section explains the application of the rough D’LMAW methodology for determining the weight coefficients of the criteria.
Step 1. Expert assessment consisting of a survey, the assessment of concordance of expert preferences, and the results obtained were used in the analysis. Methods of determining the weights of the criteria to describe risk management are considered to be subjective if they are evaluated by respondents or experts. The estimate of one highly qualified expert may be more important than the estimates made by a number of inexperienced specialists. The research involved four experts (two traffic engineers and two civil engineers, road experts) represented by a set. Eight experts were interviewed, and the result of the interviews for four experts were accepted. Based on experts’ estimates, priority vectors for each expert were defined, as shown in
Table 2. The experts’ estimates of criteria in the priority vector were defined on the basis of a seven-point scale: very low (VL)—1; low (L)—2; medium low (ML)—3; medium (M)—4; medium high (MH)—5; high (H)—6; very high (VH)—7.
Using Equations (1)–(4), the sequences of the priority vectors from
Table 2 are transformed into the rough sequences shown in
Table 3.
An example of the transformation of experts’ estimates from
Table 3 for criterion
C1 is presented in the following section. The experts’ estimates for criterion
C1 in the priority vector (
Table 2) yielded the following values:
and
. Using Equations (1)–(4), and provided that ε1 = ε
2 = 1, we can define the lower and upper limit of rough numbers according to the following:
Based on the defined limit values, we can define a rough number, Equation (5):
The aggregated rough priority vector is obtained by applying Equation (14):
Based on the aggregated rough values, we obtain an aggregated priority vector:
Step 2. By applying the condition that
, the absolute anti-ideal point
= [0.4,0.6] is defined. The vector of relations is obtained by applying Equation (15), as follows:
Step 3. By applying Equation (16), we obtain a vector of weight coefficients:
The weight coefficient for criterion
C1 is obtained by applying Equation (16) as follows:
The values of the remaining weight coefficients of the criteria are obtained in a similar way.
5.2. Determination of Risk on Road Sections Using Rough Dombi–Bonferroni MARCOS Methodology
The application of rough Dombi–Bonferroni MARCOS methodology is presented through a multi-criteria model for determining the risk on the measuring road sections. The applied multi-criteria framework is presented through the steps given below.
Step 1: In the case study, six measuring sections of the given road sections are considered, which are marked with the following codes:
D1—Vrhovi-Šešlije I;
D2—Vrhovi-Šešlije II;
D3—Rudanka-Doboj;
D4—Šepak-Karakaj 3;
D5—Donje Caparde-Karakaj 1;
D6—Border (RS/FBIH)-Donje Caparede. The data shown in
Table 4 are used to determine the level of risk.
Based on the data presented in
Table 4, we can see that the data for criteria
C4–
C8 are given for a five-year period. The values
C1–
C3 are obtained by empirical research, and the values
C4–
C8 are taken from AADT (Annual Average Daily Traffic) and accident databases. Therefore, the values for criteria
C4–
C8 are transformed into rough numbers, while the values of criteria
C1–
C3 are shown as crisp values. Using Equations (1)–(4), the sequences of criteria
C4,
C5,
C6,
C7, and
C8 are transformed into interval rough sequences. The rough sequences are averaged using the rough Bonferroni operator [
3,
31,
32,
33,
34], and an aggregated rough initial matrix is obtained, as shown in
Table 5.
Step 2: Using Equation (20), the ideal alternative (
IA) and the anti-ideal alternative (
AIA) are defined for each criterion
j. Since all criteria are of benefit type (B), the first part of Equation (20) is used to determine
IA and
AIA.
IA and
AIA are shown below:
Step 3: Normalization of initial matrix elements (
Table 5) is performed using Equation (21). The normalized rough matrix is shown in
Table 6.
The normalization of the element at position
C5–
D4 in
Table 6 is obtained by applying Equation (21) as follows:
The remaining elements from
Table 6 are obtained in a similar way.
Step 4: Using Equations (22) and (23), the utility degrees of alternatives in relation to
IA and
AIA are calculated. The utility levels of the alternatives are shown below:
The calculation of the utility degree of alternative D1 is explained below:
- (1)
By applying Equation (24), we obtain the sum of the weighted elements of the normalized matrix:
The sum calculation of the weighted elements of the normalized matrix for alternative
D1 is obtained as follows:
- (2)
Then, by applying Equations (22) and (23), we obtain the utility degrees of alternative
D1 in relation to
IA and
AIA:
Steps 5 and 6: Ranking of alternatives is performed on the basis of the values of the utility function
, Equation (25). The utility functions of alternatives are presented below:
It is desirable that the road section has the lowest possible value of risk, so the following rank is obtained:
D6 >
D5 >
D2 >
D1 >
D3 >
D4. Further in the paper, the stability of the solution is tested in the case of a change in the stabilization parameters of the Dombi–Bonferroni hybrid function Equation (24), which has been used to calculate the utility degree of the alternatives. The Dombi–Bonferroni function has three stabilization parameters,
,
, and α. During the risk calculation of the considered road sections, the values of the parameter
=
= α = 1 are obtained. Further, the application of parameters
,
, and α are simulated through 100 scenarios (1 ≤
,
, α ≤ 100).
Figure 1 shows the dependence of the Dombi–Bonferroni function on the change of parameters
,
, and α; for alternative
D1 (
Figure 1a), for alternative
D2 (
Figure 1b), for alternative
D3 (
Figure 1c), for alternative
D4 (
Figure 1d), for alternative
D5 (
Figure 1e) and for alternative
D6 (
Figure 1f).
The results from
Figure 1 indicate that an increase in the values of parameters 1 ≤ α ≤ 100 causes a decrease in the value of the Dombi–Bonferroni function of all alternatives. However, the question arises as to whether these changes affect the change in the ranks of alternatives. In order to consider the influence of the mentioned parameters on the road section ranks,
Figure 2 provides a comparative overview of the changes in the utility functions of alternatives
depending on the changes in parameters
,
, and α.
The increase in the value of the considered parameters leads to a change in the utility functions; however, these changes are not sufficient to lead to the change of the ranks of alternatives. From the presented analysis, we can conclude that the initial rank D6 > D5 > D2 > D1 > D3 > D4 has been confirmed, and that the road section D6 has the lowest value of risk, and therefore represents the best solution from the considered set.