V5: Increase in revenue from PSO services and reduction in operating costs (Figure <sup>8</sup> left): from Public Service Obligation (PSO) services.

$$P\_2 = f(P\_{21}, P\_{22}, P\_{23}, \dots, P\_{2n}) \to \max. \land \ T = f(T\_1, T\_2, T\_3, \dots, T\_n) \to \min \tag{5}$$

o V6: Increase in revenue from ticket sales and PSO services (Figure 8 right): # V6: Increase in revenue from ticket sales and PSO services (Figure <sup>8</sup> right): *P2 = f(P21, P22, P23, ….. , P2n)→*max. ˄ *T = f(T1, T2, T3, ….. , Tn)→*min (5)

o V6: Increase in revenue from ticket sales and PSO services (Figure 8 right):

in revenue from ticket sales and PSO services.

from Public Service Obligation (PSO) services.

3.4.2. Identification of Evaluation Criteria

3.4.2. Identification of Evaluation Criteria

increase in revenues under the PSO contract.

increase in revenues under the PSO contract.

$$P\_1 + P\_2 = f(P\_{11}, P\_{12}, P\_{13}, \dots, P\_{1n}) + \\ = f(P\_{21}, P\_{22}, P\_{23}, \dots, P\_{2n}) + \\ \max \tag{6}$$

*P1 + P2= f(P11, P12, P13, …, P1n) + = f(P21, P22, P23, …, P2n)→*max (6)

o V7: Increase in revenue from ticket sales and PSO services and reduction in costs (Figure 9): **Figure 8.** V5—increase in revenue from PSO services and reduction in operating costs, V6—increase in revenue from ticket sales and PSO services. **Figure 8.** V5—increase in revenue from PSO services and reduction in operating costs, V6—increase in revenue from ticket sales and PSO services.

*P1 + P2= f(P11, P12, P13, …, P1n) + = f(P21, P22, P23, …, P2n)→*max (7) o V7: Increase in revenue from ticket sales and PSO services and reduction in costs (Figure 9): # V7: Increase in revenue from ticket sales and PSO services and reduction in costs (Figure 9):

$$P\_1 + P\_2 = f(P\_{11}, P\_{12}, P\_{13}, \dots, P\_{1n}) + \\ = f(P\_{21}, P\_{22}, P\_{23}, \dots, P\_{2n}) + \\ \max \tag{7}$$

*P1 + P2= f(P11, P12, P13, ….. , P1n) + = f(P21, P22, P23, ….. , P2n)→*max (7)

**Figure 9.** V7—increase in revenue from ticket sales and PSO services and reduction in costs.

**Figure 9.** V7—increase in revenue from ticket sales and PSO services and reduction in costs.

Identification and quantification of the criteria for evaluating the manner of implementing the principles and concluding a PSO contract were carried out over four steps: Defining the required level or volume of service, reduction in business costs, increase in revenues from ticket sales, and

Identification and quantification of the criteria for evaluating the manner of implementing the principles and concluding a PSO contract were carried out over four steps: Defining the required level or volume of service, reduction in business costs, increase in revenues from ticket sales, and

The selection of an optimal variant depends on many factors. Therefore, in the proposed methodology, as the criteria, the following values have been adopted: The reality of the feasibility of

The selection of an optimal variant depends on many factors. Therefore, in the proposed methodology, as the criteria, the following values have been adopted: The reality of the feasibility of

**Figure 8.** V5—increase in revenue from PSO services and reduction in operating costs, V6—increase

o V7: Increase in revenue from ticket sales and PSO services and reduction in costs (Figure 9):

*P1 + P2= f(P11, P12, P13, ….. , P1n) + = f(P21, P22, P23, ….. , P2n)→*max (7)

**Figure 7.** V3—increase in ticket revenue and reduction in operating costs, V4—increase in revenue

o V5: Increase in revenue from PSO services and reduction in operating costs (Figure 8 left):

o V6: Increase in revenue from ticket sales and PSO services (Figure 8 right):

*P2 = f(P21, P22, P23, ….. , P2n)→*max. ˄ *T = f(T1, T2, T3, ….. , Tn)→*min (5)

*P1 + P2= f(P11, P12, P13, ….. , P1n) + = f(P21, P22, P23, ….. , P2n)→*max (6)

from Public Service Obligation (PSO) services.

in revenue from ticket sales and PSO services.

**Figure 9.** V7—increase in revenue from ticket sales and PSO services and reduction in costs. **Figure 9.** V7—increase in revenue from ticket sales and PSO services and reduction in costs.

### 3.4.2. Identification of Evaluation Criteria 3.4.2. Identification of Evaluation Criteria

Identification and quantification of the criteria for evaluating the manner of implementing the principles and concluding a PSO contract were carried out over four steps: Defining the required level or volume of service, reduction in business costs, increase in revenues from ticket sales, and increase in revenues under the PSO contract. Identification and quantification of the criteria for evaluating the manner of implementing the principles and concluding a PSO contract were carried out over four steps: Defining the required level or volume of service, reduction in business costs, increase in revenues from ticket sales, and increase in revenues under the PSO contract.

The selection of an optimal variant depends on many factors. Therefore, in the proposed methodology, as the criteria, the following values have been adopted: The reality of the feasibility of The selection of an optimal variant depends on many factors. Therefore, in the proposed methodology, as the criteria, the following values have been adopted: The reality of the feasibility of the proposed variant, means available to the public authority - budget, the ability of the operator, the effect of realization, and the period of realization (Table 1).


**Table 1.** Criteria for evaluation of identified variants.

### **4. Results**

As for obtaining the weight value of criteria, we have used the F-PIPRECIA method; after identifying the criteria on which the ranking of potential variants will be made, it is required that we compare the criteria by using the scale presented in [17]. In order to derive the relative importance of the criteria, a team of three experts had been established; for many years, they have been performing managerial functions in the field of railway transport. As this is an already exploited method, detailed procedures for calculating the values of criteria will not be shown, but rather the summed results by each step (Table 2).


**Table 2.** Calculation and results of applying the fuzzy PIvot Pairwise RElative Criteria Importance Assessment (F-PIPRECIA) method for determining the criteria weights.

Where *s j* represents the group matrix obtained by expert's assessment, starting from the second criterion, and *k j* is the coefficient obtained when *s j* is subtracted from number 2, except for *s* 1 . *q j* is the fuzzy weight, *w<sup>j</sup>* is the relative weight of the criterion, and DF is the defuzzified value.

Based on the aggregation of the values wj shown in Table 2, the final criterion values are obtained:*w*<sup>1</sup> = 0.423; *w*<sup>2</sup> = 0.230; *w*<sup>3</sup> = 0.141; *w*<sup>4</sup> = 0.242; *w*<sup>5</sup> = 0.098. After calculating the weight value of criteria, we then begin the selection of the optimal variant by using the F-EDAS method. On the basis of the linguistic scale, the experts evaluate variants according to each criterion individually (Table 3).



Table 4 also shows, apart from the values of the average decision matrix, the values of an average solution according to all the criteria.


**Table 4.** The elements of the average decision-matrix and the average solution matrix.

Next, we need to calculate positive distances (PDA) and negative distances (NDA) from the average solutions depending on the criteria type. In this case, only the fifth criterion is useless, while the others are useful criteria. First, we obtain the values of the positive distance (PDA) and the values of the negative distance from the average solution. In order to obtain the values shown in Table 5, it is necessary to first apply step 5 of the F-EDAS method, and this represents the sum of the weighted matrix for positive *sp*f*<sup>i</sup>* and negative distance *sn*f*<sup>i</sup>* for all variants. Further, it is necessary to normalize previous values in order to obtain *nsp* <sup>g</sup>*<sup>i</sup>* and *nsn*g*<sup>i</sup>* . Finally, it is necessary to calculate the assessment of the results, the appraisal score (*as*e*<sup>i</sup>* ), and make the defuzzification appraisal score (*as*e*<sup>i</sup>* ) (Table 5).

**Table 5.** The weighted sum of distances, the normalized values of them, and the appraisal scores.


Based on the performed analysis, and in accordance with the task, implementation of Variant A<sup>7</sup> is recommended as the most acceptable solution. As good enough solutions, we might accept variants A<sup>6</sup> and A5; Variant A<sup>3</sup> could possibly represent a satisfactory solution. Thus, it is evident that the most acceptable variant is essentially the scenario in which the positive result stems from joint "efforts" of the operator (decreased costs and increased revenues from the ticket sales) and public authorities through increased subsidies for PSO. Another acceptable variant is a scenario where, because of the limitations of the market (low flow and low purchasing power of the population-passengers), there lacks any significant increase in revenue from ticket sales; the solution is then sought through reduction

in costs and increase in PSO subsidies. The variants where the problem is solved only by increased PSO subsidies by the public authorities and the combined approach based on the increase in revenues from ticket sales and operator's cost reduction are not favorable.

### **5. Validation Tests**

### *5.1. Changing the Significance of Criteria*

In this phase of validation test, the impact of changing the three most important criteria C1, C2, and C<sup>4</sup> on the ranking results was analyzed. Using Equation (8), a total of 18 scenarios were formed.

$$\mathcal{W}\_{\eta\beta} = (1 - \mathcal{W}\_{n\alpha}) \frac{\mathcal{W}\_{\beta}}{(1 - \mathcal{W}\_{\eta})} \tag{8}$$

In scenarios S1–S6, the first criterion was changed, criterion C<sup>2</sup> was changed in scenarios S7−S12, and criterion C<sup>4</sup> was changed in scenarios S13–S18. In Equation (8), *W*e*n*<sup>β</sup> represents the new value of criteria C2–C<sup>5</sup> for scenarios S1−S6; then, C1, C3–C<sup>5</sup> for scenarios S7–S12, i.e., C1–C3, and C<sup>5</sup> for scenarios S13–S18. *W*e*n*<sup>α</sup> represents the corrected value of criteria C1, C2, and C<sup>3</sup> respectively by groups of scenarios, *W*e<sup>β</sup> represents the original value of the criterion considered, and *W*e*<sup>n</sup>* represents the original value of the criterion whose value is reduced, in this case, C1, C2, and C4.

In all scenarios, the value of criteria was reduced by 15%, while the values of the remaining criteria were proportionally corrected by applying Equation (8). After forming 18 new vectors of the weight coefficients of the criteria (Table 6), new model results were obtained, as presented in Figure 10.


**Table 6.** New criterion values across 18 scenarios.

In most scenarios, there is no change in initial rank, as shown in Figure 10. However, it is important to emphasize that the model is very sensitive to the change in the most important criterion, and in scenarios S1–S6, significant changes occur. With a slight decrease in the value of the first criterion, the ranks slightly change; for example, variants V<sup>1</sup> and V<sup>4</sup> change their positions in the second scenario. As the value of the first criterion decreases drastically, the ranks also change drastically. In the fourth scenario, V<sup>7</sup> loses the first position, while in the sixth scenario, it comes in last place. Practically, the most important role is played by the first criterion in the set decision conditions. In accordance with the rank changes in the mentioned scenarios, a statistical check of the rank correlation was performed using Spearman's correlation coefficient, as shown in Figure 11.

10.

weight coefficients of the criteria (Table 6), new model results were obtained, as presented in Figure

**Table 6.** New criterion values across 18 scenarios.

 **w1 w2 w3 w4 w5 S1** 0.360 0.255 0.157 0.269 0.108 **S2** 0.296 0.281 0.172 0.296 0.119 **S3** 0.233 0.306 0.188 0.322 0.130 **S4** 0.169 0.331 0.203 0.349 0.140 **S5** 0.106 0.357 0.219 0.376 0.151 **S6** 0.042 0.382 0.235 0.402 0.162 **S7** 0.442 0.196 0.148 0.253 0.102 **S8** 0.461 0.161 0.154 0.264 0.106 **S9** 0.480 0.127 0.160 0.275 0.111 **S10** 0.499 0.092 0.167 0.286 0.115 **S11** 0.518 0.058 0.173 0.297 0.119 **S12** 0.537 0.023 0.179 0.307 0.124 **S13** 0.444 0.241 0.148 0.206 0.102 **S14** 0.464 0.252 0.155 0.170 0.107 **S15** 0.484 0.263 0.162 0.133 0.112 **S16** 0.504 0.274 0.168 0.097 0.116 **S17** 0.525 0.285 0.175 0.061 0.121 **S18** 0.545 0.296 0.182 0.024 0.126

**Figure 10.** Comparison of obtained results by F-PIPRECIA-Fuzzy Evaluation based on Distance from Average Solution (F-EDAS) model with all formed scenarios S1–S18. **Figure 10.** Comparison of obtained results by F-PIPRECIA-Fuzzy Evaluation based on Distance from Average Solution (F-EDAS) model with all formed scenarios S1–S18. conditions. In accordance with the rank changes in the mentioned scenarios, a statistical check of the rank correlation was performed using Spearman's correlation coefficient, as shown in Figure 11.

**Figure 11.** Spearman's correlation coefficient (SCC) through 18 formed scenarios.

The calculated Spearman's correlation coefficient (Figure 11), despite significant deviations in some scenarios, shows a high correlation of ranks in total, 0.821. Generally, in 13 out of 18 scenarios, variants have a full correlation. The correlation between the initial results obtained by the F-PIPRECIA-F-EDAS model and the S2 and S3 scenarios is 0.964, while in the S4 scenario, it is 0.786. The biggest deviation in the rankings is in the fifth and sixth scenarios when the negative correlations are −0.071 and –0.857, respectively. proposed F-PIPRECIA-F-EDAS model is stable and gives good results. **Figure 11.** Spearman's correlation coefficient (SCC) through 18 formed scenarios. The calculated Spearman's correlation coefficient (Figure 11), despite significant deviations in some scenarios, shows a high correlation of ranks in total, 0.821. Generally, in 13 out of 18 scenarios, variants have a full correlation. The correlation between the initial results obtained by the F-PIPRECIA-F-EDAS model and the S<sup>2</sup> and S<sup>3</sup> scenarios is 0.964, while in the S<sup>4</sup> scenario, it is 0.786. The biggest deviation in the rankings is in the fifth and sixth scenarios when the negative correlations are −0.071 and –0.857, respectively.

### *5.2. Impact of Reverse Rank Matrices 5.2. Impact of Reverse Rank Matrices*

One of the ways to test the validity of the obtained results is to construct dynamic matrices that analyze the solutions that the model provides under new conditions. A change in the number of variants is made for each scenario, eliminating the worst variant from further consideration. In the test, six scenarios are formed in which the change in elements of the decision matrix is simulated. One of the ways to test the validity of the obtained results is to construct dynamic matrices that analyze the solutions that the model provides under new conditions. A change in the number of variants is made for each scenario, eliminating the worst variant from further consideration. In the test, six scenarios are formed in which the change in elements of the decision matrix is simulated.

As can be seen in Figure 12, there is no change in ranks for any variant. That means that the As can be seen in Figure 12, there is no change in ranks for any variant. That means that the proposed F-PIPRECIA-F-EDAS model is stable and gives good results.

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**Figure 12.** Results of the test of reverse rank matrix. **Figure 12.** Results of the test of reverse rank matrix.

### *5.3. Comparison with Other fuzzy MCDM Methods 5.3. Comparison with other Fuzzy MCDM Methods 5.3. Comparison with Other fuzzy MCDM Methods*

In this part, a validation test is performed, including comparison with three other fuzzy methods: F-MARCOS, F-SAW, and the F-TOPSIS method. Obtained results are presented in Figure In this part, a validation test is performed, including comparison with three other fuzzy methods: F-MARCOS, F-SAW, and the F-TOPSIS method. Obtained results are presented in Figure 13. In this part, a validation test is performed, including comparison with three other fuzzy methods: F-MARCOS, F-SAW, and the F-TOPSIS method. Obtained results are presented in Figure 13.

**Figure 13.** Results of comparison with fuzzy Measurement Alternatives and Ranking according to the COmpromise Solution (F-MARCOS), fuzzy Simple Additive Weighing (F-SAW), and fuzzy **Figure 13.** Results of comparison with fuzzy Measurement Alternatives and Ranking according to the COmpromise Solution (F-MARCOS), fuzzy Simple Additive Weighing (F-SAW), and fuzzy Technique for Order of Preference by Similarity to Ideal Solution (F-TOPSIS) methods. **Figure 13.** Results of comparison with fuzzy Measurement Alternatives and Ranking according to the COmpromise Solution (F-MARCOS), fuzzy Simple Additive Weighing (F-SAW), and fuzzy Technique for Order of Preference by Similarity to Ideal Solution (F-TOPSIS) methods.

Technique for Order of Preference by Similarity to Ideal Solution (F-TOPSIS) methods. As can be seen in Figure 13, there is no change in ranks for any variant. In Figure 13, in addition to the rankings of variants, values for each variant are given so that a cross-sectional comparison can As can be seen in Figure 13, there is no change in ranks for any variant. In Figure 13, in addition to the rankings of variants, values for each variant are given so that a cross-sectional comparison can be made. As can be seen in Figure 13, there is no change in ranks for any variant. In Figure 13, in addition to the rankings of variants, values for each variant are given so that a cross-sectional comparison canbe made.

### be made. *5.4. Determining Criteria Weights with F-AHP and FUCOM Methods 5.4. Determining Criteria Weights with F-AHP and FUCOM Methods*

14.

14.

*5.4. Determining Criteria Weights with F-AHP and FUCOM Methods*  In this part of the paper, the criteria weights were re-determined using the F-AHP and FUCOM In this part of the paper, the criteria weights were re-determined using the F-AHP and FUCOM methods, and the results compared to the original F-PIPRECIA-F-EDAS model are shown in Figure In this part of the paper, the criteria weights were re-determined using the F-AHP and FUCOM methods, and the results compared to the original F-PIPRECIA-F-EDAS model are shown in Figure 14.

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**Figure 14.** Results obtained using different methods for determining criteria weights. **Figure 14.** Results obtained using different methods for determining criteria weights.

Applying the above methods for determining the significance of the criteria and including them into the F-EDAS method yield the results presented in Figure 14. In addition to the ranks shown on the left, values of variants on the right are defined. It can be observed that F-PIPRECIA and FUCOM give identical ranks, while, applying F-AHP, there are deviations in the ranks of the first and fourth Applying the above methods for determining the significance of the criteria and including them into the F-EDAS method yield the results presented in Figure 14. In addition to the ranks shown on the left, values of variants on the right are defined. It can be observed that F-PIPRECIA and FUCOM give identical ranks, while, applying F-AHP, there are deviations in the ranks of the first and fourth variants.

### variants. *5.5. Additional Correction of Criteria Weights Obtained Using F-AHP*

*5.5. Additional Correction of Criteria Weights Obtained Using F-AHP*  After presenting the previous results, the stability of the model is additionally determined as changing the significance of particular criteria. Therefore, a sensitivity analysis has been performed, which is presented throughout two parts in this subsection. Figure 15 shows the ranking of variants in all ten scenarios, while Figure 16 shows Spearman's coefficient of correlation for the ranking of variants. In the first set, the three most important criteria reduced the values by 10%, while the others increased by 15%. In the second set, the two most important criteria reduced the values by 15%, while the others increased by 10%. In the third set, the first criterion reduced by 20%, while the others increased by 5%. In next set, the second criterion reduced by 20%, while the others increased by 5%. In the fifth set, the fourth criterion reduced by 20%, while the others increased by 5%. In the next set, the first three criteria have values of 0.25, the fourth has 0.15, and the fifth has 0.1. In the seventh set, the criteria have values as follows: C1 = C2 = C4 = 0.30, C3 = 0.10, and the last criterion has a value of zero. In set 8: C1 = C2 = C4 = 0.30, C5 = 0.10, and the third criterion has a value of zero. In set 9: C1 = 0.34, After presenting the previous results, the stability of the model is additionally determined as changing the significance of particular criteria. Therefore, a sensitivity analysis has been performed, which is presented throughout two parts in this subsection. Figure 15 shows the ranking of variants in all ten scenarios, while Figure 16 shows Spearman's coefficient of correlation for the ranking of variants. In the first set, the three most important criteria reduced the values by 10%, while the others increased by 15%. In the second set, the two most important criteria reduced the values by 15%, while the others increased by 10%. In the third set, the first criterion reduced by 20%, while the others increased by 5%. In next set, the second criterion reduced by 20%, while the others increased by 5%. In the fifth set, the fourth criterion reduced by 20%, while the others increased by 5%. In the next set, the first three criteria have values of 0.25, the fourth has 0.15, and the fifth has 0.1. In the seventh set, the criteria have values as follows: C<sup>1</sup> = C<sup>2</sup> = C<sup>4</sup> = 0.30, C<sup>3</sup> = 0.10, and the last criterion has a value of zero. In set 8: C<sup>1</sup> = C<sup>2</sup> = C<sup>4</sup> = 0.30, C<sup>5</sup> = 0.10, and the third criterion has a value of zero. In set 9: C<sup>1</sup> = 0.34, C<sup>2</sup> = 0.27, C<sup>3</sup> = 0.20, C<sup>4</sup> = 0.13, C<sup>5</sup> = 0.06. In the last set, C<sup>1</sup> = 0.30, C<sup>2</sup> = 0.20, C<sup>3</sup> = 0.15, C<sup>4</sup> = 0.20, C<sup>5</sup> = 0.15.

C2 = 0.27, C3 = 0.20, C4 = 0.13, C5 = 0.06. In the last set, C1 = 0.30, C2 = 0.20, C3 = 0.15, C4 = 0.20, C5 = 0.15. As it can be seen in Figure 15, the seventh variant in seven, from ten formed sets, represents the best solution, while in the other scenarios, the best solution is variant six. The fifth variant is stable in all formed scenarios and has a third position. Variant three and two are also very stable and, only in the first and sixth sets, changing the position. Variant three has position five in the first set, while variant two has position six in the first and sixth sets. The ranking of the first variant varies from the fourth to seventh position in different scenarios, while the fourth variant varies from the fifth to seventh position. We can conclude that with the decrease, the three most important criteria by the 10% results and ranking of variants are very sensitive.

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**Figure 15.** Results of sensitivity analysis of Fuzzy Analytic Hierarchy Process (F-AHP)-F-EDAS changing the significance of criteria. **Figure 15.** Results of sensitivity analysis of Fuzzy Analytic Hierarchy Process (F-AHP)-F-EDAS changing the significance of criteria. considered stable. The average SCC value for all ten formed scenarios in relation to the initial rank is 0.948.

**Figure 16.** SCC through ten formed scenarios of F-AHP-F-EDAS model. **Figure 16.** SCC through ten formed scenarios of F-AHP-F-EDAS model.

**6. Discussion and Conclusion**  In certain cases, there is risk of insufficient financial resources for the execution of the PSO. The costs for the realization of PSO by the operators that are in a state or local ownership may also affect the possibility of implementing the model. Within the framework of the realization of this model, Figure 16 shows the SCC throughout all scenarios. From Figure 15, it can be seen that the model is sensitive to changes in the weight of the criteria and that each criterion can play an important role in the variant ranking. Spearman's coefficient of correlation has the range of 0.786–1.00, which represents a high degree of correlation, and the results obtained using the integrated fuzzy model are considered stable. The average SCC value for all ten formed scenarios in relation to the initial rank is 0.948.

### **6. Discussion and Conclusions**

0.948.

**Figure 16.** SCC through ten formed scenarios of F-AHP-F-EDAS model. **6. Discussion and Conclusion**  In certain cases, there is risk of insufficient financial resources for the execution of the PSO. The costs for the realization of PSO by the operators that are in a state or local ownership may also affect the possibility of implementing the model. Within the framework of the realization of this model, there are several possible sensitive situations that can appear from the moment of planning to the realization: Poor implementation of "business cost reduction" activities, especially with operators owned by government authorities; the lack of interest in "increasing revenue" in the gross contract, especially with operators owned by the authorities, regardless of whether they are revenue from the sale of tickets or other effects; incomplete and untimely realization of the fee for the execution of the PSO; lack of sufficient financial resources from the authorities to increase the fee for the execution of

In certain cases, there is risk of insufficient financial resources for the execution of the PSO. The

the PSO; the weakness of the state operators in the realization of other effects that can be realized on the basis of the granted right to perform PSO.

In this paper, a dynamic model for optimal application of the PSO system in the PPT process is proposed, which can contribute to the development of appropriate systems for the implementation of PPT services. In addition, it contributes to raising the service quality with the achievement of minimal costs of the functioning of these systems from the aspect of state and local government. By applying this model, it is possible to achieve a large number of effects (increase in passenger transport volume, higher and more stable quality of transport services, reduction in travel costs, better and more efficient cost control, etc.) and achieve significant savings in the functioning of the PPT system. Optimization of the PPT system has an indirect influence on the optimization of transport capacities and improvement in the quality of the transport service with economic quantification and cost savings.

The model was tested in the case of the organization of passenger traffic in the RRS (B&H). Based on the performed analysis, and in accordance with the task, the implementation of Variant V<sup>7</sup> is recommended as the most acceptable solution. As good enough solutions, we might accept variants V<sup>6</sup> and V5. The contribution of this research represents the possibility for rationalization of the PTT system in RRS. The new F-PIPRECIA-F-EDAS model developed in this research uses the strengths of fuzzy logic and multicriteria decision-making methods. One of the reasons for the F-PIPRECIA method application is its ability to equally handle quantitative and qualitative criteria. One of the reasons for using the F-EDAS method is a mathematical apparatus that assumes the evaluation of variants on the basis of positive and negative deviations from the average solution. The development of the new F-PIPRECIA-F-EDAS model based on TFNs represents the main scientific novelty of this paper. Future research related to this paper should be the implementation of the best variant and post-analysis of PPT systems.

**Author Contributions:** Conceptualization, S.V. and D.K.; methodology, Ž.S. and D.K.; validation, G.S. and S.M.; investigation, S.V. and S.R.; data curation, S.R.; writing—original draft preparation, Ž.S. and S.M.; writing—review and editing, S.V. and G.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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