# Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA)

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Preliminaries—Operations with Fuzzy Numbers

#### 3.2. Preliminaries—Dombi Aggregator

_{j}represents the weight of each considered year of data separately.

#### 3.3. Preliminaries—Bonferroni Aggregator

## 4. Materials and Methods

#### 4.1. The First Phase

_{1}, O

_{2}, O

_{3}, and O

_{4}, respectively.

#### 4.2. The Second Phase

#### 4.2.1. Improved Fuzzy SWARA Method (IMF SWARA)

_{j}represents the fuzzy relative weight of the criteria j, and m represents the total number of criteria.

#### 4.2.2. DEA Model

_{ij}, through s which represents outputs for each alternative y

_{ij}. Additionally, the weights of the parameters w

_{i}are taken into account [43]. The total number of DMUs is denoted by n.

#### 4.2.3. Fuzzy MARCOS Method

_{i}matrix:

_{i}

_{.}

#### 4.3. The Third Phase

## 5. Case Study

#### 5.1. Formation of Input-Output Parameters and Averaging Using Dombi and Bonferroni Aggregators

_{1}, O

_{2}, O

_{3}, and O

_{4}, respectively. These parameters are defined on the basis of the authors’ practical experiences, similar studies, and dialogue with other experts. Table 6 shows the values of all DMUs according to the input-output parameters.

**—**traffic accidents with fatalities:

**–**2019, while p, q ≥ 0 are a set of non-negative numbers. In the same way, averaged values are obtained for all outputs shown in Figure 4.

#### 5.2. Determination of Weight Values Using the Improved Fuzzy SWARA (IMF SWARA) Method

#### 5.3. Application of DEA Model

#### 5.4. Application of the Fuzzy MARCOS Method in Order to Make Final Ranking of Road Sections

## 6. Sensitivity Analysis

#### 6.1. Testing the Change in Weights of Inputs

#### 6.2. Comparison with Other MCDM Methods in a Fuzzy Form

#### 6.3. Influence of Dynamic Initial Matrix Formation

## 7. Conclusions

- (1)
- Using the fuzzy SWARA method, it is impossible to obtain results in which two criteria have equal fuzzy weights. By applying the improved fuzzy SWARA method, two or more criteria can have equal values.
- (2)
- On the contrary, applying the inadequate TFN scale shown in Table 2, where decision-makers indicate that two criteria have the same value by assigning TFN (1,1,1), the criterion Cj in relation to Cj−1 received a value that is twice less than Cj. By applying the improved fuzzy SWARA method, assigning the value (0,0,0), equal values are obtained and not values twice as large.
- (3)
- By increasing the number of criteria in the model, the least significant criteria receive values that can be negligible, i.e., with a tendency to zero. By applying the improved fuzzy SWARA method, less significant criteria have higher values and can play a greater role in the decision-making process.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mayora, J.P.; Rubio, R.L. Relevant Variables for Crash Rate Prediction in Spain´s Two Lane Rural Roads. In Proceedings of the 82nd Transportation Research Board Annual Meeting, Washington, DC, USA, 12–16 January 2003. [Google Scholar]
- Qiao, J.; Wen, Y.; Yang, N.; Song, J. The research of two-lane highway longitudinal slope based on the running speed in the plateau areas. In Proceedings of the 2011 International Conference on Consumer Electronics, Communications and Networks (CECNet), Xianning, China, 16–18 April 2011; pp. 1827–1830. [Google Scholar] [CrossRef]
- Hashim, I.H. Analysis of speed characteristics for rural two-lane roads: A field study from Minoufiya Governorate, Egypt. Ain Shams Eng. J.
**2011**, 2, 43–52. [Google Scholar] [CrossRef] [Green Version] - D’Andrea, A.; Carbone, F.; Salviera, S.; Pellegrino, O. The Most Influential Variables in the Determination of V85 Speed. Procedia Soc. Behav. Sci.
**2012**, 53, 633–644. [Google Scholar] [CrossRef] [Green Version] - Porter, R.J.; Donnell, E.T.; Mason, J.M. Geometric Design, Speed, and Safety. Transp. Res. Rec.
**2012**, 2309, 39–47. [Google Scholar] [CrossRef] - Highway Capacity Manual 2016 (HCM 2016); TRB. National Research Council: Washington, DC, USA, 2016.
- Handbuch für die Bemessung von Straßenverkehrsanlagen (HBS) (German Highway Capacity Manual); Forschungsgesellschaft für Straßen-und Verkehrswesen: Köln, Germany, 2005.
- AASHTO. The Highway Safety Manual; American Association of State Highway Transportation Professionals: Washington, DC, USA, 2010. [Google Scholar]
- Aarts, L.; van Schagen, I. Driving speed and the risk of road crashes: A review. Accid. Anal. Prev.
**2006**, 38, 215–224. [Google Scholar] [CrossRef] [PubMed] - Cruzado, I.; Donnell, E.T. Factors Affecting Driver Speed Choice along Two-Lane Rural Highway Transition Zones. J. Transp. Eng.
**2010**, 136, 755–764. [Google Scholar] [CrossRef] [Green Version] - Cheng, G.; Cheng, R.; Pei, Y.; Xu, L. Probability of Roadside Accidents for Curved Sections on Highways. Math. Probl. Eng.
**2020**, 2020, 1–18. [Google Scholar] [CrossRef] - Zheng, Y.; Guo, H.; Wei, X. The Evaluation Analysis of Design Code About the Road Design of longitudinal gradient in the Mountain Road. In Proceedings of the 7th International Conference on Education, Management, Computer and Society (EMCS 2017), Shenyang, China, 17–19 March 2017; pp. 693–699. [Google Scholar] [CrossRef] [Green Version]
- Srnová, B. A Case of Road Design in Mountainous Terrain with an Evaluation of Heavy Vehicles Performance Stockholm. Master’s Thesis, KTH Royal Institute of Technology, Stockholm, Sweden, 2017. [Google Scholar]
- Fitzpatrick, K.; Schneider, W.H.; Park, E.S. Comparisons of Crashes on Rural Two-Lane and Four-Lane Highways in Texas; FHWA/TX-06/0-4618-1; Report 0-4618-1; Texas Transportation Institute: Bryan, TX, USA, 2005. [Google Scholar]
- Calvo-Poyo, F.; de Oña, J.; Garach Morcillo, L.; Navarro-Moreno, J. Influence of Wider Longitudinal Road Markings on Vehicle Speeds in Two-Lane Rural Highways. Sustainability
**2020**, 12, 8305. [Google Scholar] [CrossRef] - Yue, L.; Wang, H. An Optimization Design Method of Combination of Steep Slope and Sharp Curve Sections for Mountain Highways. Math. Probl. Eng.
**2019**, 2019, 1–13. [Google Scholar] [CrossRef] [Green Version] - Kazemzadehazad, S.; Monajjem, S.; Larue, G.; King, M.J. Driving simulator validation for speed research on curves of two-lane rural roads. In Proceedings of the Institution of Civil Engineers—Transport; Thomas Telford Ltd.: London, UK, 2018; pp. 1–18. [Google Scholar] [CrossRef]
- Abdollahzadeh Nasiri, A.S.; Rahmani, O.; Abdi Kordani, A.; Karballaeezadeh, N.; Mosavi, A. Evaluation of Safety in Horizontal Curves of Roads Using a Multi-Body Dynamic Simulation Process. Int. J. Environ. Res. Public Health
**2020**, 17, 5975. [Google Scholar] [CrossRef] - Khabiri, M.M.; Ghaforifard, Z. The Effect of Low Friction in Pavement Due to Floods and High-speed Vehicles in Increasing the Number of Rescue Vehicles’ Driving Accidents. Health Emerg. Disasters
**2020**, 6, 29–38. [Google Scholar] [CrossRef] - Sil, G.; Maji, A.; Nama, S.; Maurya, A.K. Operating speed prediction model as a tool for consistency based geometric design of four-lane divided highways. Transport
**2019**, 34, 425–436. [Google Scholar] [CrossRef] [Green Version] - Alper, D.; Sinuany-Stern, Z.; Shinar, D. Evaluating the efficiency of local municipalities in providing traffic safety using the Data Envelopment Analysis. Accid. Anal. Prev.
**2015**, 78, 39–50. [Google Scholar] [CrossRef] - Podvezko, V.; Sivilevičius, H. The use of AHP and rank correlation methods for determining the significance of the interaction between the elements of a transport system having a strong influence on traffic safety. Transport
**2013**, 28, 389–403. [Google Scholar] [CrossRef] - Pirdavani, A.; Brijs, T.; Wets, G. A Multiple Criteria Decision-Making Approach for Prioritizing Accident Hotspots in the Absence of Crash Data. Transp. Rev.
**2010**, 30, 97–113. [Google Scholar] [CrossRef] - Barić, D.; Pilko, H.; Strujić, J. An analytic hierarchy process model to evaluate road section design. Transport
**2016**, 31, 312–321. [Google Scholar] [CrossRef] [Green Version] - Pilko, H.; Mandžuka, S.; Barić, D. Urban single-lane roundabouts: A new analytical approach using multi-criteria and simultaneous multi-objective optimization of geometry design, efficiency and safety. Transp. Res. Part C Emerg. Technol.
**2017**, 80, 257–271. [Google Scholar] [CrossRef] - Zak, J. The methodology of multiple criteria decision making/aiding in public transportation. J. Adv. Transp.
**2010**, 45, 1–20. [Google Scholar] [CrossRef] - Krstić, M.D.; Tadić, S.R.; Brnjac, N.; Zečević, S. Intermodal Terminal Handling Equipment Selection Using a Fuzzy Multi-criteria Decision-making Model. Promet Traffic Transp.
**2019**, 31, 89–100. [Google Scholar] [CrossRef] - Vakilipour, S.; Sadeghi-Niaraki, A.; Ghodousi, M.; Choi, S.-M. Comparison between Multi-Criteria Decision-Making Methods and Evaluating the Quality of Life at Different Spatial Levels. Sustainability
**2021**, 13, 4067. [Google Scholar] [CrossRef] - Yannis, G.; Kopsacheili, A.; Dragomanovits, A.; Petraki, V. State-of-the-art review on multi-criteria decision-making in the transport sector. J. Traffic Transp. Eng. (Engl. Ed.)
**2020**, 7, 413–431. [Google Scholar] [CrossRef] - Memiş, S.; Demir, E.; Karamaşa, Ç.; Korucuk, S. Prioritization of road transportation risks: An application in Giresun province. Oper. Res. Eng. Sci. Theory Appl.
**2020**, 3, 111–126. [Google Scholar] [CrossRef] - Đalić, I.; Ateljević, J.; Stević, Ž.; Terzić, S. An integrated swot–fuzzy piprecia model for analysis of competitiveness in order to improve logistics performances. Facta Univ. Ser. Mech. Eng.
**2020**, 18, 439–451. [Google Scholar] - Vesković, S.; Milinković, S.; Abramović, B.; Ljubaj, I. Determining criteria significance in selecting reach stackers by applying the fuzzy PIPRECIA method. Oper. Res. Eng. Sci. Theory Appl.
**2020**, 3, 72–88. [Google Scholar] [CrossRef] - Pamucar, D. Normalized weighted Geometric Dombi Bonferoni Mean Operator with interval grey numbers: Application in multicriteria decision making. Rep. Mech. Eng.
**2020**, 1, 44–52. [Google Scholar] [CrossRef] - Yager, R.R. On generalized Bonferroni mean operators for multi-criteria aggregation. Int. J. Approx. Reason.
**2009**, 50, 1279–1286. [Google Scholar] [CrossRef] [Green Version] - Mavi, R.K.; Goh, M.; Zarbakhshnia, N. Sustainable third-party reverse logistic provider selection with fuzzy SWARA and fuzzy MOORA in plastic industry. Int. J. Adv. Manuf. Technol.
**2017**, 91, 2401–2418. [Google Scholar] [CrossRef] - Chang, D.Y. Applications of the extent analysis method on fuzzy AHP. Eur. J. Oper. Res.
**1996**, 95, 649–655. [Google Scholar] [CrossRef] - Zarbakhshnia, N.; Soleimani, H.; Ghaderi, H. Sustainable third-party reverse logistics provider evaluation and selection using fuzzy SWARA and developed fuzzy COPRAS in the presence of risk criteria. Appl. Soft Comput.
**2018**, 65, 307–319. [Google Scholar] [CrossRef] - Ulutaş, A.; Karakuş, C.B.; Topal, A. Location selection for logistics center with fuzzy SWARA and CoCoSo methods. J. Intell. Fuzzy Syst.
**2020**, 38, 4693–4709. [Google Scholar] [CrossRef] - Ansari, Z.N.; Kant, R.; Shankar, R. Evaluation and ranking of solutions to mitigate sustainable remanufacturing supply chain risks: A hybrid fuzzy SWARA-fuzzy COPRAS framework approach. Int. J. Sustain. Eng.
**2020**, 13, 473–494. [Google Scholar] [CrossRef] - Ghasemi, P.; Mehdiabadi, A.; Spulbar, C.; Birau, R. Ranking of Sustainable Medical Tourism Destinations in Iran: An Integrated Approach Using Fuzzy SWARA-PROMETHEE. Sustainability
**2021**, 13, 683. [Google Scholar] [CrossRef] - Mitrović Simić, J.; Stević, Ž.; Zavadskas, E.K.; Bogdanović, V.; Subotić, M.; Mardani, A. A Novel CRITIC-Fuzzy FUCOM-DEA-Fuzzy MARCOS Model for Safety Evaluation of Road Sections Based on Geometric Parameters of Road. Symmetry
**2020**, 12, 2006. [Google Scholar] [CrossRef] - Blagojević, A.; Vesković, S.; Kasalica, S.; Gojić, A.; Allamani, A. The application of the fuzzy AHP and DEA for measuring the efficiency of freight transport railway undertakings. Oper. Res. Eng. Sci. Theory Appl.
**2020**, 3, 1–23. [Google Scholar] [CrossRef] - Andrejić, M.; Kilibarda, M.; Pajić, V. Measuring efficiency change in time applying malmquist productivity index: A case of distribution centres in Sserbia. Facta Univ. Ser. Mech. Eng.
**2021**. [Google Scholar] - Stanković, M.; Stević, Ž.; Das, D.K.; Subotić, M.; Pamučar, D. A new fuzzy MARCOS method for road traffic risk analysis. Mathematics
**2020**, 8, 457. [Google Scholar] [CrossRef] [Green Version] - Bakır, M.; Atalık, Ö. Application of Fuzzy AHP and Fuzzy MARCOS Approach for the Evaluation of E-Service Quality in the Airline Industry. Decis. Mak. Appl. Manag. Eng.
**2021**, 4, 127–152. [Google Scholar] [CrossRef] - Zahir, S. Normalisation and rank reversals in the additive analytic hierarchy process: A new analysis. Int. J. Oper. Res.
**2009**, 4, 446–467. [Google Scholar] [CrossRef] - Petrović, G.; Mihajlović, J.; Ćojbašić, Ž.; Madić, M.; Marinković, D. Comparison of three fuzzy MCDM methods for solving the supplier selection problem. Facta Univ. Ser. Mech. Eng.
**2019**, 17, 455–469. [Google Scholar] [CrossRef] - Roszkowska, E.; Kacprzak, D. The fuzzy saw and fuzzy TOPSIS procedures based on ordered fuzzy numbers. Inf. Sci.
**2016**, 369, 564–584. [Google Scholar] [CrossRef] - Hassanpour, M. Evaluation of Iranian wood and cellulose industries. Decis. Mak. Appl. Manag. Eng.
**2019**, 2, 13–34. [Google Scholar] [CrossRef] - Sałabun, W.; Wątróbski, J.; Shekhovtsov, A. Are MCDA Methods Benchmarkable? A Comparative Study of TOPSIS, VIKOR, COPRAS, and PROMETHEE II Methods. Symmetry
**2020**, 12, 1549. [Google Scholar] [CrossRef] - Andrejić, M.M.; Kilibarda, M.J. Measuring global logistics efficiency using PCA-DEA approach. Tehnika
**2016**, 71, 733–740. [Google Scholar] [CrossRef] - Zhang, Z.; Gao, Y.; Li, Z. Consensus reaching for social network group decision making by considering leadership and bounded confidence. Knowl. Based Syst.
**2020**, 204, 106240. [Google Scholar] [CrossRef] - Zhang, Z.; Li, Z.; Gao, Y. Consensus reaching for group decision making with multi-granular unbalanced linguistic information: A bounded confidence and minimum adjustment-based approach. Inf. Fusion
**2021**, 74, 96–110. [Google Scholar] [CrossRef] - Faizi, S.; Sałabun, W.; Nawaz, S. Best-Worst method and Hamacher aggregation operations for intuitionistic 2-tuple linguistic sets. Expert Syst. Appl.
**2021**, 115088. [Google Scholar] [CrossRef] - Badi, I.; Pamucar, D. Supplier selection for steelmaking company by using combined Grey-MARCOS methods. Decis. Mak. Appl. Manag. Eng.
**2020**, 3, 37–48. [Google Scholar] [CrossRef] - Zhang, Z.; Kou, X.; Yu, W.; Gao, Y. Consistency improvement for fuzzy preference relations with self-confidence: An application in two-sided matching decision making. J. Oper. Res. Soc.
**2020**, 1–14. [Google Scholar] [CrossRef]

**Table 1.**New linguistics and the TFN scale for the evaluation of the criteria in the improved fuzzy SWARA (IMF SWARA) method.

Linguistic Variable | Abbreviation | TFN Scale | ||
---|---|---|---|---|

Absolutely less significant | ALS | 1.000 | 1.000 | 1.000 |

Dominantly less significant | DLS | ½ | 2/3 | 1.000 |

Much less significant | MLS | 2/5 | 1/2 | 2/3 |

Really less significant | RLS | 1/3 | 2/5 | 1/2 |

Less significant | LS | 2/7 | 1/3 | 2/5 |

Moderately less significant | MDLS | ¼ | 2/7 | 1/3 |

Weakly less significant | WLS | 2/9 | 1/4 | 2/7 |

Equally significant | ES | 0.000 | 0.000 | 0.000 |

**Table 2.**Inadequate scale used by the authors in [35] in fuzzy SWARA.

Linguistic Scale | Response Scale |
---|---|

Equally important | (1, 1, 1) |

Moderately less important | (2/3, 1, 3/2) |

Less important | (2/5, 1/2, 2/3) |

Very less important | (2/7, 1/3, 2/5) |

Much less important | (2/9, 1/4, 2/7) |

**Table 3.**Calculation process and weights of the main criteria applying fuzzy SWARA in [35].

$\overline{{\mathit{s}}_{\mathit{j}}}$ | $\overline{{\mathit{k}}_{\mathit{j}}}$ | $\overline{{\mathit{q}}_{\mathit{j}}}$ | $\overline{{\mathit{w}}_{\mathit{j}}}$ | Crisp Value | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C1 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.377 | 0.405 | 0.444 | 0.407 | |||

C2 | 0.400 | 0.500 | 0.667 | 1.400 | 1.500 | 1.667 | 0.600 | 0.667 | 0.714 | 0.226 | 0.270 | 0.317 | 0.271 |

C3 | 0.222 | 0.250 | 0.286 | 1.222 | 1.250 | 1.286 | 0.467 | 0.533 | 0.584 | 0.176 | 0.216 | 0.259 | 0.217 |

C4 | 0.667 | 1.000 | 1.500 | 1.667 | 2.000 | 2.500 | 0.187 | 0.267 | 0.351 | 0.070 | 0.108 | 0.156 | 0.110 |

SUM | 2.253 | 2.467 | 2.649 |

**Table 4.**Calculation process and weights of the main criteria applying improved fuzzy SWARA (IMF SWARA) in the same example.

$\overline{{\mathit{s}}_{\mathit{j}}}$ | $\overline{{\mathit{k}}_{\mathit{j}}}$ | $\overline{{\mathit{q}}_{\mathit{j}}}$ | $\overline{{\mathit{w}}_{\mathit{j}}}$ | Crisp Value | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C1 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.360 | 0.379 | 0.406 | 0.380 | |||

C2 | 2/7 | 1/3 | 2/5 | 1.286 | 1.333 | 1.400 | 0.714 | 0.750 | 0.778 | 0.257 | 0.284 | 0.316 | 0.285 |

C3 | 2/5 | 1/2 | 2/3 | 1.400 | 1.500 | 1.667 | 0.429 | 0.500 | 0.556 | 0.154 | 0.189 | 0.225 | 0.190 |

C4 | 1/4 | 2/7 | 1/3 | 1.250 | 1.286 | 1.333 | 0.321 | 0.389 | 0.444 | 0.116 | 0.147 | 0.180 | 0.148 |

SUM | 2.464 | 2.639 | 2.778 |

**Table 5.**Comparative analysis of fuzzy SWARA and improved fuzzy SWARA (IMF SWARA) in another example.

Fuzzy SWARA | |||||||||||||

$\overline{{s}_{j}}$ | $\overline{{k}_{j}}$ | $\overline{{q}_{j}}$ | $\overline{{w}_{j}}$ | Crisp Value | |||||||||

C1 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.292 | 0.319 | 0.351 | 0.320 | |||

C2 | 0.286 | 0.333 | 0.400 | 1.286 | 1.333 | 1.400 | 0.714 | 0.750 | 0.778 | 0.209 | 0.239 | 0.273 | 0.240 |

C3 | 0.222 | 0.250 | 0.286 | 1.222 | 1.250 | 1.286 | 0.556 | 0.600 | 0.636 | 0.162 | 0.191 | 0.223 | 0.192 |

C4 | 0.400 | 0.500 | 0.667 | 1.400 | 1.500 | 1.667 | 0.333 | 0.400 | 0.455 | 0.097 | 0.127 | 0.160 | 0.128 |

C5 | 0.667 | 1.000 | 1.500 | 1.667 | 2.000 | 2.500 | 0.133 | 0.200 | 0.273 | 0.039 | 0.064 | 0.096 | 0.065 |

C6 | 1.000 | 1.000 | 1.000 | 2.000 | 2.000 | 2.000 | 0.067 | 0.100 | 0.136 | 0.019 | 0.032 | 0.048 | 0.032 |

C7 | 0.667 | 1.000 | 1.500 | 1.667 | 2.000 | 2.500 | 0.027 | 0.050 | 0.082 | 0.008 | 0.016 | 0.029 | 0.017 |

C8 | 0.286 | 0.333 | 0.400 | 1.286 | 1.333 | 1.400 | 0.019 | 0.038 | 0.064 | 0.006 | 0.012 | 0.022 | 0.013 |

SUM | 2.849 | 3.138 | 3.423 | ||||||||||

IMF SWARA | |||||||||||||

$\overline{{s}_{j}}$ | $\overline{{k}_{j}}$ | $\overline{{q}_{j}}$ | $\overline{{w}_{j}}$ | Crisp Value | |||||||||

C1 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.243 | 0.263 | 0.292 | 0.265 | |||

C2 | 2/7 | 1/3 | 2/5 | 1.286 | 1.333 | 1.400 | 0.714 | 0.750 | 0.778 | 0.174 | 0.198 | 0.227 | 0.199 |

C3 | 2/9 | 1/4 | 2/7 | 1.222 | 1.250 | 1.286 | 0.556 | 0.600 | 0.636 | 0.135 | 0.158 | 0.186 | 0.159 |

C4 | 2/5 | 1/2 | 2/3 | 1.400 | 1.500 | 1.667 | 0.333 | 0.400 | 0.455 | 0.081 | 0.105 | 0.133 | 0.106 |

C5 | 1/4 | 2/7 | 1/3 | 1.250 | 1.286 | 1.333 | 0.250 | 0.311 | 0.364 | 0.061 | 0.082 | 0.106 | 0.082 |

C6 | 0 | 0 | 0 | 1.000 | 1.000 | 1.000 | 0.250 | 0.311 | 0.364 | 0.061 | 0.082 | 0.106 | 0.082 |

C7 | 1/4 | 2/7 | 1/3 | 1.250 | 1.286 | 1.333 | 0.188 | 0.242 | 0.291 | 0.046 | 0.064 | 0.085 | 0.064 |

C8 | 2/7 | 1/3 | 2/5 | 1.286 | 1.333 | 1.400 | 0.134 | 0.181 | 0.226 | 0.033 | 0.048 | 0.066 | 0.048 |

SUM | 3.425 | 3.796 | 4.113 |

I_{1} | I_{2} | I_{3} | I_{4} | O_{1} | O_{2} | O_{3} | O_{4} | |
---|---|---|---|---|---|---|---|---|

DMU_{1} | 14.07 | 5.00 | 10.16 | 4578.95 | 0.63 | 2.49 | 3.70 | 7.26 |

DMU_{2} | 14.07 | 1.92 | 11.67 | 4578.95 | 0.63 | 2.49 | 3.70 | 7.26 |

DMU_{3} | 7.41 | 0.02 | 4.88 | 13,179.39 | 1.38 | 4.81 | 14.42 | 49.72 |

DMU_{4} | 20.95 | 1.00 | 9.61 | 5988.48 | 1.38 | 5.67 | 19.38 | 53.18 |

DMU_{5} | 15.35 | 3.00 | 6.33 | 3367.41 | 0.55 | 3.16 | 6.26 | 21.09 |

DMU_{6} | 3.14 | 7.00 | 6.29 | 3871.79 | 0.00 | 0.32 | 0.84 | 1.95 |

**Table 7.**Values of input and output significance in the considered model obtained by applying the IMF SWARA method.

IMF SWARA | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\overline{{\mathit{s}}_{\mathit{j}}}$ | $\overline{{\mathit{k}}_{\mathit{j}}}$ | $\overline{{\mathit{q}}_{\mathit{j}}}$ | $\overline{{\mathit{w}}_{\mathit{j}}}$ | Crisp Value | |||||||||

C5 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.203 | 0.213 | 0.225 | 0.213 | |||

C6 | 2/9 | 1/4 | 2/7 | 1.222 | 1.250 | 1.286 | 0.778 | 0.800 | 0.818 | 0.158 | 0.170 | 0.184 | 0.170 |

C3 | 0.000 | 0.000 | 0.000 | 1.000 | 1.000 | 1.000 | 0.778 | 0.800 | 0.818 | 0.158 | 0.170 | 0.184 | 0.170 |

C4 | 2/9 | 1/4 | 2/7 | 1.222 | 1.250 | 1.286 | 0.605 | 0.640 | 0.669 | 0.123 | 0.136 | 0.151 | 0.136 |

C1 | 1/4 | 2/7 | 1/3 | 1.250 | 1.286 | 1.333 | 0.454 | 0.498 | 0.536 | 0.092 | 0.106 | 0.121 | 0.106 |

C2 | 1/4 | 2/7 | 1/3 | 1.250 | 1.286 | 1.333 | 0.340 | 0.387 | 0.428 | 0.069 | 0.082 | 0.096 | 0.082 |

C7 | 2/7 | 1/3 | 2/5 | 1.286 | 1.333 | 1.400 | 0.243 | 0.290 | 0.333 | 0.049 | 0.062 | 0.075 | 0.062 |

C8 | 0.000 | 0.000 | 0.000 | 1.000 | 1.000 | 1.000 | 0.243 | 0.290 | 0.333 | 0.049 | 0.062 | 0.075 | 0.062 |

SUM | 4.441 | 4.706 | 4.936 |

DEA-Input | DEA-Output | DEA-Final | ||
---|---|---|---|---|

DMU1 | Vrhovi-Šešlije I | 1.00 | 1.00 | 1.00 |

DMU2 | Vrhovi-Šešlije II | 1.00 | 1.00 | 1.00 |

DMU3 | Rudanka-Doboj | 0.23 | 4.42 | 19.54 |

DMU4 | Šepak-Karakaj 3 | 0.38 | 2.66 | 7.064 |

DMU5 | Donje Caparde-Karakaj 1 | 0.65 | 1.54 | 2.362 |

DMU6 | Border (RS/FBIH)-Donje Caparde | 1.00 | 1.00 | 1.00 |

**Table 9.**Fuzzy linguistic scale for evaluating alternatives [41].

Benefit | Cost | ||
---|---|---|---|

Linguistic term | Mark | TFN | TFN |

Extremely poor | EP | (1,1,1) | (7,9,9) |

Very poor | VP | (1,1,3) | (7,7,9) |

Poor | P | (1,3,3) | (5,7,7) |

Medium poor | MP | (3,3,5) | (5,5,7) |

Medium | M | (3,5,5) | (3,5,5) |

Medium good | MG | (5,5,7) | (3,3,5) |

Good | G | (5,7,7) | (1,3,3) |

Very good | VG | (7,7,9) | (1,1,3) |

Extremely good | EG | (7,9,9) | (1,1,1) |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
---|---|---|---|---|---|---|---|---|

AAI | (1,3,3) | (3,3,5) | (5,5,7) | (3,5,5) | (3,3,5) | (5,5,7) | (5,5,7) | (7,7,9) |

DMU1 | (7,9,9) | (5,7,7) | (3,5,5) | (5,7,7) | (3,3,5) | (5,5,7) | (5,5,7) | (7,7,9) |

DMU2 | (7,9,9) | (3,3,5) | (5,5,7) | (5,7,7) | (3,3,5) | (5,5,7) | (5,5,7) | (7,7,9) |

DMU6 | (1,3,3) | (7,7,9) | (1,1,3) | (3,5,5) | (1,1,3) | (1,1,3) | (1,3,3) | (3,3,5) |

ID | (7,9,9) | (7,7,9) | (1,1,3) | (5,7,7) | (1,1,3) | (1,1,3) | (1,3,3) | (3,3,5) |

C1 | C2 | C3 | C4 | |

AAI | (0.111,0.333,0.333) | (0.333,0.333,0.556) | (0.143,0.2,0.2) | (0.429,0.714,0.714) |

DMU1 | (0.778,1,1) | (0.556,0.778,0.778) | (0.2,0.2,0.333) | (0.714,1,1) |

DMU2 | (0.778,1,1) | (0.333,0.333,0.556) | (0.143,0.2,0.2) | (0.714,1,1) |

DMU6 | (0.111,0.333,0.333) | (0.778,0.778,1) | (0.333,1,1) | (0.429,0.714,0.714) |

ID | (0.778,1,1) | (0.778,0.778,1) | (0.333,1,1) | (0.714,1,1) |

C5 | C6 | C7 | C8 | |

AAI | (0.2,0.333,0.333) | (0.143,0.2,0.2) | (0.143,0.2,0.2) | (0.333,0.429,0.429) |

DMU1 | (0.2,0.333,0.333) | (0.143,0.2,0.2) | (0.143,0.2,0.2) | (0.333,0.429,0.429) |

DMU2 | (0.2,0.333,0.333) | (0.143,0.2,0.2) | (0.143,0.2,0.2) | (0.333,0.429,0.429) |

DMU6 | (0.333,1,1) | (0.333,1,1) | (0.333,0.333,1) | (0.6,1,1) |

ID | (0.333,1,1) | (0.333,1,1) | (0.333,0.333,1) | (0.6,1,1) |

C1 | C2 | C3 | C4 | |

AAI | (0.02,0.07,0.07) | (0.06,0.06,0.09) | (0.02,0.03,0.03) | (0.06,0.1,0.1) |

DMU1 | (0.17,0.21,0.21) | (0.09,0.13,0.13) | (0.03,0.03,0.06) | (0.1,0.14,0.14) |

DMU2 | (0.17,0.21,0.21) | (0.06,0.06,0.09) | (0.02,0.03,0.03) | (0.1,0.14,0.14) |

DMU6 | (0.02,0.07,0.07) | (0.13,0.13,0.17) | (0.06,0.17,0.17) | (0.06,0.1,0.1) |

ID | (0.17,0.21,0.21) | (0.13,0.13,0.17) | (0.06,0.17,0.17) | (0.1,0.14,0.14) |

C5 | C6 | C7 | C8 | |

AAI | (0.02,0.04,0.04) | (0.01,0.02,0.02) | (0.01,0.01,0.01) | (0.02,0.03,0.03) |

DMU1 | (0.02,0.04,0.04) | (0.01,0.02,0.02) | (0.01,0.01,0.01) | (0.02,0.03,0.03) |

DMU2 | (0.02,0.04,0.04) | (0.01,0.02,0.02) | (0.01,0.01,0.01) | (0.02,0.03,0.03) |

DMU6 | (0.04,0.11,0.11) | (0.03,0.08,0.08) | (0.02,0.02,0.06) | (0.04,0.06,0.06) |

ID | (0.04,0.11,0.11) | (0.03,0.08,0.08) | (0.02,0.02,0.06) | (0.04,0.06,0.06) |

$\mathit{f}\left({\tilde{\mathit{K}}}_{\mathit{i}}^{-}\right)$ | $\mathit{f}\left({\tilde{\mathit{K}}}_{\mathit{i}}^{+}\right)$ | K- | K+ | fK- | fK+ | Ki | Rank | |
---|---|---|---|---|---|---|---|---|

DMU1 | (0.148,0.214,0.358) | (0.382,0.565,0.91) | 1.816 | 0.697 | 0.227 | 0.592 | 0.494 | 2 |

DMU2 | (0.132,0.188,0.324) | (0.342,0.495,0.822) | 1.606 | 0.616 | 0.201 | 0.524 | 0.378 | 3 |

DMU6 | (0.128,0.262,0.468) | (0.33,0.692,1.187) | 2.189 | 0.840 | 0.274 | 0.714 | 0.748 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Vrtagić, S.; Softić, E.; Subotić, M.; Stević, Ž.; Dordevic, M.; Ponjavic, M.
Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA). *Axioms* **2021**, *10*, 92.
https://doi.org/10.3390/axioms10020092

**AMA Style**

Vrtagić S, Softić E, Subotić M, Stević Ž, Dordevic M, Ponjavic M.
Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA). *Axioms*. 2021; 10(2):92.
https://doi.org/10.3390/axioms10020092

**Chicago/Turabian Style**

Vrtagić, Sabahudin, Edis Softić, Marko Subotić, Željko Stević, Milan Dordevic, and Mirza Ponjavic.
2021. "Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA)" *Axioms* 10, no. 2: 92.
https://doi.org/10.3390/axioms10020092