# A Novel Fuzzy SIMUS Multicriteria Decision-Making Method. An Application in Railway Passenger Transport Planning

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Stage 1: Forming the Parameters of Multi-Criteria Model

#### 3.1.1. Step 1. Forming the Initial Decision-Making Matrices

#### 3.1.2. Determination of the Normalized Matrices

#### 3.1.3. Step 3. Determination of the Threshold Values of the Criteria ($RHS$), (RHS Is Right Hand Side of Inequations)

#### 3.2. Stage 2 Forming the Fuzzy SIMUS Model for Each Objective

#### 3.2.1. Step 1. Solving SIMUS Procedure for Upper and Lower Initial Decision-Making Matrices

#### 3.2.2. Step 2. Solving Fuzzy Linear Optimization Models

#### 3.3. Stage 3 Ranking the Alternatives

## 4. Results and Discussion

- C1—Frequency of services, pair trains/day.
- C2—Frequency of train stops.
- C3—Average distance travelled, km.
- C4—Average operating speed, km/h.
- C5—Reliability. A coefficient accounting for the average delay of trains is determined.
- C6—Directness. If the alternative includes direct service: C6 = 1, otherwise: C6 = 0.
- C7—Train capacity, seats/day.
- C8—Direct operational costs, EUR/day.

- A1—Three categories of trains—category 1, category 2 and category 3. The train composition consists 4 wagons.
- A2—Three categories of trains—category 1, category 2 and category 3. The train composition consists 3 wagons.
- A3—Three categories of trains—category 1, category 2 and category 3. Category 1 are composed with 3 wagons, the other two categories—with 4 wagons.
- A4—Two categories of trains—category 1 and category 3. The other two categories are composed with 4 wagons.
- A5—Two categories of trains—category 1 and category 3. The other two categories are composed with 3 wagons.
- A6—Two categories of trains—category 1 and category 3. Category 1 are composed with 3 wagons, other—with 4 wagons.
- A7—Two categories of trains—category 2 and category 3. Both are composed with 4 wagons.
- A8—Two categories of trains—category 2 and category 3. The train composition consists 4 wagons.
- A9—two categories of trains—category 2 and category 3. Category 2 are composed with 3 wagons, other—with 4 wagons.

#### 4.1. Stage 1: Determination the Parameters of Multi-Criteria Model

#### 4.2. Stage 2: Fuzzy SIMUS Procedure

_{1}is:

#### 4.3. Stage 3: Ranking the Alternatives

#### 4.4. Comparison with Classical SIMUS Approach

#### 4.5. Verification of the Results

#### 4.6. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Membership function of ${Z}_{i}$. (

**a**)—case of the minimum of objectives; (

**b**)—case of the maximum of objectives.${Z}_{i,U}$, ${Z}_{i,L}$ are the upper and lower values of the optimization function received by linear optimization models.

Type | Fuzzy Method Used | Area of Evaluation | Authors |
---|---|---|---|

Pair-wise based | Fuzzy AHP | Urban transport development | [1] |

Transport planning | [2] | ||

Transport planning | [3] | ||

Railway timetable planning | [4] | ||

Location of parking selection | [5] | ||

Railway lines analysis | [6] | ||

Assessment metro system | [7] | ||

Fuzzy BWM | Road safety | [8] | |

Fuzzy FUCOM | Ranking transport demand | [9] | |

Distance based | Fuzzy CP | Supplier selection | [10] |

Fuzzy EDAS | Evaluation public transportation | [11] | |

Fuzzy TOPSIS | Transport projects selection | [12] | |

Evaluating transport systems | [13] | ||

Assessment of public transport | [14] | ||

Fuzzy VIKOR | Ranking railway transit lines | [15] | |

Railway infrastructure planning | [16] | ||

Risk assessment of the trains | [17] | ||

Selection hazardous waste transport | [18] | ||

Utility based | Fuzzy WASPAS | Information infrastructure | [19] |

Fuzzy PIPRECIA | Warehouse system | [20] | |

Logistics performance | [21] | ||

Fuzzy MARCOS | Road traffic analysis | [22] | |

Picture Fuzzy MARCOS | Railway infrastructure safety | [23] | |

Fuzzy COPRAS | Evaluating sustainable mobility | [24] | |

Fuzzy MOORA | Supplier selection | [25] | |

Fuzzy MULTIMOORA | Passenger satisfaction | [26] | |

Outranking approach | Fuzzy PROMETHEE | Logistics freight centre location | [27] |

Railway passenger service evaluation | [28] | ||

Combined approaches | Fuzzy AHP, DEA | Efficiency of railway undertaking | [29] |

Fuzzy AHP, Fuzzy TOPSIS | Public transport accessibility | [30] | |

Fuzzy AHP, TOPSIS | Railway supplier selection | [31] | |

Fuzzy Delphi, Fuzzy ELECTRE I | Intermodal route selection | [32] | |

Fuzzy DEMATEL, Fuzzy ANP, Fuzzy VIKOR | City logistics assessment | [33] | |

Fuzzy SWARA, Fuzzy TOPSIS, Fuzzy ARAS, Fuzzy EDAS | Supplier selection | [34] | |

Fuzzy PIPRECIA, Fuzzy EDAS | Railway passenger planning | [35] | |

Fuzzy AHP, Fuzzy EDAS | Supplier selection | [36] |

Criterion | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | Action | Type | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

C1 | C1, l | 36 | 42 | 38 | 31 | 40 | 35 | 34 | 41 | 37 | max | ≤ |

C1, m | 38 | 48 | 40 | 35 | 43 | 38 | 37 | 46 | 39 | |||

C1, u | 40 | 51 | 42 | 36 | 48 | 41 | 38 | 47 | 40 | |||

C2 | C2, l | 15.36 | 14.88 | 15.21 | 15.94 | 16.30 | 15.63 | 16.32 | 16.02 | 15.62 | min | ≥ |

C2, m | 15.45 | 14.94 | 14.68 | 16.17 | 16.79 | 15.55 | 16.19 | 15.72 | 15.62 | |||

C2, u | 14.85 | 15.08 | 14.36 | 16.21 | 16.18 | 15.63 | 16.23 | 15.78 | 15.79 | |||

C3 | C3, l | 327.00 | 344.00 | 335.00 | 350.00 | 361.00 | 355.00 | 335.00 | 335.00 | 331.00 | max | ≤ |

C3, u | 336.47 | 342.10 | 333.88 | 347.17 | 357.63 | 349.16 | 330.14 | 333.72 | 330.82 | |||

C3, m | 339.95 | 337.22 | 344.33 | 349.92 | 354.33 | 358.61 | 335.82 | 336.11 | 336.20 | |||

C4 | C4, l | 63 | 64 | 63 | 63 | 63 | 63 | 63 | 62 | 63 | max | ≤ |

C4, m | 63 | 64 | 63 | 63 | 63 | 63 | 63 | 63 | 63 | |||

C4, u | 64 | 64 | 64 | 63 | 63 | 64 | 63 | 63 | 63 | |||

C5 | - | 0.13 | 0.13 | 0.13 | 0.14 | 0.13 | 0.12 | 0.13 | 0.12 | 0.13 | min | ≥ |

C6 | - | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 1.00 | max | ≤ |

C7 | C7, l | 10,080 | 8820 | 10,360 | 8680 | 8400 | 9240 | 9520 | 8610 | 9660 | max | ≤ |

C7, m | 10,640 | 10,080 | 10,850 | 9800 | 9030 | 10,010 | 10,360 | 9660 | 10,150 | |||

C7, u | 11,200 | 10,710 | 11,410 | 10,080 | 10,080 | 10,780 | 10,640 | 9870 | 10,430 | |||

C8 | C8, l | 47,730 | 50,211 | 50,415 | 43,416 | 50,194 | 47,793 | 46,163 | 48,516 | 47,960 | min | ≥ |

C8, m | 51,583 | 57,234 | 52,615 | 48,772 | 53,639 | 51,159 | 49,716 | 54,073 | 50,232 | |||

C8, u | 54,336 | 60,144 | 56,472 | 50,687 | 59,177 | 56,583 | 51,969 | 55,688 | 52,485 |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 0.105 | 0.130 | 0.111 | 0.094 | 0.121 | 0.105 | 0.101 | 0.124 | 0.107 |

C2 | 0.108 | 0.106 | 0.105 | 0.114 | 0.117 | 0.111 | 0.115 | 0.112 | 0.111 |

C3 | 0.109 | 0.111 | 0.110 | 0.113 | 0.116 | 0.115 | 0.108 | 0.109 | 0.108 |

C4 | 0.111 | 0.113 | 0.111 | 0.111 | 0.111 | 0.111 | 0.111 | 0.110 | 0.111 |

C5 | 0.111 | 0.114 | 0.113 | 0.112 | 0.120 | 0.112 | 0.105 | 0.109 | 0.104 |

C6 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.000 | 0.000 | 0.000 |

C7 | 0.119 | 0.110 | 0.121 | 0.106 | 0.102 | 0.112 | 0.113 | 0.105 | 0.112 |

C8 | 0.110 | 0.120 | 0.114 | 0.102 | 0.117 | 0.111 | 0.106 | 0.113 | 0.108 |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 0.108 | 0.126 | 0.114 | 0.093 | 0.120 | 0.105 | 0.102 | 0.123 | 0.111 |

C2 | 0.109 | 0.105 | 0.108 | 0.113 | 0.115 | 0.111 | 0.116 | 0.113 | 0.111 |

C3 | 0.106 | 0.112 | 0.109 | 0.114 | 0.117 | 0.116 | 0.109 | 0.109 | 0.108 |

C4 | 0.111 | 0.113 | 0.111 | 0.111 | 0.111 | 0.111 | 0.111 | 0.109 | 0.111 |

C5 | 0.111 | 0.114 | 0.113 | 0.112 | 0.120 | 0.112 | 0.105 | 0.109 | 0.104 |

C6 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.000 | 0.000 | 0.000 |

C7 | 0.121 | 0.106 | 0.124 | 0.104 | 0.101 | 0.111 | 0.114 | 0.103 | 0.116 |

C8 | 0.110 | 0.116 | 0.117 | 0.100 | 0.116 | 0.111 | 0.107 | 0.112 | 0.111 |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | |
---|---|---|---|---|---|---|---|---|---|

C1 | 0.104 | 0.133 | 0.110 | 0.094 | 0.125 | 0.107 | 0.099 | 0.123 | 0.104 |

C2 | 0.106 | 0.108 | 0.102 | 0.116 | 0.115 | 0.112 | 0.116 | 0.113 | 0.113 |

C3 | 0.110 | 0.109 | 0.111 | 0.113 | 0.115 | 0.116 | 0.109 | 0.109 | 0.109 |

C4 | 0.112 | 0.112 | 0.112 | 0.110 | 0.110 | 0.112 | 0.110 | 0.110 | 0.110 |

C5 | 0.111 | 0.114 | 0.113 | 0.112 | 0.120 | 0.112 | 0.105 | 0.109 | 0.104 |

C6 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.167 | 0.000 | 0.000 | 0.000 |

C7 | 0.118 | 0.113 | 0.120 | 0.106 | 0.106 | 0.113 | 0.112 | 0.104 | 0.110 |

C8 | 0.109 | 0.121 | 0.114 | 0.102 | 0.119 | 0.114 | 0.104 | 0.112 | 0.105 |

Objective | Restrictive Conditions |
---|---|

Z2 | $0.109{x}_{11}+0.105{x}_{12}+0.108{x}_{13}+0.113{x}_{14}+0.115{x}_{15}+0.111{x}_{16}+0.116{x}_{17}+0.113{x}_{18}+0.111{x}_{19}\ge 0.105$ |

Z3 | $0.106{x}_{11}+0.112{x}_{12}+0.109{x}_{13}+0.114{x}_{14}+0.117{x}_{15}+0.116{x}_{16}+0.109{x}_{17}+0.109{x}_{18}+0.108{x}_{19}\le 0.117$ |

Z4 | $0.111{x}_{11}+0.113{x}_{12}+0.111{x}_{13}+0.111{x}_{14}+0.111{x}_{15}+0.111{x}_{16}+0.111{x}_{17}+0.109{x}_{18}+0.111{x}_{19}\le 0.113$ |

Z5 | $0.111{x}_{11}+0.114{x}_{12}+0.113{x}_{13}+0.112{x}_{14}+0.120{x}_{15}+0.112{x}_{16}+0.105{x}_{17}+0.109{x}_{18}+0.104{x}_{19}\ge 0.104$ |

Z6 | $0.167{x}_{11}+0.167{x}_{12}+0.167{x}_{13}+0.167{x}_{14}+0.167{x}_{15}+0.167{x}_{16}+0{x}_{17}+0{x}_{18}+0{x}_{19}\le $0.167 |

Z7 | $0.121{x}_{11}+0.106{x}_{12}+0.124{x}_{13}+0.104{x}_{14}+0.101{x}_{15}+0.111{x}_{16}+0.114{x}_{17}+0.103{x}_{18}+0.116{x}_{19}\le 0.124$ |

Z8 | $0.110{x}_{11}+0.116{x}_{12}+0.117{x}_{13}+0.100{x}_{14}+0.116{x}_{15}+0.111{x}_{16}+0.107{x}_{17}+0.112{x}_{18}+0.111{x}_{19}\ge 0.100$ |

${x}_{11},\dots ,{x}_{19}\geqq 0$ |

Objective | $\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{U}}$ | ${\mathit{Z}}_{\mathit{i},\mathit{U}}$ | $\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{L}}$ | ${\mathit{Z}}_{\mathit{i},\mathit{L}}$ | $\frac{{\mathit{Z}}_{\mathit{i},\mathit{U}}}{{\mathit{Z}}_{\mathit{i},\mathit{U}}-{\mathit{Z}}_{\mathit{i},\mathit{L}}}$ | $\frac{{\mathit{Z}}_{\mathit{i},\mathit{L}}}{{\mathit{Z}}_{\mathit{i},\mathit{U}}-{\mathit{Z}}_{\mathit{i},\mathit{L}}}$ | $\frac{\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{U}}}{\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{U}}-\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{L}}}$ | $\frac{\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{L}}}{\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{U}}-\mathit{R}\mathit{H}{\mathit{S}}_{\mathit{j},\mathit{L}}}$ |
---|---|---|---|---|---|---|---|---|

Z1 | 0.133 | 0.133 | 0.126 | 0.127 | 20.661 | 19.66 | 17.97 | 16.97 |

Z2 | 0.102 | 0.095 | 0.105 | 0.097 | 51.979 | 52.98 | 36.19 | 37.19 |

Z3 | 0.116 | 0.116 | 0.117 | 0.119 | 39.696 | 40.70 | 76.63 | 77.63 |

Z4 | 0.112 | 0.119 | 0.113 | 0.121 | 58.232 | 59.23 | 141.75 | 142.75 |

Z5 | 0.104 | 0.099 | 0.104 | 0.097 | - | - | - | - |

Z6 | 0.167 | 0.169 | 0.167 | 0.169 | - | - | - | - |

Z7 | 0.120 | 0.120 | 0.124 | 0.126 | 19.172 | 20.17 | 27.16 | 28.16 |

Z8 | 0.102 | 0.095 | 0.100 | 0.094 | 100.824 | 99.82 | 69.42 | 68.42 |

**Table 8.**Values of Coefficients of Membership Functions for Optimization Functions $\frac{{Z}_{i}}{{Z}_{i,U}-{Z}_{i,L}}$.

Objective | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 |
---|---|---|---|---|---|---|---|---|---|

Z1 | 16.36 | 20.24 | 17.22 | 14.64 | 18.80 | 16.36 | 15.65 | 19.23 | 16.65 |

Z2 | 59.29 | 58.30 | 57.46 | 62.75 | 63.98 | 60.78 | 63.29 | 61.71 | 61.07 |

Z3 | 37.12 | 37.86 | 37.48 | 38.73 | 39.69 | 39.31 | 37.03 | 37.17 | 36.92 |

Z4 | 54.55 | 55.13 | 54.55 | 54.27 | 54.27 | 54.55 | 54.27 | 53.98 | 54.27 |

Z5 | - | - | - | - | - | - | - | - | - |

Z6 | - | - | - | - | - | - | - | - | - |

Z7 | 18.97 | 17.60 | 19.39 | 16.97 | 16.35 | 17.85 | 18.14 | 16.72 | 17.97 |

Z8 | 116.97 | 127.58 | 121.43 | 108.77 | 124.10 | 118.41 | 112.56 | 120.49 | 114.71 |

**Table 9.**Values of Coefficients of Membership Functions for Restrictive Conditions $\frac{{\sum}_{j=1}^{J}{b}_{ij,s}{x}_{ij}}{RH{S}_{j,U}-RH{S}_{j,L}}$.

Objective | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 |
---|---|---|---|---|---|---|---|---|---|

Z1 | 14.23 | 17.60 | 14.98 | 12.73 | 16.35 | 14.23 | 13.61 | 16.73 | 14.48 |

Z2 | 38.16 | 37.53 | 36.98 | 40.39 | 41.18 | 39.12 | 40.74 | 39.72 | 39.31 |

Z3 | 71.87 | 73.29 | 72.57 | 75.00 | 76.85 | 76.12 | 71.69 | 71.97 | 71.48 |

Z4 | 140.85 | 142.33 | 140.85 | 140.11 | 140.11 | 140.85 | 140.11 | 139.37 | 140.11 |

Z5 | - | - | - | - | - | - | - | - | - |

Z6 | - | - | - | - | - | - | - | - | - |

Z7 | 26.88 | 24.93 | 27.47 | 24.05 | 23.16 | 25.29 | 25.70 | 23.69 | 25.46 |

Z8 | 74.84 | 81.63 | 77.69 | 69.59 | 79.40 | 75.76 | 72.02 | 77.09 | 73.39 |

Objective | Restrictive Conditions |
---|---|

Z2 | ${\mu}_{NALH{S}_{2}}:38.161{x}_{11}+37.529{x}_{12}+36.982{x}_{13}+40.387{x}_{14}+41.179{x}_{15}+39.124{x}_{16}+40.737{x}_{17}+39.717{x}_{18}+39.309{x}_{19}-37.192\ge {\lambda}_{1}$ |

Z3 | ${\mu}_{NALH{S}_{3}}:-71.869{x}_{11}-73.298{x}_{12}-72.568{x}_{13}-74.997{x}_{14}-76.853{x}_{15}-76.118{x}_{16}-71.692{x}_{17}-71.971{x}_{18}-71.482{x}_{19}+76.633\ge {\lambda}_{1}$ |

Z4 | ${\mu}_{NALH{S}_{4}}:-140.847{x}_{11}-142.333{x}_{12}-140.847{x}_{13}-140.109{x}_{14}-140.109{x}_{15}-140.847{x}_{16}-140.109{x}_{17}-139.365{x}_{18}-140.109{x}_{19}+141.750\ge {\lambda}_{1}$ |

Z5 | ${Z}_{5}:0.111{x}_{1}+0.114{x}_{12}+0.113{x}_{13}+0.112{x}_{14}+0.120{x}_{15}+0.112{x}_{16}+0.105{x}_{17}+0.109{x}_{18}+0.104{x}_{19}\ge 0.120$ |

Z6 | ${Z}_{6}:0.167{x}_{11}+0.167{x}_{12}+0.167{x}_{13}+0.167{x}_{14}+0.167{x}_{15}+0.167{x}_{16}+0{x}_{17}+0{x}_{18}+0{x}_{19}\le $0.167 |

Z7 | ${\mu}_{NALH{S}_{7}}:26.88{x}_{11}-24.89{x}_{12}-27.49{x}_{13}-24.04{x}_{14}-23.14{x}_{15}-25.278{x}_{16}-25.71{x}_{17}-23.69{x}_{18}-25.49{x}_{19}+27.163\ge {\lambda}_{1}$ |

Z8 | ${\mu}_{NALH{S}_{8}}:74.85{x}_{11}+81.55{x}_{12}+77.74{x}_{13}+69.56{x}_{14}+79.35{x}_{15}+75.71{x}_{16}+72.05{x}_{17}+77.09{x}_{18}+73.48{x}_{19}-68.420\ge {\lambda}_{1}$ |

$0\le {\lambda}_{1}\le 1$ | |

${x}_{11},\dots ,{x}_{19}\geqq 0$ |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | $\mathbf{\sum}_{\mathit{j}=1}^{\mathit{J}}{\mathit{x}}_{\mathit{i}\mathit{j}}$ | ${\mathit{\lambda}}_{\mathit{i}}$ | ${\mathit{Z}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Z1 | 0.000 | 0.000 | 0.701 | 0.000 | 0.298 | 0.000 | 0.000 | 0.000 | 0.000 | 0.999 | 1.000 | - |

Z2 | 0.000 | 0.000 | 0.000 | 0.000 | 0.997 | 0.000 | 0.000 | 0.000 | 0.000 | 0.997 | 0.000 | - |

Z3 | 0.000 | 0.000 | 0.626 | 0.372 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.998 | 1.000 | - |

Z4 | 0.000 | 0.000 | 0.626 | 0.372 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.998 | 1.000 | - |

Z5 | 0.000 | 0.048 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.895 | 0.943 | - | 0.098 |

Z6 | 0.237 | 0.000 | 0.000 | 0.000 | 0.779 | 0.000 | 0.000 | 0.000 | 0.000 | 1.016 | - | 0.169 |

Z7 | 0.000 | 0.000 | 0.976 | 0.000 | 0.024 | 0.000 | 0.053 | 0.000 | 0.000 | 1.053 | 0.129 | - |

Z8 | 0.000 | 0.000 | 0.000 | 0.927 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.927 | 1.000 | - |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | |
---|---|---|---|---|---|---|---|---|---|

Z1 | 0.000 | 0.000 | 0.701 | 0.000 | 0.299 | 0.000 | 0.000 | 0.000 | 0.000 |

Z2 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Z3 | 0.000 | 0.000 | 0.627 | 0.373 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Z4 | 0.000 | 0.000 | 0.627 | 0.373 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Z5 | 0.000 | 0.051 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.949 |

Z6 | 0.233 | 0.000 | 0.000 | 0.000 | 0.767 | 0.000 | 0.000 | 0.000 | 0.000 |

Z7 | 0.000 | 0.000 | 0.976 | 0.000 | 0.024 | 0.000 | 0.053 | 0.000 | 0.000 |

Z8 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

$S{C}_{j}$ | 0.233 | 0.051 | 2.932 | 1.746 | 2.089 | 0.000 | 0.053 | 0.000 | 0.949 |

$P{F}_{j}$ | 1 | 1 | 4 | 3 | 4 | 0 | 1 | 0 | 1 |

$NP{F}_{j}$ | 0.125 | 0.125 | 0.500 | 0.375 | 0.500 | 0.000 | 0.125 | 0.000 | 0.125 |

$NP{F}_{j}\xb7S{C}_{j}$ | 0.029 | 0.006 | 1.466 | 0.655 | 1.045 | 0.000 | 0.007 | 0.000 | 0.119 |

ranking | 5 | 7 | 1 | 3 | 2 | 8 or 9 | 6 | 9 or 8 | 4 |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | $\mathit{S}{\mathit{R}}_{\mathit{j}}$ | $\mathit{S}{\mathit{C}}_{\mathit{j}}$ | $\mathit{S}{\mathit{R}}_{\mathit{j}}$$\mathbf{-}\mathit{S}{\mathit{C}}_{\mathit{j}}$ | Rank | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A1 | - | - | - | - | - | - | - | - | - | 0 | 9 | −9 | 6 |

A2 | - | - | - | - | - | - | - | - | - | 0 | 9 | −9 | 7 |

A3 | 4 | 4 | - | 4 | 4 | 4 | 4 | 4 | 4 | 32 | 5 | 27 | 1 |

A4 | 1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 0 | 3 |

A5 | 2 | 2 | 2 | 2 | - | 2 | 2 | 2 | 2 | 16 | 7 | 9 | 2 |

A6 | - | - | - | - | - | - | - | - | - | 0 | 9 | −9 | 8 |

A7 | 1 | 1 | 1 | 1 | 1 | 1 | - | 1 | 1 | 8 | 8 | 0 | 4 |

A8 | - | - | - | - | - | - | - | - | - | 0 | 9 | −9 | 9 |

A9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | - | 8 | 8 | 0 | 5 |

$S{C}_{j}$ | 9 | 9 | 5 | 8 | 7 | 9 | 8 | 9 | 8 | ||||

PDM Ranking 3-5-4-7-9-1-2-6-8 |

Alternative | Lower Values | Average Values | Upper Values | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SC | PF | NPF | SC × NPF | Rank | SC | PF | NPF | SC × NPF | Rank | SC | PF | NPF | SC × NPF | Rank | |

A1 | 0.73 | 2 | 0.25 | 0.18 | 4 | 0.00 | 0 | 0.00 | 0.00 | 8 | 0.08 | 1 | 0.13 | 0.01 | 6 |

A2 | 1.38 | 2 | 0.25 | 0.35 | 2 | 1.03 | 2 | 0.25 | 0.26 | 5 | 2.11 | 3 | 0.38 | 0.79 | 1 |

A3 | 0.98 | 1 | 0.13 | 0.12 | 7 | 2.18 | 3 | 0.38 | 0.82 | 1 | 2.00 | 2 | 0.25 | 0.50 | 2 |

A4 | 1.00 | 1 | 0.13 | 0.13 | 6 | 1.00 | 1 | 0.13 | 0.13 | 7 | 1.54 | 2 | 0.25 | 0.38 | 3 |

A5 | 1.79 | 2 | 0.25 | 0.45 | 1 | 1.75 | 2 | 0.25 | 0.44 | 2 | 1.45 | 2 | 0.25 | 0.36 | 4 |

A6 | 0.00 | 0 | 0.00 | 0.00 | 9 | 0.00 | 0 | 0.00 | 0.00 | 9 | 0.00 | 0 | 0.00 | 0.00 | 8 |

A7 | 0.60 | 1 | 0.13 | 0.08 | 8 | 0.02 | 1 | 0.13 | 0.00 | 7 | 0.02 | 1 | 0.13 | 0.00 | 7 |

A8 | 1.02 | 2 | 0.25 | 0.25 | 3 | 0.69 | 3 | 0.38 | 0.26 | 4 | 0.00 | 0 | 0.00 | 0.00 | 9 |

A9 | 0.49 | 3 | 0.38 | 0.18 | 5 | 1.33 | 2 | 0.25 | 0.33 | 3 | 0.81 | 2 | 0.25 | 0.20 | 5 |

Alternative | Score | $\mathit{E}{\mathit{V}}_{\mathit{j}},$ [47] | Rank Fuzzy SIMUS | Rank [47] | $\mathit{d}$ | ${\mathit{d}}^{\mathbf{2}}$ |
---|---|---|---|---|---|---|

A1 | 0.029 | 0.005 | 5 | 7 | 2 | 4 |

A2 | 0.006 | 0.030 | 7 | 3 | 4 | 16 |

A3 | 1.466 | 0.041 | 1 | 1 | 0 | 0 |

A4 | 0.655 | 0.013 | 3 | 6 | 3 | 9 |

A5 | 1.045 | 0.031 | 2 | 2 | 0 | 0 |

A6 | 0.000 | 0.000 | 9 | 9 | 0 | 0 |

A7 | 0.007 | 0.002 | 6 | 8 | 2 | 4 |

A8 | 0.000 | 0.015 | 8 | 5 | 3 | 9 |

A9 | 0.119 | 0.018 | 4 | 4 | 0 | 0 |

Spearman Rank Correlation coefficient ${r}_{s}$ | 0.65 |

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## Share and Cite

**MDPI and ACS Style**

Stoilova, S.; Munier, N.
A Novel Fuzzy SIMUS Multicriteria Decision-Making Method. An Application in Railway Passenger Transport Planning. *Symmetry* **2021**, *13*, 483.
https://doi.org/10.3390/sym13030483

**AMA Style**

Stoilova S, Munier N.
A Novel Fuzzy SIMUS Multicriteria Decision-Making Method. An Application in Railway Passenger Transport Planning. *Symmetry*. 2021; 13(3):483.
https://doi.org/10.3390/sym13030483

**Chicago/Turabian Style**

Stoilova, Svetla, and Nolberto Munier.
2021. "A Novel Fuzzy SIMUS Multicriteria Decision-Making Method. An Application in Railway Passenger Transport Planning" *Symmetry* 13, no. 3: 483.
https://doi.org/10.3390/sym13030483