# Ranking Startups Using DEMATEL-ANP-Based Fuzzy PROMETHEE II

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}has the highest comprehensive potential, followed by startup project A

_{3}.

## 1. Introduction

## 2. Literature Review

#### 2.1. Accelerators and the Startup Selection Approach

#### 2.2. DEMATEL

#### 2.3. AHP

#### 2.4. Fuzzy PROMETHEE II

#### 2.5. Ranking Fuzzy Numbers

## 3. Model Establishment

#### 3.1. Fuzzy Set Theory

- Fuzzy Sets

- Fuzzy Numbers

- α-Cuts

- Arithmetic Operations on Fuzzy Numbers

- Linguistic Values

#### 3.2. Relative Maximizing and Minimizing Sets

_{i}is denoted as in Equation (8).

#### 3.3. Spread Area-Based RMMS

#### 3.4. The Hybrid DEMATEL-ANP Based Fuzzy PROMETHEE II Model

#### 3.4.1. DEMATEL

_{i}has on factor (criterion) C

_{j}in a system with m factors (criteria) $C=\left\{{C}_{1},{C}_{2},\dots ,{C}_{m}\right\}$ using an integer scale of No Effect (0), Low Effect (1), Medium Low Effect (2), Medium Effect (3), Medium High Effect (4), High Effect (5) and Extremely Strong Effect (6). Next, the individual direct-influence matrix ${Z}_{e}={\left[{z}_{ij}^{e}\right]}_{m\times m}$ provided by the eth expert can be constructed, where all main diagonal components are equal to zero and ${z}_{ij}^{e}$ represent the respondent’s evaluation of DM on the degree to which criterion C

_{i}affects C

_{j}.

_{j}influences other criteria and can be grouped into a causal group; if (D + R) is negative, the criterion C

_{j}is being influenced by the other criteria and can be grouped into an effect group. A causal diagram can be produced by mapping the (D + R, D − R) dataset, yielding valuable assessment perception. A threshold value can be defined to screen out the negligible factors [68,69]. In this work, factors that have a value higher than the average value of the “Prominence” (D + R) and/or (D − R) is positive are selected to use in the next step.

#### 3.4.2. ANP

_{i}is obtained through Equation (25), which is computed by each row’s average.

_{j}is the corresponding row of the comparison matrix, E is Eigenvector and E

_{j}represents the corresponding component in E.

_{max}is obtained by the average of the CM vector. The CI is calculated as shown in Equation (27).

#### 3.4.3. Fuzzy PROMETHEE-Based Ranking Method

_{ij}can be as

_{i}over A

_{i’}is obtained by Equations (32) and (33), based on Equation (3)

_{j}is the fuzzy preference function for the jth criterion and C

_{j}(A

_{i}) is the evaluation of alternative A

_{i}corresponding to criterion C

_{j}.

_{i}is determined as

_{i}is determined as

## 4. Numerical Comparison and Consistency Test

## 5. Numerical Example

_{6}) demand validation” has the greatest (D + R) value and is the most critical factor, followed by “(C

_{7}) customer affordability” and “(C

_{8}) market demographic”. All these factors need to be evaluated in the initial steps when building a product or service. Additionally, the (D − R) values of “(C

_{3}) prior startup experience”, “(C

_{1}) sales”, and “(C

_{2}) product development cost” demonstrate that these criteria have net influences on other factors. Other medium value factors that are selected when proceeding to the next steps are “(C

_{9}) concept maturity”, “(C

_{10}) product maturity”, “(C

_{11}) value proposition”, “(C

_{13}) technology experience”, “(C

_{15}) growth strategy”, “(C

_{18}) creativity”, and “(C

_{19}) negotiation”.

_{8}) market demographics” has the highest value with 0.1253, followed by “(C

_{6}) demand validation” with 0.1196 and “(C

_{3}) prior startup experience” with 0.0940. The lowest weight value is “(C

_{11}) value proposition” with 0.0215.

_{1}(−0.0519), A

_{2}(0.0905), A

_{3}(0.0594) and A

_{4}(−0.0980) as presented in Table 16. The final ranking of four startup projects ${A}_{4}<{A}_{1}<{A}_{3}<{A}_{2}$ indicates that startup project A

_{2}has the highest comprehensive potential, followed by startup project A

_{3}.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | C_{16} | C_{17} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 0 | 2 | 1.5 | 4 | 5 | 1 | 1 | 1 | 1.5 | 1 | 2.75 | 4 | 4 | 3 | 2 | 5 | 3 | 3 | 3 |

C_{2} | 5.5 | 0 | 1.25 | 5 | 4 | 1.25 | 1 | 1.75 | 1.25 | 2 | 1 | 4 | 2 | 3 | 1 | 4 | 2 | 2 | 2 |

C_{3} | 6 | 6 | 0 | 6 | 6 | 4 | 5 | 4 | 3 | 3 | 4 | 6 | 5 | 6 | 6 | 5 | 6 | 5 | 5 |

C_{4} | 4 | 3 | 1 | 0 | 4 | 1 | 1.75 | 1 | 2 | 1 | 2 | 3 | 1 | 5 | 2 | 3 | 3 | 4 | 2 |

C_{5} | 3 | 4 | 1 | 4 | 0 | 1.75 | 2 | 2 | 1 | 2 | 1 | 4 | 4 | 6 | 1 | 2 | 3 | 5 | 4 |

C_{6} | 6 | 6 | 4 | 5.75 | 6 | 0 | 4.25 | 4.5 | 6 | 6 | 6 | 6 | 4 | 5.75 | 6 | 6 | 5.75 | 5.25 | 3.75 |

C_{7} | 6 | 6 | 3 | 6 | 6 | 3.75 | 0 | 3.25 | 6 | 5.75 | 5.75 | 6 | 6 | 6 | 5.25 | 5.5 | 6 | 4.75 | 3.75 |

C_{8} | 5.75 | 6 | 4 | 5.75 | 5.25 | 3.5 | 4.75 | 0 | 5.5 | 5.25 | 6 | 5.5 | 4 | 6 | 5.75 | 6 | 5.75 | 5 | 3.75 |

C_{9} | 6 | 6 | 5 | 6 | 6 | 1.5 | 1 | 2.5 | 0 | 6 | 5 | 6 | 3.25 | 6 | 6 | 6 | 5 | 4 | 4 |

C_{10} | 6 | 6 | 5 | 6 | 5.75 | 1 | 1 | 2.25 | 2 | 0 | 5 | 6 | 3 | 6 | 5 | 5 | 6 | 3 | 4 |

C_{11} | 5.25 | 6 | 4 | 6 | 6 | 2 | 2 | 1.75 | 2.75 | 3 | 0 | 6 | 5 | 5 | 3 | 6 | 4 | 3 | 3 |

C_{12} | 4 | 4 | 1 | 4.75 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 0 | 2 | 3 | 2 | 4 | 4 | 2 | 3 |

C_{13} | 4 | 6 | 3 | 6 | 4 | 4 | 2 | 4 | 4.75 | 5 | 3 | 6 | 0 | 6 | 6 | 5 | 5 | 6 | 6 |

C_{14} | 5 | 5 | 2 | 3 | 2 | 2.25 | 2 | 1.5 | 1.75 | 1.75 | 3 | 5 | 1 | 0 | 1 | 2 | 2 | 1 | 3 |

C_{15} | 6 | 6 | 2 | 6 | 6 | 1 | 2.75 | 2 | 1.75 | 3 | 5 | 6 | 2 | 6 | 0 | 6 | 6 | 2 | 3 |

C_{16} | 3 | 4 | 3 | 5 | 6 | 2 | 1.75 | 1 | 2 | 3 | 2 | 4 | 3 | 6 | 1 | 0 | 3 | 2 | 2 |

C_{17} | 5 | 6 | 2 | 5 | 5 | 1.75 | 2 | 2 | 3 | 2 | 4 | 4 | 3 | 6 | 2 | 5 | 0 | 3 | 2 |

C_{18} | 4.75 | 5.75 | 3 | 4 | 3 | 2.75 | 3 | 3 | 4 | 5 | 5 | 6 | 2 | 6 | 6 | 6 | 5 | 0 | 5 |

C_{19} | 5 | 6 | 3 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 2 | 5 | 5 | 6 | 6 | 3 | 0 |

## Appendix D

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | C_{16} | C_{17} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 0 | 0.0206 | 0.0155 | 0.0412 | 0.0515 | 0.0103 | 0.0103 | 0.0103 | 0.0155 | 0.0103 | 0.0284 | 0.0412 | 0.0412 | 0.0309 | 0.0206 | 0.0515 | 0.0309 | 0.0309 | 0.0309 |

C_{2} | 0.0567 | 0 | 0.0129 | 0.0515 | 0.0412 | 0.0129 | 0.0103 | 0.0180 | 0.0129 | 0.0206 | 0.0103 | 0.0412 | 0.0206 | 0.0309 | 0.0103 | 0.0412 | 0.0206 | 0.0206 | 0.0206 |

C_{3} | 0.0619 | 0.0619 | 0 | 0.0619 | 0.0619 | 0.0412 | 0.0515 | 0.0412 | 0.0309 | 0.0309 | 0.0412 | 0.0619 | 0.0515 | 0.0619 | 0.0619 | 0.0515 | 0.0619 | 0.0515 | 0.0515 |

C_{4} | 0.0412 | 0.0309 | 0.0103 | 0 | 0.0412 | 0.0103 | 0.0180 | 0.0103 | 0.0206 | 0.0103 | 0.0206 | 0.0309 | 0.0103 | 0.0515 | 0.0206 | 0.0309 | 0.0309 | 0.0412 | 0.0206 |

C_{5} | 0.0309 | 0.0412 | 0.0103 | 0.0412 | 0 | 0.0180 | 0.0206 | 0.0206 | 0.0103 | 0.0206 | 0.0103 | 0.0412 | 0.0412 | 0.0619 | 0.0103 | 0.0206 | 0.0309 | 0.0515 | 0.0412 |

C_{6} | 0.0619 | 0.0619 | 0.0412 | 0.0593 | 0.0619 | 0 | 0.0438 | 0.0464 | 0.0619 | 0.0619 | 0.0619 | 0.0619 | 0.0412 | 0.0593 | 0.0619 | 0.0619 | 0.0593 | 0.0541 | 0.0387 |

C_{7} | 0.0619 | 0.0619 | 0.0309 | 0.0619 | 0.0619 | 0.0387 | 0 | 0.0335 | 0.0619 | 0.0593 | 0.0593 | 0.0619 | 0.0619 | 0.0619 | 0.0541 | 0.0567 | 0.0619 | 0.0490 | 0.0387 |

C_{8} | 0.0593 | 0.0619 | 0.0412 | 0.0593 | 0.0541 | 0.0361 | 0.0490 | 0 | 0.0567 | 0.0541 | 0.0619 | 0.0567 | 0.0412 | 0.0619 | 0.0593 | 0.0619 | 0.0593 | 0.0515 | 0.0387 |

C_{9} | 0.0619 | 0.0619 | 0.0515 | 0.0619 | 0.0619 | 0.0155 | 0.0103 | 0.0258 | 0 | 0.0619 | 0.0515 | 0.0619 | 0.0335 | 0.0619 | 0.0619 | 0.0619 | 0.0515 | 0.0412 | 0.0412 |

C_{10} | 0.0619 | 0.0619 | 0.0515 | 0.0619 | 0.0593 | 0.0103 | 0.0103 | 0.0232 | 0.0206 | 0 | 0.0515 | 0.0619 | 0.0309 | 0.0619 | 0.0515 | 0.0515 | 0.0619 | 0.0309 | 0.0412 |

C_{11} | 0.0541 | 0.0619 | 0.0412 | 0.0619 | 0.0619 | 0.0206 | 0.0206 | 0.0180 | 0.0284 | 0.0309 | 0 | 0.0619 | 0.0515 | 0.0515 | 0.0309 | 0.0619 | 0.0412 | 0.0309 | 0.0309 |

C_{12} | 0.0412 | 0.0412 | 0.0103 | 0.0490 | 0.0412 | 0.0103 | 0.0103 | 0.0103 | 0.0103 | 0.0206 | 0.0206 | 0 | 0.0206 | 0.0309 | 0.0206 | 0.0412 | 0.0412 | 0.0206 | 0.0309 |

C_{13} | 0.0412 | 0.0619 | 0.0309 | 0.0619 | 0.0412 | 0.0412 | 0.0206 | 0.0412 | 0.0490 | 0.0515 | 0.0309 | 0.0619 | 0 | 0.0619 | 0.0619 | 0.0515 | 0.0515 | 0.0619 | 0.0619 |

C_{14} | 0.0515 | 0.0515 | 0.0206 | 0.0309 | 0.0206 | 0.0232 | 0.0206 | 0.0155 | 0.0180 | 0.0180 | 0.0309 | 0.0515 | 0.0103 | 0 | 0.0103 | 0.0206 | 0.0206 | 0.0103 | 0.0309 |

C_{15} | 0.0619 | 0.0619 | 0.0206 | 0.0619 | 0.0619 | 0.0103 | 0.0284 | 0.0206 | 0.0180 | 0.0309 | 0.0515 | 0.0619 | 0.0206 | 0.0619 | 0 | 0.0619 | 0.0619 | 0.0206 | 0.0309 |

C_{16} | 0.0309 | 0.0412 | 0.0309 | 0.0515 | 0.0619 | 0.0206 | 0.0180 | 0.0103 | 0.0206 | 0.0309 | 0.0206 | 0.0412 | 0.0309 | 0.0619 | 0.0103 | 0 | 0.0309 | 0.0206 | 0.0206 |

C_{17} | 0.0515 | 0.0619 | 0.0206 | 0.0515 | 0.0515 | 0.0180 | 0.0206 | 0.0206 | 0.0309 | 0.0206 | 0.0412 | 0.0412 | 0.0309 | 0.0619 | 0.0206 | 0.0515 | 0 | 0.0309 | 0.0206 |

C_{18} | 0.0490 | 0.0593 | 0.0309 | 0.0412 | 0.0309 | 0.0284 | 0.0309 | 0.0309 | 0.0412 | 0.0515 | 0.0515 | 0.0619 | 0.0206 | 0.0619 | 0.0619 | 0.0619 | 0.0515 | 0 | 0.0515 |

C_{19} | 0.0515 | 0.0619 | 0.0309 | 0.0619 | 0.0412 | 0.0412 | 0.0412 | 0.0412 | 0.0412 | 0.0412 | 0.0515 | 0.0515 | 0.0206 | 0.0515 | 0.0515 | 0.0619 | 0.0619 | 0.0309 | 0 |

## Appendix E

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | C_{16} | C_{17} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 0.0681 | 0.0911 | 0.0507 | 0.1117 | 0.1165 | 0.0406 | 0.0418 | 0.0417 | 0.0532 | 0.0539 | 0.0754 | 0.1097 | 0.0820 | 0.1040 | 0.0655 | 0.1147 | 0.0891 | 0.0780 | 0.0781 |

C_{2} | 0.1152 | 0.0620 | 0.0442 | 0.1134 | 0.1000 | 0.0391 | 0.0381 | 0.0450 | 0.0462 | 0.0580 | 0.0532 | 0.1018 | 0.0581 | 0.0954 | 0.0504 | 0.0979 | 0.0727 | 0.0631 | 0.0629 |

C_{3} | 0.1930 | 0.1963 | 0.0692 | 0.1986 | 0.1895 | 0.0979 | 0.1110 | 0.1010 | 0.1050 | 0.1151 | 0.1353 | 0.1951 | 0.1320 | 0.2009 | 0.1486 | 0.1774 | 0.1747 | 0.1419 | 0.1419 |

C_{4} | 0.1022 | 0.0939 | 0.0424 | 0.0646 | 0.1001 | 0.0372 | 0.0459 | 0.0383 | 0.0542 | 0.0495 | 0.0642 | 0.0936 | 0.0488 | 0.1156 | 0.0607 | 0.0893 | 0.0830 | 0.0821 | 0.0635 |

C_{5} | 0.1045 | 0.1161 | 0.0487 | 0.1166 | 0.0708 | 0.0507 | 0.0543 | 0.0545 | 0.0524 | 0.0675 | 0.0635 | 0.1154 | 0.0841 | 0.1374 | 0.0606 | 0.0910 | 0.0938 | 0.1004 | 0.0917 |

C_{6} | 0.2028 | 0.2062 | 0.1154 | 0.2064 | 0.1997 | 0.0614 | 0.1072 | 0.1095 | 0.1383 | 0.1502 | 0.1615 | 0.2051 | 0.1286 | 0.2088 | 0.1554 | 0.1962 | 0.1806 | 0.1503 | 0.1363 |

C_{7} | 0.1983 | 0.2019 | 0.1034 | 0.2044 | 0.1952 | 0.0969 | 0.0626 | 0.0957 | 0.1360 | 0.1451 | 0.1556 | 0.2009 | 0.1449 | 0.2067 | 0.1453 | 0.1873 | 0.1790 | 0.1430 | 0.1338 |

C_{8} | 0.1952 | 0.2009 | 0.1124 | 0.2009 | 0.1873 | 0.0941 | 0.1096 | 0.0627 | 0.1307 | 0.1396 | 0.1577 | 0.1950 | 0.1254 | 0.2056 | 0.1493 | 0.1912 | 0.1759 | 0.1442 | 0.1326 |

C_{9} | 0.1810 | 0.1836 | 0.1125 | 0.1862 | 0.1783 | 0.0671 | 0.0657 | 0.0797 | 0.0652 | 0.1341 | 0.1349 | 0.1828 | 0.1071 | 0.1881 | 0.1389 | 0.1748 | 0.1541 | 0.1226 | 0.1237 |

C_{10} | 0.1690 | 0.1713 | 0.1054 | 0.1737 | 0.1641 | 0.0575 | 0.0606 | 0.0718 | 0.0791 | 0.0670 | 0.1258 | 0.1703 | 0.0975 | 0.1753 | 0.1205 | 0.1537 | 0.1528 | 0.1050 | 0.1152 |

C_{11} | 0.1562 | 0.1659 | 0.0934 | 0.1685 | 0.1614 | 0.0655 | 0.0677 | 0.0652 | 0.0846 | 0.0953 | 0.0727 | 0.1653 | 0.1140 | 0.1607 | 0.0988 | 0.1582 | 0.1293 | 0.1028 | 0.1029 |

C_{12} | 0.1037 | 0.1051 | 0.0431 | 0.1140 | 0.1026 | 0.0378 | 0.0393 | 0.0391 | 0.0452 | 0.0596 | 0.0648 | 0.0649 | 0.0595 | 0.0985 | 0.0614 | 0.1006 | 0.0942 | 0.0645 | 0.0739 |

C_{13} | 0.1672 | 0.1894 | 0.0965 | 0.1911 | 0.1631 | 0.0941 | 0.0785 | 0.0975 | 0.1172 | 0.1301 | 0.1213 | 0.1880 | 0.0765 | 0.1933 | 0.1446 | 0.1709 | 0.1593 | 0.1453 | 0.1463 |

C_{14} | 0.1139 | 0.1143 | 0.0532 | 0.0973 | 0.0834 | 0.0500 | 0.0491 | 0.0441 | 0.0528 | 0.0576 | 0.0749 | 0.1143 | 0.0505 | 0.0667 | 0.0526 | 0.0819 | 0.0752 | 0.0545 | 0.0738 |

C_{15} | 0.1589 | 0.1607 | 0.0711 | 0.1633 | 0.1571 | 0.0529 | 0.0721 | 0.0642 | 0.0714 | 0.0909 | 0.1184 | 0.1600 | 0.0822 | 0.1648 | 0.0632 | 0.1535 | 0.1435 | 0.0885 | 0.0981 |

C_{16} | 0.1067 | 0.1181 | 0.0692 | 0.1289 | 0.1331 | 0.0533 | 0.0525 | 0.0452 | 0.0619 | 0.0774 | 0.0734 | 0.1175 | 0.0770 | 0.1404 | 0.0606 | 0.0716 | 0.0952 | 0.0737 | 0.0738 |

C_{17} | 0.1379 | 0.1492 | 0.0659 | 0.1415 | 0.1357 | 0.0558 | 0.0599 | 0.0598 | 0.0781 | 0.0755 | 0.1011 | 0.1297 | 0.0846 | 0.1527 | 0.0777 | 0.1331 | 0.0749 | 0.0909 | 0.0815 |

C_{18} | 0.1638 | 0.1759 | 0.0910 | 0.1614 | 0.1443 | 0.0770 | 0.0827 | 0.0822 | 0.1032 | 0.1224 | 0.1324 | 0.1772 | 0.0915 | 0.1816 | 0.1359 | 0.1703 | 0.1496 | 0.0782 | 0.1285 |

C_{19} | 0.1689 | 0.1808 | 0.0923 | 0.1833 | 0.1570 | 0.0904 | 0.0939 | 0.0932 | 0.1056 | 0.1148 | 0.1343 | 0.1701 | 0.0941 | 0.1755 | 0.1281 | 0.1727 | 0.1613 | 0.1115 | 0.0810 |

## Appendix F

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 2 | 3 | 3 | 1/4 | 1/5 | 1/4 | 1/5 | 3 | 1/7 | 1/4 | 1/3 | 1/2 |

C_{2} | 1/2 | 1 | 1/3 | 3 | 1/5 | 1/8 | 1/4 | 1/5 | 1/4 | 1/8 | 1/3 | 1/8 | 1/5 |

C_{3} | 1/3 | 3 | 1 | 3 | 1/5 | 1/4 | 2 | 1/2 | 3 | 1/6 | 1/3 | 1/3 | 1/6 |

C_{6} | 1/3 | 1/3 | 1/3 | 1 | 1/5 | 1/8 | 1/2 | 1/5 | 1/3 | 1/8 | 1/6 | 1/8 | 1/3 |

C_{7} | 4 | 5 | 5 | 5 | 1 | 3 | 6 | 3 | 6 | 1/2 | 4 | 2 | 3 |

C_{8} | 5 | 8 | 4 | 8 | 1/3 | 1 | 5 | 2 | 5 | 1/3 | 2 | 2 | 3 |

C_{9} | 4 | 4 | 1/2 | 2 | 1/6 | 1/5 | 1 | 1/2 | 2 | 1/8 | 1/2 | 1/7 | 1/5 |

C_{10} | 5 | 5 | 2 | 5 | 1/3 | 1/2 | 2 | 1 | 3 | 1/6 | 1/2 | 1/2 | 2 |

C_{11} | 1/3 | 4 | 1/3 | 3 | 1/6 | 1/5 | 1/2 | 1/3 | 1 | 1/8 | 1/2 | 1/5 | 1/4 |

C_{13} | 7 | 8 | 6 | 8 | 2 | 3 | 8 | 6 | 8 | 1 | 4 | 2 | 4 |

C_{15} | 4 | 3 | 3 | 6 | 1/4 | 1/2 | 2 | 2 | 2 | 1/4 | 1 | 1/2 | 1/2 |

C_{18} | 3 | 8 | 3 | 8 | 1/2 | 1/2 | 7 | 5 | 5 | 1/2 | 2 | 1 | 2 |

C_{19} | 2 | 5 | 6 | 3 | 1/3 | 1/3 | 5 | 4 | 4 | 1/4 | 2 | 1/2 | 1 |

Inconsistency: 0.08328 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 1/6 | 1/5 | 1/7 | 1/8 | 1/8 | 1/6 | 1/8 | 3 | 1/2 | 3 | 1/4 | 3 |

C_{2} | 6 | 1 | 1/3 | 1/5 | 1/2 | 1/4 | 1/3 | 1/2 | 3 | 1/2 | 4 | 1/2 | 4 |

C_{3} | 5 | 3 | 1 | 3 | 3 | 1/2 | 2 | 4 | 7 | 2 | 5 | 2 | 6 |

C_{6} | 7 | 5 | 1/3 | 1 | 5 | 3 | 2 | 3 | 9 | 5 | 7 | 3 | 7 |

C_{7} | 8 | 2 | 1/3 | 1/5 | 1 | 1/6 | 3 | 1/2 | 5 | 2 | 4 | 1/2 | 4 |

C_{8} | 8 | 4 | 2 | 1/3 | 6 | 1 | 3 | 2 | 8 | 6 | 4 | 2 | 6 |

C_{9} | 6 | 3 | 1/2 | 1/2 | 1/3 | 1/3 | 1 | 1/2 | 4 | 2 | 4 | 1/3 | 5 |

C_{10} | 8 | 2 | 1/4 | 1/3 | 2 | 1/2 | 2 | 1 | 6 | 1/3 | 4 | 1/2 | 4 |

C_{11} | 1/3 | 1/3 | 1/7 | 1/9 | 1/5 | 1/8 | 1/4 | 1/6 | 1 | 1/3 | 1/2 | 1/7 | 1/3 |

C_{13} | 2 | 2 | 1/2 | 1/5 | 1/2 | 1/6 | 1/2 | 3 | 3 | 1 | 3 | 1/2 | 2 |

C_{15} | 1/3 | 1/4 | 1/5 | 1/7 | 1/4 | 1/4 | 1/4 | 1/4 | 2 | 1/3 | 1 | 1/6 | 2 |

C_{18} | 4 | 2 | 1/2 | 1/3 | 2 | 1/2 | 3 | 2 | 7 | 2 | 6 | 1 | 4 |

C_{19} | 1/3 | 1/4 | 1/6 | 1/7 | 1/4 | 1/6 | 1/5 | 1/4 | 3 | 1/2 | 1/2 | 1/4 | 1 |

Inconsistency: 0.09659 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 3 | 3 | 1/2 | 3 | 2 | 8 | 5 | 6 | 3 | 2 | 4 | 3 |

C_{2} | 1/3 | 1 | 3 | 1/2 | 2 | 2 | 4 | 3 | 3 | 2 | 4 | 2 | 1/2 |

C_{3} | 1/3 | 1/3 | 1 | 1/3 | 3 | 1/2 | 2 | 1/3 | 2 | 1/3 | 1/2 | 3 | 1/4 |

C_{6} | 2 | 2 | 3 | 1 | 1/2 | 2 | 4 | 2 | 4 | 3 | 2 | 5 | 2 |

C_{7} | 1/3 | 1/2 | 1/3 | 2 | 1 | 1/2 | 2 | 1/3 | 2 | 1/2 | 3 | 2 | 1/3 |

C_{8} | 1/2 | 1/2 | 2 | 1/2 | 2 | 1 | 3 | 1/2 | 3 | 1/2 | 2 | 3 | 1/2 |

C_{9} | 1/8 | 1/4 | 1/2 | 1/4 | 1/2 | 1/3 | 1 | 1/4 | 1/2 | 1/3 | 1/2 | 1/2 | 1/4 |

C_{10} | 1/5 | 1/3 | 3 | 1/2 | 3 | 2 | 4 | 1 | 2 | 3 | 1/2 | 4 | 1/3 |

C_{11} | 1/6 | 1/3 | 1/2 | 1/4 | 1/2 | 1/3 | 2 | 1/2 | 1 | 1/2 | 1/3 | 1/2 | 1/5 |

C_{13} | 1/3 | 1/2 | 3 | 1/3 | 2 | 2 | 3 | 1/3 | 2 | 1 | 3 | 1/2 | 2 |

C_{15} | 1/2 | 1/4 | 2 | 1/2 | 1/3 | 1/2 | 2 | 2 | 3 | 1/3 | 1 | 3 | 1/3 |

C_{18} | 1/4 | 1/2 | 1/3 | 1/5 | 1/2 | 1/3 | 2 | 1/4 | 2 | 2 | 1/3 | 1 | 1/2 |

C_{19} | 1/3 | 2 | 4 | 2 | 3 | 2 | 4 | 3 | 5 | 1/2 | 3 | 2 | 1 |

Inconsistency: 0.09391 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 1/6 | 1/5 | 1/7 | 1/6 | 1/7 | 1/4 | 1/5 | 3 | 2 | 3 | 3 | 4 |

C_{2} | 6 | 1 | 1/3 | 1/4 | 1/2 | 1/4 | 1/3 | 1/3 | 3 | 1/2 | 4 | 1/2 | 3 |

C_{3} | 5 | 3 | 1 | 4 | 3 | 1/2 | 2 | 4 | 6 | 3 | 4 | 6 | 7 |

C_{6} | 7 | 4 | 1/4 | 1 | 4 | 1/2 | 2 | 2 | 7 | 8 | 6 | 6 | 7 |

C_{7} | 6 | 2 | 1/3 | 1/4 | 1 | 1/2 | 1/3 | 2 | 5 | 3 | 4 | 3 | 2 |

C_{8} | 7 | 4 | 2 | 2 | 2 | 1 | 3 | 2 | 8 | 5 | 8 | 6 | 7 |

C_{9} | 4 | 3 | 1/2 | 1/2 | 3 | 1/3 | 1 | 2 | 7 | 3 | 7 | 3 | 5 |

C_{10} | 5 | 3 | 1/4 | 1/2 | 1/2 | 1/2 | 1/2 | 1 | 5 | 3 | 4 | 3 | 5 |

C_{11} | 1/3 | 1/3 | 1/6 | 1/7 | 1/5 | 1/8 | 1/7 | 1/5 | 1 | 1/3 | 1/2 | 1/4 | 1/2 |

C_{13} | 1/2 | 2 | 1/3 | 1/8 | 1/3 | 1/5 | 1/3 | 3 | 3 | 1 | 4 | 1/2 | 1/2 |

C_{15} | 1/3 | 1/4 | 1/4 | 1/6 | 1/4 | 1/8 | 1/7 | 1/4 | 2 | 1/4 | 1 | 1/3 | 1/2 |

C_{18} | 1/3 | 2 | 1/6 | 1/6 | 1/3 | 1/6 | 1/3 | 1/3 | 4 | 2 | 3 | 1 | 3 |

C_{19} | 1/4 | 1/3 | 1/7 | 1/7 | 1/2 | 1/7 | 1/5 | 1/5 | 2 | 2 | 2 | 1/3 | 1 |

Inconsistency: 0.09784 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 1/5 | 1/5 | 1/4 | 1/6 | 1/7 | 1/4 | 1/5 | 3 | 2 | 3 | 3 | 4 |

C_{2} | 5 | 1 | 1/4 | 1/4 | 1/2 | 1/4 | 1/3 | 1/4 | 3 | 1/2 | 4 | 1/2 | 3 |

C_{3} | 5 | 4 | 1 | 4 | 2 | 1/2 | 2 | 4 | 6 | 3 | 4 | 6 | 7 |

C_{6} | 4 | 4 | 1/4 | 1 | 4 | 1/2 | 2 | 2 | 7 | 8 | 6 | 6 | 7 |

C_{7} | 6 | 2 | 1/2 | 1/4 | 1 | 1/2 | 1/3 | 2 | 5 | 3 | 4 | 3 | 2 |

C_{8} | 7 | 4 | 2 | 2 | 2 | 1 | 3 | 2 | 8 | 5 | 8 | 6 | 7 |

C_{9} | 4 | 3 | 1/2 | 1/2 | 3 | 1/3 | 1 | 2 | 7 | 3 | 7 | 3 | 5 |

C_{10} | 5 | 4 | 1/4 | 1/2 | 1/2 | 1/2 | 1/2 | 1 | 5 | 3 | 4 | 3 | 5 |

C_{11} | 1/3 | 1/3 | 1/6 | 1/7 | 1/5 | 1/8 | 1/7 | 1/5 | 1 | 1/3 | 1/2 | 1/4 | 1/2 |

C_{13} | 1/2 | 2 | 1/3 | 1/8 | 1/3 | 1/5 | 1/3 | 3 | 3 | 1 | 4 | 1/2 | 1/2 |

C_{15} | 1/3 | 1/4 | 1/4 | 1/6 | 1/4 | 1/8 | 1/7 | 1/4 | 2 | 1/4 | 1 | 1/3 | 1/2 |

C_{18} | 1/3 | 2 | 1/6 | 1/6 | 1/3 | 1/6 | 1/3 | 1/3 | 4 | 2 | 3 | 1 | 3 |

C_{19} | 1/4 | 1/3 | 1/7 | 1/7 | 1/2 | 1/7 | 1/5 | 1/5 | 2 | 2 | 2 | 1/3 | 1 |

Inconsistency: 0.09426 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 1/5 | 1/5 | 1/4 | 1/6 | 1/7 | 1/4 | 1/5 | 3 | 2 | 3 | 3 | 4 |

C_{2} | 5 | 1 | 1/4 | 1/4 | 1/2 | 1/4 | 1/3 | 1/4 | 3 | 1/2 | 4 | 1/2 | 3 |

C_{3} | 5 | 4 | 1 | 4 | 2 | 1/2 | 2 | 4 | 6 | 3 | 4 | 6 | 7 |

C_{6} | 4 | 4 | 1/4 | 1 | 4 | 1/2 | 2 | 2 | 7 | 5 | 5 | 4 | 5 |

C_{7} | 6 | 2 | 1/2 | 1/4 | 1 | 1/2 | 1/3 | 2 | 5 | 3 | 4 | 3 | 2 |

C_{8} | 7 | 4 | 2 | 2 | 2 | 1 | 3 | 2 | 8 | 5 | 8 | 6 | 7 |

C_{9} | 4 | 3 | 1/2 | 1/2 | 3 | 1/3 | 1 | 2 | 7 | 3 | 7 | 3 | 5 |

C_{10} | 5 | 4 | 1/4 | 1/2 | 1/2 | 1/2 | 1/2 | 1 | 5 | 3 | 4 | 3 | 5 |

C_{11} | 1/3 | 1/3 | 1/6 | 1/7 | 1/5 | 1/8 | 1/7 | 1/5 | 1 | 1/3 | 1/2 | 1/4 | 1/2 |

C_{13} | 1/2 | 2 | 1/3 | 1/5 | 1/3 | 1/5 | 1/3 | 3 | 3 | 1 | 4 | 1/2 | 1/2 |

C_{15} | 1/3 | 1/4 | 1/4 | 1/5 | 1/4 | 1/8 | 1/7 | 1/4 | 2 | 1/4 | 1 | 1/2 | 1/3 |

C_{18} | 1/3 | 2 | 1/6 | 1/4 | 1/3 | 1/6 | 1/3 | 1/3 | 4 | 2 | 2 | 1 | 2 |

C_{19} | 1/4 | 1/3 | 1/7 | 1/5 | 1/2 | 1/7 | 1/5 | 1/5 | 2 | 2 | 3 | 1/2 | 1 |

Inconsistency: 0.09230 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 3 | 3 | 1/2 | 3 | 2 | 6 | 3 | 4 | 2 | 2 | 4 | 2 |

C_{2} | 1/3 | 1 | 3 | 1/2 | 2 | 2 | 5 | 3 | 3 | 2 | 4 | 2 | 1/2 |

C_{3} | 1/3 | 1/3 | 1 | 1/3 | 1/2 | 1/2 | 2 | 1/3 | 2 | 1/3 | 1/2 | 3 | 1/4 |

C_{6} | 2 | 2 | 3 | 1 | 1/2 | 2 | 4 | 2 | 4 | 3 | 2 | 5 | 2 |

C_{7} | 1/3 | 1/2 | 2 | 2 | 1 | 1/2 | 2 | 1/3 | 2 | 1/2 | 3 | 2 | 1/3 |

C_{8} | 1/2 | 1/2 | 2 | 1/2 | 2 | 1 | 3 | 1/2 | 3 | 1/2 | 2 | 3 | 1/2 |

C_{9} | 1/6 | 1/5 | 1/2 | 1/4 | 1/2 | 1/3 | 1 | 1/4 | 1/2 | 1/3 | 1/2 | 1/2 | 1/4 |

C_{10} | 1/3 | 1/3 | 3 | 1/2 | 3 | 2 | 4 | 1 | 2 | 3 | 1/2 | 4 | 1/3 |

C_{11} | 1/4 | 1/3 | 1/2 | 1/4 | 1/2 | 1/3 | 2 | 1/2 | 1 | 1/2 | 1/3 | 1/2 | 1/5 |

C_{13} | 1/2 | 1/2 | 3 | 1/3 | 2 | 2 | 3 | 1/3 | 2 | 1 | 3 | 1/2 | 2 |

C_{15} | 1/2 | 1/4 | 2 | 1/2 | 1/3 | 1/2 | 2 | 2 | 3 | 1/3 | 1 | 3 | 1/3 |

C_{18} | 1/4 | 1/2 | 1/3 | 1/5 | 1/2 | 1/3 | 2 | 1/4 | 2 | 2 | 1/3 | 1 | 1/2 |

C_{19} | 1/2 | 2 | 4 | 2 | 3 | 2 | 4 | 3 | 5 | 1/2 | 3 | 2 | 1 |

Inconsistency: 0.09890 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 3 | 3 | 1/2 | 3 | 2 | 7 | 3 | 4 | 3 | 2 | 4 | 2 |

C_{2} | 1/3 | 1 | 3 | 1/2 | 2 | 2 | 5 | 3 | 3 | 2 | 4 | 2 | 1/2 |

C_{3} | 1/3 | 1/3 | 1 | 1/3 | 1/2 | 1/2 | 2 | 1/3 | 2 | 1/3 | 1/2 | 3 | 1/4 |

C_{6} | 2 | 2 | 3 | 1 | 1/2 | 2 | 4 | 2 | 4 | 3 | 2 | 4 | 2 |

C_{7} | 1/3 | 1/2 | 2 | 2 | 1 | 1/2 | 2 | 1/3 | 2 | 1/2 | 3 | 2 | 1/3 |

C_{8} | 1/2 | 1/2 | 2 | 1/2 | 2 | 1 | 3 | 1/2 | 3 | 1/2 | 2 | 3 | 1/2 |

C_{9} | 1/7 | 1/5 | 1/2 | 1/4 | 1/2 | 1/3 | 1 | 1/4 | 1/2 | 1/3 | 1/2 | 1/2 | 1/4 |

C_{10} | 1/3 | 1/3 | 3 | 1/2 | 3 | 2 | 4 | 1 | 2 | 3 | 1/2 | 4 | 1/3 |

C_{11} | 1/4 | 1/3 | 1/2 | 1/4 | 1/2 | 1/3 | 2 | 1/2 | 1 | 1/3 | 1/3 | 1/2 | 1/5 |

C_{13} | 1/3 | 1/2 | 3 | 1/3 | 2 | 2 | 3 | 1/3 | 3 | 1 | 3 | 1/2 | 2 |

C_{15} | 1/2 | 1/4 | 2 | 1/2 | 1/3 | 1/2 | 2 | 2 | 3 | 1/3 | 1 | 3 | 1/3 |

C_{18} | 1/4 | 1/2 | 1/3 | 1/4 | 1/2 | 1/3 | 2 | 1/4 | 2 | 2 | 1/3 | 1 | 1/4 |

C_{19} | 1/2 | 2 | 4 | 2 | 3 | 2 | 4 | 3 | 5 | 1/2 | 3 | 4 | 1 |

Inconsistency: 0.09964 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 2 | 3 | 3 | 1/4 | 1/2 | 1/2 | 1/4 | 1/3 | 1/3 | 1/2 | 1/3 | 1/2 |

C_{2} | 1/2 | 1 | 1/2 | 2 | 1/4 | 1/3 | 1/2 | 1/4 | 1/2 | 1/7 | 1/4 | 1/5 | 1/3 |

C_{3} | 1/3 | 2 | 1 | 3 | 1/5 | 1/4 | 2 | 1/2 | 3 | 1/6 | 1/3 | 1/3 | 1/6 |

C_{6} | 1/3 | 1/2 | 1/3 | 1 | 1/5 | 1/8 | 1/2 | 1/5 | 1/3 | 1/8 | 1/6 | 1/4 | 1/3 |

C_{7} | 4 | 4 | 5 | 5 | 1 | 2 | 5 | 3 | 3 | 1/2 | 3 | 2 | 2 |

C_{8} | 2 | 3 | 4 | 8 | 1/2 | 1 | 5 | 2 | 1/2 | 1/3 | 2 | 2 | 3 |

C_{9} | 2 | 2 | 1/2 | 2 | 1/5 | 1/5 | 1 | 1/2 | 2 | 1/8 | 1/2 | 1/7 | 1/5 |

C_{10} | 4 | 4 | 2 | 5 | 1/3 | 1/2 | 2 | 1 | 1/3 | 1/6 | 1/2 | 1/2 | 2 |

C_{11} | 3 | 2 | 1/3 | 3 | 1/3 | 2 | 1/2 | 3 | 1 | 1/5 | 1/2 | 1/3 | 1/2 |

C_{13} | 3 | 7 | 6 | 8 | 2 | 3 | 8 | 6 | 5 | 1 | 4 | 2 | 4 |

C_{15} | 2 | 4 | 3 | 6 | 1/3 | 1/2 | 2 | 2 | 2 | 1/4 | 1 | 1/2 | 1/2 |

C_{18} | 3 | 5 | 3 | 4 | 1/2 | 1/2 | 7 | 5 | 3 | 1/2 | 2 | 1 | 2 |

C_{19} | 2 | 3 | 6 | 3 | 1/2 | 1/3 | 5 | 4 | 2 | 1/4 | 2 | 1/2 | 1 |

Inconsistency: 0.09801 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 4 | 4 | 1/2 | 3 | 2 | 7 | 4 | 6 | 3 | 2 | 4 | 3 |

C_{2} | 1/4 | 1 | 3 | 1/2 | 2 | 2 | 4 | 3 | 3 | 2 | 4 | 2 | 1/2 |

C_{3} | 1/4 | 1/3 | 1 | 1/3 | 3 | 1/2 | 2 | 1/3 | 2 | 1/2 | 1/2 | 2 | 1/2 |

C_{6} | 2 | 2 | 3 | 1 | 3 | 2 | 4 | 2 | 4 | 3 | 2 | 5 | 2 |

C_{7} | 1/3 | 1/2 | 1/3 | 1/3 | 1 | 1/2 | 2 | 1/3 | 2 | 1/2 | 3 | 2 | 1/3 |

C_{8} | 1/2 | 1/2 | 2 | 1/2 | 2 | 1 | 3 | 1/2 | 3 | 1/2 | 2 | 3 | 1/2 |

C_{9} | 1/7 | 1/4 | 1/2 | 1/4 | 1/2 | 1/3 | 1 | 1/4 | 1/2 | 1/3 | 1/2 | 1/2 | 1/4 |

C_{10} | 1/4 | 1/3 | 3 | 1/2 | 3 | 2 | 4 | 1 | 2 | 3 | 1/2 | 4 | 1/3 |

C_{11} | 1/6 | 1/3 | 1/2 | 1/4 | 1/2 | 1/3 | 2 | 1/2 | 1 | 1/2 | 1/3 | 1/2 | 1/5 |

C_{13} | 1/3 | 1/2 | 2 | 1/3 | 2 | 2 | 3 | 1/3 | 2 | 1 | 3 | 1/2 | 2 |

C_{15} | 1/2 | 1/4 | 2 | 1/2 | 1/3 | 1/2 | 2 | 2 | 3 | 1/3 | 1 | 3 | 1/3 |

C_{18} | 1/4 | 1/2 | 1/2 | 1/5 | 1/2 | 1/3 | 2 | 1/4 | 2 | 2 | 1/3 | 1 | 1/2 |

C_{19} | 1/3 | 2 | 2 | 2 | 3 | 2 | 4 | 3 | 5 | 1/2 | 3 | 2 | 1 |

Inconsistency: 0.09084 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 5 | 4 | 3 | 1/4 | 1/4 | 1/3 | 1/4 | 4 | 1/5 | 1/4 | 1/3 | 1/2 |

C_{2} | 1/5 | 1 | 1/3 | 3 | 1/5 | 1/7 | 1/3 | 1/5 | 1/4 | 1/8 | 1/4 | 1/6 | 1/4 |

C_{3} | 1/4 | 3 | 1 | 3 | 1/5 | 1/4 | 2 | 1/2 | 3 | 1/6 | 1/3 | 1/3 | 1/6 |

C_{6} | 1/3 | 1/3 | 1/3 | 1 | 1/5 | 1/8 | 1/2 | 1/5 | 1/3 | 1/8 | 1/6 | 1/8 | 1/3 |

C_{7} | 4 | 5 | 5 | 5 | 1 | 3 | 6 | 3 | 6 | 1/2 | 4 | 2 | 3 |

C_{8} | 4 | 7 | 4 | 8 | 1/3 | 1 | 5 | 2 | 5 | 1/3 | 2 | 2 | 3 |

C_{9} | 3 | 3 | 1/2 | 2 | 1/6 | 1/5 | 1 | 1/2 | 2 | 1/8 | 1/2 | 1/7 | 1/5 |

C_{10} | 4 | 5 | 2 | 5 | 1/3 | 1/2 | 2 | 1 | 3 | 1/6 | 1/2 | 1/2 | 2 |

C_{11} | 1/4 | 4 | 1/3 | 3 | 1/6 | 1/5 | 1/2 | 1/3 | 1 | 1/8 | 1/2 | 1/5 | 1/4 |

C_{13} | 5 | 8 | 6 | 8 | 2 | 3 | 8 | 6 | 8 | 1 | 4 | 2 | 4 |

C_{15} | 4 | 4 | 3 | 6 | 1/4 | 1/2 | 2 | 2 | 2 | 1/4 | 1 | 1/2 | 1/2 |

C_{18} | 3 | 6 | 3 | 8 | 1/2 | 1/2 | 7 | 5 | 5 | 1/2 | 2 | 1 | 2 |

C_{19} | 2 | 4 | 6 | 3 | 1/3 | 1/3 | 5 | 4 | 4 | 1/4 | 2 | 1/2 | 1 |

Inconsistency: 0.08784 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 2 | 3 | 3 | 1/4 | 1/5 | 1/5 | 1/4 | 4 | 1/5 | 1/4 | 1/3 | 1/2 |

C_{2} | 1/2 | 1 | 1/3 | 3 | 1/5 | 1/7 | 1/3 | 1/5 | 1/4 | 1/8 | 1/4 | 1/6 | 1/4 |

C_{3} | 1/3 | 3 | 1 | 3 | 1/5 | 1/4 | 2 | 1/2 | 3 | 1/6 | 1/3 | 1/3 | 1/6 |

C_{6} | 1/3 | 1/3 | 1/3 | 1 | 1/4 | 1/7 | 1/2 | 1/5 | 1/3 | 1/7 | 1/5 | 1/6 | 1/4 |

C_{7} | 4 | 5 | 5 | 4 | 1 | 3 | 6 | 3 | 6 | 1/2 | 4 | 2 | 3 |

C_{8} | 5 | 7 | 4 | 7 | 1/3 | 1 | 5 | 2 | 5 | 1/3 | 2 | 2 | 3 |

C_{9} | 5 | 3 | 1/2 | 2 | 1/6 | 1/5 | 1 | 1/2 | 2 | 1/8 | 1/2 | 1/7 | 1/5 |

C_{10} | 5 | 5 | 2 | 5 | 1/3 | 1/2 | 2 | 1 | 3 | 1/6 | 1/2 | 1/2 | 2 |

C_{11} | 1/4 | 4 | 1/3 | 3 | 1/6 | 1/5 | 1/2 | 1/3 | 1 | 1/8 | 1/2 | 1/5 | 1/4 |

C_{13} | 5 | 8 | 6 | 7 | 2 | 3 | 8 | 6 | 8 | 1 | 4 | 2 | 4 |

C_{15} | 4 | 4 | 3 | 5 | 1/4 | 1/2 | 2 | 2 | 2 | 1/4 | 1 | 1/2 | 1/2 |

C_{18} | 3 | 6 | 3 | 6 | 1/2 | 1/2 | 7 | 5 | 5 | 1/2 | 2 | 1 | 2 |

C_{19} | 2 | 4 | 6 | 4 | 1/3 | 1/3 | 5 | 4 | 4 | 1/4 | 2 | 1/2 | 1 |

Inconsistency: 0.08708 |

C_{1} | C_{2} | C_{3} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{13} | C_{15} | C_{18} | C_{19} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | 1 | 4 | 4 | 2 | 3 | 2 | 5 | 4 | 6 | 3 | 2 | 4 | 3 |

C_{2} | 1/4 | 1 | 3 | 1/2 | 2 | 2 | 5 | 3 | 3 | 2 | 4 | 2 | 1/2 |

C_{3} | 1/4 | 1/3 | 1 | 1/3 | 3 | 1/2 | 2 | 1/3 | 2 | 1/3 | 1/2 | 2 | 1/2 |

C_{6} | 1/2 | 2 | 3 | 1 | 3 | 2 | 4 | 2 | 4 | 3 | 2 | 5 | 2 |

C_{7} | 1/3 | 1/2 | 1/3 | 1/3 | 1 | 1/2 | 2 | 1/3 | 2 | 1/2 | 3 | 2 | 1/3 |

C_{8} | 1/2 | 1/2 | 2 | 1/2 | 2 | 1 | 3 | 1/2 | 3 | 1/2 | 2 | 3 | 1/2 |

C_{9} | 1/5 | 1/5 | 1/2 | 1/4 | 1/2 | 1/3 | 1 | 1/4 | 1/2 | 1/3 | 1/2 | 1/2 | 1/4 |

C_{10} | 1/4 | 1/3 | 3 | 1/2 | 3 | 2 | 4 | 1 | 2 | 3 | 1/2 | 4 | 1/3 |

C_{11} | 1/6 | 1/3 | 1/2 | 1/4 | 1/2 | 1/3 | 2 | 1/2 | 1 | 1/2 | 1/3 | 1/2 | 1/5 |

C_{13} | 1/3 | 1/2 | 3 | 1/3 | 2 | 2 | 3 | 1/3 | 2 | 1 | 3 | 1/2 | 2 |

C_{15} | 1/2 | 1/4 | 2 | 1/2 | 1/3 | 1/2 | 2 | 2 | 3 | 1/3 | 1 | 3 | 1/3 |

C_{18} | 1/4 | 1/2 | 1/2 | 1/5 | 1/2 | 1/3 | 2 | 1/4 | 2 | 2 | 1/3 | 1 | 1/2 |

C_{19} | 1/3 | 2 | 2 | 2 | 3 | 2 | 4 | 3 | 5 | 1/2 | 3 | 2 | 1 |

Inconsistency: 0.09114 |

## Appendix G

DMs | Alternatives | Qualitative Criteria | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | C_{13} | C_{14} | C_{15} | ||

D_{1} | A_{1} | H | EH | H | M | H | H | VP | VH | H | H |

A_{2} | VH | H | H | VH | H | H | VH | H | VH | M | |

A_{3} | H | H | VH | VH | H | VH | VH | H | VH | H | |

A_{4} | M | H | M | M | H | M | M | VP | EP | M | |

D_{2} | A_{1} | H | EH | M | M | H | H | P | VH | M | H |

A_{2} | H | VH | M | VH | H | H | VH | H | VH | H | |

A_{3} | M | H | H | VH | H | M | VH | H | VH | M | |

A_{4} | H | M | M | M | M | M | M | VP | P | M | |

D_{3} | A_{1} | H | EH | M | H | VH | H | P | VH | M | H |

A_{2} | EH | H | H | VH | H | H | VH | H | M | M | |

A_{3} | VH | H | H | VH | H | P | VH | H | H | H | |

A_{4} | M | M | M | M | M | H | M | P | P | M | |

D_{4} | A_{1} | H | EH | H | H | M | M | P | VH | M | H |

A_{2} | H | H | M | H | EH | P | VH | H | M | M | |

A_{3} | H | VH | H | VH | H | P | VH | H | VH | M | |

A_{4} | M | M | M | P | M | H | M | P | P | M |

## Appendix H

Alternatives | Quantitative Criteria | ||||||||
---|---|---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | |||||||

A_{1} | 2001 | 2500 | 3000 | 101 | 150 | 200 | 3 | 4 | 5 |

A_{2} | 4001 | 4500 | 5000 | 401 | 450 | 500 | 9 | 10 | 11 |

A_{3} | 3001 | 3500 | 4000 | 201 | 250 | 300 | 6 | 7 | 8 |

A_{4} | 1001 | 1500 | 2000 | 101 | 150 | 200 | 6 | 7 | 8 |

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Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

R.I | 0 | 0 | 0.52 | 0.89 | 1.11 | 1.25 | 1.35 | 1.40 | 1.45 | 1.49 |

**Table 2.**Modified comparison based on Examples 2, 3, and 4 from Nejad and Mashinchi [59].

Situations | Methods | Results | Results after Adding New FNs | |
---|---|---|---|---|

(1) | (1.1) ${A}_{4}=(-3,-2,-1)$ | (1.2) ${A}_{4}=(8.75,9.5,11)$ | ||

${A}_{1}=(2,3,5,6)$ ${A}_{2}=(1,4,7)$ ${A}_{3}=(4,5,7)$ | [54] | ${A}_{1}={A}_{2}\prec {A}_{3}$ | ${A}_{2}\prec {A}_{1}\prec {A}_{3}$ | ${A}_{1}={A}_{2}\prec {A}_{3}$ |

[59] | ${A}_{1}={A}_{2}\prec {A}_{3}$ | ${A}_{1}={A}_{2}\prec {A}_{3}$ | ${A}_{1}={A}_{2}\prec {A}_{3}$ | |

Proposed method | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | |

(2) | (2.1) ${A}_{4}=(-1.5,-0.8,-0.6)$ | (2.2) ${A}_{4}=(1.15,2.5,3.15)$ | ||

${A}_{1}=(0.2,0.5,0.8)$ ${A}_{2}=(0.4,0.5,0.6)$ | [54] | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}={A}_{2}$ |

[59] | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | |

Proposed method | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | |

(3) | ${A}_{1}\succ {A}_{2}$ | (3.1) ${A}_{4}=(-5,-4,-3,-1)$ | (3.2) ${A}_{4}=(6,6,7,8)$ | |

${A}_{1}=(1,2,5)$ ${A}_{2}=(1,2,2,4)$ | [54] | ${A}_{1}\succ {A}_{2}$ | ${A}_{1}\succ {A}_{2}$ | ${A}_{1}={A}_{2}$ |

[59] | ${A}_{1}\succ {A}_{2}$ | ${A}_{1}\succ {A}_{2}$ | ||

Proposed method | ${A}_{1}\succ {A}_{2}$ | ${A}_{1}\succ {A}_{2}$ | ${A}_{1}\succ {A}_{2}$ |

Situations | Methods | Results | Results after Adding New FNs | |
---|---|---|---|---|

(1) | (1.1) ${A}_{5}=(-5,-4,-3)$ | (1.2) ${A}_{5}=(8,9,10)$ | ||

${A}_{1}=(3,3,3)$ ${A}_{2}=(3,3,6)$ ${A}_{3}=(3,3,8)$ ${A}_{4}=(3,3,6,8)$ | [54] | ${A}_{1}={A}_{2}={A}_{3}={A}_{4}$ | ${A}_{1}={A}_{2}={A}_{3}={A}_{4}$ | ${A}_{1}={A}_{2}={A}_{3}={A}_{4}$ |

[59] | ${A}_{1}={A}_{2}={A}_{3}={A}_{4}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}\prec {A}_{4}$ | ${A}_{1}={A}_{2}={A}_{3}={A}_{4}$ | |

Proposed method | ${A}_{1}\prec {A}_{2}\prec {A}_{3}\prec {A}_{4}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}\prec {A}_{4}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}\prec {A}_{4}$ | |

(2) | (2.1) ${A}_{4}=(-7,-5,-3,-2)$ | (2.2) ${A}_{4}=(7,9,11,12)$ | ||

${A}_{1}=(3,3,3)$ ${A}_{2}=(3,3,6)$ ${A}_{3}=(3,3,5,6)$ | [54] | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ |

[59] | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ | |

Proposed method | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | |

(3) | (3.1) ${A}_{3}=(-4,-2.5,-1.5)$ | (3.2) ${A}_{3}=(6,7.8,8.5)$ | ||

${A}_{1}=(2,2,7)$ ${A}_{2}=(2,4,4)$ | [54] | ${A}_{1}={A}_{2}$ | ${A}_{1}={A}_{2}$ | ${A}_{1}={A}_{2}$ |

[59] | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\succ {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | |

Proposed method | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ |

**Table 4.**Numerical comparison with Chu and Nguyen [63].

Situations | Methods | Results | Results after Adding New FNs | |
---|---|---|---|---|

(1) | (1.1) ${A}_{3}=(1,4,5)$ | (1.2) ${A}_{3}=(-3,-2,-1)$ | ||

${A}_{1}=(1,3,5)$ ${A}_{2}=(2,3,4)$ | [63] | ${A}_{1}={A}_{2}$ | ${A}_{1}={A}_{2}$ | ${A}_{1}={A}_{2}$ |

Proposed method | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | |

(2) | (2.1) ${A}_{3}=(2,3,7)$ | (2.2) ${A}_{3}=(-4,-2,-2)$ | ||

${A}_{1}=(2,2,4)$ ${A}_{2}=(2,2,6)$ | [63] | ${A}_{1}={A}_{2}$ | ${A}_{1}={A}_{2}$ | ${A}_{1}={A}_{2}$ |

Proposed method | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ | ${A}_{1}\prec {A}_{2}$ |

µ | Examples | ||
---|---|---|---|

(1) Three FNs ${\mathit{A}}_{1}=(2,3,5,6)$, ${\mathit{A}}_{2}=(1,4,7)$ ${\mathit{A}}_{3}=(4,5,7)$ | (2) Add an FN to the Right Side ${\mathit{A}}_{1}=(2,3,5,6)$, ${\mathit{A}}_{2}=(1,4,7)$ ${\mathit{A}}_{3}=(4,5,7)$, ${\mathit{A}}_{4}=(8,9,10)$ | (3) Add an FN to the Left Side ${\mathit{A}}_{1}=(2,3,5,6)$, ${\mathit{A}}_{2}=(1,4,7)$ ${\mathit{A}}_{3}=(4,5,7)$, ${\mathit{A}}_{4}=(-3,-2,-1)$ | |

0.1 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.2 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.3 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.4 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.5 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.6 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.7 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.8 | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{3}$ |

0.9 | $$ |