# Item Response Theory Models for the Fuzzy TOPSIS in the Analysis of Survey Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fuzzy TOPSIS Method

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Materials and Methods

#### 3.1. Ordinal Items and Threshold Values

#### 3.2. Polytomous Rasch Model

#### 3.3. Rating Scale Model

#### 3.4. Partial Credit Model and Generalized Partial Credit Model

#### 3.5. Graded Response Model

#### 3.6. Nominal Response Model

#### 3.7. Method of Constructing Fuzzy Conversion Scales for the Fuzzy TOPSIS Method

## 4. Empirical Example

_{1}—job security, C

_{2}—salary, C

_{3}—promotion opportunities, C

_{4}—development opportunities, C

_{5}—chances of finding a new comparable or better job, C

_{6}—relations with colleagues, C

_{7}—relations with superiors, C

_{8}—understanding of the employer, C

_{9}—justice of the superior, C

_{10}—communication in the workplace, C

_{11}—material working conditions, C

_{12}—the flexibility of working time. In the evaluation of communes, in terms of the adopted criteria, an ordinal measurement scale was used with the following categories: 1—very low (VL), 2—low (L), 3—medium (M), 4—high (H), 5—very high (VH). The research did not consider answers such as “don’t know”, “I have no opinion”; therefore, the final analysis included the opinions of 939 respondents. According to the methodology presented in Section 3.7, the measurement results using the ordinal measurement scale are used to determine the thresholds. The thresholds were established for each of the criteria. All IRT models described in Section 3 were used to estimate thresholds, which allowed for a comparative analysis of the obtained results. The thresholds for the models are shown in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15 and Table 16.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Behzadian, M.; Otaghsara, S.K.; Yazdani, M.; Ignatius, J. A state-of the-art survey TOPSIS applications. Expert Syst. Appl.
**2012**, 39, 13051–13069. [Google Scholar] - Nădăban, S.; Dzitac, S.; Dzitac, I. Fuzzy TOPSIS: General View. Procedia Comput. Sci.
**2016**, 91, 823–831. [Google Scholar] [CrossRef] [Green Version] - Palczewski, P.; Sałabun, W. The Fuzzy TOPSIS applications in the last decade. Procedia Comput. Sci.
**2019**, 159, 2294–2303. [Google Scholar] [CrossRef] - Salih, M.M.; Zaidan, B.B.; Zaidan, A.A.; Ahmed, M.A. Survey on Fuzzy TOPSIS State of-the-Art between 2007–2017. Comput. Oper. Res.
**2019**, 104, 207–227. [Google Scholar] - Ziemba, P.; Becker, A.; Becker, J. A Consensus Measure of Expert Judgment in the Fuzzy TOPSIS Method. Symmetry
**2020**, 12, 204. [Google Scholar] - Chen, C.-T. Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst.
**2000**, 114, 1–9. [Google Scholar] [CrossRef] - Zhang, X.; Gao, L.; Barrett, D.; Chen, Y. Evaluating Water Management Practice for Sustainable Mining. Water
**2014**, 6, 413–443. [Google Scholar] [CrossRef] [Green Version] - Solangi, Y.A.; Tan, Q.; Mirjat, N.H.; Valasai, G.D.; Khan, M.W.A.; Ikram, M. An Integrated Delphi-AHP and Fuzzy TOPSIS toward Ranking and Selection of Renewable Energy Resources in Pakistan. Processes
**2019**, 7, 118. [Google Scholar] [CrossRef] [Green Version] - Falqi, I.I.; Ahmed, M.; Mallick, J. Siliceous Concrete Materials Management for Sustainability Using Fuzzy-TOPSIS Approach. Appl. Sci.
**2019**, 9, 3457. [Google Scholar] [CrossRef] [Green Version] - Zhao, H.; Li, N. Performance Evaluation for Sustainability of Strong Smart Grid by Using Stochastic AHP and Fuzzy TOPSIS Methods. Sustainability
**2016**, 8, 129. [Google Scholar] [CrossRef] [Green Version] - Zhao, H.; Guo, S. Selecting Green Supplier of Thermal Power Equipment by Using a Hybrid MCDM Method for Sustainability. Sustainability
**2014**, 6, 217–235. [Google Scholar] [CrossRef] [Green Version] - Feng, Y.; Zhang, Z.; Tian, G.; Fathollahi-Fard, A.M.; Hao, N.; Li, Z.; Wang, W.; Tan, J. A Novel Hybrid Fuzzy Grey TOPSIS Method: Supplier Evaluation of a Collaborative Manufacturing Enterprise. Appl. Sci.
**2019**, 9, 3770. [Google Scholar] [CrossRef] [Green Version] - Husin, S.; Fachrurrazi, F.; Rizalihadi, M.; Mubarak, M. Implementing Fuzzy TOPSIS on Project Risk Variable Ranking. Adv. Civ. Eng.
**2019**, 2019, 9283409. [Google Scholar] [CrossRef] - Kabassi, K.; Amelio, A.; Komianos, V.; Oikonomou, K. Evaluating Museum Virtual Tours: The Case Study of Italy. Information
**2019**, 10, 351. [Google Scholar] [CrossRef] [Green Version] - Prato, T. Conceptual Framework for Assessing the Sustainability of Forest Fuel Reduction Treatments and Their Adaptation to Climate Change. Sustainability
**2015**, 7, 3571–3591. [Google Scholar] [CrossRef] [Green Version] - Chou, Y.-C.; Yen, H.Y.; Dang, V.T.; Sun, C.-C. Assessing the Human Resource in Science and Technology for Asian Countries: Application of Fuzzy AHP and Fuzzy TOPSIS. Symmetry
**2019**, 11, 251. [Google Scholar] [CrossRef] [Green Version] - He, Y.; Wang, X.; Lin, Y.; Zhou, F. Optimal Partner Combination for Joint Distribution Alliance using Integrated Fuzzy EW-AHP and TOPSIS for Online Shopping. Sustainability
**2016**, 8, 341. [Google Scholar] [CrossRef] [Green Version] - Kahraman, C. Multi-Criteria Decision Making Methods and Fuzzy Sets. In Fuzzy Multi-Criteria Decision Making. Theory and Applications with Recent Developments; Kahraman, C., Ed.; Springer: New York, NY, USA, 2008; pp. 1–18. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Zimmerman, H.J. Fuzzy Sets, Decision Making, and Expert Systems; Kluwer Academic Publishers: Boston, MA, USA, 1987; p. 11. [Google Scholar]
- Olsson, U. Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika
**1979**, 44, 443–460. [Google Scholar] - Borboom, D. Measuring the Mind Conceptual Issues in Contemporary Psychometrics; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Lord, F.M.; Novick, M.R. Statistical Theories of Mental Test Scores; Addison-Wesley: Menlo Park, CA, USA, 1968. [Google Scholar]
- Nering, M.L.; Ostini, R. Handbook of Polytomous Item Response Theory Models; Routledge: New York, NY, USA, 2010. [Google Scholar]
- Andrich, D. A rating formulation for ordered response categories. Psychometrika
**1978**, 43, 561–573. [Google Scholar] [CrossRef] - Masters, G.N. A Rasch model for partial credit scoring. Psychometrika
**1982**, 47, 149–174. [Google Scholar] [CrossRef] - Muraki, E. A generalized partial credit model: Application of an EM algorithm. Appl. Psychol. Meas.
**1992**, 16, 159–176. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) Item response category characteristic curves for C

_{1}; (

**b**) Item operation characteristic curves for C

_{1}.

**Figure 4.**(

**a**) Item response category characteristic curves for C

_{2}; (

**b**) Item operation characteristic curves for C

_{2}.

**Figure 5.**(

**a**) Item response category characteristic curves for C

_{3}; (

**b**) Item operation characteristic curves for item C

_{3}.

**Figure 6.**(

**a**) Item response category characteristic curves for C

_{4}; (

**b**) Item operation characteristic curves for C

_{4}.

**Figure 7.**(

**a**) Item response category characteristic curves for C

_{5}; (

**b**) Item operation characteristic curves for C

_{5}.

**Figure 8.**(

**a**) Item response category characteristic curves for C

_{6}; (

**b**) Item operation characteristic curves for C

_{6}.

**Figure 9.**(

**a**) Item response category characteristic curves for C

_{7}; (

**b**) Item operation characteristic curves for C

_{7}.

**Figure 10.**(

**a**) Item response category characteristic curves for C

_{8}; (

**b**) Item operation characteristic curves for C

_{8}.

**Figure 11.**T(

**a**) Item response category characteristic curves for C

_{9}; (

**b**) Item operation characteristic curves for C

_{9}.

**Figure 12.**(

**a**) Item response category characteristic curves for C

_{10}; (

**b**) Item operation characteristic curves for C

_{10}.

**Figure 13.**(

**a**) Item response category characteristic curves for C

_{11}; (

**b**) Item operation characteristic curves for C

_{11}.

**Figure 14.**(

**a**) Item response category characteristic curves for C

_{12}; (

**b**) Item operation characteristic curves for C

_{12}.

Category | Parameters of Triangular Fuzzy Numbers | ||
---|---|---|---|

$\mathit{a}$ | $\mathit{b}$ | $\mathit{c}$ | |

VL | −4 | −4 | ${\tau}_{i1}$ |

L | ${\tau}_{i1}$ | $\frac{{\tau}_{i1}+{\tau}_{i2}}{2}$ | ${\tau}_{i2}$ |

M | ${\tau}_{i2}$ | $\frac{{\tau}_{i2}+{\tau}_{i3}}{2}$ | ${\tau}_{i3}$ |

H | ${\tau}_{i3}$ | $\frac{{\tau}_{i3}+{\tau}_{i4}}{2}$ | ${\tau}_{i4}$ |

VH | ${\tau}_{i4}$ | 4 | 4 |

Planned Sample Size | Realized Sample Size | |
---|---|---|

Maximum Sampling Error Assumed = 3% | Maximum Sampling Error Obtained = 3% | |

Total | 1047 | 1054 |

Commune A | 97 | 95 |

Commune B | 107 | 110 |

Commune C | 97 | 98 |

Commune D | 242 | 283 |

Commune E | 504 | 468 |

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

1→2 | −1.270 | −1.110 | −0.980 | −1.280 | −0.820 | −1.970 |

2→3 | −0.597 | −0.352 | −0.231 | −0.529 | 0.032 | −1.291 |

3→4 | 0.323 | 0.574 | 0.714 | 0.445 | 0.830 | −0.430 |

4→5 | 1.310 | 1.560 | 1.560 | 1.480 | 1.730 | 0.620 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −1.790 | −1.640 | −1.460 | −2.100 | −1.750 | −1.120 |

2→3 | −1.125 | −0.985 | −0.894 | −1.317 | −1.059 | −0.552 |

3→4 | −0.311 | −0.126 | −0.029 | −0.314 | 0.087 | 0.397 |

4→5 | 0.780 | 0.860 | 0.980 | 0.770 | 1.060 | 1.310 |

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

1→2 | −1.468 | −1.356 | −1.16 | −1.599 | −1.018 | −2.549 |

2→3 | −1.09 | 0.675 | −0.523 | −0.968 | 0.033 | −1.991 |

3→4 | 0.424 | 0.808 | 1.102 | 0.625 | 1.103 | −0.771 |

4→5 | 1.946 | 2.380 | 2.215 | 2.284 | 2.610 | 0.800 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −2.268 | −2.000 | −1.589 | −2.911 | −2.225 | −1.081 |

2→3 | −1.707 | −1.584 | −1.545 | −2.121 | −1.764 | −1.157 |

3→4 | −0.651 | −0.287 | −0.145 | −0.549 | −0.244 | 0.598 |

4→5 | 1.114 | 1.175 | 1.384 | 1.088 | 1.632 | 1.868 |

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

1→2 | −1.463 | −1.142 | −0.986 | −1.342 | −0.724 | −2.503 |

2→3 | −0.927 | −0.606 | −0.450 | −0.806 | −0.188 | −1.967 |

3→4 | 0.386 | 0.707 | 0.864 | 0.508 | 1.125 | −0.653 |

4→5 | 1.839 | 2.160 | 2.316 | 1.960 | 2.578 | 0.799 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −2.284 | −2.073 | −1.911 | −2.375 | −2.020 | −1.368 |

2→3 | −1.748 | −1.537 | −1.376 | −1.839 | −1.484 | −0.832 |

3→4 | −0.435 | −0.224 | −0.062 | −0.525 | −0.171 | 0.481 |

4→5 | 1.018 | 1.229 | 1.391 | 0.927 | 1.282 | 1.934 |

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

1→2 | −1.365 | −1.258 | −1.075 | −1.484 | −0.941 | −2.377 |

2→3 | −0.997 | −0.613 | −0.471 | −0.883 | 0.041 | −1.838 |

3→4 | 0.405 | 0.763 | 1.035 | 0.592 | 1.041 | −0.702 |

4→5 | 1.830 | 2.239 | 2.091 | 2.147 | 2.461 | 0.761 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −2.113 | −1.863 | −1.483 | −2.711 | −2.072 | −1.009 |

2→3 | −1.573 | −1.458 | −1.419 | −1.955 | −1.622 | −1.056 |

3→4 | −0.589 | −0.254 | −0.122 | −0.498 | −0.213 | 0.566 |

4→5 | 1.051 | 1.110 | 1.305 | 1.027 | 1.532 | 1.761 |

**Table 11.**Thresholds for the Generalized Partial Credit Model (GPCM) model (for criteria C

_{1}–C

_{6}).

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

1→2 | −1.366 | −1.283 | −1.107 | −1.583 | −1.076 | −2.317 |

2→3 | −1.026 | −0.661 | −0.542 | −1.045 | 0.084 | −1.788 |

3→4 | 0.408 | 0.800 | 1.144 | 0.673 | 1.306 | −0.686 |

4→5 | 1.865 | 2.351 | 2.238 | 2.434 | 3.086 | 0.747 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −1.916 | −1.747 | −1.455 | −2.560 | −2.026 | −0.960 |

2→3 | −1.320 | −1.253 | −1.120 | −1.815 | −1.577 | −1.456 |

3→4 | −0.444 | −0.212 | −0.086 | −0.467 | −0.210 | 0.730 |

4→5 | 0.887 | 0.996 | 1.108 | 0.970 | 1.495 | 2.131 |

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

1→2 | −1.739 | −1.648 | −1.506 | −2.036 | −1.462 | −2.798 |

2→3 | −0.795 | −0.523 | −0.365 | −0.820 | 0.049 | −1.777 |

3→4 | 0.450 | 0.826 | 1.040 | 0.676 | 1.448 | −0.575 |

4→5 | 1.842 | 2.380 | 2.435 | 2.410 | 3.247 | 0.838 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −2.139 | −2.000 | −1.701 | −2.916 | −2.457 | −1.839 |

2→3 | −1.308 | −1.168 | −1.030 | −1.766 | −1.439 | −0.942 |

3→4 | −0.335 | −0.129 | −0.037 | −0.422 | −0.111 | 0.545 |

4→5 | 0.897 | 1.021 | 1.109 | 1.021 | 1.454 | 2.088 |

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

1→2 | −1.371 | −1.559 | −1.280 | −1.533 | −1.130 | −2.374 |

2→3 | −1.203 | −0.973 | −0.791 | −1.416 | −0.454 | −2.505 |

3→4 | −0.598 | −0.318 | −0.158 | −0.635 | 0.038 | −1.518 |

4→5 | −0.096 | 0.389 | 0.539 | 0.197 | 0.891 | −0.924 |

Thresholds | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} |
---|---|---|---|---|---|---|

1→2 | −2.053 | −1.855 | −1.541 | −2.707 | −2.788 | 0.692 |

2→3 | −1.628 | −1.504 | −1.370 | −2.577 | −2.059 | −1.898 |

3→4 | −1.136 | −0.993 | −0.779 | −1.622 | −1.220 | −0.455 |

4→5 | −0.732 | −0.608 | −0.357 | −0.859 | −0.472 | 0.438 |

**Table 17.**Parameters of triangular fuzzy numbers estimated on the basis of the PM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.730 | 2.890 | 3.020 | 2.720 | 3.180 | 2.030 | |

2—L | a | 2.730 | 2.890 | 3.020 | 2.720 | 3.180 | 2.030 |

b | 3.067 | 3.269 | 3.395 | 3.096 | 3.606 | 2.370 | |

c | 3.403 | 3.648 | 3.769 | 3.471 | 4.032 | 2.709 | |

3—M | a | 3.403 | 3.648 | 3.769 | 3.471 | 4.032 | 2.709 |

b | 3.863 | 4.111 | 4.242 | 3.958 | 4.431 | 3.140 | |

c | 4.323 | 4.574 | 4.714 | 4.445 | 4.830 | 3.570 | |

4—H | a | 4.323 | 4.574 | 4.714 | 4.445 | 4.830 | 3.570 |

b | 4.817 | 5.067 | 5.137 | 4.963 | 5.280 | 4.095 | |

c | 5.310 | 5.560 | 5.560 | 5.480 | 5.730 | 4.620 | |

5—VH | a | 5.310 | 5.560 | 5.560 | 5.480 | 5.730 | 4.620 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 18.**Parameters of triangular fuzzy numbers estimated on the basis of the PM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.210 | 2.360 | 2.540 | 1.900 | 2.250 | 2.880 | |

2—L | a | 2.210 | 2.360 | 2.540 | 1.900 | 2.250 | 2.880 |

b | 2.543 | 2.688 | 2.823 | 2.292 | 2.596 | 3.164 | |

c | 2.875 | 3.015 | 3.106 | 2.683 | 2.941 | 3.448 | |

3—M | a | 2.875 | 3.015 | 3.106 | 2.683 | 2.941 | 3.448 |

b | 3.282 | 3.445 | 3.539 | 3.185 | 3.514 | 3.923 | |

c | 3.689 | 3.874 | 3.971 | 3.686 | 4.087 | 4.397 | |

4—H | a | 3.689 | 3.874 | 3.971 | 3.686 | 4.087 | 4.397 |

b | 4.235 | 4.367 | 4.476 | 4.228 | 4.574 | 4.854 | |

c | 4.780 | 4.860 | 4.980 | 4.770 | 5.060 | 5.310 | |

5—VH | a | 4.780 | 4.860 | 4.980 | 4.770 | 5.060 | 5.310 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 19.**Parameters of triangular fuzzy numbers estimated on the basis of the PRM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.532 | 2.644 | 2.840 | 2.401 | 2.982 | 1.451 | |

2—L | a | 2.532 | 2.644 | 2.840 | 2.401 | 2.982 | 1.451 |

b | 2.721 | 3.660 | 3.159 | 2.717 | 3.508 | 1.730 | |

c | 2.910 | 4.675 | 3.477 | 3.032 | 4.033 | 2.009 | |

3—M | a | 2.910 | 4.675 | 3.477 | 3.032 | 4.033 | 2.009 |

b | 3.667 | 4.742 | 4.290 | 3.829 | 4.568 | 2.619 | |

c | 4.424 | 4.808 | 5.102 | 4.625 | 5.103 | 3.229 | |

4—H | a | 4.424 | 4.808 | 5.102 | 4.625 | 5.103 | 3.229 |

b | 5.185 | 5.594 | 5.659 | 5.455 | 5.857 | 4.015 | |

c | 5.946 | 6.380 | 6.215 | 6.284 | 6.610 | 4.800 | |

5—VH | a | 5.946 | 6.380 | 6.215 | 6.284 | 6.610 | 4.800 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 20.**Parameters of triangular fuzzy numbers estimated on the basis of the PRM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 1.732 | 2.00 | 2.411 | 1.089 | 1.775 | 2.919 | |

2—L | a | 1.732 | 2.00 | 2.411 | 1.089 | 1.775 | 2.919 |

b | 2.013 | 2.208 | 2.433 | 1.484 | 2.006 | 2.881 | |

c | 2.293 | 2.416 | 2.455 | 1.879 | 2.236 | 2.843 | |

3—M | a | 2.293 | 2.416 | 2.455 | 1.879 | 2.236 | 2.843 |

b | 2.821 | 3.065 | 3.155 | 2.665 | 2.996 | 3.721 | |

c | 3.349 | 3.713 | 3.855 | 3.451 | 3.756 | 4.598 | |

4—H | a | 3.349 | 3.713 | 3.855 | 3.451 | 3.756 | 4.598 |

b | 4.232 | 4.444 | 4.620 | 4.270 | 4.694 | 5.233 | |

c | 5.114 | 5.175 | 5.384 | 5.088 | 5.632 | 5.868 | |

5—VH | a | 5.114 | 5.175 | 5.384 | 5.088 | 5.632 | 5.868 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 21.**Parameters of triangular fuzzy numbers estimated on the basis of the RSM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.537 | 2.858 | 3.014 | 2.658 | 3.276 | 1.497 | |

2—L | a | 2.537 | 2.858 | 3.014 | 2.658 | 3.276 | 1.497 |

b | 2.805 | 3.126 | 3.282 | 2.926 | 3.544 | 1.765 | |

c | 3.073 | 3.394 | 3.550 | 3.194 | 3.812 | 2.033 | |

3—M | a | 3.073 | 3.394 | 3.550 | 3.194 | 3.812 | 2.033 |

b | 3.730 | 4.050 | 4.207 | 3.851 | 4.469 | 2.690 | |

c | 4.386 | 4.707 | 4.864 | 4.508 | 5.125 | 3.347 | |

4—H | a | 4.386 | 4.707 | 4.864 | 4.508 | 5.125 | 3.347 |

b | 5.113 | 5.433 | 5.590 | 5.234 | 5.852 | 4.073 | |

c | 5.839 | 6.160 | 6.316 | 5.960 | 6.578 | 4.799 | |

5—VH | a | 5.839 | 6.160 | 6.316 | 5.960 | 6.578 | 4.799 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 22.**Parameters of triangular fuzzy numbers estimated on the basis of the RSM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 1.716 | 1.927 | 2.089 | 1.625 | 1.980 | 2.632 | |

2—L | a | 1.716 | 1.927 | 2.089 | 1.625 | 1.980 | 2.632 |

b | 1.984 | 2.195 | 2.357 | 1.893 | 2.248 | 2.900 | |

c | 2.252 | 2.463 | 2.624 | 2.161 | 2.516 | 3.168 | |

3—M | a | 2.252 | 2.463 | 2.624 | 2.161 | 2.516 | 3.168 |

b | 2.908 | 3.119 | 3.281 | 2.818 | 3.172 | 3.825 | |

c | 3.565 | 3.776 | 3.938 | 3.475 | 3.829 | 4.481 | |

4—H | a | 3.565 | 3.776 | 3.938 | 3.475 | 3.829 | 4.481 |

b | 4.291 | 4.502 | 4.664 | 4.201 | 4.555 | 5.208 | |

c | 5.018 | 5.229 | 5.391 | 4.927 | 5.282 | 5.934 | |

5—VH | a | 5.018 | 5.229 | 5.391 | 4.927 | 5.282 | 5.934 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 23.**Parameters of triangular fuzzy numbers estimated on the basis of the PCM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.635 | 2.742 | 2.925 | 2.516 | 3.059 | 1.623 | |

2—L | a | 2.635 | 2.742 | 2.925 | 2.516 | 3.059 | 1.623 |

b | 2.819 | 3.065 | 3.227 | 2.816 | 3.550 | 1.893 | |

c | 3.003 | 3.387 | 3.529 | 3.117 | 4.041 | 2.162 | |

3—M | a | 3.003 | 3.387 | 3.529 | 3.117 | 4.041 | 2.162 |

b | 3.704 | 4.075 | 4.282 | 3.854 | 4.541 | 2.730 | |

c | 4.405 | 4.763 | 5.035 | 4.592 | 5.041 | 3.298 | |

4—H | a | 4.405 | 4.763 | 5.035 | 4.592 | 5.041 | 3.298 |

b | 5.118 | 5.501 | 5.563 | 5.369 | 5.751 | 4.029 | |

c | 5.830 | 6.239 | 6.091 | 6.147 | 6.461 | 4.761 | |

5—VH | a | 5.830 | 6.239 | 6.091 | 6.147 | 6.461 | 4.761 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 24.**Parameters of triangular fuzzy numbers estimated on the basis of the PCM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 1.887 | 2.137 | 2.517 | 1.289 | 1.928 | 2.991 | |

2—L | a | 1.887 | 2.137 | 2.517 | 1.289 | 1.928 | 2.991 |

b | 2.157 | 2.340 | 2.549 | 1.667 | 2.153 | 2.968 | |

c | 2.427 | 2.542 | 2.581 | 2.045 | 2.378 | 2.944 | |

3—M | a | 2.427 | 2.542 | 2.581 | 2.045 | 2.378 | 2.944 |

b | 2.919 | 3.144 | 3.230 | 2.774 | 3.082 | 3.755 | |

c | 3.411 | 3.746 | 3.878 | 3.502 | 3.787 | 4.566 | |

4—H | a | 3.411 | 3.746 | 3.878 | 3.502 | 3.787 | 4.566 |

b | 4.231 | 4.428 | 4.592 | 4.265 | 4.660 | 5.163 | |

c | 5.051 | 5.110 | 5.305 | 5.027 | 5.532 | 5.761 | |

5—VH | a | 5.051 | 5.110 | 5.305 | 5.027 | 5.532 | 5.761 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 25.**Parameters of triangular fuzzy numbers estimated on the basis of the GPCM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.634 | 2.717 | 2.893 | 2.417 | 2.924 | 1.683 | |

2—L | a | 2.634 | 2.717 | 2.893 | 2.417 | 2.924 | 1.683 |

b | 2.804 | 3.028 | 3.176 | 2.686 | 3.504 | 1.948 | |

c | 2.974 | 3.339 | 3.458 | 2.955 | 4.084 | 2.212 | |

3—M | a | 2.974 | 3.339 | 3.458 | 2.955 | 4.084 | 2.212 |

b | 3.691 | 4.070 | 4.301 | 3.814 | 4.695 | 2.763 | |

c | 4.408 | 4.800 | 5.144 | 4.673 | 5.306 | 3.314 | |

4—H | a | 4.408 | 4.800 | 5.144 | 4.673 | 5.306 | 3.314 |

b | 5.137 | 5.576 | 5.691 | 5.553 | 6.196 | 4.030 | |

c | 5.865 | 6.351 | 6.238 | 6.434 | 7.086 | 4.747 | |

5—VH | a | 5.865 | 6.351 | 6.238 | 6.434 | 7.086 | 4.747 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 26.**Parameters of triangular fuzzy numbers estimated on the basis of the GPCM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.084 | 2.253 | 2.545 | 1.440 | 1.974 | 3.040 | |

2—L | a | 2.084 | 2.253 | 2.545 | 1.440 | 1.974 | 3.040 |

b | 2.382 | 2.500 | 2.713 | 1.812 | 2.198 | 2.792 | |

c | 2.680 | 2.747 | 2.880 | 2.185 | 2.423 | 2.544 | |

3—M | a | 2.680 | 2.747 | 2.880 | 2.185 | 2.423 | 2.544 |

b | 3.118 | 3.268 | 3.397 | 2.859 | 3.107 | 3.637 | |

c | 3.556 | 3.788 | 3.914 | 3.533 | 3.790 | 4.730 | |

4—H | a | 3.556 | 3.788 | 3.914 | 3.533 | 3.790 | 4.730 |

b | 4.221 | 4.392 | 4.511 | 4.251 | 4.643 | 5.430 | |

c | 4.887 | 4.996 | 5.108 | 4.970 | 5.495 | 6.131 | |

5—VH | a | 4.887 | 4.996 | 5.108 | 4.970 | 5.495 | 6.131 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 27.**Parameters of triangular fuzzy numbers estimated on the basis of the NRM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.629 | 2.441 | 2.720 | 2.467 | 2.870 | 1.626 | |

2—L | a | 2.629 | 2.441 | 2.720 | 2.467 | 2.870 | 1.626 |

b | 2.713 | 2.734 | 2.964 | 2.526 | 3.208 | 1.560 | |

c | 2.797 | 3.027 | 3.209 | 2.584 | 3.546 | 1.495 | |

3—M | a | 2.797 | 3.027 | 3.209 | 2.584 | 3.546 | 1.495 |

b | 3.099 | 3.355 | 3.526 | 2.975 | 3.792 | 1.988 | |

c | 3.402 | 3.682 | 3.842 | 3.365 | 4.038 | 2.482 | |

4—H | a | 3.402 | 3.682 | 3.842 | 3.365 | 4.038 | 2.482 |

b | 3.653 | 4.035 | 4.191 | 3.781 | 4.465 | 2.779 | |

c | 3.904 | 4.389 | 4.539 | 4.197 | 4.891 | 3.076 | |

5—VH | a | 3.904 | 4.389 | 4.539 | 4.197 | 4.891 | 3.076 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 28.**Parameters of triangular fuzzy numbers estimated on the basis of the NRM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 1.947 | 2.145 | 2.459 | 1.293 | 1.212 | 4.692 | |

2—L | a | 1.947 | 2.145 | 2.459 | 1.293 | 1.212 | 4.692 |

b | 2.160 | 2.321 | 2.545 | 1.358 | 1.576 | 3.397 | |

c | 2.372 | 2.496 | 2.630 | 1.423 | 1.941 | 2.102 | |

3—M | a | 2.372 | 2.496 | 2.630 | 1.423 | 1.941 | 2.102 |

b | 2.618 | 2.751 | 2.926 | 1.900 | 2.360 | 2.824 | |

c | 2.864 | 3.007 | 3.221 | 2.378 | 2.780 | 3.545 | |

4—H | a | 2.864 | 3.007 | 3.221 | 2.378 | 2.780 | 3.545 |

b | 3.066 | 3.199 | 3.432 | 2.760 | 3.154 | 3.992 | |

c | 3.268 | 3.392 | 3.643 | 3.141 | 3.528 | 4.438 | |

5—VH | a | 3.268 | 3.392 | 3.643 | 3.141 | 3.528 | 4.438 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 29.**Parameters of triangular fuzzy numbers estimated on the basis of the GRM (for criteria C

_{1}–C

_{6}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

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

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 2.261 | 2.352 | 2.494 | 1.964 | 2.538 | 1.202 | |

2—L | a | 2.261 | 2.352 | 2.494 | 1.964 | 2.538 | 1.202 |

b | 2.733 | 2.914 | 3.065 | 2.572 | 3.293 | 1.713 | |

c | 3.205 | 3.477 | 3.635 | 3.180 | 4.049 | 2.223 | |

3—M | a | 3.205 | 3.477 | 3.635 | 3.180 | 4.049 | 2.223 |

b | 3.827 | 4.151 | 4.338 | 3.928 | 4.749 | 2.824 | |

c | 4.450 | 4.826 | 5.040 | 4.676 | 5.448 | 3.425 | |

4—H | a | 4.450 | 4.826 | 5.040 | 4.676 | 5.448 | 3.425 |

b | 5.146 | 5.603 | 5.737 | 5.543 | 6.348 | 4.132 | |

c | 5.842 | 6.380 | 6.435 | 6.410 | 7.247 | 4.838 | |

5—VH | a | 5.842 | 6.380 | 6.435 | 6.410 | 7.247 | 4.838 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 30.**Parameters of triangular fuzzy numbers estimated on the basis of the GRM (for criteria C

_{7}–C

_{12}).

Category | Fuzzy Number Parameters | Criteria | |||||
---|---|---|---|---|---|---|---|

C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | ||

1—VL | a | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

b | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

c | 1.861 | 2.000 | 2.299 | 1.084 | 1.543 | 2.161 | |

2—L | a | 1.861 | 2.000 | 2.299 | 1.084 | 1.543 | 2.161 |

b | 2.277 | 2.416 | 2.634 | 1.659 | 2.052 | 2.609 | |

c | 2.692 | 2.832 | 2.970 | 2.234 | 2.561 | 3.058 | |

3—M | a | 2.692 | 2.832 | 2.970 | 2.234 | 2.561 | 3.058 |

b | 3.179 | 3.352 | 3.467 | 2.906 | 3.225 | 3.801 | |

c | 3.665 | 3.871 | 3.963 | 3.578 | 3.889 | 4.545 | |

4—H | a | 3.665 | 3.871 | 3.963 | 3.578 | 3.889 | 4.545 |

b | 4.281 | 4.446 | 4.536 | 4.300 | 4.671 | 5.316 | |

c | 4.897 | 5.021 | 5.109 | 5.021 | 5.454 | 6.088 | |

5—VH | a | 4.897 | 5.021 | 5.109 | 5.021 | 5.454 | 6.088 |

b | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | |

c | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |

**Table 31.**The values of the fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS).

Communes | Fuzzy TOPSIS | Rating | ||||||
---|---|---|---|---|---|---|---|---|

PM | PRM | RSM | PCM | GPCM | GRM | NRM | ||

A | 0.626 | 0.604 | 0.604 | 0.607 | 0.607 | 0.606 | 0.611 | 4 |

B | 0.644 | 0.623 | 0.625 | 0.624 | 0.624 | 0.625 | 0.623 | 3 |

C | 0.667 | 0.648 | 0.651 | 0.650 | 0.649 | 0.651 | 0.645 | 2 |

D | 0.689 | 0.674 | 0.676 | 0.675 | 0.674 | 0.676 | 0.670 | 1 |

E | 0.600 | 0.578 | 0.582 | 0.581 | 0.580 | 0.578 | 0.589 | 5 |

Fuzzy Conversion Scale CS1 [12] | Fuzzy Conversion Scale CS2 [10] | ||||||

Triangular fuzzy number | Triangular fuzzy number | ||||||

Category | a | b | c | Category | a | b | c |

Terrible | 0.1 | 0.1 | 0.2 | Very low | 0 | 0 | 0.2 |

Bad | 0.2 | 0.3 | 0.4 | Low | 0 | 0.2 | 0.4 |

Normal | 0.4 | 0.5 | 0.6 | Fair | 0.3 | 0.5 | 0.7 |

Good | 0.6 | 0.7 | 0.8 | High | 0.6 | 0.8 | 1 |

Superb | 0.8 | 0.9 | 0.9 | Very high | 0.8 | 1 | 1 |

Fuzzy conversion scale CS3 [14] | Fuzzy conversion scale CS4 [16] | ||||||

Category | a | b | c | Category | a | b | c |

Very poor | 0 | 0 | 1 | Very poor | 0 | 1 | 3 |

Poor | 0 | 1 | 3 | Poor | 1 | 3 | 5 |

Fair | 3 | 5 | 7 | Fair | 3 | 5 | 7 |

Good | 7 | 9 | 10 | Good | 5 | 7 | 9 |

Very good | 9 | 10 | 10 | Very good | 7 | 9 | 10 |

Fuzzy conversion scale CS5 [17] | |||||||

Category | a | b | c | ||||

Very low | 1 | 1 | 3 | ||||

Low | 1 | 3 | 5 | ||||

Medium | 3 | 5 | 7 | ||||

High | 5 | 7 | 9 | ||||

Very high | 7 | 9 | 9 |

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## Share and Cite

**MDPI and ACS Style**

Jefmański, B.; Sagan, A.
Item Response Theory Models for the Fuzzy TOPSIS in the Analysis of Survey Data. *Symmetry* **2021**, *13*, 223.
https://doi.org/10.3390/sym13020223

**AMA Style**

Jefmański B, Sagan A.
Item Response Theory Models for the Fuzzy TOPSIS in the Analysis of Survey Data. *Symmetry*. 2021; 13(2):223.
https://doi.org/10.3390/sym13020223

**Chicago/Turabian Style**

Jefmański, Bartłomiej, and Adam Sagan.
2021. "Item Response Theory Models for the Fuzzy TOPSIS in the Analysis of Survey Data" *Symmetry* 13, no. 2: 223.
https://doi.org/10.3390/sym13020223