A Novel Approach to the Vectorial Redefinition of Ordered Fuzzy Numbers for Improved Arithmetic and Directional Representation
Abstract
1. Introduction
2. Vectorial Ordered Fuzzy Numbers (VOFNs)
- VOFNs of a positive order—the ordering direction conforms to the increasing values along the OX axis.
- VOFNs of a negative order—the ordering direction is opposite to the increasing values along the OX axis.
- and are the graphs of the vectors;
- represents the order;
- is the membership curve.
- A is the ordered fuzzy number represented by the vector pair ;
- B is the outcome of the vOFN order reversal operation;
- represents the reversal of the vOFN order.
- A = [-2, -4, -4, -7] B = [8,3,3,1] C = A + B
- C = [6,1,1,-6]
- A = [-1,0,1,4]
- B = [5,3,2,0]
- C = A - B = A + (B)
- C = [-1,0,1,4]+[-5,-3,-2,0]
- C = [-6, -3, -1, 4]
- A = [5,4,3,2]
- -A = [-5, -4, -3, -2] A-A = A+(-A)=[0,0,0,0]
- There is an ordered fuzzy number
- Determine for .
- .
- A = [3, 5, 5, 8]
- B = [1.5, 1.5, 2, 2.5]
- C = A * B = [4.5, 7.5, 10, 20.0]
- A = [3, 6, 6, 8]
- B = [1.5, 1.5, 2, 2.5]
- C = A/B = [2.0, 4.0, 3.0, 3.2]
- and
- A = [5,4,2,1], n = 2
- C = [25,16,4,1]
3. Solution to the Problem of Comparing OFN/vOFNs
- If the condition or is encountered, an affine transformation (translation) along the positive axis is required. This can be achieved by adding a constant value of 1 to the coordinates of both fuzzy numbers. The process should be repeated until no denominator becomes zero.
3.1. Experiment and Evaluation
3.2. Discussion and Comparison with Alternative Methods
- GR (General Ranking Method)—A classical ranking method based on centroid calculations and distance metrics, originally proposed to compare triangular and trapezoidal fuzzy numbers in a scalarized manner [41].
- ML (Modified Linguistic Ranking)—Introduced to align numerical results with linguistic preferences by using weighting functions and ranking indices [41].
- Adamo’s 0.5-cut, 0.9-cut methods—Utilize alpha-cuts to reduce fuzzy sets to intervals and then compare representative crisp values [31].
- Baldwin and Guild proposed several comparative measures based on the concept of distance between fuzzy sets [39].
- F1, F2, F3 (Yager’s Methods)—A series of fuzzy comparison techniques using score functions and area-based evaluations to handle non-normal and overlapping fuzzy numbers [29].
- CH (Chang’s index)—Uses a distance-based approach derived from the expected values of fuzzy numbers and their support intervals [9].
- PCM (Partial Comparison Method)—It was developed as a response to ambiguity in comparing fuzzy numbers, especially when their membership functions overlap significantly [33].
- K (Kerre’s Method)—Based on t-norms and fuzzy implications to determine dominance relationships [42].
- BK (Baas–Kwakkernaak Method)—Uses centroid-based distance and ranking functions [43].
- PSD (Possibility-Based Dominance)—Proposed by Dubois and Prade, this approach compares fuzzy numbers based on possibility theory, evaluating dominance through -cut analysis [6].
3.3. Summary
4. Solution to the Problem of OFN/vOFNs with Non-Convex Shapes
4.1. Mathematical Foundations of the Shape Normalization Operator (SNO)
- A number has positive order if , otherwise it has negative order;
- A number has positive order if , otherwise it has negative order;
- A number has positive order if , otherwise it has negative order.
- is the ascending permutation of ;
- is the descending permutation of .
4.2. Normalization of vOFNs with Unique Shapes
4.3. Summary
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
OFN | ordered fuzzy numbers proposed by Kosiński |
vOFN | vectorial ordered fuzzy numbers |
LR | fuzzy arithmetic proposed by Dubois and Prade |
CFA | constrained fuzzy arithmetic proposed by Klir |
RMD | fuzzy arithmetic proposed by Piegat and Pluciński |
SNO | shape normalization operator |
JC | metric dedicated to comparing ordered fuzzy numbers |
CFA | constrained fuzzy arithmetic proposed by Klir |
Appendix A
Appendix B
Appendix C
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Author | Method | A, B | C, D | E, F | F, G | G, E | H, I | J, K | K, L | L, J |
---|---|---|---|---|---|---|---|---|---|---|
Czerniak | JC | A < B | C > D | E > F | F > G | G < E | H < I | J > K | K > L | L < J |
Dobrosielski | GR | A < B | C > D | E < F | F = G | G > E | H > I | J = K | K > L | L < J |
ML | A > B | C > D | E > F | F = G | G < E | H = I | J > K | K > L | L < J | |
TR | A < B | C < D | E < F | F > G | G > E | H < I | J > K | K < L | L < J | |
Yager | F1 | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J |
F2 | A < B | C < D | E > F | F > G | G < E | H > I | J > K | K > L | L < J | |
F3 | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J | |
Chang | CH | A < B | C > D | E > F | F > G | G < E | H > I | J > K | K > L | L < J |
Adamo | 0.9 M | A < B | C < D | E > F | F > G | G < E | H > I | J > K | K > L | L < J |
0.9 m | A < B | C < D | E > F | F > G | G < E | H > I | J > K | K > L | L < J | |
0.5 | A < B | C > D | E > F | F > G | G < E | H > I | J > K | K > L | L < J | |
Baas-Kwakkernaak | BK | A < B | C < D | E > F | F > G | G < E | H = I | J = K | K = L | L = J |
Baldwin-Guild | lap | A < B | C > D | E > F | F > G | G < E | H < I | J > K | K > L | L < J |
g | A < B | C > D | E > F | F > G | G < E | H > I | J > K | K > L | L < J | |
ra | A < B | C > D | E > F | F > G | G < E | H < I | J > K | K > L | L < J | |
Kerre | K | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J |
Jain | k = 1 | A < B | C < D | E > F | F > G | G < E | H > I | J > K | K > L | L < J |
k = 2 | A < B | C > D | E > F | F > G | G < E | H > I | J > K | K > L | L < J | |
k = 0.5 | A < B | C < D | E < F | F > G | G < E | H > I | J > K | K > L | L < J | |
PD | A < B | C < D | E > F | F > G | G < E | H = I | J = K | K = L | L = J | |
Dubois-Prade | PSD | A < B | C > D | E > F | F > G | G < E | H > I | J > K | K > L | L < J |
ND | A < B | C > D | E > F | F > G | G < E | H < I | J = K | K = L | L = J | |
NSD | A < B | C < D | E > F | F = G | G < E | H = I | J = K | K = L | L = J | |
Lee-Li | Um | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J |
Pm | A < B | C < D | E > F | F > G | G < E | H = I | J > K | K > L | L < J | |
Dadgostar-Kerr | PCM | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J |
Dorohonceanu-Marin | B2 | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J |
B2x | A < B | C > D | E > F | F > G | G < E | H = I | J > K | K > L | L < J |
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Zarzycki, H.; Żak, A.; Czerniak, J.M. A Novel Approach to the Vectorial Redefinition of Ordered Fuzzy Numbers for Improved Arithmetic and Directional Representation. Appl. Sci. 2025, 15, 7427. https://doi.org/10.3390/app15137427
Zarzycki H, Żak A, Czerniak JM. A Novel Approach to the Vectorial Redefinition of Ordered Fuzzy Numbers for Improved Arithmetic and Directional Representation. Applied Sciences. 2025; 15(13):7427. https://doi.org/10.3390/app15137427
Chicago/Turabian StyleZarzycki, Hubert, Andrzej Żak, and Jacek M. Czerniak. 2025. "A Novel Approach to the Vectorial Redefinition of Ordered Fuzzy Numbers for Improved Arithmetic and Directional Representation" Applied Sciences 15, no. 13: 7427. https://doi.org/10.3390/app15137427
APA StyleZarzycki, H., Żak, A., & Czerniak, J. M. (2025). A Novel Approach to the Vectorial Redefinition of Ordered Fuzzy Numbers for Improved Arithmetic and Directional Representation. Applied Sciences, 15(13), 7427. https://doi.org/10.3390/app15137427