TY  - JOUR
AU  - Mandal, Wasim Akram
PY  - 2026
DA  - 2026/04/21
TI  - Bipolar Spherical Fuzzy Sets and Their Application in Multi-Attribute Decision Making Problems
JO  - Journal of Numerical Simulations in Physics and Mathematics
T2  - Journal of Numerical Simulations in Physics and Mathematics
JF  - Journal of Numerical Simulations in Physics and Mathematics
VL  - 2
IS  - 1
SP  - 25
EP  - 38
DO  - 10.62762/JNSPM.2025.484692
UR  - https://www.icck.org/article/abs/JNSPM.2025.484692
KW  - fuzzy sets
KW  - picture fuzzy sets
KW  - spherical fuzzy sets
KW  - bipolar spherical fuzzy sets
KW  - multi attribute decision making
AB  - In this paper, the concept of the bipolar spherical fuzzy set (BSFS) and its operations is developed. It is proposed as a generalization of fuzzy sets, bipolar fuzzy sets, picture fuzzy sets, and spherical fuzzy sets, allowing uncertain information to be handled more flexibly in the decision-making process. The key objective of this research paper is to introduce a new version of the spherical fuzzy set so called bipolar spherical fuzzy set (BSFS). The bipolar spherical fuzzy set is a new extension of spherical fuzzy sets and picture fuzzy sets. In bipolar spherical fuzzy sets, membership degrees are satisfying the condition \( 0 \leq (s^+)^2(x) + (i^+)^2(x) + (d^+)^2(x) \leq 1 \) and \( 0 \leq (s^-)^2(x) + (i^-)^2(x) + (d^-)^2(x) \leq 1 \) instead of \( 0 \leq s^2(x) + i^2(x) + d^2(x) \leq 1 \) as is in spherical fuzzy sets and \( 0 \leq s(x) + i(x) + d(x) \leq 1 \) as is in the picture fuzzy set. Here, negative membership degree denotes the implicit counter-property corresponding to a bipolar spherical fuzzy set. Also, the BSF-set weighted average operator and BSF-set weighted geometric operator are given to aggregate the BSF-sets. Further a multi attribute decision making (MADM) technique is developed and the proposed aggregation operators are utilized. Finally, a numerical approach for implementation of the proposed technique is developed.
SN  - 3068-9082
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Mandal2026Bipolar,
  author = {Wasim Akram Mandal},
  title = {Bipolar Spherical Fuzzy Sets and Their Application in Multi-Attribute Decision Making Problems},
  journal = {Journal of Numerical Simulations in Physics and Mathematics},
  year = {2026},
  volume = {2},
  number = {1},
  pages = {25-38},
  doi = {10.62762/JNSPM.2025.484692},
  url = {https://www.icck.org/article/abs/JNSPM.2025.484692},
  abstract = {In this paper, the concept of the bipolar spherical fuzzy set (BSFS) and its operations is developed. It is proposed as a generalization of fuzzy sets, bipolar fuzzy sets, picture fuzzy sets, and spherical fuzzy sets, allowing uncertain information to be handled more flexibly in the decision-making process. The key objective of this research paper is to introduce a new version of the spherical fuzzy set so called bipolar spherical fuzzy set (BSFS). The bipolar spherical fuzzy set is a new extension of spherical fuzzy sets and picture fuzzy sets. In bipolar spherical fuzzy sets, membership degrees are satisfying the condition \( 0 \leq (s^+)^2(x) + (i^+)^2(x) + (d^+)^2(x) \leq 1 \) and \( 0 \leq (s^-)^2(x) + (i^-)^2(x) + (d^-)^2(x) \leq 1 \) instead of \( 0 \leq s^2(x) + i^2(x) + d^2(x) \leq 1 \) as is in spherical fuzzy sets and \( 0 \leq s(x) + i(x) + d(x) \leq 1 \) as is in the picture fuzzy set. Here, negative membership degree denotes the implicit counter-property corresponding to a bipolar spherical fuzzy set. Also, the BSF-set weighted average operator and BSF-set weighted geometric operator are given to aggregate the BSF-sets. Further a multi attribute decision making (MADM) technique is developed and the proposed aggregation operators are utilized. Finally, a numerical approach for implementation of the proposed technique is developed.},
  keywords = {fuzzy sets, picture fuzzy sets, spherical fuzzy sets, bipolar spherical fuzzy sets, multi attribute decision making},
  issn = {3068-9082},
  publisher = {Institute of Central Computation and Knowledge}
}