Fostering linguistic decision-making under uncertainty: A proportional interval type-2 hesitant fuzzy TOPSIS approach based on Hamacher aggregation operators and andness optimization models

Abstract Interval type-2 fuzzy sets (IT2 FS) have played a prominent role in the development of type-2 (T2) fuzzy logic and fuzzy systems for application to linguistic approximation transformations. Although there have been a number of studies of individual linguistic perception understanding based on T2 fuzzy logic, few of these have paid attention to the computational manipulation of group linguistic perceptions based on IT2 FS theory and methods. Proportional hesitant fuzzy linguistic term sets (PHFLTSs) allow conceptualization of group linguistic perceptions with the inclusion of generalized linguistic terms and their associated proportions. Interpreting PHFLTSs from the viewpoint of T2 fuzzy logic instead of its type-1 counterpart has the potential to provide useful results because of the ability of IT2 FSs to accurately model individual comprehension in the presence of linguistic uncertainty. This paper brings a novel perspective to encoding PHFLTSs based on the concept of a proportional interval T2 hesitant fuzzy set (PIT2 HFS). PIT2 HFSs combine IT2 FSs translated from generalized linguistic terms together with their proportional information. To facilitate computing with PIT2 HFSs, basic operations satisfying the closure property are defined for PIT2 HFSs based on Archimedean t-norms and s-norms. Information measurements such as a distance and a score function for PIT2 HFS are also defined. Two instrumental aggregation operators for PIT2 HFSs, namely, the Hamacher proportional interval T2 hesitant fuzzy power weighted average (Ham-PIT2HPWA) operator and the Hamacher proportional interval T2 hesitant fuzzy power weighted geometric (Ham-PIT2HPWG) operator, are investigated, and their limiting cases with respect to various parameters are discussed. Furthermore, two andness optimization models with the andness measure being the attitudinal character are constructed in a bid to determine reasonable parameter values associated with the Ham-PIT2HPWA and Ham-PIT2HPWG operators. On the basis of the Hamacher aggregation operators and the andness optimization models, a proportional interval T2 hesitant fuzzy TOPSIS approach is developed to provide linguistic decision making under uncertainty. The proposed approach represents a novel paradigm for linguistic group decision making under the umbrella of T2 fuzzy logic and systems for computing with words that has potential for application in real-life scenarios.