The problem of ordered clustering in the context of decision‐making with multiple criteria has garnered significant interest from researchers in the field of management science and operational research. In real‐world scenarios, the datasets often exhibit imprecision or uncertainty, which can lead to suboptimal ordered‐clustering outcomes. However, the intuitionistic fuzzy c‐means (IFCM) clustering algorithm enhances the accuracy and effectiveness of decision‐making processes by effectively handling uncertain dataset information for clustering. Therefore, we propose a new clustering algorithm, called the generalized ordered intuitionistic fuzzy c‐means (G‐OIFCM), based on PROMETHEE and the IFCM clustering algorithm. Different from the classical IFCM clustering algorithm, we use positive flow (φ+(si) ∈ [0, 1]) and negative flow (φ−(si) ∈ [0, 1]) of PROMETHEE to generate ordered clusters within the intuitionistic environment. We define a new objective function based on the positive and negative flow of the PROMETHEE and IFCM clustering algorithm, whose properties are mathematically justified in terms of convergence and optimization. The performance of the proposed algorithm is evaluated using two different real‐world datasets to assess both the ordered clustering and the quality of partitioning. To demonstrate the effectiveness of G‐OIFCM, a comparison is conducted with three other algorithms: fuzzy c‐means (FCM), ordered fuzzy c‐means (OFCM), and an adaptive generalized intuitionistic fuzzy c‐means (G‐IFCM). The results demonstrate the effectiveness of G‐OIFCM in enhancing optimal ordered clustering and utility when dealing with uncertainty in datasets.

Best-worst method (BWM) is extended to uncertain situations, hesitant fuzzy best-worst method (HFBWM) is proposed by using hesitant fuzzy multiplicative preference relation for multiple-criteria group decision-making problems. The reference comparison of the best criterion and the worst criterion are described by the linguistic terms, which are expressed in hesitant fuzzy elements, of the decision makers. Weights of criteria are calculated by using score function. Using the concept of BWM, nonlinearly constrained optimization problems are formed to obtain hesitant fuzzy weights (HFWs) of different criteria and alternatives. To check the reliability of the HFBWM, consistency ratio is proposed. The advantage and suitability of the proposed HFBWM are determined by three case studies. The results indicate that the HFBWM, due to higher comparison consistency as compared to BWM, obtain plausible preference ranking for alternatives.

Dealing with uncertainty is always a challenging problem, and different tools have been proposed to deal with it. Fuzzy sets was presented to manage situations in which experts have some membership value to assess an alternative. The fuzzy linguistic approach has been applied successfully to many problems. The linguistic information expressed by means of 2-tuples, which were composed by a linguistic term and a numeric value assessed in [ − 0.5, 0.5). Linguistic values was used to assess an alternative and variable in qualitative settings. Intuitionistic fuzzy sets were presented to manage situations in which experts have some membership and nonmembership value to assess an alternative. In this paper, the concept of an I2LI model is developed to provide a linguistic and computational basis to manage the situations in which experts assess an alternative in possible and impossible linguistic variable and their translation parameter. A method to solve the group decision making problem based on intuitionistic 2-tuple linguistic information (I2LI) by the group of experts is formulated. Some operational laws on I2LI are introduced. Based on these laws, new aggregation operators are introduced to aggregate the collective opinion of decision makers. An illustrative example is given to show the practicality and feasibility of our proposed aggregation operators and group decision making method.

The group decision-making problem with linguistic information evaluation values of decision makers are used based on 2-tuple interval-valued. Operational laws on interval value 2-tuple are introduced. On the basis of these laws, new aggregation operators are introduced by using the Choquet integral. A multiple attribute decision-making method based on these aggregation operators is proposed. An example is given to illustrate the efficiency, practicality, and feasibility of our method.

A hierarchical structure is proposed for the performance evaluation of vague, complicated humanistic systems. An improved fuzzy clustering algorithm is developed to produce several partition trees with different levels and clusters according to different triangular norm compositions. Additionally, a fuzzy clustering algorithm is given to produce a partition tree without using the transitive closure composition. The usefulness of the proposed algorithm is illustrated by an example of actual academic data.

We propose a new method to aggregate the opinion of experts or decision makers on different criteria, regarding a set of alternatives, where the opinion of the experts is represented by hesitant fuzzy linguistic term sets. An illustrative example is provided to elaborate the proposed method for selection of the best alternative.