**1. Introduction**

Multi-attribute group decision-making (MAGDM) is an interesting and complex day-to-day problem which involves implicit uncertainty and vagueness [1]. Hesitant fuzzy set (HFS) [2] is a powerful extension of the fuzzy set [3] that allows multiple degrees of truth to be associated with each preference information for better handling uncertainty and vagueness. Attracted by the strength of HFS, many researchers used HFS for di fferent MAGDM applications viz., supplier selection [4,5], plant location selection [6], hospital site selection [7] and pattern recognition [8]. Recently, Rodriguez et al. [9] conducted a thorough analysis of HFS and some of its variants and identified its usefulness in MAGDM.

Though HFS is powerful, it lacks the ability to consider occurrence probability for each hesitant fuzzy element (HFE). To alleviate the issue, Xu and Zhou [10] put forward the probabilistic hesitant fuzzy set (PHFS), which associates occurrence probability value for each HFE. Motivated by the power of PHFS, researchers widely explored the idea for multi-attribute decision-making (MADM) [11–17]. Though PHFS alleviates the weakness of HFS to some extent, still the elicitation of occurrence probability is prone to imprecision and inaccuracy. To circumvent the weakness, a generalized model called interval-valued PHFS (IVPHFS) [18] is put forward which associates a range of values as occurrence probability to each HFE with a constraint that the sum of upper limit probability equals unity. As a generalization of Reference [18], Krishankumar et al. [19] proposed an IVPHFS concept which associates a range of values as occurrence probability for each HFE (for flexible elicitation of occurrence probability values). This mitigates the problem of imprecision and inaccuracy in the elicitation of occurrence probability values by providing multiple choices of values as occurrence probability for each HFE.

Based on the literature analysis of PHFS, we can infer that IVPHFS is a recent extension to PHFS which needs to be better explored for e ffective MAGDM. Group decision-making (GDM) [20] is a widely studied problem which obtains preference information from multiple DMs and aggregates them into single preference information without much loss of information. Mesiar et al. [21] made an interesting analysis of aggregation functions and we can infer that the realization of the interrelationship between attributes is a key factor for aggregation operators. Most of the state-of-the-art operators ignore this theme and are hence, not very suitable for MAGDM.

The MM [22] operator is an aggregation operator that reflects the interrelationship between attributes in a better way by considering risk appetite (refer Section 3 for details)s. MM is a generalized operator which can easily represent other operators viz., arithmetic average, geometric average, generalized arithmetic average, Bonferroni mean [23] and Maclaurin symmetric mean [24] as special cases.

Some challenges that can be encountered from the literature analysis made above are:


Motivated by these challenges and to circumvent the same, in this paper, some key contributions are made:


The rest of the paper is constructed as follows. Section 2 describes some basic concepts of HFS, PHFS and IVPHFS. Section 3 presents the core idea of the research in which the proposed aggregation operator along with some desirable properties is discussed. Further, a new programming model is put forward for calculating attributes' weight values and finally, a systematic procedure is presented for MAGDM using proposed aggregation operator. In Section 4, a numerical example for green supplier selection is put forward to validate the applicability of the proposed method. In Section 5, some superiority and weakness of the proposal are discussed by comparison with other methods and finally, in Section 6, concluding remarks with future research directions are presented.
