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Uncertainties in decision making problems (DMPs) are by reason of either randomness or fuzziness, or by both and can be classified into stochastic and non-stochastic uncertainty (Meghdadi and Akbarzadeh, 2001). Valavanis and Saridis (1991), Pidre et al. (2003) show that stochastic uncertainties in the system may be captured very well by the probabilistic modeling (Zadeh (1968). Although a number of theories have been developed to deal with non-stochastic uncertainties (De Luca and Termini (1972), Laviolette and Seaman (1994), Liang and Song (1996), Meghdadi and Akbarzadeh, (2001)) but fuzzy set theory developed by Zadeh (1965, 1975) is extensively researched and successfully applied in DMPs by various researchers Yoon and Hwang (1995), Liu et al. (2015), Lee and Chen (2015), Liu et al. (2016). Type-2 fuzzy sets (Zadeh, 1975), interval-valued fuzzy set (Zadeh, 1965), intuitionistic fuzzy sets (Atanassov, 1986) and interval-valued intuitionistic fuzzy sets (Atanassov and Gargov, 1989) are some other extensions of fuzzy sets practiced in MCDM problems to take in non-stochastic hesitation and uncertainty.
Very often decision makers in DMPs are not in favor on the same assessment of decision attributes and give diverse assessment information on each attribute. Difficulty of agreeing on a common result is not because of margin of error or some possible distribution as in case of IFS and Type-2 fuzzy sets. To deal with this concern in MCDM problems Tora and Narukawa (2009) and Tora (2010) introduced hesitant fuzzy set (HFS) and applied in MCGDM problems by Xu and Xia (2011), Xia and Xu (2011), Farhadinia (2014). Triangular hesitant fuzzy set, generalized hesitant fuzzy set, multi hesitant fuzzy set, interval valued hesitant fuzzy set, dual hesitant fuzzy set and interval valued intuitionistic hesitant fuzzy set are few extension of HFS which were used in decision making problems by various researchers Zhang (2013), Qian et al. (2013), Chen and Cai (2013), Yu and Li (2016), Ye et al. (2016).
Fuzzy and probabilistic approach based decision making method process only either probabilistic or non probabilistic uncertainty. One major limitation is not to handle both types of uncertainties concurrently. Comprehensive concurrence of stochastic and non stochastic uncertainty in real world problems concerned researchers to incorporate theory of probability with fuzzy logic theory. Idea of merging fuzzy theory with probabilistic theory was initiated by Zadeh (1995), Liang and Song (1996) and Meghdadi and Akbarzadeh (2001). In 2005, Liu and Li (2005) defined probabilistic fuzzy set (PFS) to handle both stochastic and non stochastic uncertainties in a single framework. To tackle simultaneous occurrence of both statistical and non statistical uncertainties with hesitation, Xu and Zhou (2017) introduced probabilistic hesitant fuzzy set (PHFS). PHFS permits more than one membership degree of an element with different probabilities. Recently many applications of PHFS are found in MCDM problems (Xu and Zhou (2017), Hao et al. (2017), Zhou and Xu (2017, 2017a), Ding et al. (2017), Li and Wang (2017), Joshi et al. (2018), Garg and Kaur (2020), Li et al. (2020), Li and Chen (2020).