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Top1. Introduction
Multi-criteria decision making (MCDM) is an activity of selecting the best alternative with highest priority among finite available alternatives on the basis of their performance against finite decision criteria. To include the multiple preferences in selection of the best possible alternative, multiple decision makers are preferred in decision making process rather than a single decision maker and this decision making is known as multi criteria group decision making (MCGDM) process. In group decision making, individual preference of each decision maker is studied and then a collective decision about ranking of alternatives is made. Profound applications of MCGDM are found in various fields of investment, human resource development (HRD), resource allocation, supplier selection etc. In real life problem of MCGDM, very often uncertainty come across due to vague or imprecise information, incomplete preference information because of limited knowledge of decision makers. Fuzzy set theory (Zadeh, 1965) was integrated in decision making problems by the various researchers to cope uncertainty in real life problem of decision making.
In fuzzy set, both membership and non-membership grades are computed using single function. In real life problem of decision making fuzzy set fail to handle non-determinacy may arises because of presence of “can’t say” factor. To deal with non-determinacy which is caused by single membership function in fuzzy set, Atanassov (1986, 1989) introduced intuitionistic fuzzy set (IFS). An IFS is characterized by a membership function and a non- membership function. As IFS coincides with fuzzy set in the absence of non-determinacy, IFS based decision making includes an added advantage to coincide with fuzzy set-based decision making method when there non-determinacy is not there in criteria and decision makers’ preference information.
Very frequently decision makers do not agree on the same evaluation of alternative against certain decision criteria and provide different evaluation information. This particular type of non-determinacy or hesitancy cannot be handled by IFS. Torra & Narukawa (2009) and Torra (2010) defined hesitant fuzzy set (HFS) in which each element is assigned by set of possible membership degree. Prominent characteristic of HFSs of handling non-determinism that is caused by descent of decision makers, made it very popular in MCGDM problems.
Further, to deal with vagueness and ambiguity, in the environment of hesitancy of possible membership grades and non-determinacy, Zhu et al. (2012) defined dual hesitant fuzzy set (DHFS). In DHFS set of possible membership grades and non-membership grades are associated with each element. Fuzzy set, IFS and HFS can be regarded as particular case of DHFS. As DHFS is expressed by several determined numbers rather a single number, it makes the description of fuzziness of real world problems more accurately than other extensions of fuzzy set theory. Generally, DHFS is very helpful in MCGDM problem when membership and non-membership grades are difficult to compute and are provided intuitively by decision makers.
When it comes to orderings that describe human preferences for one thing over another, the referred term is called preference relation. A decision maker has to face choices between different preference orders over finite objects and chooses the one which is “closer” to his own preference order. Several comparison rules are considered and are employed by decision makers in decision making problems. Recently, computational approaches are also proposed to avoid hectic and complicated calculations in decision making methods. In these methods, decision matrix provided by the decision maker is idealized using fuzzy ideal over the semi ring and computational algorithms are proposed to find the preferences of one alternative over other against decision criteria.
In this paper, we mention the limitation of fuzzy ideal defined over semi ring of positive integers in decision making problem. We propose use of fuzzy or intuitionistic fuzzy ideal over the semi ring 
to idealize fuzzy or intuitionistic fuzzy group decision matrix and extended computational approach for hesitant fuzzy group decision making problems (Joshi & Kumar, 2017). In this paper we also propose a DHFS based MCDM method using semi ring 
to include both non-determinacy and hesitancy in decision making problems.