Q-Neutrosophic Soft Expert Set and its Application in Decision Making

Introduction

The theory of probability, fuzzy set (Zadeh, 1965), intuitionistic fuzzy set (Atanassov, 1986), rough set (Pawlak, 1982), and soft set (Molodtsov, 1999) are utilized as tools to deal with diverse types of uncertainties and imprecision. It was further extended to vague sets (Gau & Buehrer, 1993) which were shown to be intuitionistic fuzzy sets (Bustince & Burillo, 1996). The vague sets were extended to the interval-valued form, generalized form and later to its multisets by Alhazaymeh and Hassan (2012, 2013a, 2014a). However, the above theories are inadequate to deal with information of indeterminate and inconsistent nature. Smarandache (1998) developed neutrosophic set (NS) generalizing probability, fuzzy and intuitionistic fuzzy sets. NS can be described by membership degrees of truth, indeterminacy and and non-membership (Smarandache, 2005). This theory and its variants has been applied to varied areas of topology (Lupianez, 2008), control theory (Aggarwal et al., 2010), databases (Arora & Biswas, 2010; Arora et al., 2011), diagnosis (Ansari et al., 2011), decision making (Broumi et al., 2014; Deli, 2015; Deli & Broumi, 2015; Kharal, 2014), and multiple attribute decision-making (Chi & Liu, 2013; Liu & Shi, 2015; Liu & Wang, 2014; Sahin & Liu, 2016; Ye, 2015). Maji (2013) introduced neutrosophic soft set, followed by fuzzy soft expert set (Alkhazaleh & Salleh, 2014) which were later extended to generalized fuzzy soft expert set (Hazaymeh et al., 2012), vague soft expert set (Hassan & Alhazaymeh, 2013), generalized vague soft expert set by Alhazaymeh and Hassan (2013b, 2014b, 2014c) and multi Q-fuzzy soft expert set (Adam & Hassan, 2016). Sahin et al. (2015) introduced neutrosophic soft expert sets, while Al-Quran and Hassan (2016a, 2016b) extended it further to neutrosophic vague soft expert set and fuzzy parameterised single valued neutrosophic soft expert set, which were later enlarged to possibility neutrosophic vague soft expert set (Hassan & Al-Quran, 2017). Q -fuzzy soft sets were on the other hand, proposed by Adam and Hassan (2014a, 2014b, 2014c), followed by multi Q -fuzzy soft sets (2014d, 2014e, 2015).

Sahin et al. (2015) introduced neutrosophic soft expert sets using a single universe U. This can be seen from the example provided by Sahin et al. (2015) in their article with a single universe U regarding a set of products. In some real-life problems, we need to deal with two dimensional universal set X × Q, hence the need for Q-neutrosophic soft expert sets. Most decisions are not simply based on the products, but also to suppliers of these products or their manufacturers too. We therefore extend the single dimension neutrosophic soft expert sets to Q-neutrosophic soft expert sets by adding an extra dimension. This can be seen by our illustrated examples where we take the consideration of suppliers besides the products that they supply.