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Top1. Introduction
In many practical management problems, information is uncertain, decision makers are limited rational and decision making behavior is complex. There are often ambiguities and uncertainties when decision makers make judgments. Fuzzy numbers can be used to indicate this uncertain information (Dubois & Prade, 1987). Establishing fuzzy metric space is an important basic research on fuzzy theory. In recent decades, many researchers have studied the distance of fuzzy numbers (Cheng, 1998; Chakraborty, 2006; Guha & Chakraborty, 2010a; Rouhparvar et al., 2011; Wu et al., 2012; Sadinezhad et al., 2013; Akbari & Hesamian, 2018; Chia & Yu, 2018). However, as an extension of fuzzy numbers, the intuitionistic fuzzy (IF) numbers (IFNs) can describe the fuzzy nature of uncertain information more comprehensively (Li & Liu, 2015). Several scholars have focused on studying the distance between IFNs (Wang & Xin, 2005; Grzegorzewski, 2004; Zeng & Su, 2011; Guha & Chakraborty, 2010b; Zeng, 2013; Singh & Grag, 2017; Grag & Kumar, 2018; Fe et al., 2019; Muharrem, 2016). These distance measures between IFS or IFNs were expressed as crisp values. However, the distance measure of the IFNs that have been proposed has not escaped the constraint of real numbers, that is, it uses the real number as a distance measure between two IFNs and discusses their properties. As we all know, logically, the distance of IFNs is also an IFN instead of a real number. Therefore, we need to establish analysis theories of IFNs that are independent of real numbers. This needs to establish a metric space that the distance measure of IFNs is still an IFN. It can be called an IF distance measure. There is a few investigation on IF distance measure. Guha and Chakraborty, (2010b) studied the IF distance measure. But in some cases, the distance of the IFN which is proposed by Guha and Chakraborty (2010b) is a negative number. In order to overcome this shortcoming, Nan et al. (2016) proposed a novel IF distance measure for triangular IF numbers (TIFNs) based on interval difference.
Most real decision making problems involve multiple criteria and uncertain information at the same time. In this case, relying on individual decision maker is not enough. Hence, group decision making is getting more and more attention due to its unique advantages (Wang et al., 2016; Li, 2007b; Zuo et al., 2020; Liu et al., 2017; Meng et al., 2014; Gong, 2013; Liu et al., 2014; Boran et al., 2009; Liu et al., 2018; Li & Liu, 2020; Morente-Molinera et al., 2019). There is an effective and widely used method to solve multiple attribute group decision making(MAGDM), which is called the compromise ratio method (CRM) (Li, 2007a). The basic idea of the CRM is that the chosen alternative should be as far away from the negative ideal solution as possible and as close as possible to the positive ideal solution. Therefore, the use of an effective scientific distance measure becomes the key to scientific decision making. Initially, Li (2007a) introduced the CRM to solve the fuzzy multiple attribute decision making (FMADM) problems. Afterwards, the CRM was extended to solve the FMADM problems with a group of decision makers, which are abbreviated as FMAGDM problems (Li, 2007b). However, the distance measure of fuzzy quantities was expressed as a real number (Li, 2007a; Li, 2007b). As mentioned earlier, such distance measures are illogical. In fact, there is no research on CRM with the IF distance measure to solve the FMAGDM. Hence, based on the IF distance measure of (TIFNs) proposed by Nan et al. (2016), we study the CRM with IF distance for solving FMAGDM in which the ratings of alternatives on attributes and the attribute weights are expressed by using linguistic variables, which are transformed to TIFNs.