## 2 Related Work
### Three-Way Decisions  
Decision-theoretic rough sets (DTRSs) [14; 15] were put forward to deal with fuzzy, rough, imprecise, and uncertain concepts. The classification rules derived from DTRSsdivided the universe into the positive region, negative region, and boundary region [16; 17]. Their corresponding decisions are acceptance decision, rejection decision, and delay decision, respectively. This is usually called a three-way decision (3 WD) [18]. Due to the constraints of decision-makers' professional abilities, knowledge, and experience in the actual decision-making process, it is often difficult to use a crisp value loss function for evaluation [19; 20]. DTRSs provide fuzzy or uncertain evaluation information, such as interval numbers [21] and fuzzy linguistic variables [22; 23]. In recent years, 3 WD theories with uncertain evaluation loss functions have been widely studied [24; 25]. The Pythagorean fuzzy set (PFS) [26; 27; 28] is an important mathematical tool to deal with uncertain evaluations. Recently, some cost-sensitive loss functions have been expressed by PFSs. According to Bayesian theory, [PERSON] and [PERSON] [29] calculated the loss function matrix represented by PFSs, constructed a Pythagorean fuzzy 3 WD (PF-3 WD) model, and obtained the corresponding decision knowledge. [PERSON] et al. [30] further studied a 3 WD model based on Pythagorean fuzzy ideal TOPSIS solutions. [PERSON] et al. [31] constructed probability measures and belief and plausibility functions of a PFS and applied them to attribute reduction methods. It is necessary to combine DTRSs and PFSs for an in-depth study of the 3 WD theory of cost-sensitive loss functions with Pythagorean fuzzy evaluation [32]. To obtain complete PF-3 WD knowledge, it is necessary to determine the conditional probability \(\Pr(X|[x])\) and probability thresholds \(\alpha\) and \(\beta\). The probability threshold is determined according to the cost-sensitive loss matrix, and the conditional probability is obtained from the Pythagorean fuzzy decision system. To measure the similarity of two objects about the conditional attribute set and obtain the similarity class, we aimed to establish a new Pythagorean fuzzy similarity measure. At present, some similarity measures of PFS have been developed. [PERSON] et al. [33] constructed the concept of a fuzzy similarity matrix, whose value is a Pythagorean fuzzy number (PFN), and applied it to clustering analysis. Some other Pythagorean fuzzy similarity measures were investigated [34]. However, the existing Pythagorean fuzzy similarity measures have the following two problems. (1) There may be \"counter intuitive\" situations. For example, let \(A=\{(x,0.31,0.94)|\)\(x\in U\}\), \(B=\{(x,0.70,0.54)|x\in U\}\) and \(C=\{(x,0.77,0.54)|x\in U\}\) be three PFSs on \(U=\{x_{1}\}\). Obviously, \(A\subseteq B\subseteq C\). The Pythagorean fuzzy similarity measures are calculated as \(\theta(A,B)=(0.6879,0.2960)\) and \(\theta(A,C)=(0.7002,0.2960)\). Then, \(\theta(A,B)\subset\theta(A,C)\). Intuitively, this is obviously inconsistent with \(A\subseteq B\subseteq C\). There are similar problems in other similarity measures [35]. In addition, it can be calculated as \(\theta(A,B)=(0.4080,0.5920)\) and \(\theta(A,C)=(0.4080,0.5920)\) by [PERSON] et al. 's method [36]; that is, \(\theta(A,B)=\theta(A,C)\). Although \(B\
eq C\), it does not meet the property of Pythagorean fuzzy similarity measures in some cases [37]. (2) The practical semantics of membership degree and non-membership degree are not considered. Most of the existing Pythagorean fuzzy similarity measures are constructed based on Pythagorean fuzzy similarity relations. In the construction process, the practical semantics of membership degree and non-membership degree are rarely considered. To solve the above problems, a novel Pythagorean fuzzy closeness measure based on the similarity-divergence measure was constructed.