# A Credit Rating Model in a Fuzzy Inference System Environment
## 2 Literature Review and Basic Definitions
### Fuzzy Inference System  
One of the advantages of fuzzy sets is their ability to translate qualitative and vague information into deterministic and quantitative data. This method has been applied in many different industries worldwide in spite of some conflicting opinions about its methodology ([PERSON], 2015). The most common application of this method is decision-making, especially in an uncertain environment. To implement this method, we introduce the definition and notation of sets below. The first definition related to the membership function is as follows:  
**Definition 1**.: Fuzzy membership: \(\mu_{A}\) is defined as a membership function or characteristic function with values \(\mu_{A}(x)\in[0;1]\) for x\(\in\)X. If A\(\subseteq\)X indicates a crisp (traditional) set, then \(\mu_{A}\) assigns a value 0 or 1 to each member of X. \(\mu_{A}(x)=1\) if x \(\in\) A; this means that x has full membership. \(\mu_{A}(X)=0\) if x \(\
otin\) A; this means that X does not have any membership in X ([PERSON], 2015).  
The membership function of \(\widetilde{\mathrm{A}}\) can be specified, for instance, as a triangular, a trapezoidal, a Gaussian function, or a sigmoid function. Moreover, logical operations can be used including AND, OR, and NOT [24].  
**Definition 2.** Triangular fuzzy numbers: \(\widetilde{\mathrm{A}}=\left\{\mathrm{x},\mu_{\widetilde{\mathrm{A}}}^{-}| \mathrm{x}\in\mathrm{X}\right\}\). There are three parameters of a triangular fuzzy membership function, \(\mathrm{a}\), \(\mathrm{m}\), and \(\mathrm{b}\). The corresponding function is defined in Equation (1) [1]:  
\[\mu_{\widetilde{\mathrm{A}}}^{-}(x)=\begin{cases}\frac{x-x}{m-a},&a\leq x\leq m \\ \frac{b-x}{b-m},&m\leq x\leq b\\ 0,&\text{otherwise}\end{cases} \tag{1}\]  
The third definition relates to the product of fuzzy numbers based on a t-norm operator.  
**Definition 3.** The product of fuzzy numbers: The fuzzy numbers of \(\widetilde{\mathrm{A}}\) and \(\widetilde{\mathrm{B}}\) are produced by t-norm operators, as shown in Equation (2) [1]:  
\[\mu_{\widetilde{\mathrm{A}}}(\mathrm{x})\text{ AND }\mu_{\widetilde{\mathrm{B}}}( \mathrm{y})=-\mu_{\widetilde{\mathrm{A}}}^{-}(\mathrm{x})\times\mu_{ \widetilde{\mathrm{B}}}(\mathrm{y}) \tag{2}\]  
Based on this method, one of the most important applications is the fuzzy inference system (FIS), which uses IF-THEN rules based on fuzzy membership functions. In FIS, all inputs based on a membership function change to an output membership function according to IF-THEN rules. There are various systems that translate the inputs to output membership functions in the FIS. However, we discuss only the two most essential [25; 26; 27]. [PERSON]'s output membership function is based on defuzzification. After computation with fuzzy numbers, these numbers must be transferred into crisp numbers to make decisions easier. There are many methods available for this. In our study, we use the mean method for changing fuzzy numbers to crisp numbers.  
The FIS consists of four steps. First, the inputs and their degree of fuzziness are defined. Second, we set up some fuzzy operators. Third, we determine the weights of each IF-THEN rule and use them to obtain the decision. Fourth, all rules are entered either as inputs or operators.