## 2 Materials and Methods
### Model Development
#### 2.3.2 AnFIS  
The ANFIS model is a soft computation model that serves as a combination of neural networks and fuzzy systems [20]. Hence, it provides a synergetic effect that incorporates the advantages of both systems in linear and non-linear processes. ANFIS constitutes an integration of fuzzy inference systems (FIS) and ANNs, maintaining both advantageous features in a single structure, such as the trainability, the optimizable connectionist structure, the human expert knowledge in linguistic terms using if-then rules. In ANFIS, the main problem is determining the type of membership function and their degree of membership, for which there is no specific solution, and by using trial and error, the best structure of the model is determined. For this purpose, in training the structure of ANFIS, the hybrid method was used, which is a combination of the least-squares method and the back propagation method. ANFIS can be potently employed to find the optimum relationship between a set of input and output data effectively. Fuzzy rules can be generated from the given input-output data by implementing the Takagi-Sugeno fuzzy inference system, thus this system is always interpretable in terms of fuzzy if-then rules in contrast with ANNs where the knowledge gained by training is split in every network weight and is not possible to be interpreted by humans [20]. The inputs and outputs parameters are related together using some fuzzy rules known as IF-THEN rules. In the present study, fuzzy IF-THEN rules were used to evaluate the _MR_ of parboiled hulls. ANFIS is a five-layered model that utilizes fuzzy logic controllers in solving problems at a high rate with a minimum steady-state error. The ANFIS model consists of five layers whose functions are fuzzification, multiplication, normalization, defuzzification, and summation, respectively [42; 47]. The designed ANFIS system consists of three inputs (inlet air temperature (T), infrared power (P), and drying time (DT)), and from the other side one outlet (_MR_ (f)) (Figure 5). Fuzzy if-then rules are written for this structure based on Equations (7) and (8). Figure 4: Activation functions of Tansig, Logsig, and Purlin. In this study, fuzzy if-then rules consisted of four rules, three of them for inputs (T, P, and DT) and one rule for output (f), as in the following forms [30]. \[Rule\ 1:\text{if }T\text{ is }A_{1}\text{, }P\text{ is }B_{1}\text{ and }DT\text{ is }C_{1} \tag{7}\]  
\[\text{then }f_{1}=p_{1}T+q_{1}P+r_{1}DT+s_{1} \tag{8}\]  
\[Rule\ 1:\text{if }T\text{ is }A_{2}\text{, }P\text{ is }B_{2}\text{ and }DT\text{ is }C_{2}\] \[\text{then }f_{2}=p_{2}T+q_{2}P+r_{2}DT+s_{2} \tag{9}\]  
where T is the input air temperature (\"C), P is the infrared power (W/cm\({}^{2}\)), DT is the drying time (min), f is the \(MR\), A\({}_{1}\), A\({}_{2}\), B\({}_{1}\), B\({}_{2}\), C\({}_{1}\) and C\({}_{2}\) are the language indicators, r\({}_{1}\), r\({}_{2}\), q\({}_{1}\), q\({}_{2}\), p\({}_{1}\), p\({}_{2}\), s\({}_{1}\), and s\({}_{2}\) are the linear coefficients of the output function and f\({}_{1}\) and f\({}_{2}\) are first-degree polynomials. The first layer (fuzzification): each node in the first layer is an adaptive node and the corresponding node function is written as follows [39]:  
\[O^{1}{}_{Ai}=\mu_{Ai}(T)\hskip 28.452756 pti=1,2 \tag{10}\]  
\[O^{1}{}_{Bj}=\mu_{Bj}(P)\hskip 28.452756 pti=1,2 \tag{11}\]  
\[O^{1}{}_{Ai}=\mu_{Ci}(DT)\hskip 28.452756 pti=1,2 \tag{12}\]  
where, \(\mu_{Ai}\), \(\mu_{Bj}\), and \(\mu_{Ci}\) are the membership functions from the language variables of A, B, and C.  
In the second layer; known as multiplication; each rule functions as a node in the ANFIS network using soft-min or product to determine the binding factor of the rule. The input signal in Equation (12) is multiplied by this layer and exits the product.