*3.3. Third Layer*

The third layer of the fuzzy neural network is composed of the neural network that aggregates all the fuzzy rules generated in the first layer. Its inputs are the fuzzy neurons with their respective weights generated in the training process, to be explained later. An artificial neuron aggregates these fuzzy rules and processes them with weights to generate model outputs. This singleton neural network (composed of a single neuron) has a linear activation function, and a signal function transforms the values obtained by the model into

the expected outputs (−1 and 1), indicating whether or not there was fraud in the auction. This defuzzification process can be mathematically described by [38]:

$$y = \Omega\left(\sum\_{j=0}^{l} f\_{\Gamma}(z\_j, v\_j)\right) \tag{39}$$

where *z*0 = 1, *v*0 is the bias, and *zj* and *vj*, *j* = 1, . . . , *l* are the output of each fuzzy neuron of the second layer and their corresponding weight, respectively. *f*Γ represents the linear activation function and Ω the signal function [38].

The linear and signal function are represented, respectively, by:

$$f\_{\Gamma}(\mathbf{z}) = \mathbf{z}^\* \mathbf{w} \tag{40}$$

$$\Omega = \begin{cases} -1, \text{ if } \sum\_{j=0}^{l} f\_{\Gamma}(z\_{j\prime} v\_{j}) > 0 \\ -1, \text{ if } \sum\_{j=0}^{l} f\_{\Gamma}(z\_{j\prime} v\_{j}) < 0 \end{cases} \tag{41}$$
