## 4 Discussion and Examples  
It is not surprising that biotechnology-inspired collective intelligence methods are intensively applied in the field of engineering design to study various multi-agent systems. In particular, such methods are also applied in product form design problems [5; 6; 7; 8; 9; 10; 11]. Using different neural network models, researchers have attempted to better understand the connections between an initial design idea and the desired output to achieve the goals of modern design. The implementation of the fractional calculus approach in neural network modeling leads to the design of more flexible models. However, this method has not been well studied for neural network models used in the area of product form design [31]. Given the advantages of discrete fractional-order neural network models, in this study, a generalization of some existing models applied in the product form design is proposed by using the nabla fractional operator. Moreover, different from the model proposed in [31], impulsive controllers are proposed. Examining such a control technique enables the use of its well-known advantages [40; 42; 44; 45]. There are many cases where the impulse control technique can provide an efficient means to deal with systems that cannot undergo continuous control inputs. The advanced properties of the hybrid fractional-order impulsive model system (5) make it an appropriate extension of some integer-order neural network models used in product form design analysis. For example, model (5) can be described as follows. * A fractional-order generalization of the neural network model designed in [5], in which the state variables \(\mathcal{U}_{h}(\eta)\), \((h=1,2,\ldots,n)\) represent the form parameters at the discrete time \(\eta\). Using the generalized model (5), it is possible to study the dependence of the product image on the configuration parameters and element shapes using the fractional-order differences between their current and previous values. * A fractional-order generalization of the form design model introduced in [6], in which the state variables \(\mathcal{U}_{h}(\eta)\), \((h=1,2,\ldots,n)\) are defined as word pairs or a set of target image values and their respective weights, and there is a set of 21 output-node data. * A fractional-order generalization of the neural network model considered in [7], aimed at achieving a desirable product image depending on the best combination of form elements used to study the mobile phone's form, in which the state variables \(\mathcal{U}_{h}(\eta)\), \((h=1,2,\ldots,n)\) are defined as 27 form elements (input neurons) extracted by morphological analysis, namely the top shape (\(\mathcal{U}_{1}\)), body shape (\(\mathcal{U}_{2}\)), bottom shape (\(\mathcal{U}_{3}\)), body length and width ratio (\(\mathcal{U}_{4}\)), function button style (\(\mathcal{U}_{5}\)), number button arrangement (\(\mathcal{U}_{6}\)), screen size (\(\mathcal{U}_{7}\)), screen mask and function buttons (\(\mathcal{U}_{8}\)) and outline division style (\(\mathcal{U}_{9}\)), and the outputs are the evaluation values for the desirable product image. * A fractional-order generalization of the neural network model considered in [11], which generates forms that have a greater possibility of satisfying the architect's personal design taste and consists of 21 input variables \(\mathcal{U}_{0}\) that represent numbers that are sufficient to identify a form from others, as well as two hidden layers of 50 neuronal variables \(\mathcal{U}_{h}(\eta)\) each and one output variable, which denotes the score of the form. In all above-discussed generalizations, \(w_{hj}\) are the weights between the neurons (states) in the input and output layers, \(f_{j}\) is the activation function (in most practical cases, sigmoid) and \({}^{C}\
abla_{0}^{a}\mathcal{U}_{h}(\eta)\) is the Caputo nabla fractional difference, which generalizes the integer-order difference between the current and previous states of the state variable \(\mathcal{U}_{h}(\eta)\).