## 5 An Example of Using the Proposed Method  
### Cost Estimation of a New Product  
The relationships between the input and output variables (\(V_{1}\), \(V_{2}\), \(V_{3}\)) have been identified with the use of parametric estimation techniques such as the artificial neuron networks (ANNs), linear regression (LR), and compared with the average. The use of ANNs offers advantages over ability to learn and identification of complex nonlinear relationships. The dataset used in the analysis included 25 completed NPD projects that featured products in the same line as the investigated project. The data were divided into two sets--learning (20 cases) and testing (five cases)--to evaluate the quality of the estimating model. The relationships between the input and output variables are determined using the data related to the previous NPD projects to estimate the cost of product development. The set of input variables is obtained taking into account the significant impact of an input variable on an output variable, as well as the possibility of their controllability by a company and the access to the data through the past specifications of NPD projects. There are the following relationships between  
Figure 4: A framework for solving the inverse form of the project prototyping problem.  
input variables and costs: the NPD cost (\(V_{1}\)) depends on \(\{V_{4},\ldots,V_{9}\}\), the unit production cost (\(V_{2}\)) depends on \(\{V_{9},\ldots,V_{14}\}\), and the advertising and promotional cost (\(V_{3}\)) depends on \(\{V_{15},\)\(V_{16}\}\).  
In this study, a multilayer feed-forward ANN has been trained according to the back-propagation algorithm and weights optimized according to the Levenberg-Marquardt algorithm (LM) and gradient descent momentum with adaptive learning rate algorithm (GDX). The neural network structure has been determined in an experimental way, by the comparison of a learning set and testing set for the different number of layers and hidden neurons. The root mean square errors (RMSEs) have been calculated as the average of 50 iterations for each structure of a neural network with a number to the extent of 30 hidden neurons. Table 1 presents the RMSEs calculated in the learning and testing set using the different parametric models for estimating the cost of NPD (\(V_{1}\)), production (\(V_{2}\)), and promotion (\(V_{3}\)).  
The ANNs trained according to the LM algorithm obtained the least RMSEs in the learning set. However, the least RMSEs in the testing set was calculated using the ANNs trained according to the GDX algorithm (for \(V_{2}\) and \(V_{3}\)) and LM algorithm (for \(V_{1}\)). The trained ANN is used to estimate the unit production cost for the following values of input variables: \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 55, \(V_{12}\) = 25, \(V_{13}\) = 15, \(V_{14}\) = 7. The estimated unit production cost reaches 24.5 \(\upepsilon\), and the total production cost 970 thousand \(\upepsilon\) in the two-year projected period of product life cycle. In turn, the NPD cost reaches 190 thousand \(\upepsilon\), and the advertising and promotional cost 150 thousand \(\upepsilon\).  
Let us assume that the estimated unit production cost does not satisfy the decision maker who is interested in reducing this cost to 24 \(\upepsilon\). To check whether these expectations can be fulfilled, the problem is reformulated into an inverse problem in which such values of input variables are sought to ensure the desired level of the unit production cost.