## 3 Problem Formulation
The following set of variables for estimating the NPD, production, and promotional cost were proposed:  
\(V_{1}\)--NPD cost (in thousand \(\mathfrak{C}\)),  
\(V_{2}\)--unit production cost (in \(\mathfrak{C}\)),  
\(V_{3}\)--advertising and promotional cost (in thousand \(\mathfrak{C}\)),  
\(V_{4}\)--number of R&D employees involved in product design,  
\(V_{5}\)--number of R&D employees involved in prototype tests,  
Figure 2: The project prototyping problem stated in a forward (**a**) and inverse form (**b**). \(V_{6}\)--duration of product design (in months),  
\(V_{7}\)--duration of prototype tests (in months),  
\(V_{8}\)--number of prototype tests (in hundreds),  
\(V_{9}\)--number of product components,  
\(V_{10}\)--amount of materials needed to manufacture a unit of a new product,  
\(V_{11}\)--amount of energy needed to manufacture a unit of a new product,  
\(V_{12}\)--assembly time for a unit of a new product,  
\(V_{13}\)--processing time for a unit of a new product,  
\(V_{14}\)--number of workplace units needed for assembling and processing a new product,  
\(V_{15}\)--duration of advertising and promotional campaign of a new product,  
\(V_{16}\)--number of potential receivers of advertising and promotional campaign,  
\(V_{17}\)--sales volume,  
\(V_{18}\)--production cost (in thousand \(\mathfrak{C}\)),  
\(V_{19}\)--desirable margin (in \(\mathfrak{C}\)),  
\(V_{20}\)--product price (in \(\mathfrak{C}\)),  
\(V_{21}\)--sales revenue of a new product (in thousand \(\mathfrak{C}\)). There are the following constraints regarding the available quantity of resources in a company and technical parameters of a new product: NPD project budget (\(PB\), in thousand \(\mathfrak{C}\)), total number of R&D employees involved in an NPD project (\(TE\)), deadline for launching the new product into the market (\(LD\), in months), time needed to manufacture a unit of a new product (\(MT\)), and advertising and promotional budget (\(AB\), in thousand \(\mathfrak{C}\)). The set of constraints and relationships is as follows:  
\[V_{1}\leq PB \tag{2}\]  
\[V_{4}+V_{5}\leq TE \tag{3}\]  
\[V_{6}+V_{7}\leq LD \tag{4}\]  
\[V_{12}+V_{13}\leq MT \tag{5}\]  
\[V_{3}\leq AB \tag{6}\]  
\[V_{2}\cdot V_{17}=V_{18} \tag{7}\]  
\[V_{2}+V_{19}\leq V_{20} \tag{8}\]  
\[V_{1}+V_{3}+V_{18}\leq V_{21} \tag{9}\]  
The model formulated as an CSP incorporates the technical parameters of the new product that refer to the planned project performance and the available resources. The problem is solved by searching for answers to the following questions:  
* What is the cost of product development (including unit production cost)? * What values should the variables have to reach the desirable level of the cost? A project prototyping problem can be expressed as a CSP and then be solved with the use of specific techniques such as constraint propagation and variable distribution. Constraint propagation applies constraints to prune the search space. Propagation techniques aim to reach a certain level of consistency, and accelerate the search procedures to reduce the size of the search tree [28]. The values of the variables excluded by constraints are removed from their domains. A CSP may be effectively solved with the use of CP techniques. The declarative nature of CP is particularly useful in applications where it is enough to state what has to be solved without saying how to solve it [28]. As CP uses specific search methods and constraint propagation algorithms, it allows to considerably reduce the search space. Consequently, CP is suitable for modeling and solving complex problems.