## 3 Computer-Aided Product Family Modeling
### Product Family Algebra  
Product Family Algebra (PFA) is a formal language with a mature mathematical foundation, it mainly translates it into a formal description by studying the hierarchical relationships between features and the constraints between features in the product line family model [28], PFA is defined as a (S, +, 0, \(\cdot\), 1) quintet to describe the features in the product family feature model, and mathematically proves the given theorem, making the product family. The formalization of the feature model becomes possible.  
**Definition 1**.: _A semiring is a quintuple (S, +, 0, \(\cdot\), 1) such that (S, +, 0) is a commutative monoid and (S, \(\cdot\), 1) is a monoid such that \(\cdot\) distributes over + and 0 is an annihilator._  
In the definition of PFA, S is the set of features. The symbol + can be interpreted as a choice between features of different products, while the symbol \(\cdot\) is the composition or forced presence of features. 0 represents an empty product family that does not contain any features. Element 1 represents a pseudo-product family without any features.  
We will use the definition of relationships and constraints between features to build templates for guiding the construction and evolution of feature models according to the definition of PFA, which does not require any knowledge of the PFA language. It is not necessary to know the PFA language to use but to select the template according to the requirement description and give the set of parameters to automatically form the PFA expressions of the pair, which will be used to build the visual display of the feature model and the subsequent consistency check. PFA uses mathematical language to describe the relationships and constraints between features and convert feature trees into concrete representations. In this way, we can describe product feature models and relationships between features mathematically.  
An example of an automotive product line is given to illustrate the PFA expression.  
\(\text{automotive}=((\text{Roadster}+\text{SUV}+\text{Truck})\cdot\text{opt}[ \text{Autopilot}]\cdot\text{Tire}\cdot(\text{ManualGearbox}+\text{AutomaticGearbox})\cdot\)  
\((\text{opt}[\text{leather}]+\text{opt}[\text{Audio}]+\text{opt}[\text{CarFridge }]+\text{opt}[(\text{leather}\cdot\text{Audio})]+\text{opt}[(\text{leather} \cdot\text{CarFridge})]+\)  
\(\text{opt}[(\text{Audio}\cdot\text{CarFridge})]+\text{opt}[(\text{leather} \cdot\text{Audio}\cdot\text{CarFridge})])\).  
In the definition of the feature model, features are classified into two main classes. The first class is single features, including mandatory and optional; the second class is multi-feature structure, including multiple, alternative, and or-groups. PFA describes features in the following way. In the automotive product line, including the mandatory models (roadster, SUV, truck), the optional Autopilot configuration, defined as opt-in Autopilot. When a feature can be decomposed to be represented by multiple sub-features, the correlation relationship between the sub-features needs to be considered, containing both uncorrelated multi-feature models and correlated multi-feature models. Unrelated multi-feature models are those that can be chosen arbitrarily and have no influence on each other. There are two types of correlated multi-feature models, which can be classified as or-group features and alternative features. The or-group structure means that they can be combined at will. Just like the relationship between leather, audio, and car fridge in a car product line, they can be combined in any way. Alternative features imply that one and only one feature of the group must be selected. Features have two dependent limitations. Autopilot includes automatic gearbox, which means that the automatic gearbox feature must be selected when the Autopilot feature is present. The second relationship is an exclude relationship, meaning that two features cannot exist in the same product.