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add foundations for group-based configuration

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02_Proposal/sections/10_foundations.tex
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\chapter{Foundations} \chapter{Foundations}
\label{ch:Foundations} \label{ch:Foundations}
\section{Group Decision-Making}
\label{sec:Foundation:GroupDecisionMaking}
\section{Decision Tasks}
\label{sec:Foundation:DecisionTask}
\section{Recommender System} \section{Recommender System}
\label{sec:Foundation:RecommenderSystem} \label{Foundations:RecommenderSystem}
\section{Single User Decision Scenario}
\label{sec:Foundation:SingleUserDecisionScenario}
\section{Group Decision Scenario}
\label{sec:Foundation:GroupDecisionScenario}
\section{Product Configuration} \section{Product Configuration}
\label{sec:Foundation:ProductConfiguration} \label{Foundations:ProductConfiguration}
Product configuration is a process consisting of a series of decision tasks whereby a product is constructed of components which interact with each other. During a configuration process no new components are created their interplay and specification is defined beforehand \cite[~ p. 42-43]{sabinProductConfigurationFrameworksa1998}. Product configuration is a process consisting of a series of decision tasks whereby a product is constructed of components which interact with each other. During a configuration process no new components are created. Their interplay and specification is defined beforehand \cite[~ p. 42-43]{sabinProductConfigurationFrameworksa1998}.
Formally a configuration problem can be specified as a \emph{constraint satisfaction problem (CSP)} \cite{tsangFoundationsConstraintSatisfaction1993} as Formally a configuration problem can be specified as a \emph{constraint satisfaction problem (CSP)} \cite{tsangFoundationsConstraintSatisfaction1993} as
\[ \[
@@ -27,6 +15,24 @@ Formally a configuration problem can be specified as a \emph{constraint satisfac
\] \]
where \( V = \{v_1,\dots, v_n\} \) is a set of variables, \( D = dom : V \mapsto X \) is a relation of variables and their corresponding domain definitions \( X \), and \( C = C_{PREF} \cup C_{KB} \) is a set of constraints with customer preferences \( C_{PREF} \) and configuration knowledge base \( C_{KB} \) \cite{felfernigOpenConfiguration2014, felferningGroupBasedConfiguration2016}. where \( V = \{v_1,\dots, v_n\} \) is a set of variables, \( D = dom : V \mapsto X \) is a relation of variables and their corresponding domain definitions \( X \), and \( C = C_{PREF} \cup C_{KB} \) is a set of constraints with customer preferences \( C_{PREF} \) and configuration knowledge base \( C_{KB} \) \cite{felfernigOpenConfiguration2014, felferningGroupBasedConfiguration2016}.
\section{Collaborative Product Configuration}
\label{sec:Foundation:CollaborativeProductConfiguration} \section{Group-Based Product Configuration}
\label{Foundations:GroupBasedProductConfiguration}
To define group-based product configuration we extend the definition (\ref{Foundations:ProductConfiguration}) to
\[
C_{PREF} = \bigcup
PREF_i \]
with preferences of user \( i \) as \( PREF_i \) \cite{ felferningGroupBasedConfiguration2016}.
\section{Group-Based Configuration-Solution}
\label{Foundations:GroupBasedConfigurationSolution}
\ref{Foundations:ProductConfiguration} and \ref{Foundations:GroupBasedProductConfiguration} expand to a solution of a group-based configuration with the addition of variable assignments
\[
C_{CONF} = \bigcup_{v_i \in V} \{ v_i = x_i \}, \ x_i \in dom(v_i)
\]
and where \( C_{CONF} \cup C_{PREF} \cup C_{KB} \) is consistent \cite{ felferningGroupBasedConfiguration2016}.