\chapter{Concept}
\label{ch:Concept}

\section{Foundations Extension}
\label{sec:Concept:Requirements}

The definitions described in \autoref{ch:Foundations} need to be extended for this thesis. This section adds definitions that are needed.

\subsection{Configuration State}

A \emph{configuration} $S$ will be defined as a tuple of variables (\autoref{eq:Foundations:ProductConfiguration:Variables}) and their corresponding domain value with
\begin{equation} \label{eq:Foundations:ProductConfiguration:ConfigurationState}
    S = \{ (v_i,\ d) \ |\ v_i \in V \ \land \ d \in \mathfrak{D}(i),\ i=1,\dotsc,m \}.
\end{equation}
Essentially it is a set of variables and assigned values.

\subsection{Finished Configuration}
To define what a \emph{finished configuration} is, it is required to first define what it means for a configuration to be valid. Therefore $is\_valid$ is defined as
\begin{equation} \label{eq:Foundations:ProductConfiguration:IsValid}
    is\_valid : S \to \{true, false\}; x \mapsto 
    \begin{cases}
        true, & S \in solution\_space \\
        false, & \text{otherwise}
    \end{cases},
\end{equation}
with $solution\_space$ being the solution space of the corresponding constraint satisfaction problem. A \emph{finished configuration} $S_F$ is a configuration that contains all variables and is a valid configuration:
\begin{equation} \label{eq:Foundations:ProductConfiguration:FinishedConfiguration}
    S_F \subset S,\ where \ \forall v_i \in V (\exists (v_i, d) \in S_F : d \in \mathfrak{D}(i)) \land is\_valid(S_F).
\end{equation}
In practice a finished configuration of a product (or solution) is something that is ready to be produced. For example if a care is being configured, this means that the car could be produced in the specified way that is given by the finished configuration.


\subsection{Group-Based Product Configuration}
\label{sec:Foundations:GroupBasedProductConfiguration}

Instead of a single person configuring a product, a group of people is configuring one product which can be useful in multi-stakeholder decisions. This setting needs mechanisms for describing the preferences of multiple people. Therefore to the definitions there will be added a set of users $U$ with
\begin{equation}\label{eq:Foundations:ProductConfiguration:Users}
    U = \{1, \dotsc, n\},
\end{equation}
a users \emph{utility function} that maps a domain value to a utility value and is only known to the user
\begin{equation}
    \begin{split}
        u_i(d_j), \qquad \text{where}\ & d_j \in  \mathfrak{D}(j),\\
        & 1 <= j <= m, \\
        & 1 <= i <= n .
    \end{split}
\end{equation}


\section{Requirements}
\label{sec:Concept:Requirements}

\section{Assumptions}
\label{sec:Concept:Assumptions}

\begin{itemize}
    \item Only one product/solution is supposed to be configured
    \item Features only support single value attributes
    \item Users join the system and start configuring once all group members have joined
\end{itemize}

\section{User Interaction with the System}
\label{sec:Concept:UserSystemInteraction}

The system has one main way to be used as defined in \autoref{tab:Concept:MainUseCase}. This process is also visualized in \autoref{fig:Concept:ConfigurationProcess}.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{./figures/40_concept/bpmn_configuration_process_with_continious_recommendation.pdf}
    \caption{A BPMN diagram of the configuration process.}
    \label{fig:Concept:ConfigurationProcess}
\end{figure}

\begin{table}
    \begin{center}
        \begin{tabularx}{\columnwidth}{l|X}
            \multicolumn{2}{c}{Main System Usage} \\
            \hline
            Preconditions   & 
                \begin{itemize}
                    \item The configurator is opened with the same session on each of the group member's machines
                    \item The configuration is in an unfinished state (this state is a consensus state)
                \end{itemize} \\
            \hline
            Postcondition   & 
                \begin{itemize}
                    \item All users have entered their preferences for each attribute explicitly.
                    \item The system gives a recommendation based on all preferences and the unfinished configuration state
                \end{itemize} \\
            \hline
            Basic Flow      & 
                \begin{enumerate}
                    \item A user indicates a preference for an attribute
                    \item The system generates a recommendation (based on preferences and configuration status)
                    \item If not all users have given their preferences go to step 1.
                \end{enumerate} \\
            \hline
        \end{tabularx}
        \caption{A description of the main way users will interact with the system}
        \label{tab:Concept:MainUseCase}
    \end{center}
\end{table}

\section{Case Study}
\label{sec:Concept:CaseStudy}

The case study used in this thesis is a simplified version from forestry.
The used characteristics and attributes are shown in \autoref{fig:Concept:ForestExample}. Additionally as example are given preferences, a configuration state and a finished configuration.

\begin{figure}
    \begin{mdframed}[frametitle={Example for Forest Use Case}]
        In this example there are a small group of users. The use case is a piece of forest and variables are for example harvesting activity, which trees to grow and accessibility for people.
        \begin{align}
            \begin{split}
                V = \{ & \textit{indigenous}, \textit{resilient}, \textit{usable}, \textit{effort}, \textit{quantity}, \textit{price}, \textit{accessibility} \},
            \end{split} \notag \\
            \mathfrak{D}(\textit{indigenous}) =  \{ & \text{low}, \text{moderate}, \text{high}\}, \notag \\
            \mathfrak{D}(\textit{resilient}) = \{ & \text{low}, \text{moderate}, \text{high}\}, \notag \\
            \mathfrak{D}(\textit{usable}) = \{ & \text{low}, \text{moderate}, \text{high}\}, \notag \\
            \mathfrak{D}(\textit{effort}) = \{ & \text{manual}, \text{harvester}, \text{autonomous}\}, \notag \\
            \mathfrak{D}(\textit{quantity}) = \{ & \text{low}, \text{moderate}, \text{high}\}, \notag \\
            \mathfrak{D}(\textit{price}) = \{ & \text{low}, \text{moderate}, \text{high}\}, \notag\\
            \mathfrak{D}(\textit{accessibility}) = \{ & \text{low}, \text{moderate}, \text{high}\},\notag \\
            U = \{ & 1,2\} \notag\\
            P = \{ & P_1, P_2\} \notag\\
            \begin{split}
                P_1 = \{ & (\text{manual}, 0.8), (\text{harvester}, 0.3) \} \\ 
                & \cup \{ (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V,\ i \notin \{ \text{manual}, \text{harvester}\} \ \} \ 
            \end{split} \notag \\
            P_2 = \{ & (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V \} \notag \\
            S  =  \{ & (\textit{indigenous}, \text{low}), (\textit{quantity}, \text{moderate}) \} \notag \\
            \begin{split}
            S_F  =  \{ & (\textit{indigenous}, \text{low}), (\textit{resilient}, \text{low}), (\textit{usable},\text{low}), (\textit{effort}, \text{manual}), \\
            & (\textit{quantity}, \text{low}), (\textit{price},\text{high}),(\textit{accessibility}, \text{low}) \} 
            \end{split} \notag
        \end{align}
    \end{mdframed}
    \caption{An example of a forest use case that includes two people.}
    \label{fig:Concept:ForestExample}
\end{figure}


\section{Solution Generation}
\label{sec:Concept:SolutionGeneration}

Given an unfinished configuration and preferences of all group members, rate a finished configuration on how well it reflects the configuration state and preferences. Use this to choose the best finished configuration out of a list to recommend.

\subsection{Generating a Recommendation}

Hereby the idea is there is a database of complete configurations (possibly historic from other groups or automatically generated or both).
Now the recommendation procedure looks as follows:

\begin{enumerate}
    \item Assign a score to each stored configuration according to $$score_{group}(\overline{configurationState},\ \overline{preferences}, \ configurationInStore)$$
    \item Optional: Filter out configurations that have a score below a certain value using a different scoring function. For example filter out configurations that cause a certain level of misery.
    \item Chose the configuration with the highest score as recommendation.
\end{enumerate}

\todo[inline]{move definitions that are made by me to here}



\begin{samepage}
\subsection{Scoring Function}
\label{subsec:Concept:SolutionGeneration:ScoringFunction}
\emph{Group configuration scoring function} includes preferences and current configuration state. This function gives a score for a finished configuration (while using the current configuration state and all user preferences):
\begin{equation}
    score_{group}: S \times P \times S_F \to \mathbb{R}
\end{equation}

An example group configuration scoring function is $score_{group}$ with

\begin{equation}
    \notag \alpha \in \mathbb{R}, \qquad     changed(d,\overline{s}, s) = 
    \begin{cases}
      1, & d \in \overline{s} \land d \notin s \\
      0, & \text{otherwise}
    \end{cases}
\end{equation}

\begin{equation}
    \begin{split}
        score_{group}(\overline{s},\ \overline{p},\ s)
        & = score(\overline{p},\ s) - penalty(\overline{s},\ s) \\
        & = score(\overline{p},\ s) - \sum_{d \in \overline{s}} changed(d,\overline{s}, s) \cdot \alpha
    \end{split}
\end{equation}

This thesis will use multiple scoring functions. Among those are ones for least misery, average and multiplicative. Average and multiplicative yield good results among the studies presented by \citeauthor{Masthoff2015} \cite{Masthoff2015}. Strategies can also be combined, one example here is average without misery.
\end{samepage}

\subsubsection{Preference Scoring}

All of the aggregation functions mentioned in \autoref{subsec:Concept:SolutionGeneration:ScoringFunction} use one preference per user per product. Therefore to use them in as is a score for the whole configuration per user has to be calculated. I propose to use the difference from the selected feature compared to the average rating of all characteristics. This approach includes all preferences of a user meaning a preference is also seen relative to others.

As an example a feature could be
\begin{equation}
    F = \text{ClimateResilientTrees},
\end{equation} with characteristics
\begin{equation}
    \mathfrak{D}(F)= \{\text{low}, \text{medium}, \text{high}\},
\end{equation}
preferences
\begin{equation}
    P_1 = \{(\text{low}, 0), (\text{medium},0.6), (\text{high},0.9) \}
\end{equation} 
and the configuration that is supposed to be rated
\begin{equation}
    S_F = \{(\text{ClimateResilientTrees}, \text{high})\}.
\end{equation}
The average rating for the feature is $F = 0.5$. Therefore the score given by a user $1$ is $0.9-0.5 = 0.4$.
A second user with preferences 
\begin{equation}
    P_2 = \{(\text{low}, 0), (\text{medium},0), (\text{high},0.9) \}
\end{equation} 
on the other hand results in a feature score of $0.9-0.3=0.6$. For this user characteristic \emph{high} is of higher importance.

As scores should be kept as percentages and not in the interval $[-1,1]$ a normalisation is applied by adding one and dividing by two. Therefore the respective scores are $0.7$ for user one and $0.95$ for user two. A configuration usually consists of more than one feature therefore an average rating over all features is taken to get the score one user gives to a configuration. Based on that score the in \autoref{subsec:Concept:SolutionGeneration:ScoringFunction} mentioned aggregation functions can be used.

\subsubsection{Cofiguration Change Penalty}

In this thesis a penalty function is proposed which gives the percentage of characteristics that exist in the configuration that is to be rated. This value can be tuned to be more or less strict by potentiating. Thereby allowing more deviation or less deviation from the current configuration state. The penalty function is defined as
\begin{equation}
    penalty_{proportion}(\overline{s},\ s) =  \left(\frac{\sum_{d \in \overline{s}} changed(d,\overline{s}, s)}{|\overline{s}|}\right)^\alpha.
\end{equation}

By including the current configuration into scoring the scoring function can take into account, changes that have been already implemented and therefore might be very costly to changed to make illustrations more easy.

\section{Illustration}
\label{sec:Concept:Illustration}

This section gives an example to illustrate how the recommendation works. The example in \autoref{fig:Concept:ForestExample} is used for that but the preferences are extended.

\begin{itemize}
    \item[$S_{F2}$ vs. $S_{F1}$] effect of divergence from configuration state
    \item[$S_{F3}$ vs. $S_{F1}$] preference comparison
    \item[$S_{F4}$ vs. $S_{F1}$] better preferences but divergence configuration state
\end{itemize}

\begin{table}
    \tiny
    \begin{tabularx}{\columnwidth}{C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|}
        & \multicolumn{3}{c|}{\textit{indigenous}} & \multicolumn{3}{c|}{\textit{resilient}} & \multicolumn{3}{c|}{\textit{usable}} & \multicolumn{3}{c|}{\textit{effort}} & \multicolumn{3}{c|}{\textit{quantity}} & \multicolumn{3}{c|}{\textit{price}} & \multicolumn{3}{c|}{\textit{accessibility}} \\
        \rotatebox[origin=c]{90}{\ preferences} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{manual} & \rotatebox[origin=c]{90}{harvester} & \rotatebox[origin=c]{90}{autonomous} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} \\
        \hline
        $P_1$   & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & \textbf{0.1} & \textbf{0.7} & \textbf{0.9} & \textbf{0.6} & \textbf{0.8} & \textbf{0.2} & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\
        $P_2$   & \textbf{0.1} & \textbf{0.6} & \textbf{0.9} & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & \textbf{0.8} & \textbf{0.3} & \textbf{0.1} & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 \\
    \end{tabularx}
    \caption{A table showing the preferences of an example for this section.}
    \label{tab:Concept:UseCaseRating}
\end{table}

\begin{table}
    \tiny
    \begin{tabularx}{\columnwidth}{C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|C|}
        & \multicolumn{3}{c|}{\textit{indigenous}} & \multicolumn{3}{c|}{\textit{resilient}} & \multicolumn{3}{c|}{\textit{usable}} & \multicolumn{3}{c|}{\textit{effort}} & \multicolumn{3}{c|}{\textit{quantity}} & \multicolumn{3}{c|}{\textit{price}} & \multicolumn{3}{c|}{\textit{accessibility}} \\
        \rotatebox[origin=c]{90}{\ configuration} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{manual} & \rotatebox[origin=c]{90}{harvester} & \rotatebox[origin=c]{90}{autonomous} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} & \rotatebox[origin=c]{90}{low} & \rotatebox[origin=c]{90}{moderate} & \rotatebox[origin=c]{90}{high} \\
        \hline
        $S_{\ \ }$         & - & \pmb{\checkmark} & - & - & \pmb{\checkmark} & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
        $S_{F1}$    & - & \pmb{\checkmark} & - & - & \pmb{\checkmark} & - & - & \checkmark & - & \pmb{\checkmark} & - & - & - & - & \checkmark & - & \checkmark & - & \checkmark & - & - \\
        $S_{F2}$    & - & \pmb{\checkmark} & - & - & - & \pmb{\checkmark} & - & \checkmark & - & \pmb{\checkmark} & - & - & - & - & \checkmark & - & \checkmark & - & \checkmark & - & - \\
        $S_{F3}$    & - & \pmb{\checkmark} & - & - & \pmb{\checkmark} & - & - & \checkmark & - & - & \pmb{\checkmark} & - & - & - & \checkmark & - & \checkmark & - & \checkmark & - & - \\
        $S_{F4}$    & - & - & \pmb{\checkmark} & - & \pmb{\checkmark} & - & - & \checkmark & - & \pmb{\checkmark} & - & - & - & - & \checkmark & - & \checkmark & - & \checkmark & - & - \\
    \end{tabularx}
    \caption{A table showing the current configuration state $ S $ and the stored finished configurations $ S_{Fi} $.}
    \label{tab:Concept:UseCaseConfigurations}
\end{table}