\chapter{Evaluation}
\label{ch:Evaluation}

In this chapter the prototype is evaluated in terms of its functionality and its properties.

All possible valid configurations will be generate for one use case i.e. all possible valid configurations for the forest use case.

Generate groups with preferences (explicit preferences) and configuration state (which would be for example the currently existing forest).

\section{Metric}
\label{sec:Evaluation:Metrics}

For the evaluation a metric to evaluate by is needed. The proposed metric for usage is that of satisfactions. Satisfaction is quantified in this thesis by a threshold metric. A user's preference is used to calculate a rating for each possible solution. The score will be calculated using the average of a user's rating for each characteristic that is part of the solution. The result allows that a configuration can be compared to all other configurations and ranked according to the percentage of configurations that it beats. The threshold metric consists of two parameters. First the threshold center $tc$ and second the satisfaction distance $sd$. The threshold for a person being satisfied is $tc + sd$ and of a person being dissatisfied at $tc - sd$. If a recommendation lies in between these two thresholds the person is classified to neither by satisfied nor be unsatisfied with the solution. For this thesis  $sd=5\%$ will be used. This choice is guided by the assumption that people switch from satisfied to unsatisfied rather quickly. Therefore the parameter considered in this thesis is the $tc$. An example is the choice of $tc = 60\%$. This results in a person being satisfied if recommendation is better than at lest $65\%$ of possible finished configurations. Moreover, a person is dissatisfied if the recommendation is only better than $55\%$ of possible finished configurations. A recommendation that is better than at least $55\%$ and not better than $65\%$ of possible solutions is considered neutral by the individual.

Different $tc$ values allow to model different situations. A situation where there is a low willingness to compromise is modelled by a high $tc$. A contrary situation where a group has a high willingness to compromise is modelled by a low $tc$.

\todo[inline]{add disclaimer that this metric is a new one not found in literature because no fitting metric was found}

\section{Questions to Answer During the Evaluation}
\label{sec:Evaluation:Questions}

\begin{itemize}
    \item Main question: How does the satisfaction with a group decision, guided by the recommender, differ from the decision of a single decision maker, the dictator, who does not take the other group member's opinion into account?
    \item How many group members are satisfied with the group decision on average?
    %\item Is the recommender fair, i.e. no user type is always worse off than others? (Just uses groupe preferences)
    \item How does the amount of stored finished configurations relate to recommendation satisfaction?
\end{itemize}

\section{Effect of Stored Finished Configurations}
\label{sec:Evaluation:EffectFinishedConfiguration}

When evaluating just a subset of stored finished configurations it is important to avoid outliers. This is the reason why a process inspired by cross validation is used. The configuration database is randomly ordered and sliced into sub databases of the needed size. As an example, if the evaluated stored data size is 20, a configuration database containing 100 configurations is split into five sub databases of size 20. Now the evaluation is done on each of the sub databases and as a result the average is taken.

\section{Use Case}
\label{sec:Evaluation:UseCase}

To evaluate data a given use case is needed. In this thesis a forestry use case is evaluated. This is a use case with four stakeholders. \autoref{fig:Concept:ForestExample} already presented the attributes and characteristics used in this use case but an extension is needed to fully show the whole use case. Namely rules of non valid configurations. The constraints for this use case are listed in not with form in \autoref{tab:Evaluation:UseCase}. 

\begin{table}
    \begin{center}
        \begin{tabular}{r|l}
            & not with (either of the listed) \\
            \hline
            $(\textit{indigenous}, \text{moderate})$   & $(\textit{resilient}, \text{high})$ \\
            \hline
            $(\textit{indigenous}, \text{high})$   & $(\textit{resilient}, \text{high}), (\textit{usable}, \text{moderate}), (\textit{usable}, \text{high}),$ \\
            & $(\textit{quantity}, \text{high}), (\textit{price}, \text{low})$ \\
            \hline 
            $(\textit{resilient}, \text{moderate})$   & $(\textit{usable}, \text{high})$ \\
            \hline
            $(\textit{resilient}, \text{high})$   & $(\textit{usable}, \text{high}), (\textit{usable}, \text{moderate}), (\textit{quantity}, \text{high}),$ \\
            & $(\textit{price}, \text{moderate}), (\textit{price}, \text{low})$ \\
            \hline
            $(\textit{usable}, \text{low})$ & $(\textit{quantity}, \text{high}), (\textit{price}, \text{moderate}), (\textit{price}, \text{low})$\\
            \hline
            $(\textit{usable}, \text{high})$ & $(\textit{accessibility}, \text{high})$\\
            \hline
            $(\textit{effort}, \text{manual})$ & $(\textit{quantity}, \text{high}), (\textit{price}, \text{low}), (\textit{price}, \text{moderate})$\\
            \hline
            $(\textit{effort}, \text{harvester})$ & $(\textit{accessibility}, \text{high}), (\textit{accessibility}, \text{moderate})$\\
            \hline
            $(\textit{effort}, \text{autonomous})$ & $(\textit{accessibility}, \text{high}), (\textit{accessibility}, \text{moderate})$\\
            \hline
            $(\textit{quantity}, \text{low})$ & $(\textit{price}, \text{low}), (\textit{price}, \text{moderate})$\\
            \hline
            $(\textit{quantity}, \text{moderate})$ & $(\textit{price}, \text{low}), (\textit{price}, \text{moderate})$\\
            \hline
            $(\textit{quantity}, \text{high})$ & $(\textit{accessibility}, \text{high}), (\textit{accessibility}, \text{moderate})$\\
            \hline
        \end{tabular}
        \caption{Constrains in not with form for the forest use case.}
        \label{tab:Evaluation:UseCase}
    \end{center}
\end{table}

The stakeholders of this use case are: a forest owner, an athlete, an environmentalist and a consumer. The owner sees the forest as an investment, he is interested in a high long term profit. On the other hand the consumer is interested in reasonable wood price as she uses wood for furniture and also for her fireplace. In contrast, the environmentalist is interested in a healthy forest that is not impacted negatively by human activity. Last is the athlete who is interested in good accessibility of the forest and that there is some life.  

\section{Generating Data}
\label{sec:Evaluation:GeneratingGroups}

The whole process explained in this section is visualized in \autoref{fig:Evaluation:GeneratingDataProcess}.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{./figures/60_evaluation/bpmn_evaluation_input_data_generation.pdf}
    \caption{The process used for generating data for the evaluation.}
    \label{fig:Evaluation:GeneratingDataProcess}
\end{figure}

\subsection{Generating Unfinished Configurations}

Unfinished configurations are generated using all finished configurations and taking a subset of the contained characteristics. This way all generated configurations will be valid and lead to valid solutions. For the results that are presented in this chapter around $\frac{1}{7} \approx 15\%$ of characteristics is kept.

\todo[inline]{why this paramter, elalobrate on that}

\subsection{Generating Preferences}

For the forest use case, the idea is that there are multiple types of user profiles. Each group profile is represented by a neutral, negative or positive attitude to an attribute value. Now during data generation the attitude is converted to a preference using a normal distribution. \autoref{fig:Evaluation:DataGeneration} shows how the user profile can be converted to preferences.

\pgfplotsset{height=5cm,width=\textwidth,compat=1.8}
\pgfmathdeclarefunction{gauss}{2}{%
  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}
\begin{figure}
    \begin{tikzpicture}
        \begin{axis}[
            every axis plot post/.append style={
                mark=none, domain=0:1, samples=50, smooth
            },
            axis x line*=bottom,
            xmin=0,
            xmax=1,
            ymin=0.1,
            xticklabel style={
                /pgf/number format/precision=3,
            },
            xtick={0,0.25, 0.5, 0.75,1},
            hide y axis]
          \addplot [draw=black, style={dashdotdotted}][very thick] {gauss(0.25,0.1)} node[text=black][above,pos=0.5] {negative};
          \addplot [draw=black, style={solid}][very thick] {gauss(0.5,0.05)} node[text=black][above,pos=0.48] {neutral};
          \addplot [draw=black, style={dotted}][very thick] {gauss(0.75,0.1)} node[text=black][above,pos=0.5] {positive};
        \end{axis}
        \end{tikzpicture}
 \caption{Distribution of preferences for a user type.}
\label{fig:Evaluation:DataGeneration}
\end{figure}

These user profiles can be used to generate rather homogenous groups but also to create groups that have interests that are more conflicting. The following group types are generated:

\begin{itemize}
    \item random groups (preferences are uniformly random)
    \item heterogeneous groups (people adhere to one preference profile like forest owner, athlete, consumer, environmentalist)
    \item homogeneous groups (only one preference profile for all group members which in this evaluation is the forest owner)
\end{itemize}

\text

\begin{table}
    \begin{center}
        \begin{tabular}{l|c|c|c|c}
                                                        & athlete           & forest owner      & environmentalist  & consumer          \\
            \hline
            $(\textit{indigenous}, \text{low})$         & \textbf{negative} & \textit{positive} & \textbf{negative} & neutral           \\
            $(\textit{indigenous}, \text{moderate})$    & \textit{positive} & neutral           & \textbf{negative} & neutral           \\
            $(\textit{indigenous}, \text{high})$        & \textit{positive} & \textbf{negative} & \textit{positive} & \textbf{negative} \\
            \hline
            $(\textit{resilient}, \text{low})$          & neutral           & \textit{positive} & neutral           & neutral           \\
            $(\textit{resilient}, \text{moderate})$     & \textit{positive} & neutral           & neutral           & neutral           \\
            $(\textit{resilient}, \text{high})$         & \textit{positive} & \textbf{negative} & \textbf{negative} & \textbf{negative} \\
            \hline
            $(\textit{usable}, \text{low})$             & neutral           & neutral           & neutral           & \textbf{negative} \\
            $(\textit{usable}, \text{moderate})$        & neutral           & neutral           & \textbf{negative} & neutral           \\
            $(\textit{usable}, \text{high})$            & \textbf{negative} & \textit{positive} & \textbf{negative} & \textit{positive} \\
            \hline
            $(\textit{effort}, \text{manual})$          & \textbf{negative} & neutral           & \textit{positive} & \textbf{negative} \\
            $(\textit{effort}, \text{harvester})$       & \textbf{negative} & \textit{positive} & \textbf{negative} & neutral           \\
            $(\textit{effort}, \text{autonomous})$      & \textbf{negative} & \textit{positive} & \textbf{negative} & neutral           \\
            \hline
            $(\textit{quantity}, \text{low})$           & \textit{positive} & \textbf{negative} & \textit{positive} & \textbf{negative} \\
            $(\textit{quantity}, \text{moderate})$      & neutral           & \textit{positive} & neutral           & \textbf{negative} \\
            $(\textit{quantity}, \text{high})$          & \textbf{negative} & \textit{positive} & \textbf{negative} & \textit{positive} \\
            \hline
            $(\textit{price}, \text{low})$              & neutral           & neutral           & neutral           & \textit{positive} \\
            $(\textit{price}, \text{moderate})$         & neutral           & \textit{positive} & neutral           & neutral           \\
            $(\textit{price}, \text{high})$             & neutral           & \textit{positive} & neutral           & \textbf{negative} \\
            \hline
            $(\textit{accessibility}, \text{low})$      & \textbf{negative} & \textit{positive} & \textit{positive} & neutral           \\
            $(\textit{accessibility}, \text{moderate})$ & neutral           & neutral           & neutral           & neutral           \\
            $(\textit{accessibility}, \text{high})$     & \textit{positive} & \textbf{negative} & \textbf{negative} & neutral           \\
            \hline
        \end{tabular}
        \caption{ The attitudes of each group member profile. }
        \label{tab:Evaluation:GroupMemberMappings}
    \end{center}
\end{table}

\todo[inline]{explain preference profiles}

\section{Hypotheses}
\label{sec:Evaluation:Hypotheses}

Understanding data is made easier by first posing hypotheses. This section gives an overview over the hypothesis used during data analysis.

\begin{enumerate}[font={\bfseries},label={H\arabic*}]
    \item \label{hyp:Evaluation:MaximumMinimum} Highest improvements with group recommendation are when the amount of people satisfied with the dictators decision is slightly lower than two. Respectively that holds true for dissatisfaction. 
    \item \label{hyp:Evaluation:HigherTcLessSatisfied} A higher $tc$ value results in less satisfied people and more unsatisfied people with regard to the dictator's decision.
    \item \label{hyp:Evaluation:OnlyOneSatisfied} There exists a $tc$ value which causes only one person to be satisfied with the dictator's decision and no one is satisfied with the group recommender's decision.
    \item \label{hyp:Evaluation:HomogenousMoreSatisfied} Homogeneous groups have more satisfied members with the recommender's decision but also with the dictator's decision compared to heterogeneous groups.
    \item \label{hyp:Evaluation:HeterogenousBiggerSatisfactionIncrease} More heterogeneous groups see a bigger satisfaction increase than less heterogeneous groups when comparing the dictator's decision with the recommender's decision.
    \item \label{hyp:Evaluation:StoreSizeBetterResults} A higher amount of stored finished configurations results in a higher amount of satisfied and a lower amount of dissatisfied group member.
    \item \label{hyp:Evaluation:AggregationStrategies} Multiplication and best average aggregation strategies perform better than least misery. % These strategies are listed by \citeauthor{Masthoff2015} \cite[p. 755f]{Masthoff2015} and multiplication and best average came out as the best in most studies. Least misery was in some listed as performing worst. Therefore it fares worse than the other strategies here.
\end{enumerate}

\todo[inline]{explain hypotheses}

\section{Findings}
\label{sec:Evaluation:Findings}

\subsection{Threshold Center}

To get an understanding of the data all parameters except the $tc$ will be fixed. The preference aggregation strategy looked at is multiplication. The configuration database is used with all possible solutions (which is 148 in total). This results in a bigger visible effect as the recommender has access to all possible configurations. \autoref{fig:Evaluation:tcChange} shows the satisfaction change based on choice of $tc$. Of note is that the maxima of satisfaction change precedes the minima of dissatisfaction change for all group types. Maxima and minima occur at different tc values depending on the group type. Heterogeneous groups peek earliest while homogenous groups only show a peek towards the maximum $tc$ value. Changes in dissatisfaction are minimal even with $tc$ close to its maximum value. \autoref{fig:Evaluation:tcCount} shows the amount of group members satisfied and dissatisfied with a decision. The number of satisfied people decreases with an increasing $tc$ and its downward movement accelerates. The dissatisfaction curve shows a similar trend but in contrast here the number of dissatisfied group members increases with and increase in $tc$. The curve accelerates its growth analogues to the acceleration of the satisfaction curve. The behaviour of heterogeneous groups and random groups is similar but the curve for heterogeneous groups show less happiness for a given tc and more unhappiness. Also both curves have a negative satisfaction change when $tc$ reaches a certain height. Homogeneous groups only have happy group members for most $tc$ values but they decrease rapidly for values greater $85$. Dissatisfied group members are at zero for the whole value range of $tc$ except a very slight upward tick at the end that is barely noticeable.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{./figures/60_evaluation/tc_change__multi__db-size-148.pdf}
    \caption{The average satisfaction and dissatisfaction change based on $tc$ with a database size of 148 and multiplication as aggregation strategy.}
    \label{fig:Evaluation:tcChange}
\end{figure}

\hyporef{hyp:Evaluation:MaximumMinimum} states that the highest satisfaction change is expected at places where the overall satisfaction with the dictator's decision is one. However the data shows a slightly different result. This hypothesis does not hold true. When looking at the data we see peeks in satisfaction change when $2.81, 2.51, 3$ (heterogeneous, random, homogenous). Therefore the expectation does not hold up. Moreover, valleys for dissatisfaction change are also not at the expected value of \textit{two}. They are instead at $1.19, 1.49, 0.04$ (heterogeneous, random, homogenous). Here the valleys are lower than expected. However data from homogenous groups seems cut of therefore it is not possible to say if there could be a potentially bigger decrease if the solution space is bigger.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{./figures/60_evaluation/tc_dictator__multi__db-size-148.pdf}
    \caption{The average satisfaction and dissatisfaction with the dictator's decision based on $tc$.}
    \label{fig:Evaluation:tcCount}
\end{figure}

The predicted trend that a higher $tc$ results in a lower satisfaction and a higher dissatisfaction, with the dictator's decision, as predicted by \hyporef{hyp:Evaluation:HigherTcLessSatisfied} can be clearly seen in \autoref{fig:Evaluation:tcCount} and has been described in this section already.

\hyporef{hyp:Evaluation:OnlyOneSatisfied} predicts that the satisfaction with the individual decision eventually reaches one and that no one is satisfied with the group recommender decision. This means the satisfaction change should reach minus one. \autoref{fig:Evaluation:tcCount} shows a downward trend that come close to one for heterogeneous and random groups. Also homogenous groups see a big drop but this drop does not reach one. Nonetheless, the steep drop suggests that the hypothesis holds in regards to reaching only one person satisfied with the individual decision when using quantiles that do not have to be integers. Also, satisfaction change in heterogeneous groups reaches close to minus one but this value is neither reached by random groups, nor by homogenous groups. The hypothesis therefore should not be seen as confirmed in that regard and further investigation is needed.

During a group decision it is better to make one less person dissatisfied opposed to one more person satisfied. Therefore, this thesis uses $tc$ values that are closer to minima of unhappiness reduction than to the maxima of satisfaction change. The minima for heterogeneous is at $tc = 70\%$ therefore this is the chosen value for the evaluation of other aspects. For random groups the minima of dissatisfaction change can be found at $tc = 85\%$ which is the value used for all following analysis of random groups. For homogenous group dissatisfaction change is sinking until the highest value of $tc$. Because of that $tc = 94\%$ is used for analysis.

\subsection{Analysing Data}

This subsection holds fixed parameters of $tc$. In it the satisfaction change and the total amount of satisfied people with the recommenders decision dependent on the amount of stored configurations. For clarity reasons not all graphs of the data are included. The missing graphs are in the appendix and have references to them.

\autoref{fig:Evaluation:HeteroSatisfactionIncrease} shows the relationship between the change in satisfaction and dissatisfaction and the stored number of configurations. There are three graphs each. One for multiplication, one for least misery and one for best average. The graphs for satisfaction look similar to a logarithmic curve. The increase in change satisfaction decelerates with a higher number of stored configurations. The change in satisfaction is always above zero and a satisfaction increase of more than three quarters of the maximum can already be seen with around 25 stored configurations. The curve for multiplication is above all other curves. Least misery reaches the lowest amount of change across all values. The minimum number of satisfaction change is $0$ for least misery and $0.1$ for best average and multiplications. The highest number is around $0.3$ for least misery, $0.4$ for best average and $0.5$ for multiplication
When looking at dissatisfaction change the graphs are all in the negative number range. Multiplication reaches the lowest number and best average the highest. The gap between all three functions is less than for satisfaction increase. And overall the curves are flatter meaning the change with 25 stored configurations already reaches close to five sixth of the minimum value. The highest number of satisfaction change is $-0.4$ for all strategies meanwhile the lowest number is around $-0.57$ for least misery, $-0.53$ for best average and $-0.63$ for multiplication.

The figures for homogenous (\autoref{fig:Appendix:HomoSatisfactionIncrease}) and random groups (\autoref{fig:Appendix:RandomSatisfactionIncrease}) are in the appendix. The figures have a similar shape but their values and slope vary. The satisfaction change for homogenous groups is mostly negative, starting at $-2$, and only reaches a positive level for more than $100$ stored configurations with a value of $0.04$. Multiplication and best average have higher values than least misery here too. Moreover the dissatisfaction change is positive across the bored with a value range of $[0,1]$.
Random groups as seen in \autoref{fig:Appendix:RandomSatisfactionIncrease} mostly have a positive change in satisfaction. Values range here from $-0.55$ to $0.27$ for least misery, from $-0.27$ and $-0.28$ to $0.74$ for best average and multiplication. The change is higher than the change for heterogeneous groups. dissatisfaction also changes similarly to heterogeneous groups. Here the values for random groups reach a lower level. They range from $0$ to $-0.59$ for least misery. Multiplication and best average both have as minimum value around $-0.21$ and behave similarly. The range goes down to $-0.84$ for best average and $-0.86$ for multiplication.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{./figures/60_evaluation/heterogeneous_happy_unhappy_increase_amount-1000__tc-70}
    \caption{The satisfaction and dissatisfaction change using the group recommender for heterogeneous groups with $tc = 70$.}
    \label{fig:Evaluation:HeteroSatisfactionIncrease}
\end{figure}

\autoref{fig:Evaluation:HeteroSatisfactionTotal} shows the total number of group members satisfied and dissatisfied with the recommender's decision. The horizontal black continuous line shows the value for satisfaction and dissatisfaction with the dictators decision. The graphs show the same curve as \autoref{fig:Evaluation:HeteroSatisfactionIncrease} but in absolute numbers. Satisfaction with the recommender's decision starts at $2.4$ and quickly reaches $2.65$ for least misery and $2.8$ for best average and multiplication. The highest value for multiplication is at $2.89$. Dissatisfaction also  quickly plateaus. Here values for different recommenders are closer together. They start at $0.74$ (least misery) to $0.78$ (best average) and go as low as $0.62$ for least misery, $0.66$ for best average and $0.56$ for multiplication.

As shown in \autoref{fig:Appendix:HomoSatisfactionTotal} the value range for homogenous groups is much larger but the overall shape stays the same. Here satisfaction numbers go from $0.55$ to $2.95$. Least misery performs visibly worse than multiplication and best average reaching only $2.7$. Dissatisfaction values range from $1.21$ to $0.01$ and the values are not really visibly distinguishable besides that in the range $[25,50]$ least misery seems to have the highest number of dissatisfied group members.

Random groups have less overall satisfaction with $tc = 85\%$ as seen in \autoref{fig:Appendix:RandomSatisfactionTotal}. Satisfaction numbers start from $1.33$ (least misery), $1.61$ (best average) and $1.6$ (multiplication) and go up to $2.15$ for least misery and $2.62$ for best average and multiplication. The dissatisfaction numbers start at $1.5$ for least misery and $1.27$ for best average and multiplication and level of at $0.9$ (least misery), $0.65$ (best average) and $0.63$ (multiplication). Visibly there is a big difference between least misery and the other two aggregation functions.

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{./figures/60_evaluation/heterogeneous_happy_unhappy_total_amount-1000__tc-70}
    \caption{The average satisfaction and dissatisfaction with the recommender's decision for heterogeneous groups based on $tc = 70$.}
    \label{fig:Evaluation:HeteroSatisfactionTotal}
\end{figure}

\hyporef{hyp:Evaluation:HomogenousMoreSatisfied} states that homogenous groups have more satisfied member's with regards to the dictator's and the group recommender's decision. \autoref{fig:Evaluation:tcCount} shows that this holds true for dictator's decision as for every instance satisfaction in homogeneous groups is higher than that of other groups. However \autoref{fig:Evaluation:HeteroSatisfactionTotal}, \autoref{fig:Appendix:HomoSatisfactionTotal} and \autoref{fig:Appendix:RandomSatisfactionTotal} show that for satisfaction with the recommender's decision this does not hold when looking at $tc$ values where the recommender performs best for each  segment. In those places the homogenous group only reaches the highest amount of satisfaction when the recommender has access to all stored configurations. With a decreasing number of stored configurations both random groups and heterogeneous groups perform better. It is important to note, when the same $tc$ values are used homogenous groups have a higher amount of satisfied people across the board.

\hyporef{hyp:Evaluation:HeterogenousBiggerSatisfactionIncrease} states that the increase in satisfaction should be bigger for more heterogeneous groups. However \autoref{fig:Evaluation:HeteroSatisfactionIncrease}, \autoref{fig:Appendix:HomoSatisfactionIncrease} and \autoref{fig:Appendix:RandomSatisfactionIncrease} show this to be not true. The recommendations for heterogeneous groups indeed cause a larger change in satisfaction compared to homogeneous groups but random groups cause a bigger positive of higher magnitude. Also the decrease in dissatisfaction is higher among random groups.

The data shows that having a larger configuration store causes the amount of satisfied group members to be greater than compared to recommendation's using a smaller store. With dissatisfaction the same is seen, just that here it is lower with a higher amount of stored configurations. However in some runs there have been instances of least misery that have seen a slightly lower number. This can be seen in \autoref{fig:Evaluation:HeteroSatisfactionIncrease} when comparing $74$ and $148$ as number of stored configurations. Why this happens is not entirely clear but a cause of that might be that least misery just takes into account the worst performing group member of the group. Therefore it is possible that there is a second slightly worse rated solution (by least misery) that actually has a slight advantage over the configuration chosen by least misery. Having a second best configuration can cause it to land in the second partition of the data therefore resulting in an on average less unhappiness. \hyporef{hyp:Evaluation:StoreSizeBetterResults} therefore is mostly supported by the data but it does not fully hold up when looking at least misery.

\hyporef{hyp:Evaluation:AggregationStrategies} states least misery performs worse than multiplication. For a change in satisfaction this can be seen across the board however for dissatisfaction change this is not true everywhere. \autoref{fig:Evaluation:HeteroSatisfactionIncrease} shows that least misery performs better than best average in terms of dissatisfaction reduction. However in other cases it performs visibly worse. Also of note is multiplication performs best across the board. This supports the findings by \citeauthor{Masthoff2015} \cite[p. 755f]{Masthoff2015} and also shows that the satisfaction model does show some similar results to online evaluations.  

