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try to make formulas more understandable

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hannes.kuchelmeister
2020-02-17 14:09:42 +01:00
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25_Outline/outline.tex
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@@ -157,22 +157,22 @@ The system has one main way to be used as defined in \autoref{tab:MainUseCase}.
\section{Definitions}
Set of variables:
Set of \emph{variables}:
\begin{equation}
V = \{v_1, \dotsc, v_m\}
\end{equation}
Set of their domains of values:
A corresponding \emph{domain mapping function} that maps a variable to its possible values:
\begin{equation}
\mathfrak{D} : V \to D; x \mapsto \mathfrak{D}(x) \qquad where \ D = \{d_1, \dotsc, d_o\}
\end{equation}
Set of users:
Set of \emph{users}:
\begin{equation}
U = \{1, \dotsc, n\}
\end{equation}
Users utility for a domain value of a variable:
A users \emph{utility function} for a domain value of a variable. This is the utility that a user has from a certain selected domain value. It is a function that only the user himself knows:
\begin{equation}
\begin{split}
u_i(d_j), \qquad \text{where}\ & d_j \in \mathfrak{D}(j),\\
@@ -181,18 +181,18 @@ Users utility for a domain value of a variable:
\end{split}
\end{equation}
User preferences:
\emph{User preferences} that are entered into the system by the user according to his utility function:
\begin{gather}
P = \{ P_1, \dotsc, P_n\},\ \text{where} \\
P_i = \{(d,\ u_i(d)) \ | \ \forall d \in \mathfrak{D}(i),\ i=1,\dotsc,m \} \notag
\end{gather}
Configuration state:
A \emph{configuration} has a state it is defined by a tuple of variables and their corresponding domain value. Essentially it is a set of variables and assigned values:
\begin{equation}
S = \{ (v_i,\ d) \ |\ v_i \in V \ \land \ d \in \mathfrak{D}(i),\ i=1,\dotsc,m \}
\end{equation}
Finished configuration state:
A \emph{finished configuration} is a configuration that contains all variables:
\begin{equation}
\begin{split}
S_F \subset S,\ where \ & \forall v_i \in V (\exists (v_i, d) \in S_F : d \in \mathfrak{D}(i)) \\
@@ -202,7 +202,7 @@ Finished configuration state:
% TODO: define valid configuration state
Group configuration scoring function using preferences and current configuration state:
\emph{Group configuration scoring function} using 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}
@@ -229,29 +229,29 @@ An example group configuration scoring function is $score_{group}$ with
\begin{mdframed}[frametitle={Forest Example}]
\begin{align}
\begin{split}
V = \{ & Heimisch, Klimaresilient, Verwertbar, Ernteaufwand, \\
& Menge, Preis, Walderfahrung \},
V = \{ & \textit{Heimisch}, \textit{Klimaresilient}, \textit{Verwertbar}, \textit{Ernteaufwand}, \\
& \textit{Menge}, \textit{Preis}, \textit{Walderfahrung} \},
\end{split} \notag \\
\mathfrak{D}(Heimisch) = \{ & Gering, Mittel, Hoch\}, \notag \\
\mathfrak{D}(Klimaresilient) = \{ & Gering, Mittel, Hoch\}, \notag \\
\mathfrak{D}(Verwertbar) = \{ & Gering, Mittel, Hoch\}, \notag \\
\mathfrak{D}(Ernteaufwand) = \{ & Motormanuel, Harvester, Vollautomatisch\}, \notag \\
\mathfrak{D}(Menge) = \{ & Keine, Gering, Hoch\}, \notag \\
\mathfrak{D}(Preis) = \{ & Gering, Mittel, Hoch\}, \notag\\
\mathfrak{D}(Walderfahrung) = \{ & Gering, Mittel, Intensiv\},\notag \\
\mathfrak{D}(\textit{Heimisch}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag \\
\mathfrak{D}(\textit{Klimaresilient}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag \\
\mathfrak{D}(\textit{Verwertbar}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag \\
\mathfrak{D}(\textit{Ernteaufwand}) = \{ & \text{Motormanuel}, \text{Harvester}, \text{Vollautomatisch}\}, \notag \\
\mathfrak{D}(\textit{Menge}) = \{ & \text{Keine}, \text{Gering}, \text{Hoch}\}, \notag \\
\mathfrak{D}(\textit{Preis}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag\\
\mathfrak{D}(\textit{Walderfahrung}) = \{ & \text{Gering}, \text{Mittel}, \text{Intensiv}\},\notag \\
U = \{ & 1,2\} \notag\\
P = \{ & P_1, P_2\} \notag\\
\begin{split}
P_1 = \{ & (Motormanuel, 0.5), (Harvester, -0.3) \} \\
& \cup \{ (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V,\ i \notin \{ Motormanuel, Harvester\} \ \} \
P_1 = \{ & (\text{Motormanuel}, 0.5), (\text{Harvester}, -0.3) \} \\
& \cup \{ (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V,\ i \notin \{ \text{Motormanuel}, \text{Harvester}\} \ \} \
\end{split} \notag \\
P_2 = \{ & (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V \} \notag \\
S = \{ & (Heimisch, Gering), (Menge, Gering) \} \notag \\
S = \{ & (\textit{Heimisch}, \text{Gering}), (\textit{Menge}, \text{Gering}) \} \notag \\
\begin{split}
S_F = \{ & (Heimisch, Gering), (Klimaresilient, Gering), (Verwertbar, Gering), \\
& (Ernteaufwand, Motormanuel),
(Menge, Keine), (Preis, Hoch),\\
& (Walderfahrung, Gering) \}
S_F = \{ & (\textit{Heimisch}, \text{Gering}), (\textit{Klimaresilient}, \text{Gering}), (\textit{Verwertbar}, \text{Gering}), \\
& (\textit{Ernteaufwand}, \text{Motormanuel}),
(\textit{Menge}, \text{Keine}), (\textit{Preis}, \text{Hoch}),\\
& (\textit{Walderfahrung}, \text{Gering}) \}
\end{split} \notag
\end{align}
\end{mdframed}