Commit 07a36135 authored by Markus Klinik's avatar Markus Klinik
Browse files

explain the fundamental problem of MCDM

parent d2a13ba7
......@@ -22,58 +22,82 @@ As we are discovering new hotels, we want to compare them to the ones we already
If a hotel is both cheaper and closer to the sea, it is definitely better, according to our criteria.
If a hotel is both more expensive and further from the sea, it is definitely worse.
But if we find a hotel that is closer to the sea but more expensive, or the other way around?
This is the central problem of MCDM: \emph{Objective vectors that are better in some and worse in other components are not comparable}.
This is the central problem of MCDM: \emph{It is not possible to compare objective vectors that are better in some and worse in other components}.
\definecolor{myGreen}{RGB}{203,243,146}
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\draw[fill=white,draw=none] (33,66) rectangle (66,33);
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%\begin{scope}
%\draw[draw=yellow,ultra thick] (50,0) -- (50,100);
%\draw[draw=yellow,ultra thick] (0,50) -- (100,50);
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% coordinate system
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\node[anchor=east] at (0,66) {75};
\end{tikzpicture}
\begin{figure}
\begin{minipage}{0.49\linewidth}
\begin{tikzpicture}[scale=0.03,>=stealth]
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\draw[fill=white,draw=none] (66,33) rectangle (100,0);
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%\draw[draw=yellow,ultra thick] (0,33) -- (100,33);
%\end{scope}
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\draw[dashed] (0,33) -- (100,33);
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\draw[->] (0,0) -- (100,0); % x-axis
\draw[->] (0,0) -- (0,100); % y-axis
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\node[anchor=north] at (66,0) {5.0};
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\node[anchor=east] at (0,33) {50};
\end{tikzpicture}
\subcaption{One candidate.}
\label{fig:one-candidate}
\end{minipage}%
\begin{minipage}{0.49\linewidth}
\begin{tikzpicture}[scale=0.03,>=stealth]
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\draw[fill=myRed,draw=none] (33,100) rectangle (100,33);
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\end{tikzpicture}
\subcaption{Two candidates.}
\label{fig:two-candidates}
\end{minipage}
\caption{Two incomparable hotels.}
\label{fig:incomparable-hotels}
\end{figure}
\Cref{fig:incomparable-hotels} illustrates the issue.
\Cref{fig:one-candidate} shows one hotel with a distance of 5.0 km from the sea that costs 50 EUR per night.
Discovering a hotel divides the criterion space into four quadrants.
The red area to the upper-right marks hotels that are definitely worse according to our criteria.
Should we discover a hotel that falls into this area, we can immediately disregard it.
The green area to the lower-left marks hotels that are better in both criteria.
Should we discover a hotel that falls into this area, we would immediately discard the current one.
The white areas to the upper-left and lower-right mark hotels that are closer but more expensive and cheaper but further away, respectively.
\Cref{fig:two-candidates} shows what happens when we discover a hotel that falls into a white quadrant.
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