### 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} \definecolor{myRed}{RGB}{243,146,146} \begin{tikzpicture}[scale=0.03,>=stealth] \draw[fill=myGreen,draw=none] (0,0) rectangle (66,66); \draw[fill=myRed,draw=none] (66,33) rectangle (100,100); \draw[fill=white,draw=none] (0,100) rectangle (66,33); \draw[fill=white,draw=none] (66,33) rectangle (100,0); %\begin{scope} %\draw[draw=yellow,ultra thick] (66,0) -- (66,100); %\draw[draw=yellow,ultra thick] (0,33) -- (100,33); %\end{scope} \draw[dashed] (66,0) -- (66,100); \draw[dashed] (0,33) -- (100,33); \filldraw (66,33) circle [radius=3]; % coordinate system \draw[->] (0,0) -- (100,0); % x-axis \draw[->] (0,0) -- (0,100); % y-axis \node[anchor=north] at (0, 0) {0}; \node[anchor=north] at (66,0) {50}; \node[anchor=east] at (0, 0) {0}; \node[anchor=east] at (0,33) {50}; \end{tikzpicture} \begin{tikzpicture}[scale=0.03,>=stealth] \draw[fill=myGreen,draw=none] (0,0) rectangle (66,66); \draw[fill=myRed,draw=none] (33,100) rectangle (100,33); \draw[fill=white,draw=none] (0,100) rectangle (33,66); \draw[fill=white,draw=none] (33,66) rectangle (66,33); \draw[fill=white,draw=none] (66,33) rectangle (100,0); %\begin{scope} %\draw[draw=yellow,ultra thick] (50,0) -- (50,100); %\draw[draw=yellow,ultra thick] (0,50) -- (100,50); %\end{scope} \draw[dashed] (33,100) -- (33,33) -- (100,33); \draw[dashed] (0,66) -- (66,66) -- (66,0); \filldraw (66,33) circle [radius=3]; \filldraw (33,66) circle [radius=3]; % coordinate system \draw[->] (0,0) -- (100,0); % x-axis \draw[->] (0,0) -- (0,100); % y-axis \node[anchor=north] at (0, 0) {0}; \node[anchor=north] at (33,0) {25}; \node[anchor=north] at (66,0) {50}; \node[anchor=east] at (0, 0) {0}; \node[anchor=east] at (0,33) {50}; \node[anchor=east] at (0,66) {75}; \end{tikzpicture} \begin{figure} \begin{minipage}{0.49\linewidth} \begin{tikzpicture}[scale=0.03,>=stealth] \draw[fill=myGreen,draw=none] (0,0) rectangle (66,66); \draw[fill=myRed,draw=none] (66,33) rectangle (100,100); \draw[fill=white,draw=none] (0,100) rectangle (66,33); \draw[fill=white,draw=none] (66,33) rectangle (100,0); %\begin{scope} %\draw[draw=yellow,ultra thick] (66,0) -- (66,100); %\draw[draw=yellow,ultra thick] (0,33) -- (100,33); %\end{scope} \draw[dashed] (66,0) -- (66,100); \draw[dashed] (0,33) -- (100,33); \filldraw (66,33) circle [radius=3]; % coordinate system \draw[->] (0,0) -- (100,0); % x-axis \draw[->] (0,0) -- (0,100); % y-axis \node[anchor=north] at (0, 0) {0}; \node[anchor=north] at (66,0) {5.0}; \node[anchor=east] at (0, 0) {0}; \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] \draw[fill=myGreen,draw=none] (0,0) rectangle (66,66); \draw[fill=myRed,draw=none] (33,100) rectangle (100,33); \draw[fill=white,draw=none] (0,100) rectangle (33,66); \draw[fill=white,draw=none] (33,66) rectangle (66,33); \draw[fill=white,draw=none] (66,33) rectangle (100,0); %\begin{scope} %\draw[draw=yellow,ultra thick] (50,0) -- (50,100); %\draw[draw=yellow,ultra thick] (0,50) -- (100,50); %\end{scope} \draw[dashed] (33,100) -- (33,33) -- (100,33); \draw[dashed] (0,66) -- (66,66) -- (66,0); \filldraw (66,33) circle [radius=3]; \filldraw (33,66) circle [radius=3]; % coordinate system \draw[->] (0,0) -- (100,0); % x-axis \draw[->] (0,0) -- (0,100); % y-axis \node[anchor=north] at (0, 0) {0}; \node[anchor=north] at (33,0) {2.5}; \node[anchor=north] at (66,0) {5.0}; \node[anchor=east] at (0, 0) {0}; \node[anchor=east] at (0,33) {50}; \node[anchor=east] at (0,66) {75}; \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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