@@ -110,7 +110,28 @@ The white areas to the upper-left and lower-right mark hotels that are closer bu

\Cref{fig:two-candidates} shows what happens when we discover a hotel that falls into a white area.

We have to remember it as a \emph{non-dominated} candidate, and the shape of the red and green areas changes.

Once we have discovered all hotels in the area, we end up with a set of non-dominated candidates.

Without further information about the candidates or personal preferences of the decision maker, this is all we can expect.

Without further information about the candidates and personal preferences of the decision maker, it is not possible to narrow down the selection.

\subsection{Scalarization}

In the previous section we discussed that two objective vectors are incomparable if each is better than the other in at least one objective.

\citet{Deb2011} writes that in situations where objective vectors are longer than three, most candidates become incomparable.

The example scenario in \cref{sec:example-search-and-rescue} has objective vectors of length 10, and realistic scenarios can easily have 50 or more objectives.

This means that any pure MCDM method will present the user large and diverse sets of solutions.

To give the user more useful answers, these solutions should be further ranked and evaluated.

This can be done by scalarizing the objective vector, which means all objectives are combined and turned into a single number.

This process is called \emph{scalarization}.

Scalarization discards information and must be done carefully to not dismiss favourable solutions.

Two common scalarization methods are the weighted-sum and the weighted-product method.

\citet{Tofallis2014} highlights the disadvantages of the weighted-sum method and advocates using the weighted-product method.

He argues that the weighted-sum method requires normalization of the objectives, and the choice of the normalization method can produce different rankings.

The weighted-product method does not require normalization and as such is not subject to to this source of subjectivity.

Let $\vec{q}=\tuple{p_1, p_2, \ldots, q_1, q_2, \ldots}$ be an objective vector where the $p_i$ are more-is-better objectives and the $q_i$ are less-is-better objectives.

Let $\vec{w}=\tuple{w_1, w_2, \ldots, v_1, v_2, \ldots}$ be the weights, pre-determined by the decision maker.

The weighted product of the objective vector is calculated with the following formula.