@@ -116,17 +116,17 @@ Without further information about the candidates and personal preferences of the

\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.

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

This means that any pure MCDM method will present the user with 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 can be done by scalarizing their objective vectors, which means all objectives are combined 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.

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.

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@@ -135,3 +135,21 @@ Let $\vec{q} = \tuple{p_1, p_2, \ldots, q_1, q_2, \ldots}$ be an objective vecto

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.

Some of the properties of the weighted-product method relevant for c2 scheduling are as follows.

\paragraph{Mixed units of measurement}

\paragraph{Weights allow for diminishing returns}

\paragraph{Measurement theory}

All objectives must be units of ratio scale.

For a unit of measurement to be of ratio scale, it must have a meaningful zero value that represents absence of the measured quantity, and must allow multiplication by positive constants.

This allows comparisons based on ratios of measurements, like "this is twice as long as that".

Examples of units of ratio scale are mass, length, time, money, and temperature in Kelvin.

Temperature in degrees Celsius is not of ratio scale, because it does not have a meaningful zero value.

$20^\circ$C is not twice as warm as $10^\circ$C.

Levels of measurement were first described by \citet{Stevens1951}.

A more recent exposition is given by \citet{Wohlin2012}.