@@ -128,8 +128,8 @@ Scalarization discards information and must be done carefully to not dismiss fav

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.

He argues that the weighted-sum method is unintuitive and error-prone to use.

The weighted-product method is easier to use and can capture user preferences more faithfully.

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.

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@@ -141,11 +141,12 @@ Some of the properties of the weighted-product method relevant for c2 scheduling

\paragraph{Mixed units of measurement}

This is the main reason why we chose the weighted-product method.

The units and magnitudes of the individual objectives does not matter.

One objective can be a distance in meters, another one a cost in euros, yet another one a time in seconds.

No conversion or normalization is required, the numbers can be multiplied as-is.

One objective can be a distance in meters, another one a cost in euros, yet another one a time in seconds.

\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.