Multi-criteria decision making (MCDM) is the problem of picking a preferred solution from a set of candidates that performs best in a number of criteria.

This section gives a brief introduction to the topic.

More information can be found in \todo{cite something}.

In our domain, the decision consists of a number of sub-decisions: one has to decide which resource to assign to each capability requirement.

The decisions can be represented by a decision vector $\vec{x}=\tuple{x_1, x_2, \ldots, x_n}$, where each $x_i$ stands for one capability requirement of a task.

Each decision variable $x_i$ has its own domain.

...

...

@@ -10,19 +13,21 @@ For example, the decision variable \emph{Perform rescue (Transport)} has domain

Every decision vector gives rise to an objective vector $\vec{q}=\tuple{q_1, q_2, \ldots, q_m}$.

The lengths of $\vec{x}$ and $\vec{q}$ can be different.

To illustrate the problems in MCDM, consider a much simpler problem.

To understand why MCDM is difficult, consider a much simpler problem.

Imagine we are planning a seaside vacation, for which we are looking for a hotel.

We want the hotel to be as close to the sea as possible and as cheap as possible.

We disregard the quality of hotels in this example.

For the sake of this example, these are the only two criteria of interest.

There is only one decision variable $x$, whose domain is the set of hotels we know of so far.

Every hotel has an objective vector with two elements $\tuple{q_1, q_2}$, the distance to the sea and the price, both of which should be minimized.

As we are discovering new hotels, we want to compare them to the ones we already know of.

As we discover new hotels, we want to compare them to the ones we already know of.

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?

But what 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{It is not possible to compare objective vectors that are better in some and worse in other components}.

The result of MCDM is therefore usually a set of candidates that are pairwise incomparable.

These are called the \emph{non-dominated candidates}.

\definecolor{myGreen}{RGB}{203,243,146}

\definecolor{myRed}{RGB}{243,146,146}

...

...

@@ -94,10 +99,15 @@ This is the central problem of MCDM: \emph{It is not possible to compare objecti

\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.

Discovering this hotel divides the criterion space into four areas.

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.

Should we discover a hotel that falls into this area, we can immediately disregard it, because the current one \emph{dominates}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.

Should we discover a hotel that falls into this area, we would immediately discard the current one, as it is \emph{dominated by} the new 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.

\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 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 hotels.