3.3.3. Median Rank Dictators

The median rank dictators [15] social welfare metric ranks solutions based on their median cost value as shown in Equation (13). Here, ( ↓ *q <sup>C</sup>*)*<sup>r</sup>* is the *<sup>r</sup>*th cost value in ↓ *q <sup>C</sup>*, which is the normalized cost value vector rearranged in descending order. Here, *r* is calculated as *r* = *n*/2, when the number of individuals *n* is even and *r* = (*n* + 1)/2 when the number of individuals *n* is odd.

$$f\_{MRD}(\mathbb{C}) = (\stackrel{\downarrow}{q'}\_{\mathbb{C}})\_r \tag{13}$$

The median rank dictator social welfare metric prefers a solution *C* over a solution *C* if and only if ( ↓ *q <sup>C</sup>*)*<sup>r</sup>* < ( ↓ *q <sup>C</sup>*)*<sup>r</sup>* .

#### **4. Relative Importance and Social Welfare Ordering**

While social welfare metrics do offer a way of prioritizing solutions in a Pareto set, they pose a limitation. The social welfare metrics are not able to rank objectives according to importance. To this end, Umair et al. [5] introduced the notion of relative importance and the Relative Importance Graph (RIG). By combining social welfare metrics and relative importance, we get a social welfare ordering method. Both relative importance and social welfare ordering are explained below.

#### *4.1. Relative Importance and the Relative Importance Graph*

The relative importance between objectives *c*<sup>1</sup> and *c*<sup>2</sup> can be expressed through the integer values −1, 1, and 0. Here, a value of −1 means that objective *c*<sup>1</sup> is relatively more important than *c*2. Likewise, a value of 1 reflects that objective *c*<sup>1</sup> is relatively less important than *c*2. Finally, a value of zero reflects that objective *c*<sup>1</sup> and objective *c*<sup>2</sup> are equally important. Under this assumption we define an abstraction, relation, which allows us to specify the relative importance of an objective towards another objective. When all possible relation instances have been created, a list of sorted objectives can be created by adding all objectives that are not part of a relation in which they are relatively less important than any other objective to the top of the list. This (or these in case of multiple) objective(s) are then followed by the objectives that are only relatively less important than the objectives already added to the list. This continues until all objectives have been added to the list.

An RIG can be constructed as a directed graph, *RIG* =< *N*, *E* >, based on the list of sorted objectives. Here, *N* = {*N*1, *N*2, ... , *Nn*} is a set of nodes where each node *Ni* comprises an objective or a group of objectives that are equally important, and *E* = {*E*1, *E*2, ... , *Em*} is a set of edges that contains edges between all nodes *N*, where *Ei*,*<sup>j</sup>* represents a direction between node *Ni* and node *Nj* based on the relative importance of objectives residing in nodes *Ni* and *Nj*. The graph is constructed by creating a root node containing the first objective *c*<sup>1</sup> from the sorted list of objectives. The next objective in the sorted list, *c*2, is then added to the RIG through comparison with objective *c*1: If *c*<sup>2</sup> is relatively equally important to objective *c*1, then it is added to the same node as shown in Figure 1a. If objective *c*<sup>2</sup> is relatively less important than objective *c*1, then it is added to a new empty node *N*<sup>2</sup> as illustrated in Figure 1b. In this case, a directed edge is created from node *N*<sup>1</sup> to node *N*<sup>2</sup> to represent the hierarchy between the objectives contained in the nodes. This process is repeated until all nodes in the sorted list have been added to the RIG, at which point, the RIG constitutes a graph in which nodes (and consequently objectives) can be traversed in order of relative importance. While this creates an absolute hierarchy of objectives, this voids the task of assigning an absolute importance to the individual objectives.

**Figure 1.** Examples of relative importance graphs. (**a**) Objectives *c*<sup>1</sup> and *c*<sup>2</sup> are relatively equally important. (**b**) Objective *c*<sup>1</sup> is relatively more important than objective *c*2.
