3.2.1. Egalitarian Social Welfare

The egalitarian social welfare metric defines fairness as ensuring that the worst off objective is as good off as possible. The egalitarian social welfare metric is defined in Equation (3).

$$f\_E(\mathbb{C}) = \max \{ q\_{t\_i, \mathbb{C}}' \mid i \in n \} \tag{3}$$

This means that the egalitarian social welfare metric prefers a solution *C* over solution *C* if and only if ↓ *q <sup>t</sup>*1,*<sup>C</sup>* < ↓ *q <sup>t</sup>*1,*C* where ↓ *q <sup>C</sup>* contains the values of the normalized cost vector *q <sup>C</sup>* = {*q <sup>t</sup>*1,*C*, *q <sup>t</sup>*2,*C*,..., *q tn*,*C*} rearranged in descending order.

The egalitarian social welfare metric has a weakness in that it only takes into account the normalized cost of the worst off objective while defining the ordering of solutions. Consider three normalized cost value vectors (1,1,0), (1,1,1), and (1,0,0) for solutions *C*1, *C*2, and *C*3, respectively. Each of the vectors is comprised of normalized cost values in decreasing order. The highest cost value in all the cost value vectors is one. In this case, the egalitarian social welfare metric fails to distinguish between solutions that have the same highest (worst) cost and assigns the same order to all three solutions, irrespective of the fact that *C*<sup>3</sup> is better objectively than *C*<sup>1</sup> and *C*2.
