*3.3. Overall Utility Based Metrics*

Overall utility based metrics again do not consider fairness among objectives, but rather strive to get the best overall utility from a final solution. Here, no guarantees are made towards any single objective, and one objective could be much worse than any other objective with the selected solution. However, the solution is guaranteed to yield the highest overall utility across the society.

## 3.3.1. Utilitarian Social Welfare

The utilitarian social welfare metric is one of the most simple notions of social welfare, which ranks solutions based on the sum of individual normalized cost values of objectives. The metrics are defined in Equation (10).

$$f\_{ll}(\mathbb{C}) = \sum\_{i=1}^{n} q\_{t\_i, \mathbb{C}}^{\prime} \tag{10}$$

From Equation (10), we see that the utilitarian social welfare metric prefers a solution *C* over solution *C* if and only if ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *q ti*,*<sup>C</sup>* <sup>&</sup>lt; <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *q ti*,*C* .

#### 3.3.2. Nash Product

The Nash product social welfare metric combines the features of the utilitarian and the egalitarian social welfare metrics. This notion of social welfare seeks to achieve the best combined cost value across the society of objectives while working to reduced inequality among objectives. Its definition is found in Equation (11).

$$f\_N(\mathbb{C}) = \prod\_{i=1}^n q'\_{t\_i, \mathbb{C}} \tag{11}$$

The Nash product social welfare metric ranks solutions based on the product of individual normalized cost values of objectives. Hence, the Nash product social welfare metric prefers a solution *C* over a solution *C* if and only if ∏*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *q ti*,*<sup>C</sup>* <sup>&</sup>lt; <sup>∏</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *q ti*,*C* .

The Nash product metric maximizes society wellbeing by minimizing the combined product of the normalized cost values in a particular solution. Adherence to fairness comes from the fact that the contribution from high normalized cost values contributes proportionally The Nash product social welfare metric is meaningful for cost value vectors without zero value elements. If zero elements are present, only the zero element considered as the Nash product will yield zero regardless of the other values. To account for this fact, we modified the Nash product social welfare metric according to Equation (11), where we added the same non-zero value to all vector elements. This way, we never end up in a zero product situation.

$$f\_N(\mathbb{C}) = \prod\_{i=1}^n (q\_{t\_i; \mathbb{C}}^\prime + r) \quad [r \in Z^+] \tag{12}$$
