3.2.5. Quantitative Fairness

The quantitative fairness social welfare metric is based on Jain's index or fairness index [17], which is shown in Equation (6), where *X* is a particular resource allocation across *n* individuals and *xi* is the resource allocated to individual *i* ∈ *n*.

$$j(X) = \frac{\left(\sum\_{i=1}^{n} x\_i\right)^2}{n \sum\_{i=1}^{n} x\_i^2}, \quad 0 \le j(X) \le 1 \tag{6}$$

According to Jain's index, the allocation of resources is fair and ideal when the index is equal to one. We can translate this into the quantitative fairness social welfare metric based on cost values by substituting the individual resource allocations with individual cost values and substituting the result from one to reflect minimization as defined in Equation (7). In this case, the metric value is equal to zero when all objectives have the same normalized cost value. *r* is a constant with arbitrary value added to each cost value to avoid division by zero.

$$f\_{QF}(\mathbf{C}) = 1 - \frac{\left(\sum\_{i=1}^{n} \left(q\_{t\_i, \mathbf{C}}^{\prime} + r\right)\right)^2}{n \sum\_{i=1}^{n} \left(q\_{t\_i, \mathbf{C}}^{\prime} + r\right)^2} \tag{7}$$
 
$$\text{where } [r \in \mathbb{Z}^+], \ 0 \le f\_{QF}(\mathbf{C}) \le 1$$

In Equation (7), a positive integer *r* is added to the normalized cost value of each objective to avoid cases with division by zero. The quantitative fairness social welfare metric prefers a solution *C* over solution *C* if and only if *fQF*(*C*) < *fQF*(*C* ).

### 3.2.6. Entropy

The authors in [18] introduced entropy as a fairness metric. This metric considers the proportion, *pi*, of resource *X* allocated to individual *i* ∈ *n* in a population of *n* individuals. The entropy, *H*, of the distribution of *X* can be calculated as shown in Equation (8).

$$H(X) = -\sum\_{i=1}^{n} (p\_i \log\_2 p\_i) \tag{8}$$
 
$$\text{where } p\_i = \frac{\mathbf{x}\_i}{\sum\_{i=1}^{n} (\mathbf{x}\_i)} \text{ and } \log\_2 p\_i = \frac{\ln(p\_i)}{\ln(2.0)}$$

Here, the entropy will be larger the more fair an allocation is. In Equation (9), the entropy metric is expressed as a social welfare metric defined in terms of normalized cost values, where *σ* is an arbitrary, large number 1 and *r* is a constant added to avoid division by zero.

$$f\_{EN}(\mathbb{C}) = \sigma - |\sum\_{i=1}^{n} (p\_i \log\_2 p\_i)|\tag{9}$$
 
$$\text{where } p\_i = \frac{q\_{t\_i, \mathbb{C}}^{\prime} + r}{\sum\_{i=1}^{n} (q\_{t\_i, \mathbb{C}}^{\prime} + r)} \quad [r \in Z^{+}]$$

The entropy social welfare metric prefers a solution *C* over a solution *C* if and only if *fEN*(*C*) < *fEN*(*C* ).
