3.2.2. Lexi-Min Ordering

The lexi-min ordering social welfare metric prefers a solution *C* over solution *C* if and only if an integer *r* ∈ {1, . . . , *n*} exists such that the following two conditions are satisfied:

$$1.\quad (\stackrel{\star}{q'}\_C)\_i = (\stackrel{\star}{q'}\_{C'})\_i \text{ for all } i < r$$

$$2. \quad (\stackrel{\stackrel{\star}{q'}}{q'\_{C}})\_r < (\stackrel{\stackrel{\star}{q'}}{q'\_{C'}})\_r$$

Here again, ↓ *q <sup>C</sup>* is the normalized cost vector *q <sup>C</sup>* arranged in descending order.

This means that the lexi-min ordering social welfare metric addresses the weakness of the egalitarian social welfare metric by considering the next worst off cost value in compared solutions until their values no longer coincide. The lexi-min ordering social welfare metric will successfully distinguish among three solutions used in the example of the egalitarian social welfare metric and assign different rank to them, *C*<sup>3</sup> = 1, *C*<sup>1</sup> = 2 and *C*<sup>2</sup> = 3, respectively.

While the lexi-min ordering social welfare metric offers an improvement over the egalitarian social welfare metric, it deviates from the other social welfare metrics in that it offers no collective cost value. Rather, it presents a method for direct comparison between solutions.

#### 3.2.3. Approximated Fairness

The approximated fairness social welfare metric [9] has adopted the concept of simple fair division, explained in social choice and game theory [19], in order to implement the concept of fairness. The metric is defined in Equation (4).

$$f\_{AF}(\mathbb{C}) = \sum\_{i=1}^{n} \frac{(q\_{t\_i, \mathbb{C}}' - q\_{avg, \mathbb{C}}')^2}{n} \tag{4}$$
 
$$\text{where } q\_{avg, \mathbb{C}}' = \frac{\sum\_{i=1}^{n} (q\_{t\_i, \mathbb{C}}')}{n}$$

Equation (4) shows how the approximated fairness social welfare metric ranks solutions based on the sum of the squared difference between the individual, normalized cost values of the objectives, and the average normalized cost value across all objectives. This means that the approximated fairness social welfare metric prefers a solution *C* over solution *C* if and only if *fAF*(*C*) < *fAF*(*C* ).

#### 3.2.4. Fairness Analysis

The concept behind the fairness analysis social welfare metric is very much similar to the concept behind the approximated fairness social welfare metric. This metric is used to balance the requirement fulfillment between customers of Motorola Company for hand-held communication devices [10]. The aim is to minimize the standard deviation of the number of fulfilled requirements for each

customer in order to treat customers on a fair basis. The fairness analysis social welfare metric is defined in Equation (5).

$$f\_{FA}(\mathbb{C}) = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left(q\_{t\_i, \mathbb{C}}^{\prime} - q\_{avg, \mathbb{C}}^{\prime}\right)^2} \tag{5}$$
 
$$\text{where } q\_{avg, \mathbb{C}}^{\prime} = \frac{\sum\_{i=1}^{n} (q\_{t\_i, \mathbb{C}}^{\prime})}{n}$$

Equation (5) shows that the fairness analysis social welfare metric ranks solutions based on the standard deviation of the normalized cost value of each objective in order to achieve fairness. Hence, the fairness analysis social welfare metric prefers a solution *C* over solution *C* if and only if *fFA*(*C*) < *fFA*(*C* ).
