Moreover, their formulation is defined in the parameter space to handle multi-modal function optimization, which may not be appropriate for distributing the population uniformly along the Pareto-optimal front in the objective domain. 

In contrast to existing approaches, we propose a dynamic sharing method that adaptively computes the sharing distance V*share* **to uniformly distribute all individuals along the Pareto-optimal** 
front at each generation. This requires no prior knowledge of the trade-off surface. Intuitively, the trade-offs for an m-objective optimization problem are in the form of an (m-**1) dimensional** 
hyper-volume (Tan et al. 1999), which can be approximated by the hyper-volume Vpop
(n) **of a** 
hyper-sphere as given by, 

$$V_{pop}^{\ (n)}=\frac{\pi^{(m-1)/2}}{\left(\frac{m-1}{2}\right)\ !}\times\left[\frac{d^{(n)}}{2}\right]^{m-1}\tag{9}$$

where n)(
d is the diameter of the hyper-sphere at generation n. **Note that computation of the** 
diameter n)(
d **depends on the curvature of the trade-off curve formed by the non-dominated** 
individuals in the objective space. For a two-objective optimization problem, the diameter n)(
d **is** 
equal to the interpolated distance of the trade-off curve covered by the non-dominated individuals as shown in Figure 4. Although computation of n)(
d **that accurately represents the interpolated** 
curvature of the non-dominated individuals distribution is complex, it can be estimated by the average distance between the shortest and the longest possible diameter given by dmin
(n) **and** dmax
(n)
respectively (Tan et al. 1999). Let Fx and Fy **denote the objective function of the two furthest** 
individuals in a population. Then dmin
(n) is equal to the minimum length between Fx and Fy**, and** 
dmax
(n) **can be estimated by** d1
(n) + d2
(n) **as shown in Figure 4.** 
The same procedure can also be extended to any multi-dimensional objective space. To achieve a uniformly distributed population along the trade-off set, the sharing distance V*share*
(n) **could be** 
computed as half of the distance between each individual in the (m-**1)-dimensional hyper-volume** 
Vpop
(n) covered by the population size N **at generation** n,

$$N\times\frac{\pi^{(m-1)/2}}{\left(\frac{m-1}{2}\right)!}\times\left(\sigma_{\textit{share}}^{(n)}\right)^{m-1}=V_{\textit{pop}}^{(n)}\tag{1}$$

$$(10)$$

Substituting Eq. 9 into Eq. 10 gives the sharing distance V*share*
(n) at generation n **in term of the** 
diameter n)(
d and the population size N **as given by** 

$\sigma_{\text{share}}^{(n)}=N^{1/(1-m)}\times\frac{d^{(n)}}{2}$  **a computation of $\sigma_{\text{share}}$ that is capable of distribution**
Clearly, Eq. 11 provides a simple computation of V*share* **that is capable of distributing the** 
population evenly along the Pareto front, without the need for any prior knowledge of the usually 

$$(11)$$
