Pareto-optimal front, unless prohibited by the geometry of the Pareto front. This is to achieve a smooth transition from one Pareto-optimal solution to its neighbors, thus facilitating the decision-maker in choosing his/her final solution. Mathematically, UD(X'**) for a given set of** 
non-dominated individuals X' in a population X, where X'  X**, is defined as (Tan et al. 2001a),** 

$$U D(\mathbf{X}^{\prime})={\frac{1}{1+S_{_{n c}}}}$$
$$(14)$$
X **)'( (14)** 
where Snc **is the standard deviation of niche count of the overall set of non-dominated individuals** X'.

It can be seen that larger value of UD(X**') indicates a more uniform distribution and vice versa.** 
(iii) Number of Function Evaluation (*Neval***): The computational effort required to solve an** optimization problem is often an important issue, especially when only limited computing resources are available. In the case that a fixed period of CPU time is allocated and the CPU time for each function evaluation is assumed to be equal, then more function evaluations being performed by an optimization indirectly indicates less additional computational effort is required by the algorithm. 

## 4.2.4 Simulation Settings And Comparison Results

The decimal coding scheme (Tan et al. 1999) is applied to all the evolutionary methods studied in this comparison, where each parameter is coded in 3-digit decimals and all parameters are concatenated together to form a chromosome. In all cases, two-point crossover with a probability of 0.07 and standard mutation with a probability of 0.01 are used. A reproduction scheme is applied according to the method used in the original literature of each algorithm under comparison. The population size of 100 is used in FFGA, NPGA, NGGA and MOEA, which only require a single population in the evolution. SPEA and MIGA are assigned a population size of 30 and 70 for their external/archive and evolving population size, respectively, which form an overall population size of 100. All approaches under comparison were implemented with the same common sub-functions using the same programming language in Matlab (The Math Works, 1998) on an Intel Pentium II 450 MHz computer. Each simulation is terminated automatically when a fixed simulation period of 180 seconds is reached. The simulation period is determined, after a few preliminary runs, in such a way that different performance among the algorithms could be observed. To avoid random effects, 30 independent simulation runs, with randomly initialized population, have been performed on each algorithm and the performance distributions are visualized in the box plot format (Chambers et al. 1983; Zitzler & Thiele, 1999). 

Figure 19 displays the performance of SSC **(size of space covered) for each algorithm. In general,** 
SPEA and MOEA produce a relatively high value of SSC **indicating their ability to have a more** distributed discovered Pareto-optimal front and/or to produce more non-dominated solutions that are nearer to the global trade-offs. It can also be observed that, compared to the others, FFGA, SPEA and MOEA are more consistent in the performance of SSC**. The performance of** UD (uniform distribution) for all algorithms is summarized in Figure 20. In general, the UD distributions are mostly overlapping with each other and thus there is too little evidence to draw any 