sharing distance adaptively based upon the on-line population distribution is described in Section 3. Section 4 examines the usefulness and contribution of each proposed feature in the algorithm. The performance comparison of the proposed algorithm with other evolutionary MO optimization methods is also shown in the section. Practical application of the proposed algorithm to servo control system design optimization is given in Section 5. Conclusions are drawn in Section 6. 

## 2. Advanced Goal And Priority Specifications For Mo Optimization

A multi-objective optimization problem seeks to optimize a vector of non-commensurable and often competing objectives, i.e., it tends to find a parameter set P **for Min**F P)(
P)
,n P  R **, where** P
= {p1, p2,…, pn} is a n-dimensional vector having n decision variables or parameters, and ) **defines** 
a feasible set of P. F = {f1, f2,…, fm} is an objective vector with m **objectives to be minimized. For** 
a MO optimization problem with simple goal or priority specification on each objective function, the Pareto-based ranking scheme is sufficient (Fonseca & Fleming, 1993). In practice, however, it may be difficult to define an accurate goal and priority setting in a priori to an optimization process for real-world optimization problems. Besides goal and priority information, there could also be additional supporting specifications that are useful or need to be satisfied in the evolutionary search, such as optimization constraints or feasibility of a solution. Moreover, the Pareto-based ranking scheme does not allow advanced specifications, such as logical "AND" and "OR" operations among multiple goals and priorities for better decision support in complex MO optimization. In this section, a new goal-sequence Pareto-based domination **scheme is proposed to address these** issues and to provide hard/soft goal and priority specifications for better controls in the evolutionary optimization process. 

## 2.1 Pareto-Based Domination With Goal Information

This section is about an effective two-stage Pareto-based domination scheme for MO optimization, which is then extended to incorporate advanced soft/hard goal and priority specifications. Consider a minimization problem. An objective vector Fa **is said to dominate another objective vector** Fb based on the idea of Pareto dominance, denoted by Fa S Fb, *iff* 2,1{ ,..., } f,ia d f,ib  i  m **and** 
ja jb f f
,  ,
 for some j  2,1{ ,..., m**}. Adopting this basic principle of Pareto dominance, the first** 
stage in the proposed domination approach ranks all individuals that satisfy the goal setting to minimize the objective functions as much as possible. It assigns the same smallest cost for all non-dominated individuals, while the dominated individuals are ranked according to how many individuals in the population dominate them. The second stage ranks the remaining individuals that do not meet the goal setting based upon the following extended domination scheme. Let a a

F **and** 
a b

F **denote the component of vector** Fa and Fb respectively, in which Fa does not meet the 