Multi-Objective Optimization

35.5 Multi-Objective Optimization

Modification of each searching

(MOO)

point

In many real-life applications, multiple and often conflict- No

ing objectives need to be satisfied. Satisfying these conflict- Reach maximum iterations ing objective functions is called Multi-Objective Optimization

(MOO). For example, to place more functional blocks on a

Yes

chip while minimizing that chip’s area and/or power dissipa-

Stop

tion are conflicting objectives that need performing a tradeoff analysis [9, 10]. The objective of MO optimization is to find a

FIGURE 35.3 The general flow chart of PSO search algorithm. set of acceptable solutions and present them to the user, who will then choose from them. Generally, there are two general approaches to solve MOO. The first approach lies in combin-

V K +1 is its modified velocity, ing the individual objective functions into a single composite

V pbest is the velocity considering its p best location, and function. Determination of a SO is possible with methods,

V gbest is the velocity considering its g best location. such as the weighted sum method, but the problem lies in the Using the above concept, a certain velocity, which gradually correct selection of the weights. In practice, it can be very dif-

gets close to P best and g best , can be calculated. The current posi- ficult to accurately select these weights, even for someone very tion (searching point in the solution space) can be modified by familiar with the problem domain. In addition, optimizing a the following equation:

particular solution with respect to a SO can result in unac- ceptable results with respect to the other objectives [11, 12].

X id

(35.2) =X The second general approach is to obtain the optimal solution. +V There will be a set of optimal trade-offs between the conflict-

id

id

ing objectives, but this optimal solution is called Pareto optimal

35.4.3 PSO Search Algorithm

solution set or Pareto front [13, 14]. A Pareto optimal set is

a set of solutions that are nondominated with respect to one Figure 35.3 shows the general flow chart of the PSO algorithm. another. While moving from one Pareto solution to another, The main steps in the PSO process are described as follows: there is always a certain amount of importance in one objective

1. System initialized with a population of random poten- to achieve a certain amount of gain in the other. Generating tial solutions. Each potential solution is assigned a ran- the Pareto set has several advantages. The Pareto set allows the dom “velocity” and is called a particle (It has position user to make an informed decision by seeing a wide range of in the space; i.e., it is a point in the solution space and it options. The Pareto set contains the solutions that are optimum

998 A. M. Sharaf and A. A. A. El-Gammal from an “overall” standpoint. SO optimization may ignore this usual operators are performed. In the ranking procedure, the

trade-off viewpoint. This feature is useful since it provides non-dominated individuals in the current population are first better understanding of this system in which all the conse- identified. Then, these individuals are assumed to constitute quences of a decision with respect to all the objectives can be the first non-dominated front with a large dummy fitness explored [9].

value [16]. The same fitness value is assigned to all of them. In The following definitions are used in the proposed MOO order to maintain diversity in the population, a sharing method search algorithm:

is then applied. Afterward, the individuals of the first front are Def. 1 The general MO problem requiring the optimization of N ignored temporarily and the rest of population is processed objectives may be formulated as follows:

in the same way to identify individuals for the second non- dominated front. A dummy fitness value that is kept smaller

Minimize than the minimum shared dummy fitness of the previous front

is assigned to all individuals belonging to the new front. This

(35.3) process continues until the whole population is classified into non-dominated fronts. Since the non-dominated fronts are

subject to g j ( (35.4) defined, the population is then reproduced according to the

dummy fitness values.