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M. Janga Reddy (Indian Institute of Technology Bombay, India) and D. Nagesh Kumar (Indian Institute of Science, India)

Copyright: © 2015
|Pages: 12

DOI: 10.4018/978-1-4666-5888-2.ch346

Top## Background

### Multi-Objective Problem

* (1)* where and* (2)* where *m* is number of objectives; *D* is feasible search space; is the set of *n*-dimensional decision variables (continuous, discrete or integer); *R* is the set of real numbers; *R*^{n} is *n*-dimensional hyper-plane or space; *l*_{i} and *u*_{i} are lower and upper limits of *i*-th decision variable.

*Definition*: A general multi-objective optimization problem (MOOP) can be defined as, minimize a set of functions *f(x)*, subject to *p* inequality and *q* equality constraints.

In MOOP, the desired goals are often conflicting against each other and it is not possible to satisfy all the goals at a time. Hence it gives a set of non-inferior solutions also known as *Pareto optimal* solutions. The Pareto optimal solution refers to a solution, around which there is no way of improving any objective without degrading at least one other objective (Deb et al., 2002).

*Pareto Front*: Pareto front is a set of nondominated solutions, being chosen as optimal, if no objective can be improved without sacrificing at least one other objective. On the other hand a solution *x** is referred to as dominated by another solution *x,* if and only if, *x* is equally good or better than *x** with respect to all objectives. The definition of Pareto optimality is very much useful in MOEAs to classify the population of solutions into dominated and non-dominated members, thereby helping in the selection of member solutions from one generation to next generation.

Pareto Front: A set of nondominated solutions, being chosen as optimal, if no objective can be improved without sacrificing at least one other objective. On the other hand a solution x* is referred to as dominated by another solution x if, and only if, x is equally good or better than x* with respect to all objectives.

Multi-Objective Optimization: Decision making for problems that require simultaneous optimization of several non-commensurable and often competitive/conflicting objectives.

Pareto Optimal Solution: It is to a solution around which there is no way of improving any objective without degrading at least one other objective.

PSO Algorithm: The algorithm that was inspired by co-operative intelligence of the swarm such as bird flocking, insect colonies etc.

Spacing Metric: The spread (distribution) of vectors throughout the set of non-dominated solutions, and is calculated with a relative distance measure between consecutive solutions in the obtained non-dominated set.

Elitist Mutation: A strategic mutation mechanism to effectively explore the solution space, thereby helps to improve the performance of the PSO algorithm.

Set Coverage Metric: The relative spread of solutions between two sets of solution vectors.

Crowding Distance: An estimate of the density of solutions surrounding that solution. The crowding distance value of a particular solution is the average distance of its two neighboring solutions.

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