Topics: Genetic algorithm, Mathematical optimization, Genetic algorithms Pages: 8 (2067 words) Published: December 30, 2012
Vector Evaluated Genetic Algorithm|
Abhishek Rehan(16/301)Ankit Garg(16/308)Sanchit Garg(16/339)Sidharth Jain(16/347)12/28/2012

Many real world problems involve two types of problem difficulty: i) multiple, conflicting objectives and ii) a highly complex search space.On the one hand, instead of a single optimal solution competing goals give rise to a set of compromise solutions, generally denoted as Pareto-optimal. In the absence of preference information, none of the corresponding trade-offs can be said to be better than the others. On the other hand, the search space can be too large and too complex to be solved by exact methods. Thus, efficient optimization strategies are required that are able to deal with both difficulties. Evolutionary algorithms possess several characteristics that are desirable for this kind of problem and make them preferable to classical optimization methods.In fact, various evolutionary approaches to multi-objective optimization have been proposed since 1985, capable of searching for multiple Pareto optimal solutions concurrently in a single simulation run.[8]

We present the classical approaches to multi-objective optimization problems in this paper as well as the evolutionary algorithm. We extend the algorithm of basic genetic algorithm to Vector Evaluated Genetic Algorithm (Schaffer, 1984). Drawbacks of VEGA have also been mentioned. We also demonstrate the formulation of Travelling Salesman Problem through Genetic Algorithm. 1. MULTIPLE OBJECTIVE FUNCTIONS

Multiobjective optimization is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Example:

* Minimizing weight while maximizing the strength of a particular component * Maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle * Design of VLSI Circuits where there is a tradeoff between performance and cost

In mathematical terms, a multi-objective optimization problem can be formulated as

M Objective Functions:
There are two goals in multi-objective optimisation problem: * To find a set as close as possible to the Pareto-optimal front * To find a set of solutions as diverse as possible

In multiobjective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to Pareto optimal solutions. 2. PARETO OPTIMAL SOLUTIONS

Pareto Optimal Solutions are solutions that cannot be improved in any of the objectives without impairment in at least one of the other objectives. In mathematical terms, a feasible solution  is said to (Pareto) dominate another solution, if 1.  for all indices  and

2.  For at least one index.
A solution  (and the corresponding outcome) is called Pareto optimal, if there does not exist another solution that dominates it. The set of Pareto optimal outcomes is often called the Pareto front. A solution x(1) is said to dominate the other solution x(2), if both conditions 1 and 2 are true: 1. The solution x(1) is no worse that x(2) in all objectives 2. The solution x(1) is strictly better than x(2) in at least one objective

Classical methods for generating pareto-optimal solutions involved aggregating multiple objective functions into a single objective function. The approach followed is decision making before search. However, the parameters of the approach are not set by the DM, but systematically varied by the optimizer. Some representatives of this class of techniques are the weighting...
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