Multi-Objective Optimization in Theory and Practice I: Classical Methods

Founding Multi-Optimization Techniques

Author(s): Andre A. Keller

Pp: 67-108 (42)

DOI: 10.2174/9781681085685117010006

* (Excluding Mailing and Handling)


Classical optimization methods solve vector optimization problems. These methods produce approximations of Pareto-optimal sets by which we analyze the performances and limitations. The research initially extended existing methods. The Zeleny’s simplex algorithm illustrates this approach in extending the simplex method to multiple linear objective functions. The simplex method is an iterative procedure that finds an optimal solution to a linear single-objective programming problem. It is one of the numerous techniques proposed to solve linear SOOP problems. The process uses a finite number of iteration steps. In the beginning, a reformulation of the program is such that slack variables introduce the inequality constraints. These slack variables are the primary basis. The process moves on from an extreme point of the feasible space to another adjacent point. Multi-objective simplex tableaus are augmented to solve linear MOO problems by using similar principles and processes. There is one row for each objective function. Numerical examples illustrate the whole computation procedure. Weighting objective method is one another class of founding the technique to solve nonlinear MOO problems. The method consists of aggregating (or making scalar) the objective functions. The objectives are normalized before the aggregation. The weighting can be a convex linear combination of objectives with different weights. These weights differ according to the method such as with the weighted sum method, the weighted metric method, and the weighted exponential method. Numerical examples illustrate each case.

Keywords: Basic feasible region, Best compromise solution, Chebychev’s problem, Excess resource variable, Ideal point, Linear programming, Neighboring nonbasic solution, Neighboring point, Non-inferior solution, Payoff table, Pivot, Scalarization, Simplex tableau, Simplex-based algorithm, Slack variable, Weakly Pareto-optimal, Weighted exponential method, Weighted metric method, Weighted sum method, Zeleny’s simplex algorithm.

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