Comprehensive Survey of the Hybrid Evolutionary Algorithms

1. Introduction

A multiobjective optimization problem (MOP) can be stated as follows:¹minimize F(x) = (f₁(x), . . ., f_m(x))^T(1) subject to x ∈ Ωwhere Ω is the decision variable space, x = (x₁, x₂, . . ., x_n)^T is a decision variable vector and x_i, i =1. . . . n are called decision variables, F(x): Ω → R^m consist of m real valued objective functions and R^m is called the objective space.

If Ω is closed and connected region in Rⁿ and all the objectives are continuous of x, a problem (1) is said to be a continuous MOP.

Very often, the objectives of the problem (1) are in conflict with one another or are incommensurable. There doesn’t exist a single solution in the search space Ω that can minimize all the objectives functions simultaneously. Instead, one has to find the best tradeoffs among the objectives. These tradeoffs can be better de-fined in terms of Pareto optimality. The Pareto optimality concept was first introduced by eminent economists Pareto and Edgeworth (Edgeworth, 1881). A formal definition of the Pareto optimality is given as follows (Coelle Coello, Lamont, & Veldhuizen, 2002); (Deb, 2002); (Deb, 2001); (Miettinien, 1999):

• Definition: Let u = (u₁, u₂, . . ., u_m)^T and v = (v₁, v₂, . . ., v_m)^T be any two given vectors in R^m. Then u is said to dominate v, denoted as u ≺ v, if and only if the following two conditions are satisfied.

  • 1.

    u_i ≤ v_i for everyi ∈ {1, 2, . . ., m}

  • 2.

    u _j < v _j for at least one index j ∈ {1, 2, . . ., m}.

• Remarks: For any two given vectors, u and v, there are two possibilities:

  • 1.

    Either u dominates v or v dominates u

  • 2.

    Neither u dominates v nor does v dominate u.

• Definition: A solution x^∗ ∈ Ω is said to be a Pareto Optimal to the problem (1) if there is no other solution x ∈ Ω such that F(x) dominates F(x^∗). F(x) is then called Pareto optimal (objective) vector.

• Remarks: Any improvement in a Pareto optimal point in one objective must lead to deterioration in at least one other objective .

• Definition: The set of all the Pareto optimal solutions is called Pareto set (PS): PS = {x ∈ Ω, F(y) ≺ F(x)}

• Definition: The image of the Pareto optimal set (PS) in the objective space is called Pareto front (PF), PF = {F(x)|x ∈ PS }.

• Weight Sum Approach: the weighted sum of the m objectivists is defined as g^ws(x, λ) = λ¹f₁(x) + λ²f₂(x) + . . . + λ^m f_m(x), where ∑ ^m_j=1 λ ^j = 1 and λ ^j ≥ 0.