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Top1. Introduction
A multiobjective optimization problem (MOP) can be stated as follows:1minimize F(x) = (f1(x), . . ., fm(x))T(1) subject to x ∈ ΩWhere Ω is the decision variable space, x = (x1, x2, . . ., xn)T is a decision variable vector and xi, i =1. . . . n are called decision variables, F(x): Ω → Rm consist of m real valued objective functions and Rm 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, . . ., um)T and v = (v1, v2, . . ., vm)T be any two given vectors in Rm. Then u is said to dominate v, denoted as u ≺ v, if and only if the following two conditions are satisfied.
- 1.
ui ≤ vi 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:
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 gws(x, λ) = λ1f1(x) + λ2f2(x) + . . . + λm fm(x), where ∑ mj=1 λ j = 1 and λ j ≥ 0.