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Wali Khan Mashwani (Kohat University of Science & Technology (KUST), Pakistan)

Copyright: © 2015
|Pages: 22

DOI: 10.4018/978-1-4666-7456-1.ch015

Chapter Preview

TopA multiobjective optimization problem (MOP) can be stated as follows:^{1}minimize *F*(*x*) = (*f*_{1}(*x*), . . ., *f _{m}*(

If Ω is closed and connected region in *R ^{n}* and all the objectives are continuous of

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*= (*u*_{1},*u*_{2}, . . .,*u*)_{m}and^{T}*v*= (*v*_{1},*v*_{2}, . . .,*v*)_{m}be any two given vectors in^{T}*R*. Then^{m}*u*is said to dominate*v*, denoted as*u*≺*v*, if and only if the following two conditions are satisfied.*1.**u*≤_{i}*v*for every_{i}*i*∈ {1, 2, . . .,*m*}*2.**u*<_{j}*v*for at least one index_{j}*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) = λ^{1}*f*_{1}(*x*) + λ^{2}*f*_{2}(*x*) + . . . + λ(^{m}f_{m}*x*), where ∑^{ m}_{j}_{=1}λ= 1 and λ^{j}≥ 0.^{j}

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