Pareto optimality concept

When trade-offs exist, no single compound will stand out uniquely as the optimum drug for the market, ranked hrst on all measures of performance. Rather, a set of compounds will be considered that, on current knowledge, span the optimal solution to the problem. These compounds are those for which there is no other compound that offers equivalent performance across all criteria and superior performance in at least one. In multicriteria decision analysis (MCDA) terminology, they are known as Pareto-optimal solutions. This concept is illustrated by the two-criteria schematic in Figure 11.3. 

The Multicriteria Decision Making (MCDM) method that is proposed here [28] is based on the Pareto Optimality (PO) concept, does not make preliminary assumptions about the weighting factors, the various responses are considered explicitly. 

In words, this definition says that x is Pareto optimal if there exists no feasible vector of decision variables, XG which would decrease some criterion without causing a simultaneous increase in at least one other criterion. Unfortunately, this concept almost always gives not just a single solution, but rather a set of solutions called the Pareto optimal set. The vectors, x, corresponding to the solutions included in the Pareto optimal set are called nondominated. The image of the Pareto optimal set under the objective functions is called the Pareto front. 

Use of relaxed forms of dominance Some researchers have proposed the use of relaxed forms of Pareto dominance as a way of regulating convergence of a MOEA. Laumanns et al (2002) proposed the so-called C-dominance. This mechanism acts as an archiving strategy to ensure both properties of convergence towards the Pareto-optimal set and properties of diversity among the solutions found. Several modem MOEAs have adopted the concept of f-dominance (see for example... 

In reference point based methods, the DM first specifies a reference point z S consisting of desirable aspiration levels for each objective and then this reference point is projected onto the Pareto optimal set. That is, a Pareto optimal solution closest to the reference point is found. The distance can be measured in different ways. Specifying a reference points is an intuitive way for the DM to direct the search of the most preferred solution. It is straightforward to compare the point specified and the solution obtained without artificial concepts. Examples of methods of this t rpe are the reference point method and the light beam search . 

There have been many surveys on evolutionary techniques for MOO (Fonseca and Fleming, 1995 Coello Coello, 1998 Van Veldhuizen and Lamont, 2000 Tan et al, 2002 Chapter 3 in this book). While conventional methods combined multiple criteria to form a composite scalar objective function, modern approach incorporates the concept of Pareto optimality or modified selection schemes to evolve a family of solutions at multiple points along the tradeoffs simultaneously (Tan et al, 2002). 

In order to bypass the need for expert opinion, the concept of Pareto optimality is used. This concept is widely used in multiobjective optimization and can be illustrated through a simple example. Consider, as in Goldberg (1989), these two quadratic functions ... 

During the last twenty years, the literature on MCDM problems has grown at a high rate where few techniques for generating the Pareto optimal have been well developed and evaluated. The idea of the Pareto optimality and generating the Pareto set, are briefly reviewed here. These concepts and techniques are dealt with in more detail by Ref. [17-22]. 

Pareto optimality is a cornerstone concept in the field of optimisation. In single objective optimisation problems, the Pareto optimal solution is unique as the focus is on the decision variable space. The multi-objective optimisation process extends the optimisation theory by allowing single objectives to be optimised simultaneously. The multi-objective optimisation is considered as a mathematical process looking for a set of alternatives that represents the Pareto optimal solution. In brief, Pareto optimal solution is defined as a set of non-inferior solutions in the objective space defining a boimdary beyond which none of the objectives can be improved without sacrificing at least one of the other objectives [17]. 

The concepts of multi-objective optimisation problems and the Pareto optimality solution can be clarified in the following general example. For a system with a number of design parameters which are under the control of the decision maker, as in Fig. 2, the particular... 

Figure 15.4 A graph illustrating the concept of Pareto optimality in two dimensions. In this case, the potency pK j and logarithm of the solubility in pM are plotted for a set of compounds on the X- and y-axes, respectively. An Ideal compound would have both high potency and high solubility, as indicated by the star on the top-right. Pareto optimal...

Note that, because /is a vector, if any of the components of / are competing, there is no unique solution to this problem. Instead, the concept of noninferiority (also called Pareto optimality) must be used to characterize the objectives. A noninferior solution is one in which an improvement in one objective requires a degradation of another. With the e-constraint method, only one of the objectives (mainly the primary one) is expressed in the cost function while the other objectives take the form of inequality constraints ... 

Since no necessary connection exists between Pareto optimality (the central concept of benefit-cost analy.sis) or Bayesian rules and socially desirable policy, it would be helpful if there were some way to avoid the tendency to assume that economic methods or Bayesian rules, alone, reveal. socially desirable policy. Alternatively weighted a.ssess-ments would enable persons to see that sound policy is not only a matter of economic calculations but also a question of epistemological and ethical analysis, as well as citizens negotiations. (pp. 26-27)... 

In contrast to single-objective problems where optimization methods explore the feasible search space to find the single best solution, in multi-objective settings, no best solution can be found that outperforms all others in every criterion (3). Instead, multiple best solutions exist representing the range of possible compromises of the objectives (11). These solutions, known as non-dominated, have no other solutions that are better than them in all of the objectives considered. The set of non-dominated solutions is also known as the Pareto-front or the trade-off surface. Figure 3.1 illustrates the concept of non-dominated solutions and the Pareto-front in a bi-objective minimization problem. 

The application of the Pareto concept, in search of the solution of the multiobjective optimization problem, allows to evaluate the optimal choice of the DV that represents a compromise solution which guarantees an acceptable level of relative displacement. An Evolutionary approach by means of a Genetic Algorithm has been used to solve the MOOP and search the population of non-inferior parallel solutions. Illustrated numerical examples show that all assessments and... 

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