Utopia point

The coordinates of the utopia point are then given by (/, /j). The meaning and optimization potentiality of the utopia point is stressed using the following simple example. Let s consider the following multiobjective optimization problem,... 

The solution of the optimization problem is depicted in a 2D plot of the involved objectives (figure 8.6). Each non-inferior attainable optimal solution is estimated at a given combination of objectives including constraint g xi,X2)- All these points define a curve of non-inferior solutions, normally referred to as Pareto curve. For the sake of transparency a linear relation between objectives is considered (. e. F = TjWi x fi), where Wi is the weighting factor of the objective function fi and Ylwi = 1. As expected, the utopia point is given by the coordinates... 

Note 8.1. Utopia point in optimization problems with more than one objective function... 

Figure 8.6. The role of utopia point in multiobjective optimization. Remarks the optimization problem is formulated according to expression 8.56 given in the explanatory note of table 8.1 same note provides the mathematical definition of the utopia point.

Figure 8.9. Pareto optimal cm ve /econ versus /exergy Legend E economic-based optimal design X exergy-based optimal design Q optimal design based equally on economic performance and exergy efficiency Remarks each point of the curve represents an optimal solution in terms of the design variables [m-cat, Q, a

This curve is termed Pareto curve cf. explanatory note 8.1) and clearly shows that one objective function can only be improved at the expense of the other objective function. This trade-off leads to significant deviations between optimal designs and the utopia point (up to 60%) for the pair /4on /exergy The decision maker should have in mind the importance of compromising both objectives at early stages of the design cycle. 

In other words, the solution of problem [7.2] can be interpreted as a compromise solution relative to a utopia point given by the collection of the ideal stakeholder weighted costs wjf. This definition of utopia point is not to be confused with the traditional definition used in multiobjective optimization [6]. 

In general, the solution which is simultaneously optimal for all objectives utopia point) is not feasible and the real purpose of multiobjective optimization is to generate the set of the so-called Pareto-optimal solutions, i.e. the set of solutions which represents the relatively best edtematives. For two objectives, this set is known as the Pareto front. Mathematically, a feasible solution jc is a Pareto-optimcd (or non-domincited, or non-inferior, or efficient) solution if there exists no jc such that Ffx) [Pg.556]

In order to understand how NBI works, it is necessary to define some terminology. The Convex Hull of Individual Minima (CHIM) is defined as the set of points that are linear combinations of F(xt ) -F for i = l,...,iw, where x, is the global optimal solution of Ffx) and F is the shadow minimum (or utopia point), i.e., the vector containing the individual global minima of the objectives. The pay-off matrix is defined as an m x m matrix whose column is F(Xi ) -F. Given a vector P, d>p defines a point on the CHIM. Mathematically, the so-called NBI subproblem is formulated as ... 

In Fig. 5 it can be seen the Pareto-optimal solutions set. An obvious conclusion is that a considerable improvement of the economic objective can be obtained with a minimum increase of the ISE, but for a value of the cost function approximately less than 1000, more economic designs can only be achieved at the expense of a large increase in the ISE. Additionally, points cy and 62 represent the designs with minimum cost and minimum ISE, respectively. Clearly, point seems to be the most desirable design, because is the closest solution to the Utopia point. 

The utopia point z, ..., Zp] is defined as comprised by the optimized value Zp of every objective function. Obviously, this ideal point is an imaginary solution for... 

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