Non-inferior solutions

There are basically two approaches for solving a multiobjective optimization problem. The first is to attempt to find the optimal or preferred solution directly, and the second is to generate the so-called non-inferior solutions (set) and then locate the preferred solution among them. The latter approach is widely adopted and will be used in this work. 

The so-called trade-off surface (curve) represents the non-inferior solution obtained in the preceding section, and the trade-off ratio between the i-th and j-th objectives is defined as... 

Figure 3 indicates that in the example, the trade-off curve (or the non-inferior solutions) is not completely concave in shape and that the feasible region is not exactly convex however, the two objectives, capital investment, in terms of the heat transfer area and available energy, f, are always in conflict with each other in the region under consideration. A reduction in the heat transfer area will always give rise to an increase in the dissipation of available energy. 

The non-inferior solution set for the two-objective problem can be solved by various methods (5,6). In this paper the e-constraint method is used by taking into account its applicability to the non-convex problems. 

The trajectory of the non-inferior solution for the two-objective problem is illustrated by the line 1-2-3-U-5-6 in Figure 2. 

The trade-off curve in the objective function space is non-convex as shown in Figure 3. Figure U shows the profiles of the sensitivities, Sj s (j=Ts,Ti,Ti), along the non-inferior solution curve. The sensitivity profile drawn in bold strokes shows the changes in the Lagrange multiplier. It is equal to the trade-off ratio along the non-inferior solution curve based on Equation (13). It is not continuous at point 3. 

Figure li. Sensitivity profiles along the non-inferior solution curve. 

The profile of the Lagrange multiplier along the non-inferior solution curve is used to obtain the optimal solution when a unit cost of exergy is specified. Using the two-objective analysis we can obtain a much better understanding of the process design under the present uncertain conditions with respect to energy. 

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... 

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]. 

In non-gel solution, if the sample channel width was narrower than the separation channel width (by five-fold), floating injection did not result in sample leakage, and no pinched injection was necessary. Moreover, a push-back voltage was not necessary during separation. In addition, a T injector, rather than a crossinjector, was sufficient to perform floating injection without leakage, but the resolution obtained using the T injector was inferior to the cross-injector [556]. If T injectors are used, the number of reservoirs in a multichannel (S) system is reduced to S + 2, which is the real theoretical limit [556], rather than S + 3 [557]. 

The results obtained show that the calibration of the ion-selective electrodes for sodium and potassium with saline phosphate buffers at 0.16 ionic strength give a different measurement if compared with non-buffered solutions and provide inferior results in respect of the flame photometry or, anyway, of the stoichiometric quantity present in the solution. 

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... 

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]

From the results, it can be seen that Tchebycheff GP produces an inferior solution compared to fuzzy GP because both the price and quality objectives for fuzzy GP are superior to those of Tchebycheff GP. The solution obtained from preemptive GP, non-preemptive GP, and fuzzy GP form a non-dominated set (i.e., it is impossible to improve on either objective without sacrificing on the other objective). 

CMP comes next. Its goal is to planarize and pattern the oxide [14]. The CMP step is discussed in more detail in the following two chapters. It should be mentioned here that CMP is not the only option for planarization. More cost-effective solutions have been studied and used, such as multilayer resist processes and spin-on glass in combination with RIE (Fig. 12.4). However, the obtained global planarity for such non-CMP processes is inferior to CMP planarity. They are therefore more suitable for ILD planarization than for STI... 

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