Weighted sum approach

This weighted-sum approach is conceptually simple but has a number of disadvantages ... 

Diversity is typically measured using a distance-based or cell-based method cost is typically given as reactant cost/gm and physiochemical properties such as AMW are typically measured as the difference in the distribution of the property in the library compared to the distribution of the same property in a collection of known drugs. The weights, wl5 w2j w3> are user-defined and are typically set so that diversity is maximized, while the cost and physicochemical properties are minimized. This weighted-sum approach leads to a single solution that represents one particular compromise in the objectives. Several other groups have also adopted this approach [67, 71 73]. 

Routine methods in solving multi-objective optimization problems are conversion of the multiple objective functions into one objective function. For this purpose, different methods are presented in scientific reports, from which the most widely used methods are Weighted sum approach, s-perturbation, Min-Max and nonsorting genetic algorithm. Genetic algorithms act well to solve the multi-objective optimization... 

Intermediate liquid 8 values are obtained by mixing liquids of known solubility parameter SPS makes use of this. The 8 value of the mixture is equal to the volume-weighted sum of the individual component liquid 8 values. Thus, the mass uptake of a miscible liquid mixture by an elastomer may be very much greater than the swelling which would occur in the presence of either one of the constituent liquids alone. The mixture could of course comprise more than two liquid components, and an analogous situation would apply MERL have applied this approach for the offshore oil-production industry to allow realistic hydrocarbon model oils to be developed,basically by mixing one simple aliphatic (paraffinic) hydrocarbon, one naphthenic, and one aromatic to proportions that meet two criteria, namely, that... 

We will now discuss the heteronuclear double-bonded systems in light of the carbene S-T gaps and the weight analysis of the contributing VB electronic structures to the total wave function. The VBSCF and OVB structure weights summed by category are set out in Tables 7 and 8. For a somewhat different approach, analysis and discussion of these aspects the reader is referred elsewhere14. 

A recent trend in library design is to optimize libraries over a number of properties simultaneously for example, whether a library is designed to be diverse, focused or some combination of the two, it is desirable that the library is cheap to synthesis and that the compounds contained within the library have drug-like physicochemical properties. Most approaches to multiobjective library design combine the different properties via a weighted-sum fitness function. For example, in the SELECT program the fitness function can have the following form ... 

Most approaches to tackling this problem make use of a weighted-sum fitness function, such as the one in Eq. (3) that is used in the SELECT program [32]. 

Several other library design programs also use a weighted-sum fitness function for the simultaneous optimisation of multiple properties [33-36] however, there are limitations associated with this approach especially when the objectives are in conflict, which is usually the case. For example, it can be difficult to determine appropriate weights for the objectives and typically several trial-and-error runs are required where the relative weights are varied. Each set of weights will result in a different solution that represents a different compromise in the ob-... 

In MoQSAR, MOGP [52, 54] is used to overcome the limitations of using a weighted-sum fitness function. The approach is similar to the MOGA approach described earlier where multiple objectives are handled independently without summation and without weights and Pareto ranking is used to identify a Pareto-optimal set of solutions. Pareto ranking was shown previously in Fig. 2. 

This discussion of mathematical modeling is limited to methods based on the assumption of error in the y (dependent) term only. The objective or minimization function will generally be restricted to the sum of the weighted residuals between the observed and calculated data, weighted sum of squared residuals (WSS). The objective of the mathematical modeling approaches is to adjust the parameter values so that a minimum value of the WSS is achieved. 

The idea of fractional occupation numbers was introduced by Slater [71], already in 1969. This approach is not limited for the JT systems, e.g. it was explored by Dunlap and Mei [32] for molecules, by Filatov and Shaik [36] for diradicals, and is also used for calculations of solids and metal clusters [8], It rests on a Arm basis in cases when the ground state density has to be represented by a weighted sum of single determinant densities [53,79], One should remember that molecular orbitals (MO) themselves have no special meaning. Thus, using partial occupation is just a way of obtainning electron density of a proper symmetry (HS). 

The standard way of devising a RTO scheme is the so-called two-step approach [1], also referred to as repeated identification and optimization in the literature. In the first step, the values of (a subset of) the adjustable model parameters 6 are estimated by using the available process measurements. This is typically done by minimizing the lack of closure in the steady-state model equations (2), such as the weighted sum of squared errors between measured outputs y and predicted outputs y [17]. 

Several attempts have been made (217, 218, 219, 220) to establish the relationship between the extrathermodynamic approach of Hansch (114) and the mathematical approaches (198, 212, 213, 215, 216). The numerical equivalence and the theoretical interrelation between the two approaches derives from the assumption that the contributions of the individual substituents to the biological activity is represented as a weighted sum of several physicochemical properties (e.g., the extrathermodynamic parameters in the Hansch equation) as expressed in equation 103... 

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