Pareto optimization

The MPO methods described above assume that an appropriate property profile for a project s objectives is known a priori. However, early in a drug discovery project, this may not be clear. In these cases, it may be appropriate to explore different property trade-offs and gather additional data before deciding on the more important properties and appropriate criteria with which to direct further optimization. 

The Pareto optimal solutions together define a Pareto front and other. 

From this, it can be clearly seen that some regions of this library have a high risk in terms of their predicted properties, while others have a higher chance of delivering a high-quality lead. 

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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. 

Elicitation of decision maker preferences may needed to reduce the set of Pareto-optimal compounds to a single candidate to be progressed. 

The direct search for a global optimum may not uncover some of the Pareto-optimal solutions close to the overall optimum, which might be good trade-off solutions of interest to the decision maker. 

Using the raw data in Fig. 3.20, we can identify the Pareto-optimal set for the HER activify/stabilify criteria. This set represents the best possible compromise between activity and stability criteria for the surface alloys that we have considered the alloys in the set are, thus, logical choices for further consideration. The presence of pure Pt on the Pareto-optimal set is, in effect, a sanity check for our computational screening procedure. Pt is well known to be the most active and stable pure metal for the HER in acidic conditions. The alloys seen on the Pareto-optimal set include RhRe and BiPt. 

Andersson MP, Bligaard T, Kustov A, Larsen KE, Greeley J, Johannessen T, Christensen CH, Nprskov JK. 2006. Towards computational screening in heterogeneous catalysis Pareto-optimal methanation catalysts. J Catal 239 501-506. 

Palladium electrocatalysts, 183 Palladium-alloy electrocatalysts, 298-300 Pareto-optimal plot, 85 Platinum-alloy electrocatalysts, 6, 70-71, 284-288, 317-337 Platinum-bismuth, 86-87, 224 Platinum chromium, 361 362 Platinum-cobalt, 71, 257-260, 319, 321-330, 334-335 Platinum-iron, 319, 321, 334-335 Platinum-molybdenum, 253, 319-320... 

Pareto efficiency, also known as Pareto optimality, is named after an Italian economist, Vilfredo Pareto (1848-1923). The definition of a Pareto efficient economic system is that no re-allocation of given goods can be made without making at least one individual worse off (there is no way to make any person better off without hurting anybody else). Pareto improvement from a non-efficient system is achieved when a change to a different allocation makes at least one individual better off without making any other individual worse off [Varian 47],... 

An introduction to one particular multi-criteria optimization method -the so called Pareto-Optimality method - is discussed in Chapter 4, where also an application of this method is given. 

There are almost always a number of criteria to which the formulation has to fulfil, and in the case of incorporating robustness aspects (as an optimisation criterion) into the optimisation the number of criteria is also increased. It is however almost impossible to fulfil all the criteria in the most optimal way at once. This means that a compromise has to be foimd between all criteria. A large number of methods is available to search for such a compromise variable setting. One of these methods is Pareto Optimality which will be explained and applied in this chapter. Pareto Optimality searches for a compromise between the optimisation of a certain tablet property and the optimisation of the robustness of this property. 

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. 

Pareto Optimality, which makes provisions about mixtures in the whole factor space, therefore cannot be used in combination with a sequential optimisation method. 

Figure 4.17 Plot of the feasible criteria space of the crushing strength and the disintegration time = Pareto-optimal point o = inferior point...

By taking every point in Figure 4.18 as point p successively, all the inferior points can be removed by applying those three rules, only the noninferior or Pareto Optimal points remain. 

A point in the feasible criteria space is a Pareto Optimal point if there exists no other point in that space which yields an improvement in one criterion without causing a degradation in the other. 

By evaluating quantitatively the pay-off between a minimal disintegration time and a maximal crushing strength, a choice can be made between the Pareto Optimal points. The method will be illustrated with an example. For an introduction to the theory of MCDM see [30]. 

In practice not only the robustness of a response at different mixture compositions is interesting for optimisation purposes but also the value of the product property itself (in our case the predicted value of the response). So an optimisation strategy directed to two criteria has to be applied. Pareto Optimality (PO) is the ideal method to consider both these optimisation goals. 

PARETO-OPTIMAL POINTS. Xi=a-LACTOSE X2=P-LACTOSE X3=RICE STARCH y,= PREDICTED VALUE OF THE CRUSHING STRENGTH (N) y2=C OF THE CRUSHING STRENGTH... 

Every point in figure 4.21 corresponds directly to a predicted value of the crushing strength and the calculated value of the robustness coefficient, at one mixture composition. These points are called Pareto Optimal (PO) points, which are listed in Table 4.5. The PO points can also be placed in the corresponding mixture triangle, which is presented in Figure 4.22. 

There are several advanced mathematical techniques available to evaluate several responses, such as the resolutions at different temperatures and relative humidities [22]. We have chosen for a technique called Pareto Optimality since this method is simple and graphical techniques can be used to display the results. 

There were 87 compositions which showed no spot crossover. To select from these 87 compositions the Pareto Optimal points [22] were calculated (maximizing all four criteria). There were nine such points, these are given in Table 6.7. Plots of the minimum resolution for all these compositions were made, and finally the composition DEA=0.08, MeOH=0, CHCl3=0.16, EtAc=0.76 was selected as resulting in the best preferred separation. In Figure 6.7 the change of minimum resolution at this mixture composition at different temperatures and relative humidities is depicted. It is clear that the resolution is reasonably well for most temperatures and relative humidities, but at real humid situations the resolution declines. 

Efficient outcomes make at least one person better off and no one worse off as the result of choice. Choices that make some better off without making others worse off are described as having gains from trade and are labeled Pareto optimal or just optimal. Neoclassical economic theory argues that, under most circumstances, a system of property rights and markets produces these efficient outcomes. But this system is efficient only if aU the effects of choices are included in market prices. If prices do not incorporate aU these effects, such situations are described as inefficient or market failures. 

When optimizing multiple objectives, usually there is no best solution that has optimal values for all, and oftentimes competing, objectives. Instead, some compromises need to be made among various objectives. If a solution A is better than another solution B for every objective, then solution UB is dominated by A. If a solution is not dominated by any other solution, then it is a nondominated solution. These nondominated solutions are called Pareto-optimal solutions, and very good compromises for a multiobjective optimization problem can be chosen among this set of solutions. Many methods have been developed and continue to be developed to find Pareto-optimal solutions and/or their approximations (see, for example, references (50-52)). Notice that solutions in the Pareto-optimal set cannot be improved on one objective without compromising another objective. 

Searching for Pareto-optimal solutions can be computationally very expensive, especially when too many objectives are to be optimized. Therefore, it is very appealing to convert a multiobjective optimization problem into a much simpler single-objective optimization problem by combining the multiple objectives into a single objective function as follows (53-55) ... 

There are two notions of efficiency that are easily confused. Changes that make everybody better off are called Pareto improvements, after the Italian economist Pareto. A state in which nobody can be made better off without someone else being made worse off is called Pareto optimal. A Pareto impnive-ment may be a move to a Pareto-optimal state, but need not be so if there is room for further Pareto improvement. A move to a Pareto-optimal state may be a Pareto improvement, but need not be so if someone is made worse off. as in the move from C to A. 

Consider the multi-criteria optimization problem defined in Eq. (11). Because of the fact that these objective functions usually conflict with each other in practice, the optimization of one objective implies the sacrifice of other targets it is thus impossible to attain their own optima, Js, s e <5 = [1,..., 5], simultaneously. Therefore, the decision maker (DM) must make some compromise among these goals. In contrast to the optimality used in single objective optimization problems, Pareto optimality characterizes the solutions in a multi-objective optimization problem [13]. 

Kalivas, J.H. and Green, R.L., Pareto optimal multivariate calibration for spectroscopic data, Appl. Spectra sc., 55, 1645-1652, 2001. 

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. 

Figure 5.15. Pareto optimality. The filled circles represent rank zero or nondominated solutions for functions fl and /2. Point C is rank 1 because it is dominated by point B. (Permission as in Fig. 5.14.)...

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