Optimal Pareto multiple solutions

Penicillin V Bioreactor Train Three cases maximization of (a) both penicillin yield and concentration at the end of fermentation, (b) penicillin yield and batch cycle time, and (c) penicillin yield and concentration at the end of fermentation as well as profit. NSGA-II Glucose feed concentration is the decision variable contributing to the Pareto-optimal front. Multiple solution sets producing the same Pareto-optimal front were observed. Lee et al (2007)... 

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. 

Pareto-optimal solutions can be represented in two spaces - objective space (e.g., /i(x) versus /2(x)) and decision variable space. Definitions, techniques and discussions in MOO mainly focus on the objective space. However, implementation of the selected Pareto-optimal solution will require some consideration of the decision variable values. Multiple solution sets in the decision variable space may give the same or comparable objectives in the objective space in such cases, the engineer can choose the most desirable solution in the decision variable space. See Tarafder et al. (2007) for a study on finding multiple solution sets in MOO of chemical processes. 

In reality, many chemical processes are defined by complex equations where the application of SOO techniques does not provide satisfactory results in the presence of multiple conflicting objectives. Instead, the solution lies with the use of MOO techniques. MOO refers to the simultaneous optimization of multiple, often conflicting objectives, which produces a set of alternative solutions called the Pareto domain (Deb, 2001). These solutions are said to be Pareto-optimal in the sense that no one solution is better than any other in the domain when compared on all criteria simultaneously and in the absence of any preferences for one criterion over another. The decision-maker s experience and knowledge are then incorporated into the optimization procedure in order to classify the available alternatives in terms of his/her preferences (Doumpos and Zopounidis, 2002). MOO techniques... 

For process optimization with respect to several economic criteria such as net present worth, payback period and operating cost, the classical Williams and Otto (WO) process and an industrial low-density polyethylene (LDPE) plant are considered. Results show that either single optimal solution or Pareto-optimal solutions are possible for process design problems depending on the objectives and model equations. Subsequently, industrial ecosystems are studied for optimization with respect to both economic and environmental objectives. Economic objective is important as companies are inherently profit-driven, and there is often a tradeoff between profit and environmental impact. Pareto-optimal fronts were successfully obtained for the 6-plant industrial ecosystem optimized for multiple objectives by NSGA-ll-aJG. The study and results reported in this chapter show the need and potential for optimization of processes for multiple economic and environmental objectives. 

Many algorithms have been developed for the generation of non-dominated solutions. A full review of theory and methods can be seen in [5-6] and the references cited therein. A common approach is to transform the original non-linear multi-objective optimization problem into a single objective optimization problem. Usually this is done by means of a characteristic parameter. The solution to this non-linear programming (NLP) problem is expected to be a Pareto-optimal solution. Multiple solutions can be obtained by changing the value of the... 

Pareto optimization, named after an Italian economist who invented the method in 1906 [33], selects multiple solutions each representing a different, optimal balance of properties. A Pareto optimal, or nondominated solution (in our context a compound), is one for which there is no other solution that is better in all properties. An illustration of this principle is shown in Figure 15.4. 

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) ... 

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. 

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). 

To use a GA for Multiobjective Optimization (MO) entails comparing two solutions with respect to the multiple objectives considered [Carlos et al., 2007], [Toshinsky et al., 2000]. In the case of a singleobjective, the comparison is trivial a vector solution X is better than another one, say y, if the corresponding objective function (fitness) value f(x) is greater than f(y).A multiobjective optimization problem, instead, deals with Nf objective functions i = 1,2,..., Nf this requires that two solutions x and y are compared in terms of dominance of one solution over the other with respect to atUV)- objectives [Sawaragi et al., 1985]. The multiobjective optimality search process, converges on a Pareto-optimal set of nondominated solutions, which provides a spectrum of possible choices for the decision-maker to a posteriori identify his or her preferred solution. 

A limitation of Pareto optimization is that the number of compounds on the Pareto front increases exponentially with the number of parameters, making it impossible to evaluate all of the optimal solutions. In practice, this limits the routine use of Pareto optimization to approximately four or less simultaneous parameters. One approach to overcome this challenge is to combine multiple, related properties into a single optimization parameter, for example, multiple, ADME-related properties can be combined using a desirability index and the balance of ADME score with potency explored with Pareto optimization to find the best balance to achieve in vivo efficacy. 

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