Multi-Objective Optimization Methods

The e-constraint and weighting methods belong to a posteriori methods using the scalarization approach. These methods convert an MOO problem into a SOO problem, which can then be solved by a suitable method to find one Pareto-optimal solution. A series of such SOO problems will have to be solved to find the other Pareto-optimal solutions. See Chapter 6 for a discussion of the weighting and -constraint methods, their properties and relative merits. 

Interactive methods, as the name implies, requires interaction with the DM during the solution of the MOO problem. After an iteration of these methods, s/he reviews the Pareto-optimal solution(s) obtained and articulates, for example, further change (either improvement, compromise or none) desired in each of the objectives. These preferences of the DM are then incorporated in formulating and solving the optimization problem in the next iteration. At the end of the iterations, the interactive methods provide one or several Pareto-optimal solutions. Examples of these methods are interactive surrogate worth trade-off method and the NIMBUS method, which have been apphed to several chemical engineering applications. 

No Preference Methods (e.g., global criterion and neutral compromise solution) These methods, as the name indicates, do not require any inputs from the decision maker either before, during or after solving the problem. Global criterion method can find a Pareto-optimal solution, close to the ideal objective vector. 

A Posteriori Methods Using Multi-Objective Approach (many based on evolutionary algorithms, simulated annealing, ant colony techniques etc.) These relatively recent methods have found many applications in chemical engineering. They provide many Pareto-optimal solutions and thus more information useful for decision making is available. Role of the DM is after finding optimal solutions, to review and select one of them. Many optimal solutions found will not be used for implementation, and so some may consider it as a waste of computational time. 

A Priori Methods (e.g., value function, lexicographic and goal programming methods) These have been studied and applied for a few decades. Their recent applications in chemical engineering are limited. These methods require preferences in advance from the DM, who may find it difficult to specify preferences with no/limited knowledge on the optimal objective values. They will provide one Pareto-optimal solution consistent with the given preferences, and so may be considered as efficient. 

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Molecular Library Design Using Multi-Objective Optimization Methods... 

As mentioned in the introduction, we here assume that a DM is able to participate in the solution process. (S)he is expected to know the problem domain and be able to specify preference information related to the objectives and/or different solutions. We assume that less is preferred to more in each objective for him/her. (In other words, all the objective functions are to be minimized.) If the problem is correctly formulated, the final solution of a rational DM is always Pareto optimal. Thus, we can restrict our consideration to Pareto optimal solutions. For this reason, it is important that the multi-objective optimization method used is able to find any Pareto op>-timal solution and produce only Pareto optimal solutions. However, weakly Pareto optimal solutions are sometimes used because they may be easier to generate than Pareto optimal ones. A decision vector x G S (and the corresponding objective vector) is weakly Pareto optimal if there does not exist another x G S such that /i(x) < /i(x ) for alH = 1,..., A . Note that Pareto optimality implies weak Pareto optimality but not vice versa. 

Finding a final solution to problem (6.1) is called a solution process. It usually involves the DM and an analyst. An analyst can be a human being or a computer program. The analyst s role is to support the DM and generate information for the DM. Let us emphasize that the DM is not assumed to know multi-objective optimization theory or methods but (s)he is supposed to be an expert in the problem domain, that is, understand the application considered and have insight into the problem. Based on that, (s)he is supposed to be able to specify preference information related to the objectives considered and different solutions. The DM can be, e.g., a designer. The task of a multi-objective optimization method is to help the DM in finding the most preferred solution as the final one. The most preferred solution is a Pareto optimal solution which is satisfactory for the DM. 

As said in the introduction, in interactive multi-objective optimization methods, a solution pattern is formed and repeated and the DM specifies preference information progressively during the solution process. In other words, the solution process is iterative and the phases of preference elicitation and solution generation alternate. In brief, the main steps of a... 

To be more specific, when classifying objective functions the DM indicates which function values should improve, which ones are acceptable and which are allowed to get worse. In addition, amounts of improvement or impairments are asked from the DM. There exist several classification-based interactive multi-objective optimization methods. They use different numbers of classes and generate new solutions in different ways. 

Interactive multi-objective optimization methods have considerable advantages over the methods mentioned above. However, they have been used very rarely in chemical engineering. For example, interactive methods can not be found in the survey of Marler and Arora (2004) and they are only briefly mentioned in Andersson (2000) and Bhaskar et al. (2000). This might be due to the lack of knowledge of interactive methods or the lack of appropriate interactive multi-objective optimization software. The few examples of interactive multi-objective optimization in chemical engineering include Grauer et al. (1984) and Umeda and Kuriyama (1980). 

Marler, R. and Arora, J. (2004). Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization 26, 6, pp. 369-395. 

Let the desired penetration rate be between 0.5 and 1 mg.h , the lag time be less than 0.75 h, and the highest acceptable irritation score be 10. Figure 6.2b shows these limits superimposed, revealing a small acceptable zone around the point 1% d-limonene and 45% ethanol. In the original paper, the authors used a multi-objective optimization method. 

Another consideration concerns the shape of the fronts. They appear convex, and would not be determinable with conventional multi-objective optimization methods like, for example, the Weighted Sum Method. This last method is a linear combination of the objectives, and the application of an Evolutionary optimization problem approach in Pareto s fronts definition would be more appropriate. 

The e-constraint method is applied in Chaps.2 and 6 for tackling multi-objective optimization problems. An extensive review of multi-objective optimization methods can be found in Ehrgott and Gandibleux (2002). 

MOPs are often characterized by vast, complex search spaces with various local optima that are difficult to explore exhaustively, largely due to the competition among the various objectives. In order to decrease the complexity of the search landscape, MOPs have traditionally been simplified, either by ignoring all objectives but one or by aggregating them. Multi-objective optimization (MOOP) methods enable the simultaneous optimization of... 

Multi-objective optimization (MOO), also known as multi-criteria optimization, particularly outside engineering, refers to finding values of decision variables which correspond to and provide the optimum of more than one objective. Unlike in SOO which gives a unique solution (or several multiple optima such as local and global optima in case of non-convex problems), there will be many optimal solutions for a multiobjective problem the exception is when the objectives are not conflicting in which case only one unique solution is expected. Hence, MOO involves special methods for considering more than one objective and analyzing the results obtained. 

On the other hand, it is not sensible, e.g., to restrict consideration to two objectives only, for the purpose of intuitive visualization. It is better to consider the problem as a whole and use as many objectives as needed instead of artificial simplifications. Furthermore, as mentioned earlier, EMO approaches do not necessarily guarantee that they generate Pareto optimal solutions. Because of the above-mentioned aspects, EMO approaches may not always be the best methods for solving multi-objective optimization problems and that is why we introduce scalarization based and interactive methods, in particular, to be considered as alternative approaches. When using them, the DM can concentrate on interesting solutions only and computational effort is not wasted. Furthermore, the DM can decide how many solutions (s)he wants to compare at a time. 

In what follows, we describe and summarize research on multi-objective optimization in chemical engineering reported in Hakanen (2006) and Haka-nen et al. (2004, 2005, 2006, 2008, 2007). These studies have focused on offering chemical engineering an efficient and practical way of handling all the necessary aspects of the problem, that is, to be able to simultaneously consider several conflicting objective functions that affect the behaviour of the problem considered. Therefore, they have been solved using the interactive NIMBUS method. 

We can say that interactive methods have not been used to optimize SMB processes and, usually, only one or two objective functions have been considered. The advantages of interactive multi-objective optimization in SMB processes has been demonstrated in Hakanen et al. (2008, 2007) for the separation of fructose and glucose (the values of the parameters in the SMB model used come from Hashimoto et al. (1983) Kawajiri and Biegler (2006b)). In Hakanen et al. (2008, 2007), the problem formulation consists of four objective functions maximize throughput (T, m/h ), minimize consumption of solvent in the desorbent stream (D, m/h ), maximize product... 

In this paper, a new computer-aided technique was presented, with which the experimental procedure of developing catalysts is scheduled sequentially. In each sequential step the neural networks model and multi-objective optimization are used to determine optimal design for the next experiment. The sequential method proved very efficient in developing catalysts for propane ammoxidation to acrylonitrile. And the yield of acrylonitrile corresponding to the best catalyst was up to 58.9%. 

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