Pareto-optimality method

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. 

MCDM methods are applied when at least two responses need to be optimized simultaneously. Different approaches can be distinguished, for example, window programming, threshold approaches, utility functions. Derringer s desirability functions, Pareto optimality methods, Electre outranking relationships, and Promethee (7). In this chapter, only the Pareto optimality methods (7, 117, 118) and Derringer s desirability functions (7, 119, 120) will be discussed. 

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. 

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. 

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. 

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. 

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. 

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. 

Interactive Methods (e.g., interactive surrogate worth tradeoff and NIMBUS methods) Decision maker plays an active role during the solution by interactive methods, which are promising for problems with many objectives. Since they find one or a few optimal solutions meeting the preferences of the DM and not many other solutions, one may consider them as computationally efficient. Time and effort from the DM are continually required, which may not always be practicable. The full range of Pareto optimal solutions may not be available. 

Fig. 1.5 Pareto-optimal solutions for maximizing profit and octane number (xq) by the -constraint method profit is shown on the x-axis in all plots.

Optimize the alkylation process for two objectives (cases A and/or B) using the weighting method. One can use the Solver tool in Excel for SOO. Try different weights to find as many Pareto-optimal solutions as possible. Compare and comment on the solutions obtained with those obtained by the -constraint method (Figures 1.5 and 1.6). Which of the two methods - the weighting and the e-constraint method, is better ... 

Fluidized bed dryer Minimization of product color deterioration and unit cost of final product. No-preference method Application is a dehydration plant for sliced potato. Pareto-optimal solutions were found from the single objective contours. Krokida and Kiranoudis (2000)... 

Cyclic adsorption processes Two examples (a) thermal swing adsorption -maximization of total adsorption efficiency and minimization of consumption rate of regeneration energy, and (b) rapid pressure swing adsorption -maximization of both purity and recovery of the desired product for RPSA. Modified Sum of Weighted Objective Function (SWOF) method Modified SWOF method is superior to the conventional SWOF as it was able to find the non-convex part of the Pareto-optimal set. Ko and Moon (2002)... 

Keywords Multiple criteria decision making (MCDM), interactive methods, scalarization, chemical engineering, Pareto optimality... 

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. 

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