Pareto-optimal solution

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. 

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. 

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) ... 

Figure 18.42 Pareto optimal solution for the multiobjective optimization of the SMB and Varicol systems. Reproduced with permission from Z. Zhang et al., AIChE 48 (2002) 2800 (Fig. 2).

Figure 18.43 Pareto optimal solution for the multiobjective optimization (maximum purity and production rate) of the SMB unit for various particle sizes. The (a) purity of the extract, (b) flow rate, (c) number of theoretical stages per column, and (d) pressure drop are plotted as the function of production rate. Reproduced with permission from Z. Zhang et ah,. Chromatogr., 989 (2003) 95 (Fig. 4).

Figure 18.47 Pareto optimal solutions and corresponding decision variables in the optimization of an existing SMBR unit. Reproduced with permission from H.. Subramani, K. Hi-...

The multiobjective nature of the activities covered by CAPE results from the fact that production processes are dependent on technical, economic, environmental, and social systems. Each has its own priorities and objectives (e.g., technical—maximizes yield and purity economic—maximizes profit environmental— minimizes the amount of wastes and use of natural resources and social—full employment for the workforce). Moreover, there are often several objectives inside the systems themselves. Usually there are conflicts between these objectives with their role being twofold. On one hand, they introduce serious constraints and limitations leading to the search of trade-offs between the objectives and finally Pareto optimal solutions. On the other, attempts to remove these conflicts could lead to the generation of highly innovative technical, economic, and organizational solutions. 

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. 

Pareto-optimal solutions are probably smooth curves except for the discontinuity in Figure 1.2b. 

Fig. 1.2 Pareto-optimal solutions for the optimization of the dual independent expander refrigeration process (a) objectives and ,0 1) to be minimized, and (b) and (c) two decision variables. See Chapter 8 for further details.

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)... 

Industrial cyclone separator Two problems maximization of overall collection efficiency while minimizing (a) pressure drop and (b) cost. NSGA Pareto-optimal solutions of the two problems are similar although their ranges are different Ravi et al. (2000)... 

Lipid production Maximizing the productivity and yield of lipid for an optimum composition of the culture medium. Diploid Genetic Algorithm (DGA) Net flow algorithm was used for ranking the Pareto-optimal solutions obtained by DGA. Muniglia et al. (2004)... 

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