Pareto-domain

Operating conditions of gluconic acid production Maximization of overall production rate and the final concentration of the gluconic acid while minimizing the final substrate concentration at the end of fermentation process. Net Flow Method (NFM) Pareto-domain was first found by a procedure which includes an evolutionary algorithm. Halsall-Whitney etd. (2003)... 

Halsall-Whitney, H. and Thibault, J. (2006). Multi-objective optimization for chemical processes and controller design Approximating and classifying the Pareto domain, Comput. Chem. Eng., 30, pp. 1155-1168. 

Net Flow and Rough Sets Two Methods for Ranking the Pareto Domain 191... 

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... 

The success of either NFM and RSM requires that a sufficiently large number of discrete solutions be identified to adequately represent the Pareto domain. The discrete solutions can originate from either experiments or simulations obtained from a model of the process. In engineering applications, the latter is preferred because of the typically large cost associated with generating experimental points. Very often, an experimental design will be used to allow modeling of the process in a manner that relates each of the objectives to all the input process variables. 

Solution Comparing solution 1 to solution 2, objectives Ci, C3, and C4 are better for solution 1, but worse for C2. The two solutions are non-dominated with respect to each other because each solution is better for at least one objective. However, the values of the four objectives of solution 3 are worse than those of solutions 1 and 2 such that solution 3 is a dominated solution with respect to the other two solutions and will be discarded. Nonetheless, it does not mean that solutions 1 and 2 will be part of the Pareto domain because they may be dominated by other solutions. In the end, the Pareto domain only contains solutions that are better for at least one of its objective criteria than all the other solutions within that domain. 

First, for each combination of solutions in the Pareto domain, the difference between the values of each objective function k is calculated by comparing solution i with solution j using the following relationship ... 

The final ranking score for each solution in the Pareto domain is obtained by summing individual outranking elements associated with each domain solution as follows ... 

The first term evaluates the extent to which element i performs relative to all the other solutions in the Pareto domain, while the second term evaluates the performance of all the other solutions relative to solution i. The solutions are then sorted from highest to lowest according to the ranking score. The solution with the highest ranking is the one that best satisfies the set of preferences provided by the decision-maker. 

Fig. 7.2 (a) Individual concordance index, and (b) discordance index calculations used in NFM algorithm to determine ranking scores for the Pareto domain solutions. 

Finally, instead of relying on the unique solution of the Pareto domain having the best ranking score, it is preferable to use the results of NFM to divide the Pareto domain into zones containing high-ranked, mid-ranked, and low-ranked domain solutions in order to identify... 

It may be tempted to believe that the NFM reduces to a simple least-squares method, where only the relative weights (are used to rank the entire Pareto domain, when the three thresholds Qu, Pk, and 14) are either made all equal to zero or to very high values. This is not the case and, in fact, the three threshold values play an important role in the ranking of the Pareto domain over the whole range of threshold values. The role of thresholds is to use the distance between two values of a given criterion to create a zone of preference around each solution of the Pareto domain and to identify the solutions that are systematically better than the other solutions. 

Table 7.3 Parameters used to rank the Pareto domain nsing Net Flow.

Finally, Eq. (7.9) is used to calculate the score of each solution. It is obtained by subtracting the sum of all elements in the column by the sum of all elements in the row for each diagonal element. The scores are 1.231, -1.623, and 0.392 for solutions 1, 2, and 3, respectively. The solution with the highest score is the best solution. In this example, the ranking in order of preference gives solution I as the best solution, followed by solutions 3 and 2. This example was performed with only three solutions. In practice, the same analysis is performed with thousands of Pareto-optimal solutions that are generated to adequately circumscribe the entire Pareto domain. 

A handful of solutions, usually 3 to 7, from different regions of the Pareto domain are selected and presented to a decision-maker who has an intimate knowledge of the process being optimized. The decision-maker is given the task of ordering the small set of solutions from the most preferred to the least preferred. This ranking procedure captures and encapsulates the expert s knowledge of the process, which RSM will use to rank all solutions in the Pareto domain. 

RSM is able to capture the knowledge that the decision-maker has of his/her process in a very straightforward manner. However, the main limitation of RSM is its reliance on the decision-maker s ranking of a very small set of solutions selected from the Pareto domain. Some of the issues associated with the generation of ranking rules and the subsequent ranking process are summarized below. 

The process of subtracting two rules in the expression (2 - 2) in Table 7.6 accounts for the removal of rules (00...00) and (11... 11) from the possible rule set, since these two rules imply a solution that suffers complete dominance in the first case and enjoys total domination in the second, and so de facto caimot be part of the Pareto domain. For example, in the case of a four-objective optimization problem, a maximum of 14 rules can be generated. It might seem reasonable in this case to present the decision-maker with a data set containing four solutions, resulting in a miitimum of two rules that will not appear in the final set. Meanwhile, if the expert s ranked data set contains five solutions, a total of 20 rules will be generated while the maximum number of possible distinct rules would be only 14, implying that some rules will be duplicated in the P and NP sets, and therefore eliminated. 

By way of illustrating this latter point, for a four-objective optimization problem, Renaud et al. (2007) used an expert s ranked data set containing seven solutions taken from the Pareto domain, resulting in a total of 42 rules. Removing the duplicate rules in each of the P and NP rule sets and identical rules appearing simultaneously in both the rule sets, only three rules remained in both rule sets (a total of 6) in one case considered, and five rules in each set (a total of 10) in the other case. Under ideal conditions, seven rules in each of the P and NP rule sets would be found. Obviously, some rules were eliminated because they appeared in both rule sets and, therefore, they will not used to rank the entire Pareto domain. 

To select solutions within the Pareto domain that are sufficiently discriminative to the decision-maker to make a clear choice. Vafaeyan et al. (2007) developed one such algorithm that ensures the... 

In a pair-wise comparison of two solutions within the Pareto domain, a rule will contain at least one zero and, as a result, a minimum of one objective function will always be sacrificed. RSM cannot be used for a two-objective optimization because the decision-maker will have to make a clear choice between one of the two objective functions and the preference rule can only be (10) or (01). For instance, choosing (01) automatically means that the optimal solution is the lowest possible value of the second objective in the case of a minimization problem, and the highest possible value for a maximization problem. RSM reduces a two-objective problem to a SOO problem. As the number of objectives increases, the overall effect of losing at least one dimension in objective space diminishes significantly. 

It may be advantageous to ask the decision-maker to classify the various rules into the P and NP sets rather than examining a series of solutions from the Pareto domain. Since all objectives do not have equal importance, the possibility of adding relative weights to the rules may further improve the ranking of the entire Pareto domain. 

The Pareto domain will first be ranked with the NFM. NFM uses four parameters for each objective criterion to express the preferences of the decision-maker the relative weight and three thresholds (indifference, preference, and veto). Table 7.10 provides, for each objective function... 

Fig. 7.4 shows the ranked Pareto domain for the two-objective optimization of gluconic acid. This plot of Pf Its versus Pf is a typical Pareto domain for a two-objective optimization where both objective functions need to be maximized. It may be tempted to believe that maximizing Pf and minimizing tg would be equivalent to this two-objective problem. However, this is not the case. Indeed, a very different Pareto domain, which would include very low and very high values of the batch time, would be obtained. Using the productivity Pfltg) is truly the best way to define the desired objective. 

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